DOCUMENT RESUME

ED 476 430 TM 034 933

AUTHOR Mumford, Karen R.; Ferron, John M.; Hines, Constance V.;Hogarty, Kristine Y.; Kromrey, Jeffery D.

TITLE Factor Retention in Exploratory Factor Analysis: A Comparisonof Alternative Methods.

PUB DATE 2003-04-00NOTE 70p.; Paper presented at the Annual Meeting of the American

Educational Research Association (Chicago, IL, April 21-25,2003) .

PUB TYPE Reports Research (143) Speeches/Meeting Papers (150)EDRS PRICE EDRS Price MF01/PC03 Plus Postage.DESCRIPTORS Comparative Analysis; *Factor Structure; Monte Carlo Methods;

SimulationIDENTIFIERS *Exploratory Factor Analysis; Parallel Analysis (Horn)

ABSTRACT

This study compared the effectiveness of 10 methods ofdetermining the number of factors to retain in exploratory common factoranalysis. The 10 methods included the Kaiser rule and a modified Kaisercriterion, 3 variations of parallel analysis, 4 regression-based variationsof the scree procedure, and the minimum average partial procedure. Theperformance of these procedures was evaluated based on the average number offactors retained by each method, the proportion of samples retaining the samenumber of factors retained by each method, the proportion of samplesretaining the same number of factors as the true number of factors in thepopulation, and the proportion of samples retaining the same number offactors when a particular rule of thumb is applied to the population. Theperformance of the 10 procedures was investigated using Monte Carlo methodsin which random samples were generated under known and controlled populationconditions. Results clearly suggest that both the choice of method and thedesign of the factor analytic study play crucial roles in retaining thecorrect number of common factors. In terms of overall accuracy across theconditions examined in this study, one of the parallel analysis approaches(Montanelli, and Humphreys, 1976) provided the largest proportion of samplesretaining the correct value. Conditions under which other approaches may workwell are discussed. (Contains 16 tables, 18 figures, 25 charts, and 44references.) (SLD)

Reproductions supplied by EDRS are the best that can be madefrom the on inal document.

Factor Retention in Exploratory Factor Analysis:

A Comparison of Alternative Methods

Karen R. Mumford

John M. Ferron

Constance V. Hines

Kristine Y. Hogarty

Jeffrey D. Kromrey

University of South Florida

U.S. DEPARTMENT OF EDUCATIONOffice of Educational Research and Improvement

EDU ATIONAL RESOURCES INFORMATIONCENTER (ERIC)

This document has been reproduced asreceived from the person or organizationoriginating it.

Minor changes have been made toimprove reproduction quality.

Points of view or opinions stated in thisdocument do not necessarily representofficial OERI position or policy.

1

Factor Retention1

PERMISSION TO REPRODUCE ANDDISSEMINATE THIS MATERIAL HAS

BEEN GRANTED BY

K. Mumford

TO THE EDUCATIONAL RESOURCESINFORMATION CENTER (ERIC)

Paper presented at the annual meeting of the American Educational Research Association, April 21 25,

2003, Chicago

2BEST COPY AVAILABLE

Factor Retention2

Factor Retention in Exploratory Factor Analysis:

A Comparison of Alternative Methods

This study extends previous investigations of the effectiveness of various

procedures for determining the number of factors to retain in exploratory common

factor analysis. In our previous Monte Carlo study on the quality of factor analytic

solutions (Ferron, Kromrey, Hogarty, Hines, & Mumford, 2002), we found that decision

rules used to make a determination of the number of factors to retain, in most cases,

yielded underestimates of the known number of factors in the population. Zwick and

Velicer (1986) suggest that the determination of the number of factors or components to

retain is likely to be the most important decision that the researcher makes in the

conduct of a factor analytic study. Researchers conducting exploratory factor analyses

in most instances do not know the true number of factors that are expected to underlie

the data, and may make decisions resulting in the retention of too few or too many

factors. Since the number-of -factors decision is made prior to the factor rotation stage,

it subsequently impacts the results of the factor analysis, such as rotated factor patterns,

factor score estimates, and the interpretability of the factors. Consequently, several

researchers (e.g., Rummel, 1970; Fava Sr Velicer, 1996; Wood, Tataryn, & Gorsuch, 1996)

have emphasized the importance of extracting the correct number of factors for rotation.

As Turner (1998) and others note, the interpretation of the factors is based on the

assumption that the researcher has extracted the correct number of factors.

The problems of underfactoring (i.e., extracting too few factors) and

overfactoring (i.e., extracting too many factors) have been addressed in the literature

(see for example, Crawford, 1975; Fava Sr Velicer, 1992, 1996; Gorsuch, 1983;

MacCallum, Widaman, Preacher & Hong, 2001; Mosier, 1939; Turner, 1998; Wood,

Tataryn & Gorusch, 1996; Zwick & Velicer, 1986). In the case of underfactoring,

retaining too few factors for inclusion in the rotation phase can result in the loss of

potentially important information (e.g., a substantive factor or factors) in the factor

solution. With underfactoring, it has been argued, the true factors in a data set cannot

be accurately portrayed (Cattell, 1958; Comrey, 1978; Fava & Velicer, 1996). Wood et al.

3

Factor Retention3

(1996) note that when underfactoring occurs, the estimated factors are likely to contain

considerable error. In their investigation of the consequences of underfactoring in both

maximum likelihood factor analysis and principal component analysis, Fava and

Velicer (1996) found "severe degradation of factor score estimates" with underfactoring.

It was noted that the principal component score degraded less rapidly than the factor

score within methods.

Most researchers concur that overfactoring is less a problem than underfactoring.

Fava and Velicer (1992) and others offer two theoretical justifications that support this

point of view. The first, they posit, is based on the fact that each subsequent factor

extracted accounts for less variance than the factor extracted prior to it. The second

relates to the notion that if too many factors are retained, after rotation, it is relatively

easy to discard trivial factors without changing the substantive factors. This

notwithstanding, there is evidence to suggest that overfactoring is none-the-less quite

problematic and should be avoided. With overfactoring, the resultant factor solution

may include factors that are not interpretable or unlikely to replicate (Zwick & Velicer,

1986). It has been suggested that overfactoring may result in factor splitting at the

rotation phase, or when rotating too many oblique factors, high interfactor correlations

may result or the factor space may collapse (Crawford, 1975). Gorsuch (1983) warns

that the extraction of too many factors may cause a common factor to be missed. In

addition, several studies lend support to the notion that overfactoring may result in

change of the overall factor structure (see for example, Keil & Wrigley, 1960; Howard &

Gordon, 1963; Dingman, Miller, & Eyman, 1964).

Wood et al. (1996) found that when overfactoring occurs, the estimated loadings

for true factors usually contain less error than in the case of underfactoring. Based on

the findings of their study, these researchers suggest that overfactoring is preferable to

underfactoring provided that factor splitting is prevented and false factors are

eventually eliminated (these authors advance methods for handling these two

conditions). Fava and Velicer (1996) also concluded from their series of studies, that

underfactoring was a much more severe problem in factor analysis than overfactoring.

4

Pardie0 Andysis

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IMF MEW AIMRegressionassed RfJethods

Cattell- Nelson- Gorsuch (CNG)

k=>3,p=>6

Uses 6 eigenvalues concurrently

Compares slopes of all possible sets of threeadjacent eigenvalues

Greatest absolute value of difference betweentwo successive slopes

Regressionassed Methods (cont'd)

Multiple Regression (MR)

k=>3,p=>6

Based on same principle as CNG

Utilizes all the eigenvalues in each comparisonof the slopes

Greatest absolute value of difference betweenthe slopes

5

Airirt" '

Regression-Based Methods (cont'd)

t-value index (t)

k=>3,p=>6A variation of the MR procedure

Slopes of the regression lines are comparedusing the usual formula for the t-test

Greatest absolute value of t

10

1111F-Airr'AIIIRegression -Based Methods (cont'd)

Standard Error of Scree (Se,e.)

Errors of estimate are calculated using a sequenceof regression analyses employing a decreasingnumber of eigenvalues.

SEy 1,2,3, ...,

SEy (2+,1 2,3,4, ..., v

SE.v (1.0 3,4,5, ..., v

If SE > 1 v

AIMSiope Comparisons

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Residual Matrix Analysis

Minimum Average Partial (MAP)

Based on matrices of partial correlations

After each of the factors is pattialed out, theaverage of the squared partial correlation iscalculated

Continued until the residual matrix mostclosely resembles an identity matrix

13

MAP Procedure

I

I

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maser al ream Pantalk0 14

Method

Number of factors (3, 5, 7)

Number of variables (p) in the correlation

matrices (3k, 5k, 10k)

Sample Size (3p, 5p, 10p, 20p, 40p)

Level of Communality (low, wide, high)

Interfactor Correlation (0, .2, .5, mixed)

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Method (continued)

AIM

For each condition (i.e., interfactorcorrelations, communality level, k, p) wegenerated a random sample of 10population R matrices

From each population R matrix generated1,000 samples of each size (N per p)

16

Example of Population Pattern Matrix(k=3 & pr-15)

High Wide Low117 -.01 21 .63 -.01 .07

27 ..08 .16

.88 .18 .18 .74 .49 .13

.76 47 .06 .68 .67 ..05 .44 ..01 .08.45 ..01 .04 .38 .23 .03

.61 -.02 58 .15 .87 .08 .38 .24 .34

.68 .51 .06 .11 .62 ..03-.01 .51 .08 -.02 .58 -.02

-.01 .66 3.15 23 ..04-64 .78 .10 -.03 .11 .70 21 -.63 .60.40 .64 35 .04 .00 54..05 .10 .88 -.04 .59 .67 .10 -.01 .64

.18 -.03 .132 .15 .03 2348 .01 .09 .16 .01 .42 .23 .14 .45

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Accuracy Criteria

Performance was evaluated based on:

Average number of factors retained by eachmethod

Proportion of samples retaining the same numberof factors as the population

Proportion of samples retaining the same numberof factors as the rule of thumb applied to thepopulation

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Nuance of Desgrt Factors

Considerable variability was evidencedacross the conditions examined for eachmethod.

To investigate which design factors wereassociated with the most variability, e2was computed and examined.

19

Distributions of RC lean dumber of Factors

Retained, k = 5

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Variable (N:p) for k = 5

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Mean Number of Factors Retained by Number of

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Winn Number of Factors Retained byCommunality Type for k = 5

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4111PIPPr .41.1Proportion of Samples Retaining KFactors by True Number of Factors

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Rule of Thumb

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Proportion of Samples Retaining K Factorsby N per Variable (3:p)

Scree

43....PA rah 13PA cl 11

- V5R 43 - WICSR

MAP

6 10 10 20 as 30 16 40 45

25N Vaddis

Proportion of Samples Retaining K Factors byNumber of Variables Per Factor (p:k)

imor- AEIProportion of Samples Retaining K Factors

by Communality TypeID

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45119 SE WI 1 PA nil PA d PA 0 NAP KSR SKSRSave

Rule of Thumb

BESTCOPYAVAILABLE

ResuOts

More variables per factor, larger samples,higher communality

Most rules underestimated true number offactors

PAMH is best overall, but MAP is slightlybetter with small samples (N = 3p)

Some methods rarely lead to correctnumber of factors

28

ice"Conclusons

No method should be used in isolation

The 'art of factor analysis'

Some methods should be avoided

What to do about low communality andsmall samples? Need a new and betterthumb

12

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10

Factor Retention4

Mac Callum et al. (2001) state "given the serious consequences of underfactoring,

...,users of factor analysis should be very rigorous in making the number-of-factors

decision and should err in the direction of overfactoring when the evidence is

ambiguous". Wood, et al. (1996), however, caution about the importance of extracting

the correct number of factors, as they found in their studies that the factor solution with

the least error is the one that extracts the correct number of factors.

Methods for Determining Number of Factors

The factor analysis literature is replete with recommendations regarding the

most appropriate procedures to employ as well as an array of criteria to consider when

addressing the number-of-factors problem (e.g., Gorusch, 1983; Hakstian, Rogers, &

Cattell, 1982; Zwick & Velicer, 1986). According to Horn and Engstrom (1979), after 50

years of work in the field, no fewer than 50 tests had been invented. There is, however,

little consistency in the results obtained from applications of these tests across data sets

and varying conditions. Thus, many applied researchers are often in doubt about the

efficiency of the procedures they use in seeking to determine reasonable estimates of the

number of factors that underlie a given data structure.

Nasser, Benson, and Wisenbaker (2002) note that although researchers

addressing the number-of-factors to retain problem have advanced numerous

approaches based on different theoretical rationales (algebraic, statistical, psychometric,

substantive importance, and interpretability), the most commonly employed methods

for determining the number of factors in applied research are Kaiser's eigenvalue-

greater-than-one rule (Kaiser, 1960) and Cattell's visual scree test (Cattell, 1966). Both

methods are readily available in commonly used statistical software packages, which

may account for their widespread use. However, several researchers have discussed the

limitations of these two procedures and have warned against their use in making

decisions regarding the number of factors to retain.

In the case of the Kaiser-greater-than-one rule (Kaiser, 1960), factor retention is

based upon the number of eigenvalues greater than one. More specifically, the rule

Factor Retention5

calls for as many components or factors to be retained as there are eigenvalues greater

than 1.0. This procedure was initially suggested by Guttman (1954) based on the

consideration that it provided a lower bound for the number of common factors that

underlie a correlation matrix having unities in the main diagonal. Turner (1998) argues

that the Kaiser rule is flawed as it is based on an assumption that a factor is not

psychometrically sound if it accounts for less variance than a single variable does. Cliff

(1988) suggests that the rule lacks a logical basis as one of the rationales for the rule was

based on a misapplication of a formula for estimating the reliability of a multi-item test.

A number of researchers (e.g., Cattell & Jaspers, 1967; Fava & Velicer, 1992; Hakstian,

Rogers, & Cattell, 1982; Lee & Comrey, 1979; Zwick & Velicer, 1986) have reported the

use of the Kaiser rule as problematic as it often leads to an overestimate of the number

of components or factors that underlie the data, thus giving rise to the potential

problems associated with overfactoring. Others report a tendency to either

overestimate or underestimate the number of components or factors depending on the

conditions present in the data, including whether it is applied to population versus

sample matrices (see for example, Cliff, 1988; Hakstian, Rogers & Cattell, 1982).

Cattell's scree test involves identifying the number of factors or components to

retain based on a visual examination of the graph of eigenvalues plotted on the vertical

axis and the factor sequence numbers plotted on the horizontal axis. The process

involves separation of the "scree" of trivial factors from the "cliff" of nontrivial factors

(Cattell, 1966). A straight edge is placed along the bottom part of the graph where the

points form an approximately straight line (Cattell & Vogelman, 1977). The points

above and to the left of the straight line correspond to the nontrivial factors; points on

or close to the straight line constitute the trivial factors. This process of identifying the

nontrivial factors and hence, the number of factors to retain, introduces a considerable

amount of subjectivity on the part of the applied researcher. Several researchers have

noted that issues of reliability often arise in the use of this procedure (e.g., Crawford &

Koopman, 1979; Zwick & Velicer, 1986). The absence of an obvious break or the

presence of multiple breaks in the eigenvalue pattern may make it difficult for one to

14

Factor Retention6

make a decision about the appropriate number of factors to retain (Jurs, Zoski, &

Mueller, 1993; Nasser, Benson & Wisenbaker, 2002). Some researchers have reported

reasonable accuracy of the scree test, however, the range of conditions examined in

their studies were limited (see for example, Tucker, Koopman & Linn, 1969; Cattell &

Jaspers, 1967; Zwick & Velicer, 1986). Zwick and Velicer (1986) indicated in their study

of the number of components to retain that the scree test "was generally accurate but

variable" and recommended against its use as the sole decision method.

In light of the subjective nature of Cattell's visual scree test, researchers have

sought to develop more objective variations of the scree test in order to overcome some

of its limitations. Four regression-based variations have been proposed, namely, the

Cattell-Nelson-Gorsuch (CNG) test (Gorsuch & Nelson, 1981); and three procedures

proposed by Zoski and Jurs (1993, 1996): a multiple regression approach (MR), a t-value

index, and the standard error of scree (SEscree). A recent Monte Carlo study conducted

by Nasser, Benson, and Wisenbaker (2002) compared the performance of these four

regression-based methods under a variety of simulated conditions. The researchers

reported the SEscree procedure to be the most accurate in terms of its performance with

both correlated and uncorrelated factors. The other three methods were found to

consistently underestimate or overestimate the number of factors to be retained. These

researchers noted several limitations of their study, including the range of conditions

examined. They suggested that this investigation should be viewed as an initial but

crucial phase in the investigation of the efficiency of regression-based procedures for

determining the number of factors, as previous work in the field was based solely on a

few correlation matrices from the published literature.

Additionally, other methods considered to be more effective than Kaiser's rule

and Cattell's scree test in estimating the number of factors that underlie a data set have

been proposed. These include parallel analysis (Horn, 1965; Montanelli & Humphreys,

1976; Lautenschlager, Lance, & Flaherty, 1989) and a minimum-average partial (MAP)

correlation procedure proposed by Velicer (1976).

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Factor Retention7

Given the need for a more comprehensive study and comparison of the

effectiveness of methods proposed for determining the number of factors to retain in

exploratory factor analysis, we undertook an investigation of ten methods that can be

classified under four broad categories of procedures: the Kaiser criterion; parallel

analysis; regression-based variations of the scree test, and residual matrix analysis. The

ten methods examined are briefly described below.

Description of Methods

Kaiser Rule

Eigenvalue Greater Than 1.0 (KSR). This is perhaps the most commonly used

method for determining the number of components to retain in principal component

analysis and is also frequently employed by more than novice researchers in the

conduct of exploratory common factor analysis. Zwick and Velicer (1986) suggest that

the popularity of its use may be due to its availability as the default option in many

statistical packages. The rationale for this method was initially proposed by Guttman

(1954) and later adapted and popularized by Kaiser (1960). Factors or components with

eigenvalues greater than 1.0 (i.e., components which evidence at least as much variance

as one of the original variables) are retained in the analysis and subjected to rotation. As

noted earlier, several researchers have indicated that Kaiser's eigenvalue-greater-than-1

rule is problematic (e.g., Yeomans, & Golden, 1982; Zwick & Velicer, 1986; Wood,

Tataryn, Sr Gorusch, 1996) and have advised against its use in factor analysis. We

include it here because of its common use, and our desire to compare this method with

more recently developed approaches.

Eigenvalue Greater Than Average Eigenvalue in Sample (M-KSR). This modification

of the Kaiser rule was first proposed by Guttman (1954). This method, which may be

more appropriate than the original KSR for common factor analysis, calls for the

retention of factors with eigenvalues greater than the average eigenvalue in the sample.

16

Factor Retention8

Parallel Analysis

This method was developed by Horn (1965). It involves the comparison of

eigenvalues of a correlation matrix of p random uncorrelated variables with those of the

eigenvalues of the correlation matrix of p variables in the actual data set, based on the

same sample size. Factors of the real data set that have eigenvalues greater than the

eigenvalue of the corresponding factor in the random data set are retained and

subjected to rotation.

Parallel Analysis (PAMH). A regression equation for predicting the eigenvalues of

random correlation matrices with squared multiple correlations on the main diagonal of

the random correlation matrix was subsequently developed by Montanelli and

Humphreys (1976):

ln(A, ) = a, + b, ln(N -1) + c, In {(.5) (k(k -1)) (i -1)k}

where N = number of observations,

k = number of variables in the correlation matrix, and

i = eigenvalue sequence number.

The regression weights (ai, bi, 0) were obtained via simulation methods and are

provided by Montanelli and Humphreys (1976).

A decision rule was suggested based on the point at which the plot of sample

eigenvalues crosses the plot of expected eigenvalues from matrices of random data

(Montanelli & Humphreys, 1976). Factors above the point at which the two lines cross,

were retained for rotation. Factors that lie below the point that the plots crossed (i.e.,

those factors whose eigenvalues were less than the eigenvalues of the corresponding

factors of the random data set) were considered trivial factors and were not retained.

Humphreys and Montanelli (1975) found the parallel analysis method to provide

accurate estimates of the number of factors to be retained in common factor analysis.

Zwick and Velicer (1986) also reported that within the context of principal components

analysis, the parallel analysis method yielded consistently accurate estimates of the

number of components to retain. In fact, they indicated that in their study in which

they compared five different rules for determining the number of components to retain,

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Factor Retention9

the parallel analysis method "was typically the most accurate method at each level of

complexity examined."

Parallel Analysis Close (PAca A modification of Horn's parallel analysis

procedure is to eliminate factors with eigenvalues that are close in magnitude but not

necessarily less than the magnitude of the corresponding eigenvalues of the random

data. Such a modified criterion is expected to retain fewer factors than the original Horn

procedure. For this study, sample eigenvalues within .10 of the random data

eigenvalues were considered close enough to stop retaining factors.

Modified Parallel Analysis (PALL). Lautenschlager, Lance and Flaherty (1989)

proposed a modification to the general regression equation first developed by

Montanelli and Humphreys (1976) and further refined by Allen and Hubbard (1986) to

predict eigenvalues of factors resulting from a random data correlation matrix. These

researchers noted that the Allen and Hubbard's equation did not yield an accurate

estimate of the first eigenvalue; this in turn directly influenced the accuracy of

subsequent eigenvalues in the series. They modified the equation proposed by Allen

and Hubbard (1986) by adding a variables-to-subjects ratio term:

ln(X) = a, + b,1n(N 1)+c, ln {(.5)(k i-1)(k + 2))+ d, ln(X, )+ e, (k / N)

The regression weights (as, bi, ci, di and ei) are provided by Lautenschlager, Lance

and Flaherty (1989).

The results of a Monte Carlo study using the newly augmented regression

equation showed that it served to significantly improve the prediction of the first, and

consequently subsequent eigenvalues (up to 48) of a random data matrix

(Lautenschlager, et al., 1989) when compared to eigenvalues generated from the Allen

and Hubbard (1986) equation. These researchers recommend use of the modified

regression equation with the added variables-to-subject ratio term in future parallel

analysis applications.

Regression-Based Variations of the Visual Scree

Is

Factor Retention10

Each of the methods described in this category is based on Cattell's (1978)

guidelines for determining the number of common factors to retain.

Cattell- Nelson Gorsuch (CNG). In an attempt to find a more objective approach to

the visual scree test, Gorsuch and Nelson (1983) developed an analytical method using

multiple linear regression to determine the number of factors to retain. In this

procedure, the slopes of all possible sets of three adjacent eigenvalues each are

compared. More specifically, the slope of the first three points (roots) in the eigenvalue

plot (1, 2, and 3) is compared with the slope of the next three points (4, 5,and 6). Then

the slope of points 2, 3, and 4 is compared with the slope of points 5, 6,and 7, and so on

until all points have been incorporated in the comparisons. The number of factors to be

retained is defined by the point at which the difference between the two successive

slopes is greatest.

Multiple Regression (MR). The CNG procedure uses only six eigenvalues (i.e., six

data points) at a time to determine each pair of slopes for comparison. Zoski and Jurs

(1993) suggest that this procedure uses only a limited amount of information to

determine the number of factors to retain. They therefore proposed a multiple linear

regression approach that would include more data points in computing the slope of the

regression lines. The MR approach is based on the same principle as the CNG, but it

utilizes all the eigenvalues in each comparison of pairs of slopes. A series of adjacent

slopes are obtained and all possible successive pairs of slopes are compared. One slope

in each paired comparison is based on an increasing number of eigenvalues and

corresponds with the major factors; the second slope is based on a decreasing number of

eigenvalues and corresponds with the trivial (scree) factors. More specifically, the slope

of the regression line developed from points 1, 2, and 3 in the eigenvalue plot is

compared to the slope of the regression line developed from points 4, 5, 6, ..., p (where

p is the number of eigenvalues in the plot). Then, the slope of the regression line for

points 1, 2, 3, and 4 is compared to the slope for points 5, 6, 7, ..., p, and so on. This

process continues until all adjacent slopes of all possible successive pairs of regression

lines are compared. The decision regarding the number of factors to retain corresponds

19

Factor Retention11

to the point at which the absolute value of the difference between the slopes is the

greatest.

t-value index (t). Another procedure examined introduced a variation of the MR

procedure known as the t-value index. Using the slopes obtained in the MR procedure,

the slopes of the regression lines are compared using the usual formula for the t-test of

the difference between slopes. The number of factors to retain is based on the largest

absolute value of t.

Standard Error of Scree (SEscred. Because the CNG, MR, and t-value index are not

applicable when the number of factors is less than three and/or the number of variables

is less than six, Zoski and Jurs (1996) developed the standard error of scree (SEscr,)

procedure which is based on the standard error of estimate for a set of points in the plot

of the eigenvalues. In this procedure, the errors of estimate are calculated using a

sequence of regression analyses employing a decreasing number of eigenvalues.

Initially, all eigenvalues are regressed onto their ordinal numbers. The procedure

continues with subsequent sets of eigenvalues and concludes when the standard error

of estimate meets the 1/p criterion. Because the error variance tends to be inversely

related to sample size, Zoski and Jurs (1996) set the value of 1/p as the criteria for

determining the number of factors to retain (i.e., the number of standard errors that

exceed 1/p is the number of factors to retain). Zoski and Jurs (1996) and Nasser, et al.

(1996) found SEscree to be more accurate in identifying the number of factors to retain

than the CNG and multiple regression (MR) procedures.

Residual Matrix Analysis

Minimum Average Partial (MAP) Correlation. The Minimum Average Partial

(MAP) method developed by Velicer (1976) is based on a matrix of partial correlations.

After each of the factors has been partialed out, the average of the squared partial

correlations is calculated. When the residual matrix most closely resembles an identity

matrix, no further factors are extracted and rotated. Using this method, at least two

variables will have high loadings on each retained factor. Zwick and Velicer (1986)

20

Factor Retention12

reported the MAP to be one of the two most accurate methods (the other being parallel

analysis) in determining the number of components to retain in their study of five

decision rules in principal components analysis.

Purpose

The purpose of this study was to compare the effectiveness of the ten

aforementioned methods (the Kaiser rule (KSR), a modified Kaiser criterion (M-KSR),

three variations of parallel analysis-- Montanelli and Humphrey's (PAmH), parallel

analysis-close (PAIL), and Lautenschlager et al. modified parallel analysis (PALL), four

regression-based variations of the scree procedure (CNG, MR, t, SEscree), and the

minimum-average partial (MAP) procedure) in determining the number of factors to

retain in exploratory common factor analysis. The performance of these procedures

was evaluated based on the average number of factors retained by each method, the

proportion of samples retaining the same number of factors as the true number of

factors in the population, and the proportion of samples retaining the same number of

factors when a particular rule of thumb is applied to the population.

Method

The performance of the ten procedures were investigated using Monte Carlo

methods, in which random samples were generated under known and controlled

population conditions. The population correlation matrices varied with respect to the

particular aspects of interest. Population correlation matrices were constructed based

on the methods described by Tucker, Koopman and Linn (1969), which have been used

in large-scale simulation studies (Mac Callum, et al., 1999; Tucker et al., 1969). For

correlated factors, a generalization of the method was utilized that was described by

Hong (1999). The true factor loading patterns underlying these matrices were

constructed to exhibit relatively clear simple structure. For these population

correlation matrices, the number of measured variables, the number of common factors,

the level of communality and the correlation between factors were controlled. Sample

21

Factor Retention13

correlation matrices were generated from each of the known population correlation

matrices. Each of the sample matrices was then analyzed using principal axis factor

extraction. The pattern of eigenvalue magnitude was used to determine the number of

factors that should be retained.

Generation of Population Matrices

In the Monte Carlo study, uncorrelated population correlation matrices were

generated using the method presented by Tucker, Koopman, and Linn (1969). These

matrices were generated under the assumption that the common factor model holds

exactly in the population. This method produces population matrices leading to

relatively clear simple structure and has been used in other simulation studies

(Mac Callum et al. 1999; Mundfrom, Shaw & Ke, 2001; Tucker, Koopman and Linn,

1969). The population R is generated based on major, minor, and unique factors,

R = A,A; +A2Al2 + A3A;

where Al is the pxk matrix of actual input factor loadings for the major factors, A2 is the

matrix of actual input factor loadings for the minor factors, and A3 is the pxp diagonal

matrix of actual input factor loadings for the unique factors. The contribution of the

minor factors (A2) was set to zero in this study so that the data generation model

matched a factor analytic model with k common factors. The common factors were

specified as orthogonal.

The process for creating A, starts with the creation of a matrix of conceptual

input factor loadings, A, . To create A, , the loading of a variable on a randomly

selected factor, j=1 thru k, is set to a value randomly chosen between 0 and k-1, (for a 3

factor model a,, could be 0,1, or 2). Next the loading on a randomly selected factor

from those remaining is set to a value randomly chosen between 0 and k-1-ilj . This

process continues until a conceptual input factor loading has been chosen for each

factor, and ensures the sum of the loadings across the factors is k-1. This process is then

repeated for each of the p variables.

Factor Retention14

The matrix of actual input factor loadings, A, , is then created from the matrix of

conceptual input factor loadings, AI , through a series of three steps: (1) normal deviates

are added to introduce error, (2) a skewing function is used to limit negative factor

loadings, and (3) the matrix is scaled to ensure desired levels of communality. The

diagonal matrix, A3, for the unique factors is also scaled to ensure the desired levels of

communality. The levels of communality (h 2) that were simulated were high (h 2 for

each variable drawn randomly from values of .6, .7, and .8), wide (h 2 for each variable

drawn randomly from values of .2, .3, .4, .5, .6, .7, and .8), or low (h 2 for each variable

drawn randomly from values of .2, .3, and .4), which are consistent with the levels

used in other simulation studies. An example matrix of input factor loadings, A, , that

was created using these methods is presented in Table 1 (with k=3 and p=15) for each

level of communality.

To construct population matrices for correlated factors, the above method was

modified following Hong (1999). More specifically,

JBJ' + A321:

where

and

j = [Al A2]

B =[(1) T

r]'where 'D is a matrix of correlations among major factors, r is a matrix of correlations

among minor factors, and I is a matrix of correlations among major and minor factors.

The simulations conducted for this study were somewhat simplified since the

contributions of minor factors (A2, r, r) were set to zero.

It should be noted that this method of generating population correlation matrices

controls the number of measured variables, the number of common factors, and the

level of communality, but that more than one population correlation matrix can be

generated having the desired number of items, factors, and communality. The

Factor Retention15

specifications define a general class of population matrices, but the random selection

involved in the creation of the conceptual input loading matrix, the introduction of

error, and the random selection of specific communality values all influence the

population correlation matrix that is obtained.

Monte Carlo Study Design

The Monte Carlo study included five factors in the design. These factors were (a)

the number of common factors, k, present in the population (populations were

simulated with k = 3, 5 and 7 factors), (b) the number of variables, p, in the correlation

matrices (with p = 3*k, 5*k and 10*k), (c) the level of communality, (high, wide, and low),

(d) the level of interfactor correlation (with ry = 0, .3, .5 and a mixed condition with

interfactor correlations ranging from 0 to .5), and (e) sample size, N (with samples of 3p,

5p, 10p, 20p, and 40p). These N:p ratios represent values that range from those generally

considered insufficient to those considered more than adequate (MacCallum et al.,

1999). The factors in the design were crossed with each other, providing a total of 540

conditions in the Monte Carlo study.

The research was conducted using SAS/IML version 8.2. Conditions for the study

were run under Windows 98 and Windows 2000. For each condition investigated, 10

population matrices were generated. Then for each population matrix, 1,000 samples

were generated. This provided the opportunity to examine the degree to which the

results varied across different population matrices within a condition. The use of 10,000

samples provided adequate precision of estimates of the sampling behavior of the factor

recovery indices. For example, 10,000 samples provide a maximum 95% confidence

interval width around an observed proportion that is ± .0098 (Robey & Barcikowski,

1992).

24

Factor Retention16

Results

Mean Number of Factors Retained

The mean number of factors obtained by each method is presented for all

combinations of k, p, N, and communality in Tables 2-5, where each table contains a

different level of correlation among the factors. A quick perusal of these tables suggests

that the mean number of factors being retained differs across methods, and for each

particular method the closeness between the mean number of factors retained and the

true number of factors, k, varies across the conditions studied, with larger sample sizes

and more variables tending to lead to better results.

To summarize the performance of the methods, a series of box plots was created.

Each box plot shows the distribution of the mean number of factors retained for a

particular method. These box plots are presented in Figures 1-3, for k = 3, 5, and 7

factors, respectively. For the three-factor condition (Figure 1), the CNG method has a

mean number of factors that is very close to 3, the true value, for all conditions. The

other methods show somewhat more variability across conditions. Interestingly, if one

examines the five-factor conditions (Figure 2), the CNG is still tending to suggest 3

factors, and it continues to suggest three factors even when the true number of factors is

seven (Figure 3). Thus its effectiveness drops off considerably as the number of factors

increases.

Two of the parallel analysis methods (PAmil and PAIL) appear to do relatively

well in the sense that the mean number of factors retained is close to the true number of

factors for a relatively large portion of the conditions. Note that when these methods

produce means that differ from k, the means tend to be smaller, suggesting too few

factors are being retained. The tendency to have too few factors is also found and

somewhat more prevalent for the PALL, MAP, and KSR methods. The distributions for

t, SE., and M-KSR show the most variability, and are the only methods that tend to

have an average number of retained factors that exceeds k for many conditions.

Since the mean number of factors retained differs across conditions for each

method, 62 was used to determine which design factors were associated with the most

25

F a c t o r Retention17

variability. These values are presented in Table 6. Note that one would hope that the

majority of the variation in the mean number of factors retained would be associated

with k, the true number of factors. The only method for which over 50% of the

variation was attributable to k was the PAci, method (a2= .61). Figure 4 contains a

graph showing the mean number of factors retained for each method as a function of k.

As k increases from 3 to 5 to 7 the mean number of factors retained also increases for

each method except the CNG, which stays level at 3 factors extracted. Ideally the bars

would reach, but not exceed, values of 3, 5, and 7, and the method that is coming closest

to this is the M-KSR.

The a2analysis also shows that the mean number of factors retained varies with

other design factors. Figure 5 shows variation as a function of sample size per variable

(N/p) for each method. This figure displays results for conditions with k = 5, but the

pattern was consistent across the values of k investigated. With increasing Nip, the

PAS, PAcL, and M-KSR tend to get close to the expected value of 5. With small N/p

ratios M-KSR tends to retain too many factors, while PAS and PAci, tend to retain too

few factors.

The effect of the number of variables per factor (p/k) is shown in Figure 6. The

methods tend to have a larger mean number of retained factors when the number of

variables per factor is greater. When the number of variables per factor is relatively

small (3), all methods tend to underestimate the number of factors. When number of

variables per factor is large (10), four methods have a mean number of retained factors

that is near the true value of 5. These are the PAmH, PAcL, MAP and KSR methods.

The effects of communality are depicted in Figure 7. As communality increases

the mean number of factors retained by the methods tends to get closer to the true value

of 5. For some methods this implies the mean number of retained factors decreases

with communality while for others it increases with communality. Phi also appears to

have some effect on the mean number of factors retained for most methods (Figure 8).

Generally the mean number of retained factors decreases with increases in the

correlations among factors.

'

Factor Retention18

Proportion of Samples Retaining the Same Number of Factors as the Population

In addition to the mean number of factors obtained, the extent to which the

number of factors retained in each sample matrix matched the number of factors

present in the population correlation matrix was examined. The proportion of

agreement between the sample matrices and the population matrices (i.e., the

proportion of samples retaining k factors) is presented for all combinations of k, p, N,

and communality in Tables 7 -10, aggregated among each level of inter-factor

correlation. A review of these tables reveals that the proportion of samples in

agreement with the population differs considerably across the methods and conditions

investigated. Generally, larger sample sizes, a greater number of variables per factor,

high communality levels, and low inter-factor correlations demonstrated better results.

Due to evidence of large variability across the conditions examined for each

method, w 2 was computed to determine which design factors were associated with the

most variability in the proportions. Table 11 displays these results. Contrary to the

expectations about a 2 in relation to the number of factors extracted, it was expected

that little variation in the proportion of agreement would be associated with k, the true

number of factors underlying the population correlation matrix. One method for which

the majority of variation was attributable to k was the CNG method (62=.98). In

addition, a considerable amount of variation that was attributable to k was the MR

procedure (62=.48). As demonstrated in Figure 9, as k increased, the proportion of

samples in agreement with the population decreased for all of the methods. Although

the CNG method demonstrated excellent results for k = 3 matrices, agreement dropped

dramatically as the number of factors increased. Additionally, for the MR procedure,

agreement dropped to zero for populations of k = 5 and k = 7.

The Vanalysis also revealed that the proportion of samples in agreement with

the population varied with other design factors. Figure 10 depicts the variation in

proportions as a function of sample size per variable (N/p) for each method. With

increasing N/p, the majority of the methods evidenced an increase in agreement, with

2%

Factor Retention19

the pAmH method exhibiting superior performance. In addition, the SE scree method

evidenced the greatest improvement in factor retention with increasing N/p.

The influence of the number of variables per factor (p/k) on the proportion of

samples in agreement with the population is shown in Figure 11. For the majority of

the methods, the proportion of samples in agreement with the population increased as

the number of variables per factor increased, with the PArviii and PAci methods

demonstrating exceptional performance. In particular, the MAP procedure showed the

most dramatic improvements in agreement as the number of variables per factor

increased. In contrast, the CNG and t procedures demonstrated the opposite trend,

declining in their agreement with the population as the number of variables per factor

increased. Interestingly, the SEscree method showed significant improvement in the level

of agreement as the number of variables per factor increased until the p/k ratio was 5.

At that point, the proportion of samples in agreement with the population declined.

The effects of communality on the proportion of samples in agreement with the

population were also examined. As shown in Figure 12, the majority of the methods

evidenced that as communality increased, the proportion of samples in agreement with

the population increased. In contrast, the MR and t methods evidenced a decrease in

performance with an increase in communality level. Phi also appeared to affect the

proportion of agreement for almost all methods (Figure 13). With the exception of the

CNG and MR methods, agreement decreased as the interfactor correlation increased.

Negligible differences were observed for the CNG and MR procedures as the

correlation among factors increased.

Proportion of Samples Retaining the Same Number of Factors as the Rule of Thumb

Applied to the Population

Lastly, an examination was made of the extent to which the number of factors

retained by each rule matched the number of factors that would have been retained if

we had used the rule on the population correlation matrix. The agreement with that

which we would have seen in the population is presented for all combinations of k, p, N,

28

Factor Retention20

and communality in Tables 12-15, with each table displaying a different level of inter-

factor correlation. An examination of these tables suggests that the proportion of

samples differs considerably across the methods, and for each method this proportion is

seen to vary across the conditions studied. In general, conditions with larger sample

sizes and more variables per factor evidenced better results.

Because the proportions evidenced variability across the conditions examined for

each method, we computed a series of a2values to investigate which design factors

were associated with the most variability in these proportions. These values are

presented in Table 16. Consistent with the results presented above, with regard to

agreement between the sample matrices and the population matrices, we would not

expect a sizeable portion of the variation in agreement to be associated with k, the true

number of factors underlying the population correlation matrix. The only method for

which a considerable amount of variation was attributable to k was the MR method

(Qv = .46). Figure 14 illustrates the proportion of samples agreeing with the rule applied

to the population R-Matrix as a function of k. As k increases, the proportion of samples

in agreement with the population decreased for most of the methods with the exception

of MR, which decreased with k = 5 but increased substantially with k = 7. Additionally,

the CNG remained zero regardless of the number of factors. Finally, for the t

procedure, agreement dropped quite dramatically as the number of factors increased.

The a2analysis revealed that the proportion of samples in agreement with the

population also varied with other design factors. Figure 15 shows variation in

proportions as a function of sample size per variable (N/p) for each method. With

increasing N/p, the majority of the methods evidenced an increase in agreement, with

the non-scree based methods exhibiting superior performance. However, the SEsave

method evidenced the biggest improvement in factor retention with increasing N/p.

The influence of the number of variables per factor (p/k) is shown in Figure 16. It

was our expectation that a considerable influence would be associated with p/k. Three

methods were observed for which a notable amount of variation was attributable to

p/k; t, PAS and PAct (cti2= .28, 62= .28 and 62= .24 respectively). The non-scree

29

Factor Retention21

based methods tended to fare slightly better or remain constant as the number of

variables per factor increased. The opposite trend was evidenced for the scree-based

methods, all of which decreased in agreement as the number of variables per factor is

increased.

Finally, the effects of communality are depicted in Figure 17. As communality

increases the proportion of samples agreeing with the population tended to increase for

the majority of methods examined, with the exception of the PALL method, which

evidenced a decrease in performance with increased communality. Phi also appeared

to have some effect on the proportion of agreement for most methods (Figure 18).

Generally, agreement decreased as the interfactor correlation increased for the parallel

analysis approaches (PAmH, PAcL and PALL) and the methods based on the Kaiser

criterion (KSR and M-KSR). Negligible differences were observed for the MAP and the

t procedures as the correlation among factors increased, however, the MAP proceddre

outperformed all other methods for the conditions examined.

Discussion

This empirical investigation of methods to determine the number of factors to

retain in common factor analysis clearly suggests that both the choice of method and the

design of the factor analytic study play crucial roles in retaining the correct number of

common factors. For example, with phenomena characterized by low communality of

variables, a study designed with few variables per factor and small sample size is

unlikely to lead to the correct number of factors, regardless of the method selected

(except for the trivial case of the success of the CNG method when the true number of

factors is 3). Investigations in such areas should include larger numbers of variables

(i.e., p = 10k) and a very large number of observations (N = 40p). With such a design, the

SEscree, parallel analysis (both PAmH and PAcL) and the M-KSR methods provided

nearly 100% accuracy across levels of k and inter-factor correlation. Phenomena that are

characterized by higher levels of communality appear to have less stringent data

requirements, especially if the correlation between factors is low.

30

Factor Retention22

The most prevalent type of error made in the number-of-factors problem appears

to be one of underfactoring if the parallel analysis methods (PAmH, PAIL, PALL), the

MAP or the Kaiser methods (KSR, M-KSR) methods are used. Turner (1998) also found

that under some circumstances, parallel analysis may underestimate the number of

factors that are in the data. For the methods based on fitting regression lines to the scree

plot, underfactoring was also characteristic of CNG and MR, while the t procedure

showed a tendency towards overfactoring (especially as the true number of factors

increased). The SEscree method appeared to offer a balance between overfactoring and

underfactoring, although it showed a tendency to estimate an exceptionally large

number of factors in conditions with small sample sizes.

In terms of overall accuracy across the conditions examined in this study, the

PAmH approach to parallel analysis provided the largest proportion of samples

retaining the correct value of k, an advantage seen across levels of communality, inter-

factor correlation and values of k (except for k = 3, in which the CNG approach was the

most accurate). Across sample sizes, the PAmH method was the most accurate except for

the smallest samples examined (N = 3p), in which the MAP evidenced a slight

advantage. Similarly, for values of p/k, the PAME was the most accurate except for the

smallest value (p/k = 3) in which the CNG provided the highest average accuracy.

Interestingly, the more recent, modified prediction equations provided by

Lautenschlager, Lance and Flaherty (1989) substantially reduced the accuracy in

number-of-factors determination. These results, of course, must be considered in the

light of the limitations of this research. Although the simulation approach we have

followed has examined a range of values for p, k and N, and a variety of communality

levels and levels of inter-factor correlation, further research is recommended to extend

these findings. Additional work is also needed in the development of methods for

accurately determining the number of factors in the more challenging conditions

revealed here (e.g., low communality, few variables per factor and small sample sizes),

conditions in which none of these rules of thumb were successful.

3i

Factor Retention23

Such additional work is important because exploratory factor analysis is a

frequently used multivariate technique in educational research. In the conduct of actual

exploratory factor analyses, researchers face critical decisions regarding the number of

factors to extract and retain. The interpretations gleaned from factor analytic solutions

depend in large part upon the appropriate use of factor retention strategies. As

exploratory factor analysis continues to enjoy a prominent position among the currently

available multivariate methods, researchers must remain mindful of the limitations of

certain procedures and methods. The results of this study underscore the need to

exercise caution in factor retention decisions and highlight the need to consider, in the

planning stages, those aspects of the factor solution that are important in the research

application. This research furnishes valuable information about the sensitivity of

commonly employed methods and provides guidance regarding the choice of

alternative strategies.

32

Factor Retention24

ReferencesAllen, S. J., & Hubbard, R. (1986). Regression equations for latent roots of random data

correlations with unities in the diagonals. Multivariate Behavioral Research, 21, 393-398.

Cattell, R. B. (1958). Extracting the correct number of factors in factor analysis.Educational and Psychological Measurement, 18, 791-838.

Cattell, R. B. (1966). The scree test for the number of factors. Multivariate BehavioralResearch, 1, 245-276.

Cattell, R. B. (1978). The scientific use of factor analysis in behavior science and life sciences.New York: Plennum.

Cattell, R. B., & Jaspers, J. (1967). A general plasmode for factor analytic exercises andresearch. Multivariate Behavioral Research Monographs, 3, 1-212.

Cattell, R. B., & Vogelman, S. (1977). A comparison trial of the scree and KG criteria fordetermining the number of factors. Multivariate Behavioral Research, 12, 289-325.

Cliff, N. (1988). The eigenvalue-greater-than-one rule and the reliability of components.Psychological Bulletin, 103(2), 276-279.

Comrey, A. L. (1978). Common methodological problems in factor analytic studies.Journal of Consulting and Clinical Psychology, 46, 648-659.

Crawford, C.B. (1975). Determining the number of interpretable factors. PsychologicalBulletin, 82(2), 226-237.

Dingman, H. F., Miller, C. R., & Eyman, R. K. (1964). A comparison between twoanalytic rotational solutions where the number of factors is indeterminate.Behavioral Science, 11, 400-404.

Fava, J. L. & Velicer, W. F. (1992). The effects of overextraction on factor and componentanalysis. Multivariate Behavioral Research, 27, 387-415.

Fava, J. L., & Velicer, W. F. (1996). The effects of underextraction in factor andcomponent analyses. Educational and Psychological Measurement, 56, 907-929.

Ferron, J. M., Kromrey, J. D., Hogarty, K. Y., Hines, C. V. & Mumford, K. R. (2002,April). Alternative Perspectives on Sample Size in Exploratory Factor Analysis: AnEmpirical Investigation of the Influence of Communality and Overdetermination. Paperpresented at the annual meeting of the American Educational ResearchAssociation, New Orleans.

Ford, K. J., MacCallum, R. C., & Tait, M. (1986). The application of exploratory factoranalysis in applied psychology: A critical review and analysis. PersonnelPsychology, 39, 291-314.

33

Factor Retention25

Gorsuch, R. L. & Nelson, J. (1981, October). CNG scree test: An objective procedure fordetermining the number of factors. Paper presented at the annual meeting of theSociety for Multivariate Experimental Psychology.

Gorsuch, R.L. (1983). Factor analysis (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum.

Guttman, L. (1954). Some necessary conditions for common factor analysis.Psychometrika, 19, 149-161.

Hakstian, A. R., Rogers, W. T., & Cattell, R. B. (1982). The behavior of number-of-factorsrules with simulated data. Multivariate Behavioral Research, 17, 193-219.

Hong, S. (1999). Generating correlation matrices with model error for simulation studiesin factor analysis: A combination of the Tucker-Koopman-Linn model andWijsman's algorithm. Behavior Research Methods, Instruments, & Computers, 31,727-730.

Horn, J. L. (1965). A rationale and test for the number of factors in factor analysis.Psychometrika, 30 (2), 179-185.

Horn, J. L. & Engstrom, R. (1979). Cattell's scree test in relation to Bartlett's chi-squaretest and other observations on the number of factors problem. MultivariateBehavioral Research, 14, 283-300.

Howard, K. J., & Gordon, R.A. (1963). Empirical note on the "number of factors"problem in factor analysis. Psychological Reports, 12, 247-250.

Humphreys, L. G., & Montanelli, R. G. (1975). An investigation of the parallel analysiscriterion for determining the number of common factors. Multivariate BehavioralResearch, 10, 193-206.

Jurs, S., Zoski, K., & Mueller, R. (1993, April). Using linear regression to determine thenumber of factors to retain in factor analysis and the number of issues to retain in delphistudies and other surveys. Paper presented at the annual meeting of the AmericanEducational Research Association, Atlanta, GA.

Kaiser, H. F. (1960). The application of electronic computers to factor analysis.Educational and Psychological Measurement, 20, 141-151.

Keil, D. F., & Wrigley, C. (1960). Effects upon the factorial solution of rotating varyingnumbers of factors. American Psychologist, 15, 487-488.

Lambert, Z. V., Wildt, A. R., & Durand, R. M. (1990). Assessing sampling variationrelative to number-of-factors criteria. Educational and Psychological Measurement,50, 33-48.

Lautenschlager, G. F., Lance, C. E., & Flaherty, V.L. (1989). Parallel analysis criteria:Revised equations for estimating the latent roots of random data correlationmatrices. Educational and Psychological Measurement, 49, 339-345.

3

Factor Retention26

Lee, H. B., & Comrey, A. L. (1979). Distortions in commonly used factor analyticprocedures. Multivariate Behavioral Research, 14, 301-321.

MacCallum, R. C., Widaman, K. F., Zhang, S., & Hong, S. (1999). Sample size in factoranalysis. Psychological Methods, 4, 84-99.

MacCallum, R. C., Widaman, K. F., Preacher, K. J. & Hong, S. (2001). Sample size infactor analysis: The role of model error. Multivariate Behavioral Research, 36(4),611- 637.

Montanelli, R. G., Jr., & Humphreys, L. G. (1976). Latent roots of random datacorrelation matrices with squared multiple correlations on the diagonal: A MonteCarlo study. Psychometrika, 41, 341-348.

Mosier, C. I. (1939). Influence of chance error on simple structure: An empiricalinvestigation of the effect of chance error and estimated communalities on simplestructure in factorial analysis. Psychometrika, 4, 33-44.

Nasser, F., Benson, J., & Wisenbaker, J. (2002). The performance of regression-basedvariations of the visual scree for determining the number of common factors.Educational and Psychological Measurement, 62, 397-419.

Robey, R. R., & Barcikowski, R. S. (1992). Type I error and the number of iterations inMonte Carlo studies of robustness. British Journal of Mathematical and StatisticalPsychology, 45, 283-288.

Rummel, R. J. (1970). Applied factor analysis. Evanston, IL: Northwestern UniversityPress.

Tucker, L. R., Koopman, R. F., & Linn, R. L. (1969). Evaluation of factor analytic researchprocedures by means of simulated correlation matrices. Psychometrika, 34, 421-459.

Turner, N. E. (1998). The effect of common variance and structure pattern on randomdata eigenvalues: Implications for the accuracy of parallel analysis. Educationaland Psychological Measurement, 58(4), 541-568.

Velicer, W.F. (1976). Determining the number of components from the matrix of partialcorrelations. Psychometrika, 41, 321-327.

Wood, J. M., Tataryn, D. J., & Gorsuch, R. L. (1996). Effects of under- and overextractionon principal axis factor analysis with varimax rotation. Psychological Methods, 1,354-365.

Yeomans, K. A., & Golden, P. A. (1982). The Guttman-Kaiser criterion as predictor ofthe number of common factors. The Statistician, 31(3), 221-229.

Zoski, K., & Jurs, S. (1993). Using multiple regression to determine the number offactors to retain in factor analysis. Multiple Linear Regression Viewpoints, 20, 5-9.

35

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Zoski, K., & Jurs, S. (1996). An objective counterpart to visual test for factor analysis:The standard error scree. Educational and Psychological Measurement, 56, 443-451.

Zwick, W. R., & Velicer, W. F. (1986). Comparison of five rules for determining thenumber of components to retain. Psychological Bulletin, 99 (3), 432-442.

36

Factor Retention28

Table 1Example Population Pattern Matrices as a Function of Communality when k=3 and p=15.

High Communality Wide Communality Low Communality.87 -.01 .21 .89 -.03 -.03 .63 -.01 .07

.87 -.05 .19 .83 .04 -.04 .55 -.02 -.03

.86 .18 .18 .74 .49 .13 .45 -.02 -.02

.76 .47 .06 .69 .57 -.05 .44 -.01 .09

.77 -.03 -.03 .45 -.01 .04 .38 .23 .03

.61 -.02 .58 .18 .87 .08 .36 .24 .34

.58 .51 -.06 .16 .69 -.01 .11 .62 -.03-.01 .89 .08 .15 .69 -.02 -.02 .55 -.02-.04 .84 -.03 -.01 .56 .3 -.02 .45 -.02.16 .82 -.04 -.03 -.04 .84 .24 .38 -.03

-.04 .78 .30 -.03 .11 .70 .21 -.03 .60.40 .64 .36 .48 -.06 .68 .04 .00 .55

-.06 .10 .89 -.04 .59 .67 .10 -.01 .54

-.03 .03 .77 .18 -.03 .52 .15 .03 .53

.48 -.01 .69 .16 -.01 .42 .23 .14 .48

Note. Largest loading for each variable is bolded.

3 7

Tab

le 2

Mea

n N

umbe

r of

Fac

tors

Ret

aine

d, P

hi--

0.0

0

Fact

or R

eten

tion 29

kp

Size

Low

Com

mun

ality

Wid

e C

omm

unal

ityH

igh

Com

mun

ality

SEC

NG

Scr

eeM

RT

M-

PAN

T},

PA

L-L

PA

LL

MA

P K

SR K

SRSE

CN

G S

cree

MR

TSE

PAS

PA 1

. PA

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MA

P K

SR M

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CN

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TM

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AL

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KSR

KSR

39

3n3.

32.

30.

00.

02.

11.

60.

51.

11.

73.

33.

12.

50.

00.

02.

32.

31.

01.

51.

91.

93.

12.

60.

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02.

42.

32.

42.

02.

22.

45p

3.2

1.8

0.0

0.0

2.2

1.8

0.5

1.1

1.3

3.1

3.1

2.3

0.0

0.0

2.5

2.5

1.1

1.5

1.7

1.7

3.1

2.6

0.0

0.0

2.5

2.5

2.5

2.1

2.1

2.4

10p

3.1

1.7

0.0

0.0

2.3

1.9

0.6

1.1

1.1

2.8

3.0

2.3

0.0

0.0

2.7

2.7

1.2

1.5

1.5

1.5

3.1

2.6

0.0

0.0

2.6

2.6

2.6

2.2

2.1

2.4

20p

3.1

1.7

0.0

0.0

2.4

2.0

0.7

1.0

1.1

2.5

3.0

2.4

0.0

0.0

2.8

2.8

1.2

1.4

1.4

1.4

3.1

2.6

0.0

0.0

2.7

2.6

2.7

2.3

2.1

2.4

40p

3.1

1.8

0.0

0.0

2.4

2.1

0.8

1.0

1.0

2.3

3.0

2.4

0.0

0.0

2.9

2.9

1.3

1.4

1.4

1.4

3.1

2.6

0.0

0.0

2.7

2.6

2.7

2.4

2.1

2.4

315

3p3.

14.

43.

14.

33.

22.

61.

21.

62.

75.

13.

14.

03.

14.

23.

03.

02.

02.

72.

92.

93.

13.

13.

24.

03.

02.

93.

02.

92.

93.

05p

3.1

3.3

3.1

4.0

3.2

2.7

1.2

1.7

2.2

4.7

3.1

3.2

3.1

4.1

3.0

3.0

2.2

2.8

2.8

2.8

3.1

3.0

3.2

3.9

3.0

3.0

3.0

3.0

2.9

3.0

10p

3.1

2.9

3.0

3.7

3.2

2.9

1.2

1.7

2.0

4.0

3.1

3.0

3.0

3.9

3.0

3.0

2.4

2.9

2.8

2.8

3.1

3.0

2.6

3.1

3.0

3.0

3.0

3.0

3.0

3.0

20D

3.1

2.9

2.1

2.5

3.1

3.0

1.3

1.7

1.8

3.2

3.1

3.0

1.7

2.2

3.0

3.0

2.5

2.9

2.8

2.8

3.1

3.0

0.3

0.4

3.0

3.0

3.0

3.0

3.0

3.0

40p

3.0

3.0

0.2

0.2

3.0

3.0

1.3

1.7

1.8

3.0

3.1

3.0

0.2

0.3

3.0

3.0

2.6

2.9

2.8

2.8

3.1

3.0

0.0

0.0

3.0

3.0

3.0

3.0

3.0

3.0

330

3p3.

210

.23.

24.

83.

23.

02.

83.

03.

68.

83.

18.

13.

14.

43.

03.

03.

03.

03.

03.

03.

14.

73.

14.

03.

03.

03.

03.

03.

03.

05p

3.2

6.6

3.3

4.1

3.1

3.0

2.9

3.0

3.0

7.6

3.1

5.7

3.1

4.1

3.0

3.0

3.0

3.0

3.0

3.0

3.1

3.2

3.2

4.0

3.0

3.0

3.0

3.0

3.0

3.0

10p

3.2

3.6

3.3

4.1

3.0

3.0

2.9

3.0

3.0

5.5

3.1

3.5

3.2

4.1

3.0

3.0

3.0

3.0

3.0

3.0

3.1

3.0

3.2

4.1

3.0

3.0

3.0

3.0

3.0

3.0

20p

3.2

3.0

3.4

4.1

3.0

3.0

3.0

3.0

3.0

3.4

3.1

3.0

3.2

4.0

3.0

3.0

3.0

3.0

3.0

3.0

3.1

3.0

3.2

4.1

3.0

3.0

3.0

3.0

3.0

3.0

40p

3.2

3.0

3.4

4.1

3.0

3.0

3.0

3.0

3.0

3.0

3.1

3.0

3.2

4.0

3.0

3.0

3.0

3.0

3.0

3.0

3.0

3.0

3.2

4.1

3.0

3.0

3.0

3.0

3.0

3.0

515

3n3.

34.

83.

24.

63.

82.

80.

81.

32.

85.

53.

24.

83.

24.

43.

83.

81.

72.

23.

13.

13.

54.

83.

35.

04.

24.

04.

23.

03.

74.

25D

3.2

3.8

3.1

4.3

4.0

3.0

0.9

1.3

2.1

5.3

3.2

4.2

3.1

4.2

4.1

4.1

1.9

2.2

2.7

2.7

3.6

4.8

3.3

4.5

4.6

4.3

4.6

3.2

3.6

4.2

10p

3.1

3.4

3.1

4.0

4.4

3.4

1.0

1.2

1.7

4.9

3.2

4.0

3.1

3.8

4.5

4.5

2.0

2.3

2.5

2.5

3.7

4.8

2.2

2.6

4.8

4.6

4.8

3.4

3.6

4.2

20p

3.1

3.5

2.4

2.9

4.6

3.8

1.0

1.1

1.4

4.5

3.2

4.0

2.1

2.5

4.7

4.7

2.1

2.3

2.4

2.4

3.9

4.8

0.2

0.2

4.9

4.8

4.9

3.4

3.6

4.2

40p

3.1

3.6

0.5

0.6

4.6

4.1

1.0

1.1

1.2

4.4

3.3

4.1

0.5

0.6

4.7

4.7

2.2

2.3

2.4

2.4

3.9

4.9

0.0

0.0

5.0

4.9

5.0

3.4

3.5

4.2

525

3p3.

29.

03.

17.

25.

14.

11.

92.

54.

38.

53.

37.

93.

16.

64.

74.

73.

24.

24.

54.

53.

95.

63.

16.

25.

04.

95.

04.

74.

95.

05p

3.2

6.6

3.1

6.5

5.2

4.4

1.9

2.6

3.6

7.8

3.3

6.2

3.0

6.2

4.9

4.9

3.4

4.4

4.3

4.3

4.0

5.0

3.1

6.4

5.0

5.0

5.0

4.8

4.9

5.0

10p

3.2

5.1

3.1

6.1

5.2

4.7

2.0

2.7

3.1

6.5

3.4

5.1

3.0

6.1

5.0

5.0

3.5

4.5

4.2

4.2

4.2

5.0

3.1

6.6

5.0

5.0

5.0

4.8

4.9

5.0

20p

3.2

4.9

3.1

6.0

5.1

4.9

2.1

2.8

2.9

5.2

3.5

5.0

3.0

6.1

5.0

5.0

3.6

4.6

4.1

4.1

4.3

5.0

3.1

6.5

5.0

5.0

5.0

4.9

4.9

5.0

40p

3.3

4.9

3.1

5.8

5.0

5.0

2.1

2.8

2.8

5.0

3.6

5.0

3.1

6.0

5.0

5.0

3.6

4.6

4.1

4.1

4.4

5.0

3.2

4.7

5.0

5.0

5.0

4.9

4.9

5.0

550

313

3.3

21.2

3.0

11.3

5.1

5.0

4.3

5.0

5.9

14.7

3.8

16.9

3.0

7.3

5.0

5.0

5.0

5.0

5.0

5.0

4.6

11.5

3.0

7.0

5.0

5.0

5.0

4.9

5.0

5.0

513

3.4

14.7

3.0

8.0

5.0

5.0

4.4

5.0

5.0

12.9

3.9

12.7

3.0

7.2

5.0

5.0

5.0

5.0

5.0

5.0

4.9

7.2

3.0

7.2

5.0

5.0

5.0

5.0

5.0

5.0

10p

3.6

8.3

3.0

7.4

5.0

5.0

4.5

5.0

5.0

9.4

4.1

8.3

3.0

7.3

5.0

5.0

5.0

5.0

5.0

5.0

5.2

5.0

3.0

7.3

5.0

5.0

5.0

4.9

5.0

5.0

20p

3.9

5.2

3.0

7.4

5.0

5.0

4.5

5.0

5.0

5.8

4.3

5.5

3.0

7.4

5.0

5.0

5.0

5.0

5.0

5.0

5.3

5.0

3.0

7.3

5.0

5.0

5.0

4.9

5.0

5.0

40p

4.1

5.0

3.0

7.4

5.0

5.0

4.4

5.0

5.0

5.0

4.5

5.0

3.0

7.4

5.0

5.0

5.0

5.0

5.0

5.0

5.4

5.0

3.0

7.3

5.0

5.0

5.0

5.0

5.0

5.0

721

3n3.

47.

83.

26.

55.

64.

01.

21.

64.

07.

73.

47.

63.

26.

95.

75.

72.

73.

04.

54.

53.

36.

73.

17.

25.

85.

45.

84.

05.

05.

75p

3.3

6.2

3.2

6.2

6.0

4.5

1.2

1.6

3.1

7.4

3.5

6.7

3.2

6.9

6.2

6.2

3.0

3.2

4.2

4.2

3.2

6.5

3.1

7.0

6.2

5.9

6.2

4.4

4.9

5.7

10D

3.3

5.4

3.1

5.9

6.3

5.0

1.2

1.5

2.4

6.8

3.5

6.3

3.2

6.7

6.5

6.5

3.1

3.3

3.9

3.9

3.1

6.5

3.1

5.9

6.5

6.3

6.5

4.7

4.8

5.7

20p

3.3

5.4

3.1

5.3

6.5

5.4

1.3

1.5

2.0

6.4

3.6

6.2

3.3

5.9

6.6

6.6

3.2

3.3

3.8

3.8

3.1

6.5

3.0

4.1

6.6

6.4

6.6

4.8

4.8

5.7

40p

3.3

5.6

3.0

4.2

6.6

5.8

1.3

1.5

1.9

6.2

3.7

6.2

3.3

5.0

6.7

6.7

3.3

3.3

3.7

3.7

3.0

6.5

3.0

3.5

6.8

6.5

6.8

4.9

4.8

5.7

735

31)

3.5

14.4

3.1

10.7

6.9

5.8

2.9

3.8

6.1

11.8

3.2

12.2

3.1

9.2

6.6

6.6

4.8

6.1

6.4

6.4

3.5

8.9

3.1

9.2

6.9

6.8

6.9

6.2

6.8

7.0

5D3.

510

.63.

110

.17.

16.

23.

04.

15.

210

.93.

29.

83.

08.

86.

86.

85.

06.

36.

26.

23.

57.

23.

09.

67.

07.

07.

06.

36.

87.

010

p3.

67.

63.

09.

77.

26.

63.

14.

24.

79.

13.

27.

63.

08.

96.

96.

95.

16.

46.

06.

03.

57.

03.

09.

87.

07.

07.

06.

46.

77.

020

p3.

76.

83.

09.

57.

06.

83.

14.

34.

57.

23.

27.

03.

09.

06.

96.

95.

16.

35.

95.

93.

57.

03.

09.

87.

07.

07.

06.

36.

77.

040

p3.

86.

83.

09.

26.

96.

83.

14.

34.

46.

83.

16.

93.

08.

97.

07.

05.

16.

35.

85.

83.

57.

03.

08.

37.

07.

07.

06.

36.

67.

07

703D

3.0

33.1

3.0

20.4

7.0

7.0

6.0

7.0

8.2

20.5

3.1

26.3

3.0

11.1

7.0

7.0

6.9

7.0

7.0

7.0

3.3

19.4

3.0

10.2

7.0

7.0

7.0

6.9

7.0

7.0

5D3.

024

.13.

013

.07.

07.

06.

17.

07.

017

.93.

120

.73.

010

.77.

07.

07.

07.

07.

07.

03.

212

.93.

010

.57.

07.

07.

07.

07.

07.

010

03.

014

.73.

011

.27.

07.

06.

07.

07.

013

.13.

114

.63.

010

.87.

07.

07.

07.

07.

07.

03.

27.

53.

010

.67.

07.

07.

07.

07.

07.

020

o3.

08.

73.

011

.17.

07.

05.

97.

07.

07.

93.

19.

83.

010

.97.

07.

06.

97.

07.

07.

03.

17.

03.

010

.67.

07.

07.

07.

07.

07.

040

o3.

07.

03.

011

.07.

07.

05.

87.

07.

07.

03.

17.

13.

011

.07.

07.

06.

97.

07.

07.

03.

07.

03.

010

.77.

07.

07.

07.

07.

07.

0

Tab

le 3

Mea

n N

umbe

r of

Fac

tors

Ret

aine

d, P

hi =

0.3

0

Fact

or R

eten

tion 30

kp

Size

Low

Com

mun

ality

Wid

e C

omm

unal

ityH

igh

Com

mun

ality

SEC

NG

Scr

eeM

RT

M-

PAS

PAL

, PA

LL

MA

P K

SR K

SRSE

CN

G S

cree

MR

TSE

PAS

PAL

. PA

u, M

AP

KSR

M-K

SR C

NG

Scr

ee M

RT

M-

PAS

PAL

I, P

AL

L M

AP

KSR

KSR

39

3n3.

12.

10.

00.

01.

81.

50.

71.

11.

53.

23.

02.

20.

00.

01.

81.

81.

01.

21.

51.

53.

02.

50.

00.

02.

12.

02.

11.

91.

92.

25o

3.0

1.6

0.0

0.0

1.9

1.5

0.8

1.0

1.1

3.0

3.0

1.9

0.0

0.0

2.0

2.0

1.0

1.2

1.3

1.3

3.0

2.5

0.0

0.0

2.4

2.3

2.4

2.0

1.8

2.2

100

3.0

1.3

0.0

0.0

2.0

1.5

0.9

1.0

1.0

2.6

3.0

1.8

0.0

0.0

2.4

2.4

1.0

1.2

1.1

1.1

3.0

2.5

0.0

0.0

2.7

2.5

2.7

2.1

1.8

2.2

20p

3.0

1.3

0.0

0.0

2.2

1.7

1.0

1.0

1.0

2.3

3.0

1.8

0.0

0.0

2.7

2.7

1.0

1.2

1.1

1.1

3.0

2.6

0.0

0.0

2.8

2.6

2.8

2.1

1.7

2.3

40D

3.0

1.3

0.0

0.0

2.3

1.9

1.0

1.0

1.0

2.2

3.0

1.8

0.0

0.0

2.8

2.8

1.0

1.2

1.0

1.0

3.0

2.6

0.0

0.0

2.9

2.7

2.9

2.2

1.7

2.2

315

3p3.

04.

13.

04.

82.

52.

01.

01.

42.

34.

93.

03.

83.

04.

62.

52.

51.

32.

22.

42.

43.

03.

13.

04.

32.

82.

72.

82.

92.

72.

95o

3.0

3.0

3.0

4.3

2.7

2.1

1.0

1.4

1.7

4.4

3.0

3.2

3.0

4.3

2.8

2.8

1.4

2.3

2.2

2.2

3.0

3.0

3.0

4.2

2.9

2.9

2.9

3.0

2.7

2.9

101,

3.0

2.5

3.0

3.9

2.8

2.4

1.0

1.3

1.4

3.7

3.0

2.9

2.9

3.9

3.0

3.0

1.5

2.4

2.1

2.1

3.0

3.0

2.5

3.4

3.0

3.0

3.0

3.0

2.7

2.9

20D

3.0

2.5

2.3

2.9

2.9

2.6

1.0

1.3

1.2

3.0

3.0

3.0

2.0

2.6

3.0

3.0

1.7

2.4

2.0

2.0

3.0

3.0

0.5

0.6

3.0

3.0

3.0

3.0

2.7

2.9

40o

3.0

2.6

0.4

0.6

3.0

2.7

1.0

1.3

1.2

2.9

3.0

3.0

0.4

0.5

3.0

3.0

1.7

2.4

2.0

2.0

3.0

3.0

0.0

0.0

3.0

3.0

3.0

3.0

2.7

2.9

330

3o3.

010

.23.

011

.73.

12.

81.

72.

63.

68.

93.

08.

13.

09.

23.

03.

02.

73.

03.

03.

03.

04.

83.

04.

53.

03.

03.

03.

03.

03.

05p

3.0

6.5

3.0

9.2

3.1

3.0

1.8

2.7

2.9

7.8

3.0

5.6

3.0

6.2

3.0

3.0

2.8

3.0

3.0

3.0

3.0

3.2

3.0

4.4

3.0

3.0

3.0

3.0

3.0

3.0

10p

3.0

3.6

3.0

6.2

3.1

3.0

1.9

2.8

2.8

5.8

3.0

3.4

3.0

4.8

3.0

3.0

2.9

3.0

3.0

3.0

3.0

3.0

3.0

4.4

3.0

3.0

3.0

3.0

3.0

3.0

20p

3.0

3.0

3.0

4.8

3.0

3.0

2.0

2.9

2.6

3.6

3.0

3.0

3.0

4.7

3.0

3.0

3.0

3.0

3.0

3.0

3.0

3.0

3.0

4.4

3.0

3.0

3.0

3.0

3.0

3.0

40o

3.0

3.0

3.0

4.5

3.0

3.0

2.0

2.9

2.5

3.0

3.0

3.0

3.0

4.7

3.0

3.0

3.0

3.0

3.0

3.0

3.0

3.0

3.0

4.4

3.0

3.0

3.0

3.0

3.0

3.0

515

3n3.

04.

23.

05.

12.

41.

71.

01.

12.

25.

23.

04.

33.

05.

22.

62.

61.

11.

72.

42.

43.

04.

23.

05.

52.

92.

62.

92.

72.

73.

35p

3.0

2.8

3.0

4.7

2.5

1.7

1.0

1.1

1.4

4.9

3.0

3.5

3.0

4.8

3.1

3.1

1.0

1.7

1.9

1.9

3.0

4.1

3.0

4.9

3.6

3.2

3.6

2.9

2.5

3.3

10D

3.0

2.0

3.0

4.2

2.9

1.9

1.0

1.0

1.0

4.3

3.0

3.2

3.0

4.2

3.7

3.7

1.0

1.8

1.6

1.6

3.0

4.2

2.6

3.5

4.2

3.8

4.2

3.1

2.4

3.3

20p

3.0

1.9

2.7

3.6

3.4

2.3

1.0

1.0

1.0

3.7

3.0

3.3

2.4

3.1

4.1

4.1

1.0

1.8

1.4

1.4

3.0

4.3

0.5

0.7

4.5

4.2

4.5

3.1

2.3

3.2

400

3.0

2.0

1.2

1.5

3.8

2.7

1.0

1.0

1.0

3.3

3.0

3.3

1.0

1.3

4.3

4.3

1.0

1.8

1.3

1.3

3.0

4.3

0.0

0.0

4.7

4.4

4.7

3.2

2.2

3.2

525

3p3.

08.

63.

011

.23.

42.

41.

01.

53.

68.

23.

07.

53.

011

.33.

63.

61.

63.

33.

73.

73.

05.

53.

010

.94.

13.

84.

14.

74.

04.

65o

3.0

6.0

3.0

10.1

3.9

2.8

1.0

1.5

2.4

7.5

3.0

6.0

3.0

10.3

4.3

4.3

1.7

3.5

3.2

3.2

3.0

5.0

3.0

10.0

4.7

4.5

4.7

4.8

3.9

4.6

10D

3.0

4.5

3.0

9.2

4.7

3.5

1.0

1.4

1.7

6.3

3.0

5.0

3.0

9.3

4.9

4.9

1.9

3.7

3.0

3.0

3.0

4.9

3.0

8.8

4.9

4.9

4.9

4.9

3.8

4.6

200

3.0

4.4

3.0

8.4

5.0

4.2

1.0

1.4

1.4

5.2

3.0

5.0

3.0

8.2

5.0

5.0

2.0

3.8

2.8

2.8

3.0

4.9

3.0

7.3

4.9

4.9

4.9

4.9

3.8

4.7

40p

3.0

4.5

3.0

7.4

5.1

4.7

1.0

1.4

1.3

4.9

3.0

5.0

3.0

7.0

5.0

5.0

2.1

3.8

2.7

2.7

3.0

4.9

3.0

5.4

5.0

4.9

5.0

4.9

3.8

4.6

550

3p3.

020

.83.

026

.94.

54.

02.

14.

15.

714

.43.

016

.83.

027

.05.

05.

03.

95.

05.

05.

03.

011

.43.

023

.85.

04.

95.

05.

05.

05.

05p

3.0

14.4

3.0

24.7

4.8

4.6

2.2

4.3

4.6

12.6

3.0

12.7

3.0

24.6

5.0

5.0

4.2

5.0

5.0

5.0

3.0

7.2

3.0

19.5

5.0

5.0

5.0

5.0

5.0

5.0

10p

3.0

8.1

3.0

22.3

5.0

4.9

2.3

4.4

4.2

9.2

3.0

8.4

3.0

21.2

5.0

5.0

4.4

5.0

5.0

5.0

3.0

5.0

3.0

15.2

5.0

5.0

5.0

5.0

5.0

5.0

20p

3.0

5.2

3.0

19.9

5.0

5.0

2.4

4.5

4.0

5.6

3.0

5.6

3.0

17.6

5.0

5.0

4.4

5.0

5.0

5.0

3.0

5.0

3.0

13.3

5.0

5.0

5.0

5.0

5.0

5.0

40p

3.0

5.0

3.0

17.3

5.0

5.0

2.5

4.5

3.9

5.0

3.0

5.0

3.0

14.7

5.0

5.0

4.4

5.0

5.0

5.0

3.0

5.0

3.0

12.4

5.0

5.0

5.0

5.0

5.0

5.0

721

3n3.

07.

03.

08.

73.

22.

11.

01.

23.

17.

33.

06.

73.

09.

03.

43.

41.

32.

33.

33.

33.

06.

33.

09.

44.

23.

64.

23.

43.

74.

751

33.

04.

93.

08.

13.

62.

21.

01.

11.

96.

93.

05.

73.

08.

34.

44.

41.

32.

42.

72.

73.

06.

13.

08.

45.

24.

65.

23.

73.

64.

710

p3.

03.

63.

07.

34.

52.

81.

01.

01.

16.

13.

05.

23.

07.

55.

55.

51.

42.

42.

22.

23.

06.

13.

06.

96.

05.

46.

03.

93.

44.

720

p3.

03.

43.

06.

75.

43.

51.

01.

01.

05.

43.

05.

23.

06.

76.

06.

01.

52.

52.

02.

03.

06.

23.

05.

16.

45.

96.

44.

03.

34.

640

p3.

03.

63.

05.

75.

84.

21.

01.

01.

05.

03.

05.

43.

05.

46.

26.

21.

62.

51.

91.

93.

06.

33.

03.

96.

76.

26.

74.

23.

34.

77

3530

3.0

13.6

3.0

17.7

4.0

3.0

1.1

2.1

4.8

11.0

3.0

12.1

3.0

17.9

5.1

5.1

2.1

4.5

5.2

5.2

3.0

8.7

3.0

17.7

5.7

5.3

5.7

6.4

5.6

6.4

5p3.

09.

73.

016

.25.

03.

71.

12.

134

10.0

3.0

9.7

3.0

16.4

6.0

6.0

2.3

4.8

4.6

4.6

3.0

7.2

3.0

16.0

6.5

6.2

6.5

6.6

5.4

6.4

10v

3.0

6.8

3.0

14.7

6.0

4.8

1.1

2.1

2.5

8.2

3.0

7.6

3.0

14.6

6.8

6.8

2.4

5.0

4.2

4.2

3.0

7.0

3.0

13.9

6.8

6.7

6.8

6.7

5.3

6.5

20D

3.0

6.1

3.0

13.2

6.6

5.5

1.1

2.1

2.1

6.6

3.0

7.0

3.0

13.1

7.0

7.0

2.5

5.2

4.0

4.0

3.0

7.0

3.0

11.5

7.0

6.9

7.0

6.8

5.3

6.5

400

3.0

6.1

3.0

11.6

6.8

6.0

1.1

2.1

1.9

6.3

3.0

7.0

3.0

11.4

7.0

7.0

2.6

5.3

3.9

3.9

3.0

7.0

3.0

8.9

7.0

7.0

7.0

6.8

5.2

6.5

770

3p3.

032

.73.

039

.36.

15.

52.

96.

08.

020

.13.

026

.33.

039

.46.

96.

95.

27.

07.

07.

03.

019

.63.

039

.16.

96.

96.

97.

07.

07.

05D

3.0

23.6

3.0

36.2

6.8

6.5

3.0

6.3

6.6

17.6

3.0

20.7

3.0

36.2

7.0

7.0

5.6

7.0

7.0

7.0

3.0

13.0

3.0

35.7

7.0

7.0

7.0

7.0

7.0

7.0

10p

3.0

14.3

3.0

33.0

7.0

6.9

3.0

6.4

6.2

12.8

3.0

14.5

3.0

32.7

7.0

7.0

5.7

7.0

6.9

6.9

3.0

7.6

3.0

31.9

7.0

7.0

7.0

7.0

7.0

7.0

20D

3.0

8.6

3029

.97.

07.

03.

06.

55.

97.

83.

09.

730

29.4

7.0

7.0

5.6

7.0

6.9

6.9

3.0

7.0

3.0

28.4

7.0

7.0

7.0

7.0

7.0

7.0

40p

3.0

7.0

3.0

26.7

7.0

7.0

3.1

6.5

5.7

7.0

3.0

7.1

3.0

26.2

7.0

7.0

5.5

7.0

6.9

6.9

3.0

7.0

3.0

24.7

7.0

7.0

7.0

7.0

7.0

7.0

Fact

or R

eten

tion 31

Tab

le 4

Mea

n N

umbe

r of

Fac

tors

Ret

aine

d, P

hi =

0.5

0

kp

Low

Com

mun

ality

Wid

e C

omm

unal

ityH

igh

Com

mun

ality

SESi

ze C

NG

Scr

eeM

RT

M-

PAM

H P

AIL

PA

LL

MA

P K

SR K

SRSE

CN

G S

cree

MR

TSE

PAm

H P

AIL

PA

u, M

AP

KSR

M-K

SR C

NG

Scr

ee M

R -

TM

-PA

mH

PA

cL P

AL

L M

AP

KSR

KSR

39

3n3.

12.

00.

00.

01.

71.

40.

81.

11.

43.

03.

01.

90.

00.

01.

51.

51.

01.

21.

31.

33.

02.

10.

00.

01.

61.

41.

61.

61.

41.

9

5v3.

01.

40.

00.

01.

61.

30.

91.

01.

12.

83.

01.

60.

00.

01.

61.

61.

01.

21.

11.

13.

02.

00.

00.

01.

81.

61.

81.

61.

31.

8

10D

3.0

1.2

0.0

0.0

1.7

1.3

1.0

1.0

1.0

2.4

3.0

1.4

0.0

0.0

1.9

1.9

1.0

1.2

1.0

1.0

3.0

2.0

0.0

0.0

2.2

2.0

2.2

1.7

1.2

1.7

20p

3.0

1.1

0.0

0.0

1.8

1.4

1.0

1.0

1.0

2.0

3.0

1.4

0.0

0.0

2.2

2.2

1.0

1.1

1.0

1.0

3.0

2.1

0.0

0.0

2.5

2.3

2.5

1.7

1.1

1.7

400

3.0

1.1

0.0

0.0

1.9

1.5

1.0

1.0

1.0

1.8

3.0

1.4

0.0

0.0

2.5

2.5

1.0

1.1

1.0

1.0

3.0

2.1

0.0

0.0

2.7

2.4

2.7

1.7

1.1

1.7

315

3D3.

03.

93.

05.

22.

01.

51.

01.

12.

04.

83.

03.

63.

05.

11.

91.

91.

11.

82.

02.

03.

03.

03.

04.

92.

11.

92.

12.

72.

12.

65p

3.0

2.5

3.0

4.7

2.0

1.5

1.0

1.1

1.2

4.4

3.0

2.9

3.0

4.6

2.3

2.3

1.1

1.9

1.7

1.7

3.0

3.0

3.0

4.5

2.6

2.4

2.6

2.8

2.0

2.6

10D

3.0

1.9

3.0

4.3

2.4

1.7

1.0

1.0

1.0

3.7

3.0

2.6

3.0

4.1

2.7

2.7

1.1

1.9

1.4

1.4

3.0

3.0

2.7

3.7

2.9

2.8

2.9

2.8

1.9

2.6

20D

3.0

1.8

2.6

3.5

2.7

2.0

1.0

1.0

1.0

3.0

3.0

2.7

2.3

2.9

2.9

2.9

1.1

2.0

1.4

1.4

3.0

3.0

0.7

0.9

3.0

3.0

3.0

2.9

1.9

2.6

40p

3.0

1.9

1.0

1.3

2.9

2.4

1.0

1.0

1.0

2.7

3.0

2.7

0.7

0.9

3.0

3.0

1.1

2.0

1.3

1.3

3.0

3.0

0.0

0.0

3.0

3.0

3.0

2.9

1.9

2.6

330

3D3.

010

.03.

014

.32.

52.

11.

12.

03.

38.

53.

08.

13.

013

.52.

82.

81.

83.

03.

03.

03.

04.

73.

08.

13.

02.

93.

03.

03.

03.

05D

3.0

6.3

3.0

12.6

2.8

2.5

1.1

2.1

2.5

7.4

3.0

5.6

3.0

10.9

2.9

2.9

2.0

3.0

2.9

2.9

3.0

3.2

3.0

5.7

3.0

3.0

3.0

3.0

3.0

3.0

10D

3.0

3.5

3.0

10.4

3.0

2.9

1.1

2.2

2.1

5.4

3.0

3.5

3.0

7.1

3.0

3.0

2.2

3.0

2.8

2.8

3.0

3.0

3.0

5.0

3.0

3.0

3.0

3.0

3.0

3.0

20D

3.0

3.0

3.0

8.0

3.0

3.0

1.1

2.2

1.9

3.3

3.0

3.0

3.0

5.3

3.0

3.0

2.3

3.0

2.8

2.8

3.0

3.0

3.0

4.9

3.0

3.0

3.0

3.0

3.0

3.0

40D

3.0

3.0

3.0

6.1

3.0

3.0

1.1

2.2

1.7

3.0

3.0

3.0

3.0

4.8

3.0

3.0

2.4

3.0

2.7

2.7

3.0

3.0

3.0

4.9

3.0

3.0

3.0

3.0

3.0

3.0

515

3n3.

03.

93.

05.

22.

01.

51.

01.

12.

05.

03.

03.

73.

05.

51.

61.

61.

01.

41.

81.

83.

03.

63.

05.

52.

01.

72.

02.

32.

02.

75D

3.0

2.5

3.0

4.7

2.0

1.4

1.0

1.0

1.2

4.6

3.0

3.0

3.0

5.0

2.0

2.0

1.0

1.3

1.3

1.3

3.0

3.5

3.0

4.9

2.7

2.2

2.7

2.5

1.8

2.6

10D

3.0

1.6

3.0

4.2

2.2

1.4

1.0

1.0

1.0

4.0

3.0

2.6

3.0

4.3

2.9

2.9

1.0

1.3

1.0

1.0

3.0

3.5

2.8

3.8

3.4

3.0

3.4

2.6

1.6

2.5

20D

3.0

1.4

2.8

3.6

2.7

1.6

1.0

1.0

1.0

3.2

3.0

2.6

2.5

3.3

3.6

3.6

1.0

1.2

1.0

1.0

3.0

3.6

1.0

1.2

3.8

3.5

3.8

2.7

1.5

2.5

40p

3.0

1.3

1.5

1.9

3.3

1.9

1.0

1.0

1.0

2.7

3.0

2.7

1.3

1.6

4.0

4.0

1.0

1.2

1.0

1.0

3.0

3.6

0.0

0.1

4.1

3.7

4.1

2.7

1.4

2.5

525

3D3.

07.

93.

011

.61.

91.

41.

01.

22.

97.

63.

07.

13.

011

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42.

41.

12.

63.

03.

03.

05.

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011

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62.

22.

64.

13.

03.

8

5p3.

05.

23.

010

.52.

21.

51.

01.

11.

66.

83.

05.

63.

010

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33.

31.

12.

72.

42.

43.

04.

93.

010

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73.

13.

74.

32.

73.

7

10p

3.0

3.3

3.0

9.3

3.0

2.0

1.0

1.1

1.1

5.4

3.0

4.7

3.0

9.6

4.3

4.3

1.1

2.9

2.1

2.1

3.0

4.9

3.0

9.2

4.6

4.2

4.6

4.5

2.5

3.7

20D

3.0

3.0

3.0

8.3

3.8

2.6

1.0

1.1

1.0

4.2

3.0

4.7

3.0

8.5

4.8

4.8

1.1

2.9

1.9

1.9

3.0

5.0

3.0

7.6

4.9

4.7

4.9

4.5

2.4

3.7

40D

3.0

3.1

3.0

7.5

4.4

3.2

1.0

1.0

1.0

3.7

3.0

4.7

3.0

7.2

4.9

4.9

1.1

3.0

1.7

1.7

3.0

5.0

3.0

5.8

5.0

4.9

5.0

4.5

2.4

3.7

550

3v3.

020

.73.

027

.12.

92.

21.

12.

75.

414

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016

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027

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04.

02.

14.

84.

84.

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011

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027

.04.

64.

44.

65.

04.

95.

05o

3.0

14.2

3.0

25.0

3.9

3.1

1.0

2.9

3.7

12.5

3.0

12.2

3.0

25.0

4.6

4.6

2.4

4.9

4.4

4.4

3.0

7.0

3.0

24.6

5.0

4.9

5.0

5.0

4.9

5.0

10D

3.0

8.0

3.0

22.8

4.8

4.3

1.0

3.0

2.8

9.1

3.0

8.0

3.0

22.5

5.0

5.0

2.7

5.0

4.1

4.1

3.0

5.0

3.0

21.9

5.0

5.0

5.0

5.0

4.9

5.0

20D

3.0

5.2

3.0

20.7

4.9

4.8

1.0

3.1

2.3

5.6

3.0

5.4

3.0

20.2

5.0

5.0

2.8

5.0

4.0

4.0

3.0

5.0

3.0

19.5

5.0

5.0

5.0

5.0

4.9

5.0

40D

3.0

5.0

3.0

18.6

5.0

4.9

1.0

3.1

2.1

5.0

3.0

5.0

3.0

18.0

5.0

5.0

2.9

5.0

3.9

193.

05.

03.

017

.05.

05.

05.

05.

04.

95.

07

213n

3.0

6.3

3.0

9.0

1.8

1.3

1.0

1.1

2.5

6.6

3.0

6.1

3.0

9.4

1.8

1.8

1.0

1.5

2.4

2.4

3.0

5.4

3.0

9.5

1.9

1.6

1.9

2.6

2.4

3.5

5p3.

04.

03.

08.

21.

91.

31.

01.

01.

36.

03.

04.

93.

08.

62.

52.

51.

01.

41.

61.

63.

05.

23.

08.

63.

12.

43.

12.

72.

03.

3

10D

3.0

2.5

3.0

7.4

2.6

1.5

1.0

1.0

1.0

5.0

3.0

4.2

3.0

7.7

4.1

4.1

1.0

1.3

1.1

1.1

3.0

5.3

3.0

7.3

4.8

3.8

4.8

2.9

1.7

3.2

20p

3.0

2.1

3.0

6.7

3.7

2.0

1.0

1.0

1.0

4.0

3.0

4.2

3.0

6.9

5.3

5.3

1.0

1.3

1.0

1.0

3.0

5.4

3.0

5.7

5.7

4.8

5.7

3.0

1.4

3.1

40D

3.0

2.2

3.0

5.9

4.7

2.6

1.0

1.0

1.0

3.4

3.0

4.4

3.0

6.0

6.0

6.0

1.0

1.2

1.0

1.0

3.0

5.5

3.0

4.4

6.2

5.4

6.2

3.0

1.3

3.1

735

313

3.0

13.0

3.0

17.9

1.8

1.3

1.0

1.3

4.1

10.6

3.0

11.7

3.0

18.1

2.7

2.7

1.0

3.0

4.1

4.1

3.0

8.4

3.0

17.9

3.5

3.0

3.5

5.5

4.2

5.2

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09.

03.

016

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03.

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03.

016

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94.

34.

95.

83.

95.

1

10p

3.0

5.9

3.0

15.0

4.4

2.4

1.0

1.1

1.1

7.8

3.0

7.1

3.0

14.9

6.0

6.0

1.0

3.3

2.4

2.4

3.0

6.8

3.0

14.2

6.2

5.6

6.2

5.9

3.7

5.1

20D

3.0

5.2

3.0

13.7

6.0

4.0

1.0

1.1

1.1

6.2

3.0

6.5

3.0

13.4

6.5

6.5

1.0

3.3

1.9

1.9

3.0

6.8

3.0

12.0

6.8

6.4

6.8

6.0

3.6

5.1

40p

3.0

5.4

3.0

12.3

6.6

5.0

1.0

1.1

1.0

5.6

3.0

6.5

3.0

11.9

6.8

6.8

1.0

3.4

1.8

1.8

3.0

6.8

3.0

9.5

6.9

6.8

6.9

6.0

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3D3.

032

.43.

039

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22.

41.

13.

77.

519

.83.

025

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039

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22.

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66.

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039

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06.

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051

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023

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014

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75.

91.

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812

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014

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033

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93.

26.

85.

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53.

032

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07.

07.

07.

06.

77.

020

D3.

08.

53.

030

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06.

81.

24.

03.

17.

73.

09.

63.

030

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07.

03.

36.

85.

75.

73.

07.

03.

029

.17.

07.

07.

07.

06.

77.

040

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07.

03.

027

.97.

07.

01.

34.

02.

87.

03.

07.

13.

027

.17.

07.

03.

46.

85.

75.

73.

07.

03.

025

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07.

07.

07.

06.

77.

0

Fact

or R

eten

tion 32

Tab

le 5

Mea

n N

umbe

r of

Fac

tors

Ret

aine

d, P

hi =

Mix

ed

kp

Low

Com

mun

ality

Wid

e C

omm

unal

ityH

igh

Com

mun

ality

SESi

ze C

NG

Scr

eeM

RT

M-

PAS

PAa.

PA

LL

MA

P K

SR K

SRSE

CN

G S

cree

MR

TSE

PAS

P&L

PA

LL

MA

P K

SR M

-KSR

CN

G S

cree

MR

TM

-PA

mH

PA

LL

PA

LL

MA

P K

SR K

SR

39

3n3.

12.

10.

00.

0L

81.

40.

71.

11.

53.

23.

02.

30.

00.

01.

91.

91.

11.

41.

71.

73.

02.

30.

00.

01.

91.

81.

91.

91.

82.

1

5o3.

01.

50.

00.

01.

81.

40.

81.

01.

13.

03.

02.

00.

00.

02.

12.

11.

11.

51.

41.

43.

02.

20.

00.

02.

22.

02.

22.

01.

72.

0

10p

3.0

1.3

0.0

0.0

1.9

1.4

0.9

1.0

1.0

2.6

3.0

1.9

0.0

0.0

2.4

2.4

1.1

1.4

1.3

1.3

3.0

2.2

0.0

0.0

2.4

2.2

2.4

2.0

1.7

2.0

20p

3.0

1.2

0.0

0.0

2.1

1.6

1.0

1.0

1.0

2.2

3.0

1.9

0.0

0.0

2.6

2.6

1.2

1.4

1.2

1.2

3.0

2.3

0.0

0.0

2.6

2.4

2.6

2.0

1.7

2.0

40o

3.0

1.2

0.0

0.0

2.2

1.7

1.0

1.0

1.0

2.0

3.0

1.9

0.0

0.0

2.7

2.7

1.2

1.4

1.2

1.2

3.0

2.3

0.0

0.0

2.8

2.5

2.8

2:0

1.7

2.0

315

3p3.

14.

33.

14.

52.

92.

31.

11.

52.

55.

03.

03.

63.

04.

32.

22.

21.

32.

02.

22.

23.

03.

13.

04.

12.

52.

42.

52.

92.

42.

8

5D3.

03.

23.

04.

13.

02.

51.

11.

52.

04.

63.

02.

93.

03.

82.

42.

41.

42.

12.

02.

03.

03.

03.

03.

92.

92.

72.

92.

92.

32.

8

10p

3.0

2.7

3.0

3.8

3.0

2.7

1.1

1.6

1.7

3.9

3.0

2.7

2.9

3.5

2.7

2.7

1.5

2.1

1.9

1.9

3.0

3.0

2.6

3.3

3.0

3.0

3.0

3.0

2.3

2.8

20p

3.0

2.7

2.2

2.7

3.0

2.8

1.1

1.6

1.6

3.1

3.0

2.7

1.9

2.2

2.9

2.9

1.6

2.1

1.8

1.8

3.0

3.0

0.6

0.7

3.0

3.0

3.0

3.0

2.2

2.8

40p

3.0

2.8

0.3

0.4

3.0

2.9

1.2

1.6

1.5

3.0

3.0

2.8

0.3

0.3

3.0

3.0

1.7

2.1

1.8

1.8

3.0

3.0

0.0

0.0

3.0

3.0

3.0

3.0

2.2

2.8

330

3p3.

010

.23.

07.

02.

92.

61.

92.

53.

58.

83.

08.

13.

05.

02.

92.

92.

33.

03.

03.

03.

04.

73.

04.

73.

03.

03.

03.

03.

03.

0

5p3.

06.

53.

04.

93.

02.

82.

02.

52.

87.

73.

05.

73.

03.

93.

03.

02.

43.

02.

92.

93.

03.

23.

04.

33.

03.

03.

03.

03.

03.

0

10v

3.0

3.6

3.0

3.7

3.0

3.0

2.0

2.5

2.4

5.6

3.0

3.5

3.0

3.7

3.0

3.0

2.5

3.0

2.9

2.9

3.0

3.0

3.0

4.3

3.0

3.0

3.0

3.0

3.0

3.0

20p

3.0

3.0

3.0

3.6

3.0

3.0

2.0

2.6

2.3

3.5

3.0

3.0

3.0

3.7

3.0

3.0

2.6

3.0

2.9

2.9

3.0

3.0

3.0

4.3

3.0

3.0

3.0

3.0

3.0

3.0

40p

3.0

3.0

3.0

3.6

3.0

3.0

2.0

2.6

2.2

3.0

3.0

3.0

3.0

3.7

3.0

3.0

2.6

3.0

2.9

2.9

3.0

3.0

3.0

4.2

3.0

3.0

3.0

3.0

3.0

3.0

515

3n3.

04.

23.

05.

02.

41.

81.

01.

22.

25.

13.

04.

23.

04.

92.

52.

51.

21.

82.

32.

33.

04.

03.

05.

02.

92.

62.

92.

72.

73.

2

5v3.

02.

93.

04.

62.

51.

81.

01.

11.

54.

73.

03.

43.

04.

42.

82.

81.

21.

81.

91.

93.

03.

83.

04.

63.

33.

03.

32.

82.

53.

1

10p

3.0

2.2

3.0

4.1

2.8

2.1

1.0

1.1

1.1

4.1

3.0

3.0

3.0

4.0

3.4

3.4

1.2

1.9

1.7

1.7

3.0

3.8

2.7

3.6

3.8

3.5

3.8

2.9

2.5

3.1

20p

3.0

2.1

2.6

3.3

3.2

2.4

1.0

1.1

1.1

3.4

3.0

3.0

2.5

3.1

3.9

3.9

1.3

1.9

1.6

1.6

3.0

3.9

0.7

0.9

4.2

3.8

4.2

3.0

2.4

3.1

40p

3.0

2.1

1.1

1.3

3.4

2.7

1.0

1.1

1.0

3.1

3.0

3.1

1.0

1.2

4.3

4.3

1.3

1.9

1.5

1.5

3.0

3.9

0.0

0.0

4.4

4.1

4.4

3.0

2.4

3.0

525

3p3.

08.

43.

010

.33.

32.

51.

21.

93.

57.

83.

07.

93.

011

.04.

24.

21.

93.

54.

04.

03.

05.

63.

010

.04.

03.

84.

04.

54.

04.

2

5p3.

05.

83.

09.

13.

62.

91.

21.

92.

57.

13.

06.

23.

010

.24.

74.

72.

13.

83.

63.

63.

05.

03.

09.

24.

24.

14.

24.

63.

94.

1

10p

3.0

4.1

3.0

7.9

4.1

3.4

1.2

1.9

2.0

5.8

3.0

5.1

3.0

9.2

4.9

4.9

2.3

4.0

3.3

3.3

3.0

4.9

3.0

8.4

4.7

4.3

4.7

4.7

4.0

4.1

20p

3.0

3.8

3.0

7.2

4.3

3.7

1.2

1.9

1.8

4.5

3.0

4.9

3.0

8.2

5.0

5.0

2.4

4.1

3.1

3.1

3.0

4.9

3.0

7.2

4.9

4.8

4.9

4.8

4.0

4.1

4013

3.0

3.9

3.0

6.5

4.5

3.9

1.2

1.9

1.7

4.0

3.0

4.9

3.0

7.1

5.0

5.0

2.5

4.1

3.0

3.0

3.0

5.0

3.0

5.5

5.0

4.9

5.0

4.8

4.0

4.0

550

3P3.

020

.83.

027

.03.

52.

91.

53.

25.

514

.43.

016

.33.

027

.14.

74.

73.

55.

05.

05.

03.

011

.53.

015

.04.

94.

84.

95.

05.

05.

0

5p3.

014

.33.

024

.94.

33.

71.

63.

33.

912

.63.

012

.33.

024

.75.

05.

03.

75.

04.

94.

93.

07.

23.

011

.15.

05.

05.

05.

05.

05.

0

1013

3.0

8.1

3.0

22.5

4.9

4.5

1.7

3.3

3.1

9.3

3.0

8.1

3.0

21.9

5.0

5.0

3.8

5.0

4.8

4.8

3.0

5.0

3.0

9.5

5.0

5.0

5.0

5.0

5.0

5.0

20p

3.0

5.2

3.0

20.3

5.0

4.9

1.8

3.4

2.8

5.7

3.0

5.4

3.0

18.9

5.0

5.0

3.9

5.0

4.8

4.8

3.0

5.0

3.0

9.2

5.0

5.0

5.0

5.0

5.0

5.0

40p

3.0

5.0

3.0

17.9

5.0

5.0

1.9

3.3

2.7

5.0

3.0

5.0

3.0

15.9

5.0

5.0

3.9

5.0

4.8

4.8

3.0

5.0

3.0

9.2

5.0

5.0

5.0

5.0

5.0

5.0

721

3n3.

06.

83.

08.

33.

12.

21.

01.

43.

17.

03.

06.

43.

08.

43.

33.

31.

42.

53.

23.

23.

05.

83.

09.

43.

32.

83.

33.

23.

24.

1

5D3.

04.

73.

07.

53.

42.

41.

01.

42.

06.

53.

05.

33.

07.

73.

93.

91.

52.

62.

72.

73.

05.

63.

08.

44.

33.

74.

33.

43.

04.

010

D3.

03.

53.

06.

74.

02.

81.

01.

31.

55.

63.

04.

73.

06.

94.

74.

71.

52.

72.

42.

43.

05.

63.

07.

15.

34.

65.

33.

52.

94.

0

20D

3.0

3.3

3.0

6.1

4,6

3.3

1.0

1.3

1.3

4.7

3.0

4.6

3.0

6.2

5.3

5.3

1.6

2.8

2.3

2.3

3.0

5.8

3.0

5.7

6.1

5.2

6.1

3.5

2.8

4.0

40D

3.0

3.4

3.0

5.4

4.9

3.7

1.0

1.3

1.2

4.2

3.0

4.6

3.0

5.4

5.7

5.7

1.7

2.8

2.3

2.3

3.0

5.9

3.0

4.3

6.3

5.8

6.3

3.5

2.8

4.0

735

3D3.

013

.63.

017

.64.

33.

41.

32.

44.

911

.13.

011

.73.

017

.74.

64.

62.

44.

74.

94.

93.

08.

43.

017

.44.

44.

14.

45.

94.

75.

4

513

3.0

9.6

3.0

16.1

5.0

4.0

1.2

2.5

3.6

10.1

3.0

9.3

3.0

16.1

5.5

5.5

2.6

5.0

4.3

4.3

3.0

6.9

3.0

15.8

5.3

4.9

5.3

6.0

4.5

5.4

10D

3.0

6.5

3.0

14.6

5.8

4.7

1.2

2.5

2.9

8.2

3.0

7.3

3.0

14.3

6.2

6.2

2.8

5.2

4.0

4.0

3.0

6.6

3.0

13.8

6.0

5.7

6.0

6.0

4.3

5.3

20p

3.0

5.8

3.0

13.0

6.1

5.3

1.2

2.5

2.5

6.3

3.0

6.7

3.0

12.7

6.7

6.7

2.9

5.4

3.9

3.9

3.0

6.6

3.0

11.9

6.3

6.0

6.3

6.0

4.2

5.3

4013

3.0

5.9

3.0

11.4

6.2

5.8

1.2

2.6

2.3

6.0

3.0

6.7

3.0

11.1

6.9

6.9

3.0

5.4

3.8

3.8

3.0

6.7

3.0

9.5

6.8

6.2

6.8

6.0

4.1

5.3

770

3o3.

032

.53.

039

.34.

74.

22.

94.

97.

619

.93.

026

.03.

039

.15.

95.

95.

16.

56.

46.

43.

019

.43.

033

.95.

75.

55.

77.

06.

46.

95D

3.0

23.5

3.0

36.2

5.7

5.0

2.9

5.0

5.6

17.3

3.0

20.2

3.0

35.2

6.1

6.1

5.2

6.7

6.0

6.0

3.0

13.0

3.0

30.1

6.3

6.1

6.3

7.0

6.2

6.8

1013

3.0

14.2

3.0

32.9

6.7

6.2

3.0

5.1

4.7

12.6

3.0

14.0

3.0

29.2

6.7

6.7

5.4

6.8

6.0

6.0

3.0

7.6

3.0

24.1

6.9

6.8

6.9

7.0

6.1

6.8

20D

3.0

8.5

3.0

29.6

7.0

6.8

3.0

5.2

4.3

7.7

3.0

9.4

3.0

24.6

6.9

6.9

5.3

6.9

5.9

5.9

3.0

7.0

3.0

20.5

7.0

6.9

7.0

7.0

6.0

6.8

40D

3.0

7.0

3.0

25.9

7.0

7.0

3.0

5.2

4.1

7.0

3.0

7.0

3.0

21.1

6.9

6.9

5.3

6.9

5.9

5.9

3.0

7.0

3.0

18.4

7.0

7.0

7.0

7.0

6.0

6.8

Tab

le 6

Om

ega

Squa

red

for

Mea

n N

umbe

r of

Fac

tors

Ret

aine

d.

Eff

ect

k p/k

N Com

mun

ality

Phi

k X

p/k

k X

Nk

X C

omm

k X

Phi

p/k

X N

p/k

X C

omm

p/k

X P

hiN

X C

omm

N X

Phi

Com

m X

Phi

k X

p/k

X N

k X

p/k

X C

omm

k X

N X

Com

mp/

k X

N X

Com

mk

X p

/k X

Phi

k X

N X

Phi

p/k

X N

X P

hiC

omm

X p

/k X

Phi

Com

m X

k X

Phi

Com

m X

n X

Phi

k X

p/k

X N

X C

omm

k X

p/k

X N

X P

hiC

omm

X k

X p

/k X

Phi

Com

m X

p/k

X n

X P

hiC

omm

X k

X n

X P

hik

X p

/k X

N X

Com

m X

phi

CN

GSE

AM

Rt

Para

llel A

naly

sis

MA

PK

SRM

-KSR

PAS

PAIL

PAL

L

0.02

70.

257

0.25

90.

320

0.61

00.

456

0.11

90.

258

0.27

10.

305

0.00

30.

230

0.23

90.

351

0.10

40.

180

0.34

00.

361

0.38

00.

183

0.00

00.

154

0.04

20.

034

0.06

20.

066

0.00

40.

001

0.03

20.

082

0.00

60.

011

0.00

30.

005

0.01

30.

045

0.14

20.

162

0.05

60.

128

0.16

00.

003

0.00

00.

060

0.05

50.

077

0.17

00.

032

0.08

70.

011

0.01

80.

055

0.25

90.

074

0.01

10.

021

0.03

30.

055

0.04

70.

022

0.00

10.

040

0.02

10.

005

0.01

80.

016

0.00

00.

000

0.00

50.

008

0.01

60.

004

0.00

20.

001

0.00

30.

007

0.01

80.

022

0.00

70.

014

0.08

70.

000

0.00

00.

024

0.01

50.

020

0.04

20.

008

0.02

00.

003

0.00

00.

126

0.02

20.

006

0.01

00.

006

0.00

00.

000

0.00

00.

034

0.00

20.

034

0.00

20.

004

0.00

20.

007

0.01

70.

030

0.00

50.

048

0.01

00.

001

0.00

00.

054

0.01

20.

013

0.02

10.

004

0.01

20.

004

0.00

00.

026

0.00

40.

000

0.00

10.

001

0.00

10.

000

0.01

70.

080

0.00

10.

000

0.00

10.

004

0.01

90.

012

0.00

00.

000

0.00

20.

000

0.01

80.

000

0.00

00.

001

0.00

50.

005

0.00

50.

006

0.00

10.

000

0.00

00.

030

0.10

60.

003

0.00

10.

001

0.00

00.

000

0.00

00.

004

0.00

20.

005

0.00

80.

001

0.00

00.

001

0.00

30.

002

0.00

00.

004

0.00

00.

004

0.00

20.

000

0.00

10.

000

0.00

00.

000

0.00

20.

008

0.00

00.

012

0.00

20.

001

0.00

30.

005

0.00

00.

001

0.00

30.

040

0.05

20.

000

0.00

10.

020

0.00

30.

003

0.00

30.

001

0.00

30.

001

0.00

30.

000

0.00

00.

001

0.00

60.

004

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

003

0.00

40.

004

0.00

10.

000

0.00

10.

000

0.00

60.

000

0.00

00.

002

0.00

10.

003

0.03

50.

016

0.01

30.

000

0.04

60.

000

0.00

00.

002

0.00

20.

002

0.00

40.

002

0.00

10.

000

0.00

00.

000

0.00

00.

000

0.00

10.

000

0.00

00.

000

0.00

00.

000

0.00

00.

002

0.01

10.

000

0.00

10.

001

0.00

00.

000

0.00

00.

004

0.00

00.

000

0.00

10.

001

0.00

10.

001

0.00

00.

000

0.00

00.

000

0.00

80.

000

0.00

00.

003

0.00

10.

001

0.00

40.

004

0.00

20.

001

0.00

00.

000

0.00

00.

000

0.00

30.

005

0.00

10.

000

0.00

20.

000

0.00

00.

000

0.00

00.

001

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

001

0.00

00.

001

0.00

00.

000

0.00

00.

000

Fact

or R

eten

tion 34

Tab

le 7

Prop

ortio

n of

Sam

ples

Ret

aini

ng K

Fac

tors

, Phi

= 0

.00

kp

Low

Com

mun

ality

Wid

e C

omm

unal

ityH

igh

Com

mun

ality

SESi

ze C

NG

Scr

eeM

RT

M-

PAM

H P

AIL

PA

LL

MA

P K

SR K

SRSE

CN

G S

cree

MR

TSE

PAS

Ma,

PA

LL

MA

P K

SR M

-KSR

CN

G S

cree

MR

TM

-PA

S PA

ct P

AL

L M

AP

KSR

KSR

39

3nill

271

022

120

02

6193

400

029

290

26

692

560

042

3542

2421

42

5088

140

025

120

00

6796

370

042

420

21

193

560

054

4754

3020

43

10D

917

00

3214

00

065

9735

00

6262

01

00

9458

00

6157

6136

2143

20D

926

00

3817

00

048

9738

00

7777

01

00

9459

00

6560

6540

2343

40D

928

00

4422

00

034

9740

00

8686

00

00

9460

00

7062

7044

2442

315

3v90

2192

3949

460

763

088

3389

2683

8317

7186

8687

8985

2296

9396

9594

99

Si,

9257

9341

6363

06

232

8975

8719

8989

2980

8080

8910

085

1810

099

100

9894

100

10D

9485

9237

8185

06

721

9095

839

9494

4387

7979

9110

069

1210

010

010

099

9610

0

20p

9693

6321

9297

06

281

9196

483

9898

5489

7979

9110

09

110

010

010

099

9710

0

40D

9698

62

9810

00

71

100

9197

70

9999

6089

7979

9210

00

010

010

010

010

098

100

330

3D82

078

1785

9778

9846

089

087

910

010

099

9997

9793

990

1510

010

010

010

010

010

0

Si,

811

7312

9299

8910

010

00

901

867

100

100

100

100

100

100

9284

8411

100

100

100

100

100

100

10D

8056

699

9710

094

100

100

091

5883

710

010

010

010

010

010

094

100

8210

100

100

100

100

100

100

20D

8010

064

899

100

9810

010

062

9110

081

910

010

010

010

010

010

095

100

8210

100

100

100

100

100

100

40p

8010

062

710

010

010

010

010

010

090

100

7910

100

100

100

100

100

100

9610

081

1010

010

010

010

010

010

0

515

3n5

303

1914

40

00

434

393

2014

140

00

011

739

3236

2236

144

24

Si,

417

328

205

00

055

527

232

2222

00

00

1175

1053

5940

5920

324

10D

37

132

336

00

066

715

I25

4040

00

00

1178

17

8162

8123

225

20D

26

07

5112

00

053

912

04

6060

00

00

1282

00

9377

9324

125

4013

18

00

6321

00

040

1214

00

7070

00

00

1386

00

9888

9825

025

525

3D4

11

1036

270

035

06

11

1063

631

4448

4810

511

197

9397

8993

99

513

519

111

5042

00

20

721

17

8282

254

3030

896

00

100

9910

092

9310

0

10D

577

111

7169

00

05

785

03

9393

361

2020

610

00

010

010

010

094

9210

0

2013

890

19

8987

00

076

796

01

9696

364

1616

310

01

110

010

010

095

9110

0

4013

1094

18

9896

00

010

07

961

299

992

6514

141

100

1159

100

100

100

9590

100

550

3p4

00

195

9737

9523

02

00

010

010

010

010

098

985

00

010

010

010

098

100

100

5D4

00

098

100

4599

100

01

00

010

010

010

010

010

010

03

30

010

010

010

099

100

100

10D

31

00

9910

047

100

100

00

00

010

010

010

010

010

010

01

990

010

010

010

099

100

100

20D

278

00

100

100

4610

010

038

057

00

100

100

100

100

100

100

010

00

010

010

010

099

100

100

40D

110

00

010

010

044

100

100

100

010

00

010

010

010

010

010

010

00

100

00

100

100

100

9910

010

0

721

3n1

240

811

20

00

331

350

1114

140

00

02

591

1212

312

40

3

Si,

124

110

173

00

050

244

016

2727

00

00

252

119

3016

307

03

10D

111

122

305

00

061

336

144

4040

00

00

150

148

4734

4710

02

20D

111

133

4710

00

043

433

039

5252

00

00

151

01

6044

6013

01

40D

115

0I

6318

00

036

632

03

6363

00

00

051

00

7850

7814

00

735

3D2

00

833

160

021

02

00

661

610

4343

435

50

093

8293

8482

99

5v3

20

1045

320

00

03

10

380

800

5325

253

790

010

099

100

8677

100

10D

548

08

6164

00

01

253

01

8989

056

1717

210

00

010

010

010

087

7110

0

20D

982

04

7679

00

071

293

00

9292

057

1616

110

00

010

010

010

085

6610

0

40D

1181

01

8880

00

082

291

00

9797

057

1616

010

00

910

010

010

085

6310

0

770

3o0

00

099

9514

9710

00

00

010

010

091

100

9999

10

00

100

100

100

9910

010

0

513

00

00

100

100

1799

100

00

00

010

010

095

100

100

100

10

00

100

100

100

9910

010

0

1013

00

00

100

100

1410

010

00

00

00

100

100

9510

010

010

01

570

010

010

010

010

010

010

0

20D

011

00

100

100

810

010

027

01

00

100

100

9310

010

010

00

100

00

100

100

100

100

100

100

40p

099

00

100

100

110

010

010

00

890

010

010

090

100

100

100

010

00

010

010

010

010

010

010

0

Fact

or R

eten

tion 35

Tab

le 8

Prop

ortio

n of

Sam

ples

Ret

aini

ng K

Fac

tors

, Phi

= 0

.30

kp

Low

Com

mun

ality

Wid

e C

omm

unal

ityH

igh

Com

mun

ality

SESi

ze C

NG

Scr

eeM

RT

M-

PAS

PAL

LL

, PA

LL

MA

P K

SR K

SRSE

CN

G S

cree

MR

TSE

PAN

ili P

Aci

. PA

LL

MA

P K

SR M

-KSR

CN

G S

cree

MR

TM

-PA

mi.,

PA

S PA

LL

MA

P K

SR K

SR3

93n

9221

10

157

00

164

9827

10

1414

00

11

100

490

025

1525

155

275p

988

00

165

00

064

9917

00

2222

00

00

100

500

045

3045

182

2410

o10

01

00

183

00

053

100

100

041

410

00

010

054

00

6552

6522

124

20D

100

00

024

30

00

3010

08

00

6767

00

00

100

580

078

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00

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00

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100

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010

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094

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00

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010

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010

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010

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0

Fact

or R

eten

tion 36

Tab

le 9

Prop

ortio

n of

Sam

ples

Ret

aini

ng K

Fac

tors

, Phi

= 0

.50

kp

Low

Com

mun

ality

Wid

e C

omm

unal

ityH

igh

Com

mun

ality

SESi

ze C

NG

Scr

eeM

RT

M-

PAN

TH

Ma,

PA

LL

MA

P K

SR K

SRSE

CN

G S

cree

MR

TSE

PAS

PAct

, PA

M. M

AP

KSR

M-K

SR C

NG

Scr

ee M

RT

M-

PAL

m P

Act

PA

LL

MA

P K

SR K

SR

39

3n96

191

011

50

01

6410

018

10

77

00

00

100

231

06

36

41

9

5v99

50

010

30

00

6110

07

00

88

00

00

100

180

014

614

20

5

10p

100

00

010

10

00

3910

01

00

1414

00

00

100

170

033

1533

20

3

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100

00

09

00

00

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00

00

3030

00

00

100

200

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10

2

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00

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00

00

210

00

00

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00

00

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10

2

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100

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100

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100

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07

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6078

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100

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00

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063

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00

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894

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100

852

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583

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00

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10

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00

00

00

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00

00

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00

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00

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00

100

100

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010

00

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010

010

010

070

99

Fact

or R

eten

tion 37

Tab

le 1

0Pr

opor

tion

of S

ampl

es R

etai

ning

K F

acto

rs, P

hi =

Mix

ed

kp

Low

Com

mun

ality

Wid

e C

omm

unal

ityH

igh

Com

mun

ality

SESi

ze C

NG

Scr

eeM

RT

M-

PAm

H P

AcL

PA

LL

MA

P K

SR K

SRSE

CN

G S

cree

MR

TSE

PAM

H P

Act

PA

LL

, MA

P K

SR M

-KSR

CN

G S

cree

MR

TM

-PA

mH

PA

IL P

AL

L M

AP

KSR

KSR

39

3n93

221

015

70

01

6499

271

013

130

1I

199

291

010

510

101

10

5D98

70

014

40

00

6710

013

00

1818

00

00

100

250

019

919

90

610

D10

01

00

162

00

055

100

60

033

330

00

010

024

00

3921

398

03

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100

00

018

20

00

2410

03

00

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00

00

100

250

062

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60

1

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100

00

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10

00

710

02

00

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00

00

100

260

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5277

50

03

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9523

9638

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05

490

100

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050

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015

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100

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5384

4480

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5297

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513

310

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310

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130

100

00

100

970

80

100

096

00

9292

088

00

010

00

010

010

010

010

04

80

Tab

le 1

1

Om

ega

Squa

red

for

Prop

ortio

n of

Sam

ples

Ret

aini

ng K

Fac

tors

.

Eff

ect

k p/k

N Com

mun

ality

Phi

k X

p/k

k X

Nk

X C

omm

k X

Phi

p/k

X N

p/k

X C

omm

p/k

X P

hiN

X C

omm

N X

Phi

Com

m X

Phi

k X

p/k

X N

k X

p/k

X C

omm

k X

N X

Com

mp/

k X

N X

Com

mk

X p

/k X

Phi

k X

N X

Phi

p/k

X N

X P

hiC

omm

X p

/k X

Phi

Com

m X

k X

Phi

Com

m X

n X

Phi

k X

p/k

X N

X C

omm

k X

p/k

X N

X P

hiC

omm

X k

X p

/k X

Phi

Com

m X

p/k

X n

X P

hiC

omm

X k

X n

X P

hik

X p

/k X

N X

Com

m X

phi

Fact

or R

eten

tion 38

Para

llel A

naly

sis

CN

GSE

scre

eM

Rt

PAm

HPA

cLPA

LL

MA

PK

SRM

-KSR

0.97

50.

043

0.47

60.

011

0.04

30.

043

0.03

60.

014

0.02

80.

033

0.00

00.

158

0.12

60.

041

0.36

60.

403

0.25

50.

473

0.46

70.

192

0.00

00.

154

0.01

10.

022

0.13

30.

082

0.00

30.

001

0.00

40.

033

0.00

00.

118

0.00

00.

005

0.04

40.

049

0.07

20.

156

0.06

20.

047

0.00

00.

023

0.00

10.

011

0.08

90.

094

0.13

00.

044

0.08

20.

044

0.00

00.

012

0.25

90.

192

0.00

30.

004

0.03

60.

004

0.01

30.

011

0.00

00.

006

0.02

20.

035

0.00

20.

001

0.00

10.

000

0.00

00.

000

0.00

00.

000

0.00

10.

015

0.00

10.

001

0.00

50.

002

0.00

20.

001

0.00

30.

001

0.00

40.

031

0.01

30.

008

0.00

20.

002

0.00

50.

002

0.00

00.

174

0.02

00.

032

0.01

40.

014

0.00

10.

000

0.00

30.

090

0.00

00.

020

0.00

10.

002

0.00

40.

005

0.06

50.

076

0.03

90.

146

0.00

00.

014

0.00

20.

015

0.00

70.

015

0.10

90.

011

0.04

00.

029

0.00

00.

013

0.00

10.

010

0.00

00.

000

0.00

10.

000

0.00

10.

034

0.00

00.

003

0.00

00.

002

0.00

70.

004

0.00

00.

000

0.00

20.

004

0.00

00.

001

0.00

00.

004

0.00

10.

002

0.01

80.

002

0.00

40.

008

0.00

00.

021

0.04

10.

110

0.00

70.

007

0.00

00.

000

0.00

10.

003

0.00

00.

002

0.00

30.

012

0.00

30.

004

0.00

30.

012

0.00

50.

003

0.00

00.

009

0.00

10.

021

0.00

00.

000

0.00

00.

000

0.00

10.

001

0.00

00.

027

0.00

10.

027

0.01

40.

011

0.00

00.

001

0.00

20.

093

0.00

00.

001

0.00

20.

022

0.00

50.

007

0.00

70.

003

0.02

30.

008

0.00

00.

000

0.00

00.

003

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

006

0.00

00.

001

0.02

00.

018

0.00

20.

000

0.01

10.

005

0.00

00.

015

0.00

00.

009

0.00

40.

009

0.03

50.

057

0.04

20.

013

0.00

00.

002

0.00

00.

010

0.00

30.

004

0.01

10.

002

0.00

40.

003

0.00

00.

001

0.00

00.

000

0.00

10.

000

0.00

00.

000

0.00

20.

002

0.00

00.

028

0.00

30.

037

0.00

30.

003

0.00

00.

000

0.00

00.

002

0.00

00.

000

0.00

00.

003

0.00

90.

006

0.00

00.

000

0.00

10.

001

0.00

00.

004

0.00

00.

014

0.00

40.

006

0.04

90.

009

0.01

60.

008

0.00

00.

004

0.00

00.

001

0.00

90.

013

0.00

10.

000

0.00

90.

008

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

001

0.00

00.

000

0.00

00.

000

0.00

30.

005

0.00

10.

000

0.00

00.

003

Fact

or R

eten

tion 39

Tab

le 1

2Pr

opor

tion

of S

ampl

es A

gree

ing

with

Rul

e A

pplie

d to

Pop

ulat

ion

R-

Mat

rix,

Phi

= 0

.00

kp

Size

Low

Com

mun

ality

Wid

e C

omm

unal

itySE

CN

G S

cree

MR

Hig

h C

omm

unal

ityM

-T

PAm

s PA

u, P

Au,

MA

P K

SR K

SRSE

CN

G S

cree

MR

TM

-PA

S PA

u, P

AL

L M

AP

KSR

KSR

SEC

NG

Scr

eeM

RT

PAS

PAL

L P

A,,,

MA

P K

SR M

-KSR

39

3n0

4099

9927

3242

8735

220

5299

9931

3162

6749

490

R7

100

100

5264

5262

7584

51,

057

100

100

3543

4890

6729

062

100

100

4747

7173

6464

093

100

100

6477

6471

7988

101,

074

100

100

4557

5794

8447

072

100

100

6868

7881

7979

097

100

100

7187

7179

8592

20p

084

100

100

5670

6696

9074

080

100

100

8585

8487

8989

099

100

100

7590

7585

8995

40p

091

100

100

6582

7398

9391

088

100

100

9595

8992

9595

010

010

010

080

9280

9093

973

1531

30

210

049

4643

5325

00

330

083

8333

7985

850

890

096

9396

9594

995p

057

00

6363

4760

542

075

00

8989

4690

8787

010

00

010

099

100

9894

100

101,

085

11

8185

5071

6821

095

44

9494

6297

9090

010

019

1910

010

010

099

9610

020

p0

9332

3292

9755

8077

810

9646

4698

9874

9993

930

100

9090

100

100

100

9997

100

40p

098

9494

9810

062

8783

100

097

9393

9999

8099

9696

010

010

010

010

010

010

010

098

100

330

3p0

00

085

9778

9846

00

00

010

010

099

9997

970

90

010

010

010

010

010

010

05p

01

00

9299

8910

010

00

01

00

100

100

100

100

100

100

084

00

100

100

100

100

100

100

100

056

00

9710

094

100

100

00

580

010

010

010

010

010

010

00

100

00

100

100

100

100

100

100

20p

010

00

099

100

9810

010

062

010

00

010

010

010

010

010

010

00

100

00

100

100

100

100

100

100

40p

010

00

010

010

010

010

010

010

00

100

00

100

100

100

100

100

100

010

00

010

010

010

010

010

010

05

153n

02R

00

158

6469

214

036

00

2222

4846

3939

075

00

3622

3635

7277

51,

033

00

2311

7373

1321

054

00

3636

5851

6363

081

00

5940

5946

7883

10p

031

11

3917

8279

4341

066

22

6060

6758

8282

086

3131

8162

8153

8589

20p

034

2222

6234

8885

6866

074

3232

8383

7765

9191

091

9595

9377

9360

8993

40p

044

8585

8157

9190

8482

082

8383

9595

8572

9494

096

100

100

9888

9865

9394

525

3p0

10

036

2749

.40

20

01

00

6363

5559

5959

051

00

9793

9789

9399

5p0

190

050

4256

4829

00

200

082

8270

7376

760

960

010

099

100

9295

100

10t

077

00

7169

6457

605

086

00

9393

8183

8585

010

00

010

010

010

094

9810

020

1,0

900

089

8774

6373

760

990

096

9686

8990

900

100

00

100

100

100

9599

100

400

094

00

9896

8168

7910

00

100

00

9999

9191

9494

010

01

110

010

010

095

100

100

550

3p0

00

095

9772

9523

00

00

010

010

010

010

098

980

00

010

010

010

098

100

100

5p0

00

098

100

8099

100

00

00

010

010

010

010

010

010

00

30

010

010

010

099

100

100

100

01

00

9910

087

100

100

00

00

010

010

010

010

010

010

00

990

010

010

010

099

100

100

20p

078

00

100

100

9210

010

038

057

00

100

100

100

100

100

100

010

00

010

010

010

099

100

100

40o

010

00

010

010

095

100

100

100

010

00

010

010

010

010

010

010

00

100

00

100

100

100

9910

010

07

213n

017

7017

146

5256

0li

022

8111

1717

4030

3030

068

9316

1219

1227

6575

5p0

3173

1822

957

648

140

5282

830

3053

3748

480

8596

1430

3830

3771

8010

p0

3176

1839

1761

7533

340

7883

645

45-

6647

6767

093

9817

4762

4750

7788

200

034

7821

6329

6583

5856

086

8210

5858

7457

8080

096

9952

6074

6063

8494

4013

045

8034

8644

7089

7468

087

8124

7171

8069

8888

099

100

8578

8078

7689

967

3531

30

092

1133

2248

358

00

095

1361

6159

5945

450

597

393

8293

7673

99Si

,0

195

845

4255

4539

00

196

1180

8065

6956

560

7999

110

099

100

7879

100

100

048

985

6181

6457

630

048

989

8989

6975

7272

010

010

00

100

100

100

8087

100

20p

090

993

7699

7262

7864

095

100

792

9275

8082

820

100

100

010

010

010

080

9210

040

p0

9110

03

8810

080

6784

980

9910

06

9797

8284

8989

010

099

010

010

010

081

9610

07

703p

00

100

299

9557

9710

00

099

110

010

094

100

9999

00

100

210

010

010

099

100

100

51,

00

100

110

010

055

9910

00

00

991

100

100

9410

010

010

00

010

01

100

100

100

9910

010

010

p0

010

00

100

100

5810

010

00

00

100

010

010

095

100

100

100

057

100

010

010

010

010

010

010

020

p0

1110

00

100

100

6710

010

027

01

100

010

010

097

100

100

100

010

010

00

100

100

100

100

100

100

40p

099

100

010

010

077

100

100

100

089

100

010

010

010

010

010

010

00

100

100

010

010

010

010

010

010

0

Fact

or R

eten

tion 40

Tab

le 1

3Pr

opor

tion

of S

ampl

es A

gree

ing

with

Rul

e A

pplie

d to

Pop

ulat

ion

R-M

atri

x, P

hi =

0.3

0

kp

Low

Com

mun

ality

Wid

e C

omm

unal

ityH

igh

Com

mun

ality

SESi

ze C

NG

Scr

eeM

RT

M-

PAN

LH

PA

ct P

AL

L M

AP

KSR

KSR

SEC

NG

Scr

eeM

RT

SEPA

S PA

L, P

AL

L M

AP

KSR

M-K

SR C

NG

Scr

ee M

RT

PAL

L, P

A.

M-

PAL

L M

AP

KSR

KSR

39

3n0

3799

9925

2766

9251

170

4899

9914

1487

7549

490

7110

010

025

2325

5659

72

5o0

5410

010

030

3277

9688

250

5610

010

022

2289

8175

750

8010

010

045

4045

6365

78

10D

068

100

100

3940

8999

9948

063

100

100

4141

9089

9090

090

100

100

6562

6571

7082

20p

074

100

100

5254

9710

010

072

070

100

100

6767

9093

9595

095

100

100

7873

7877

7587

4013

081

100

100

6571

100

100

100

860

7910

010

082

8290

9596

960

9910

010

090

7990

8280

91

315

300

210

033

2388

6911

10

380

051

5143

6654

540

910

078

6878

9075

92

5o0

480

043

2990

7639

40

750

073

7351

7569

690

990

093

8993

9679

9610

D0

661

163

5091

8571

260

922

293

9363

8378

780

100

1515

100

9710

099

8198

20D

076

2323

8070

9192

8570

096

3333

9898

7588

8484

010

085

8510

010

010

010

085

9940

130

8485

8593

8191

9690

830

9988

8899

9982

9388

880

100

100

100

100

100

100

100

8810

0

330

3D0

00

075

7444

6319

00

00

099

9967

9997

970

80

010

010

010

010

010

010

0

Si,

01

00

9094

5174

460

02

00

100

100

8410

010

010

00

830

010

010

010

010

010

010

0

1013

056

00

9510

059

8262

00

620

010

010

094

100

100

100

010

00

010

010

010

010

010

010

0

20D

010

00

098

100

6788

7848

010

00

010

010

098

100

100

100

010

00

010

010

010

010

010

010

0

40p

010

00

099

100

7492

8810

00

100

00

100

100

100

100

100

100

010

00

010

010

010

010

010

010

0

515

3n0

180

06

R97

RR

81

031

00

66

8051

1111

059

00

53

534

4063

Si,

034

00

68

100

9564

20

440

012

1281

5729

290

660

015

1115

4450

71

101,

039

00

1010

100

9998

110

521

126

2682

6557

570

7614

1438

2838

5461

82

20p

039

99

1720

100

100

100

400

5821

2142

4283

7376

760

.85

8383

6352

6362

7190

400

043

5959

2838

100

100

100

660

6466

6657

5784

8286

860

9010

010

083

6783

6981

96at

to5

253p

03

00

112

9658

00

03

00

1212

3539

2222

055

00

3117

3187

5774

tb5p

026

00

222

9866

80

031

00

3838

4547

4343

093

00

7252

7296

6880

100

043

00

4810

9976

419

087

00

8484

5558

6161

093

00

9188

9199

8087

20D

040

00

7832

100

8469

650

970

098

9866

6773

730

920

092

9192

9989

92

40D

054

00

9467

100

9281

840

990

099

9976

7383

830

920

099

9199

9993

97

550

31,

00

00

4923

2954

40

00

00

9595

5199

9898

00

00

9694

9610

010

010

0

5p0

00

082

6236

6831

00

00

010

010

063

100

100

100

03'

00

100

9910

010

099

100

lOp

02

00

9590

4379

580

00

00

100

100

7610

099

990

990

010

010

010

010

099

100

20D

083

00

100

9652

8875

490

530

010

010

084

100

9898

010

00

010

010

010

010

098

100

40D

010

00

010

010

061

9385

100

010

00

010

010

090

100

9898

010

00

010

010

010

010

098

100

117

213n

011

603

32

99R

I0

10

2570

42

259

459

90

6090

05

25

3142

55

m coSi

,10

p0 0

29 2360 60

3 3

5 102 5

100

100

91 9723 87

2 100 0

43 4970 70

3 3

6 286 28

66 7753 66

32 6732 67

0 067 75

90 900 1

12 3111 28

12 3141 52

53 6563 74

20p

017

603

2316

100

9999

330

5670

255

5588

7383

830

8590

1360

5360

6376

82

0 07

3540

03p 5D

0 0 0

210 6

60 100

100

6 0 0

42 3 9

38 0 1

100 70 72

100

45 54

100 0 3

550 0

0 0 0

66 0 2

70 100

100

10 0 0

68 4 27

68 4 27

97 40 53

82 33 43

91 9 29

919

29

0 0 0

92 11 84

91 100

100

49 0 0

85 19 59

807 44

85 19 59

72 73 81

85 58 70

89 73 82

lOp

043

100

030

1074

6431

20

5010

00

7474

6754

5656

010

010

00

8470

8486

7890

20p

047

100

054

3175

7565

420

9610

00

9595

7463

7171

010

010

00

100

8810

090

8294

40p

049

100

075

5177

8283

700

9710

00

9999

8170

8282

010

010

00

100

100

100

9385

977

703p

00

100

035

1251

504

00

010

00

8686

6210

089

890

010

00

9490

9410

010

010

0

Si,

00

100

082

5559

6924

00

010

00

9999

5110

092

920

010

00

100

100

100

100

100

100

101,

00

100

010

094

6982

460

00

100

010

010

042

100

9696

053

100

010

010

010

010

010

010

0

w r20

p40

1,

0 015

100

100

100

0 010

010

010

010

080 89

87 9065 75

42 100

0 01

9210

010

00 0

100

100

100

100

46 6210

010

099 10

099 10

00 0

100

100

100

100

0 010

010

010

010

010

010

010

010

010

010

010

010

0

PAC

CZ

)

Fact

or R

eten

tion 41

Tab

le 1

4Pr

opor

tion

of S

ampl

es A

gree

ing

with

Rul

e A

pplie

d to

Pop

ulat

ion

R-

Mat

rix,

Phi

= 0

.50

kp

Size

Low

Com

mun

ality

Wid

e C

omm

unal

ityH

igh

Com

mun

ality

SEC

NG

Scr

eeM

RT

Ni-

PAm

H P

ApL

PA

LL

MA

P K

SR K

SRSE

CN

G S

cree

MR

TSE

PAM

H P

Acr

, PA

LL

MA

P K

SR M

-KSR

CN

G S

cree

MR

TM

-PA

NH

, PA

IL P

AL

L M

AP

KSR

KSR

39

3n0

3899

9925

3177

9462

110

4699

999

999

7768

680

6199

996

136

6462

6650

062

100

100

2932

8998

9520

058

100

100

1111

100

8192

920

6710

010

014

2114

7276

73

10D

077

100

100

3636

9810

010

040

068

100

100

1919

100

8598

980

7510

010

034

4034

8287

81

20D

081

100

100

4846

100

100

100

670

7410

010

038

3810

089

100

100

081

100

100

5759

5789

9487

40D

083

100

100

6261

100

100

100

820

8010

010

059

5910

093

100

100

086

100

100

7177

7192

9691

315

3D0

190

018

1010

089

161

035

00

1919

8351

3030

085

00

2615

2671

5667

5p0

410

022

910

095

775

065

00

3737

8459

5454

095

00

6040

6083

6473

1Cto

041

11

3715

100

9999

250

792

266

6686

6974

740

989

994

8194

9072

8020

D0

4012

1263

3110

010

010

060

084

2424

8282

8978

8282

010

076

7610

098

100

9379

86

40D

049

6666

8554

100

100

100

770

8677

7795

9591

8688

880

100

100

100

100

100

100

9582

903

3031

50

00

045

2775

617

00

00

078

7838

9272

720

100

096

9296

9910

010

0

5D0

10

072

5177

6731

00

10

094

9453

9781

810

840

010

010

010

010

010

010

0

10D

061

00

9588

7777

530

059

00

100

100

6799

8787

010

00

010

010

010

010

010

010

0

20D

010

00

098

9980

8468

690

100

00

100

100

7710

092

920

100

00

100

100

100

100

100

100

40D

010

00

099

100

8491

7810

00

100

00

100

100

8610

095

950

100

00

100

100

100

100

100

100

515

3n0

90

06

1099

9115

00

280

01

I10

063

2828

058

00

42

447

3961

5p0

320

05

810

097

831

040

00

11

100

6872

720

670

011

711

5553

6810

D0

560

07

810

099

994

051

11

55

100

7497

970

798

820

2220

6567

75

2013

064

77

1312

100

100

100

180

6016

1618

1810

081

100

100

089

6868

2850

2873

7581

40D

069

4949

2719

100

100

100

420

7158

5839

3910

086

100

100

094

9999

4869

4879

7984

525

3D0

20

01

110

077

10

05

00

11

7237

77

057

00

10

152

3355

Si,

022

00

21

100

8641

00

340

06

673

4721

210

880

014

314

6746

63

10p

037

00

74

100

9192

60

750

038

3875

5837

370

910

063

3763

7961

7220

p0

280

025

1410

095

9937

085

00

7979

7768

5151

097

00

9469

9487

7279

40p

033

00

5538

100

9810

069

093

00

9090

8175

6464

010

00

010

091

100

9280

855

503D

00

00

40

9041

00

00

00

2121

2778

2525

00

00

6040

6010

093

100

5p0

00

023

491

492

00

00

066

6644

8950

500

50

096

9096

100

9510

0

10D

02

00

7742

9260

270

01

00

9999

6796

7878

099

00

100

100

100

100

9710

0

20D

081

00

9391

9268

6046

068

00

100

100

7998

9090

010

00

010

010

010

010

098

100

40D

095

00

9810

092

7582

980

100

00

100

100

8599

9595

010

00

010

010

010

010

099

100

721

3n0

510

11

110

090

30

022

800

00

100

585

50

4390

00

00

3215

445D

025

100

11

100

9771

00

3880

10

010

064

4242

045

900

00

036

3054

lfto

031

101

21

100

100

100

10

3680

13

310

073

8888

052

900

53

544

5366

20D

022

101

83

100

100

100

130

3880

119

1910

081

9999

062

906

2316

2353

7376

40D

024

101

2912

100

100

100

420

4980

449

4910

088

100

100

072

9036

4531

4561

8583

735

3D0

010

00

00

100

760

00

010

00

00

9027

00

017

100

00

00

4441

645D

09

100

00

010

089

130

03

100

04

490

358

80

7910

00

30

353

5572

10p

031

100

04

010

097

881

049

100

032

3291

4639

390

8210

00

3811

3862

7280

20D

013

100

030

010

099

9424

083

100

060

6092

5766

660

8210

00

7851

7871

8187

40D

013

100

064

810

010

097

560

8610

00

8282

9366

8383

083

100

091

7791

7986

91

770

3v0

010

00

00

5933

00

00

100

010

1035

7425

250

010

00

3924

3910

085

995p

00

100

06

061

380

00

010

00

4141

5184

5050

00

100

085

7485

100

9199

10D

00

100

070

2663

4711

00

010

00

8787

6491

7878

054

100

010

099

100

100

9599

20p

017

100

098

7867

5636

430

210

00

9696

7595

8787

010

010

00

100

100

100

100

9899

40D

010

010

00

100

9679

6251

100

093

100

010

010

080

9790

900

100

100

010

010

010

010

010

099

Fact

or R

eten

tion 42

Tab

le 1

5Pr

opor

tion

of S

ampl

es A

gree

ing

with

Rul

e A

pplie

d to

Pop

ulat

ion

R-M

atri

x, P

hi =

Mix

ed

kp

Size

Low

Com

mun

ality

Wid

e C

omm

unal

ityH

igh

Com

mun

ality

SEC

NG

Scr

eeM

RT

M-

PAS

PAc

PAL

L M

AP

KSR

KSR

SEC

NG

Scr

eeM

RT

SEPA

S Pi

ta, P

AL

L M

AP

KSR

M-K

SR C

NG

Scr

ee M

RT

M-

PAm

ii PA

ct P

AL

L M

AP

KSR

KSR

39

3n0

3899

9925

3068

9353

90

5399

9920

2084

6953

530

7099

991R

2618

7267

855D

054

100

100

2932

8098

9116

068

100

100

2626

8874

7474

077

100

100

2733

2781

7390

10D

062

100

100

3640

9310

010

037

078

100

100

4141

9278

9191

083

100

100

4746

4788

7995

20D

066

100

100

4754

9910

010

067

085

100

100

6161

9582

9797

087

100

100

7060

7092

8498

40D

068

100

100

5870

100

100

100

840

8910

010

079

7997

8599

990

9010

010

085

7685

9487

100

315

3D0

210

043

3670

5314

00

380

025

2549

7066

660

910

053

3853

8467

795D

052

00

5446

7560

343

064

00

4141

5776

8383

099

00

8570

8592

7782

10D

073

11

7267

7966

5624

072

22

7171

6881

9191

010

013

1310

097

100

9787

8720

130

8028

2887

7982

7173

780

7536

3693

9377

8396

960

100

8181

100

100

100

9894

9240

p0

8590

9097

8787

7784

920

8491

9199

9988

8699

990

100

100

100

100

100

100

9998

963

303D

00

00

6555

8864

60

00

00

8888

5793

8989

010

00

100

9910

010

010

010

05D

01

00

8476

9371

340

01

00

9696

6896

9797

084

00

100

100

100

100

100

100

1 O

D0

570

096

9697

7867

00

550

010

010

077

9799

990

100

00

100

100

100

100

100

100

20D

010

00

098

100

9983

8356

099

00

100

100

8399

9999

010

00

010

010

010

010

010

010

040

130

100

00

9910

010

088

8910

00

100

00

100

100

9010

099

990

100

00

100

100

100

100

100

100

515

3n0

190

012

1797

827

20

290

03

361

5733

330

580

03

33

5058

745D

039

00

1519

9989

564

045

00

66

6665

5858

066

00

1110

1159

6781

lOp

042

00

2329

100

9389

190

591

113

1371

7276

760

7310

1029

3529

7076

8920

130

4113

1334

4510

095

9557

067

1616

2929

7679

8686

080

7676

4363

4379

8394

40D

044

6565

4869

100

9598

830

7567

6750

5081

8590

900

8610

010

059

8659

8488

975

253D

00

00

1110

5457

20

01

00

3232

4241

2121

051

00

41

461

9379

513

013

00

1619

5562

280

023

00

6262

5655

3636

091

00

186

1874

9487

CA

IOD

067

00

2748

5672

602

082

00

8787

7369

5353

094

00

7031

7084

9594

1""

20D

087

00

4080

5880

8048

093

00

9696

8779

6868

094

00

9379

9391

9795

40D

096

00

6296

6487

9090

094

00

9999

9583

8181

096

00

100

9110

096

9996

550

313

00

00

81

4547

00

00

00

7474

5796

8181

00

00

8778

8710

099

100

5D0

00

034

851

5312

00

00

097

9771

100

8888

03

00

100

9810

010

098

100

10p

02

00

8550

6061

630

00

00

100

100

8110

094

940

990

010

010

010

010

098

100

20D

083

00

9988

7071

8242

064

00

100

100

8810

097

970

100

00

100

100

100

100

9710

0

40D

010

00

010

010

082

8091

100

010

00

010

010

093

100

9898

010

00

010

010

010

010

097

100

721

3n0

710

28

796

611

00

1710

01

I32

4921

210

4990

10

00

3938

64Si

,0

3010

210

997

6618

10

3810

04

440

6147

470

4990

12

02

5046

71

10D

038

102

2013

9973

598

052

100

1717

4772

7070

054

900

223

2264

5379

20D

035

101

3820

100

7979

310

6010

040

4052

8283

830

6590

464

2164

7759

8440

D0

4110

257

3310

083

8559

068

100

6767

6289

8989

076

9036

8152

8187

6487

735

3p0

010

01

71

3839

00

00

100

01

139

3914

140

1310

01

00

068

3662

5D0

210

00

162

3845

20

05

100

05

551

5548

480

7310

00

30

378

5368

10D

047

100

037

936

5227

10

5210

00

3636

6371

7676

075

100

012

1312

8666

7620

D0

7910

00

5740

3759

5964

080

100

069

6973

8386

860

7810

00

4222

4291

7581

40D

087

100

065

7740

6879

900

8010

00

8888

8289

9292

086

100

087

4087

9381

877

703D

00

100

01

076

450

00

010

00

00

7559

5656

00

100

06

26

9862

845D

00

100

015

282

522

00

010

00

66

7574

9191

00

100

034

1634

9972

87IO

D0

010

00

7133

8863

470

00

100

073

7374

8893

930

5110

00

9282

9210

081

8720

D0

1710

00

9977

9371

8348

02

100

090

9075

9597

970

100

100

010

092

100

100

8888

40D

010

010

00

100

9797

7895

100

096

100

092

9280

9899

990

100

100

010

010

010

010

093

89

Tab

le 1

6

Om

ega

Squa

red

for

Prop

ortio

n of

Sam

ples

Agr

eein

g w

ith R

ule

App

lied

to P

opul

atio

n R

-Mat

rix.

Eff

ect

k p/k

N Com

mun

ality

Phi

k X

p/k

k X

Nk

X C

omm

k X

Phi

p/k

X N

p/k

X C

omm

p/k

X P

hiN

X C

omm

N X

Phi

Com

m X

Phi

k X

p/k

X N

k X

p/k

X C

omm

k X

N X

Com

mp/

k X

N X

Com

mk

X p

/k X

Phi

k X

N X

Phi

p/k

X N

X P

hiC

omm

X p

/k X

Phi

Com

m X

k X

Phi

Com

m X

n X

Phi

k X

p/k

X N

X C

omm

k X

p/k

X N

X P

hiC

omm

X k

X p

/k X

Phi

Com

m X

p/k

X n

X P

hiC

omm

X k

X n

X P

hik

X p

/k X

N X

Com

m X

phi

Fact

or R

eten

tion 43

CN

GSE

s,,

MR

t

Para

llel A

naly

sis

MA

PK

SRM

-KSR

PAm

HPA

ci,

PAL

L

0.00

00.

076

0.45

80.

238

0.04

00.

055

0.00

30.

058

0.04

00.

013

0.00

00.

040

0.06

40.

281

0.27

60.

239

0.01

50.

095

0.03

60.

005

0.00

00.

359

0.02

30.

051

0.19

70.

168

0.07

10.

102

0.25

60.

255

0.00

00.

135

0.00

90.

005

0.03

30.

036

0.01

70.

006

0.07

80.

361

0.00

00.

006

0.00

20.

003

0.08

60.

099

0.01

00.

002

0.01

60.

011

0.00

00.

001

0.24

60.

213

0.00

30.

001

0.00

20.

010

0.00

40.

004

0.00

00.

009

0.01

10.

013

0.00

70.

004

0.00

00.

003

0.01

10.

000

0.00

00.

001

0.00

20.

001

0.00

10.

001

0.00

40.

000

0.00

40.

001

0.00

00.

001

0.00

30.

001

0.00

90.

013

0.00

50.

001

0.00

30.

001

0.00

00.

086

0.01

20.

026

0.01

70.

009

0.00

10.

010

0.01

30.

005

0.00

00.

009

0.00

90.

004

0.00

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170

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90.

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80.

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0.05

30.

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006

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123

0.00

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000

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004

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50.

015

0.00

60.

000

0.00

00.

026

0.05

70.

103

0.00

40.

005

0.00

00.

000

0.00

10.

000

0.00

00.

003

0.01

20.

008

0.00

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002

0.00

90.

008

0.00

20.

001

0.00

00.

012

0.00

10.

002

0.00

00.

000

0.00

00.

000

0.00

20.

003

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031

0.00

10.

005

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00.

009

0.00

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014

0.00

60.

030

0.00

00.

001

0.00

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002

0.00

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001

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000

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0.00

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0.06

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002

0.00

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001

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30.

000

0.00

20.

001

0.01

00.

004

0.00

50.

001

0.00

00.

001

0.00

00.

000

0.00

10.

002

0.00

00.

000

0.00

30.

002

0.00

00.

032

0.00

50.

011

0.00

10.

001

0.00

00.

000

0.00

00.

000

0.00

00.

001

0.00

00.

001

0.00

70.

005

0.00

00.

000

0.00

10.

000

0.00

00.

001

0.00

50.

001

0.00

50.

005

0.01

70.

011

0.00

60.

002

0.00

00.

004

0.00

00.

000

0.00

90.

012

0.00

50.

003

0.01

00.

002

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

001

0.00

20.

003

0.00

00.

000

0.00

10.

000

0 0 m

Fact

or R

eten

tion 44

16 14 12 10

0

CN

G S

E S

cree

MR

O O O O t'P

A m

h PA

cl

PA 1

1M

AP

KSR

M-K

SR

Met

hod

Figu

re 1

. Dis

trib

utio

ns o

f M

ean

Num

ber

of F

acto

rs R

etai

ned,

k =

3

Fact

or R

eten

tion 45

30 20 10

6 a 6

I1

1.

I

T

0 Pti

CN

G S

E S

cree

MR

PA m

h PA

cl

PA 1

1M

AP

KSR

M-K

SR

Met

hod

Figu

re 2

. Dis

trib

utio

ns o

f M

ean

Num

ber

of F

acto

rs R

etai

ned,

k =

5

Fact

or R

eten

tion 46

40 30 20 10

-o

ti a 8 I

It

II

I

I

'IIS

I-N

.T

, I

I I

e3

CN

G S

E S

cree

MR

tPA

mh

PA c

lPA

11

MA

PK

SRM

-KSR

Met

hod

Figu

re 3

. Dis

trib

utio

ns o

f M

ean

Num

ber

of F

acto

rs R

etai

ned,

k =

7

Figu

re 4

. Mea

n N

umbe

r of

Fac

tors

Ret

aine

d by

Tru

e N

umbe

r of

Fac

tors

16-

14-

11.

Elk

= 3

Elk

= 5

Elk

= 7

CN

GSE

Scr

eeM

Rt

PA m

hPA

cl

PA 1

1M

AP

KSR

M-K

SR

Rul

e of

Thu

mb

Fact

or R

eten

tion 47

Figu

re 5

. Mea

n N

umbe

r of

Fac

tors

Ret

aine

d by

N p

er V

aria

ble

(N:p

) fo

r k

= 5

05

1015

2025

N P

er V

aria

ble

3035

4045

Figu

re 6

. Mea

n N

umbe

r of

Fac

tors

Ret

aine

d by

Num

ber

of V

aria

bles

Per

Fac

tor

(p:k

) fo

r k

= 5

20 -

187

16-

Ti

14 -

m g 4),

12-

.F.J cz 44

100 ct .0

CA

58-

0oo

z6- 4 2

(=30

221°

CN

G0S

E S

cree

6 M

R0P

A tr

ill 0

PA c

lPA

11

0 -

KSR

°-

M-K

SR

Fact

or R

eten

tion 49

0,2

34

56

78

Num

ber

of V

aria

bles

Per

Fac

tor

910

11

Figu

re 7

. Mea

n N

umbe

r of

Fac

tors

Ret

aine

d by

Com

mun

ality

Typ

e fo

r k

= 5

12 10

Fact

or R

eten

tion 50

tt

Low

Wid

e

ID H

igh

CN

GSE

Scr

eeM

Rt

PA m

hPA

cl

PA 1

1M

AP

KSR

M-K

SR

Rul

e of

Thu

mb

Figu

re 8

. Mea

n N

umbe

r of

Fac

tors

Ret

aine

d by

Phi

for

k =

5

14-

12 10 4

Fact

or R

eten

tion 51

r =

.00

r =

.30

r =

.50

0 r

= M

ixed

CN

GSE

Scr

eeM

Rt

PA m

hPA

cl

PA 1

1M

AP

KSR

M-K

SR

Rul

e of

Thu

mb

Figu

re 9

. Pro

port

ion

of S

ampl

es R

etai

ning

K F

acto

rs b

y T

rue

Num

ber

of F

acto

rs

1.0

0.9

0.8

0.7

t 0.6

*JZ ;.

0.5

00.

40 6.1

0.3

0.2

0.1

0.0

Fact

or R

eten

tion 52

-

k=3

k=5

k=7

CN

GSE

Scr

eeM

R

111.

1.0.

tPA

mh

PA c

lPA

11

MA

PK

SRM

-KSR

Rul

e of

Thu

mb

Figu

re 1

0. P

ropo

rtio

n of

Sam

ples

Ret

aini

ng K

Fac

tors

by

N p

er V

aria

ble

(N:p

)

1.0

0.9

-

0.8

-

0.7

-

0.5

0 0 o 0.

4o.

0.3

-

0.2

0.1

-

0.0

Fact

or R

eten

tion 53

05

1015

2025

N P

er V

aria

ble

3035

4045

Figu

re 1

1. P

ropo

rtio

n of

Sam

ples

Ret

aini

ng K

Fac

tors

by

Num

ber

of V

aria

bles

Per

Fac

tor

(p:k

)

1.0

0.9

0.8

0.7

0.3

0.2

0.1

0.0

Fact

or R

eten

tion 54

CN

G0S

E S

cree

-.°9

6F M

R0

t0

PA m

h.D

PA c

l°=

PA

11

MA

P-

0 -

KSR

° M

-KSR

23

45

67

8

Num

ber

of V

aria

bles

Per

Fac

tor

910

11

Figu

re 1

2. P

ropo

rtio

n of

Sam

ples

Ret

aini

ng K

Fac

tors

by

Com

mun

ality

Typ

e

1.0

0.9

0.8

0.7

`cf.

3 E 0

.6

v-1

0.5

0 a, 0

.40 4."

0.3

-

0.2

-

0.1

0.0

Fact

or R

eten

tion 55

NIO

NM

INJ

*IV

CN

GSE

Scr

eeM

R

Low

Wid

e

Hig

h

tPA

mh

PA c

lPA

11

MA

PK

SRM

-KSR

Rul

e of

Thu

mb

Figu

re 1

3. P

ropo

rtio

n of

Sam

ples

Ret

aini

ng K

Fac

tors

by

Phi

1.0

0.9

0.8

0.7

Fact

or R

eten

tion 56

0 r

= .0

0

0 r

= .3

0

0 r

.50

0 r

= M

ixed

0.3

0.2

0.1

0.0

CN

GSE

Scr

eeM

Rt

PA m

hPA

cl

PA 1

1M

AP

KSR

M-K

SR

Rul

e of

Thu

mb

Fact

or R

eten

tion 57

Figu

re 1

4. P

ropo

rtio

n of

Sam

ples

Agr

eein

g w

ith R

ule

App

lied

to P

opul

atio

n R

-Mat

rix

by T

rue

Num

ber

of F

acto

rs

1.0

0.9

0.8

-

0.7

5 0.

6

0.5

0.4

41,4

0.3

0.2

0.1

k=3

k=5

k=7

0.0

CN

GSE

Scr

eeM

Rt

PA m

hPA

cl

Rul

e of

Thu

mb

PA 1

1M

AP

c:

KSR

M-K

SR

Fact

or R

eten

tion 58

Figu

re 1

5. P

ropo

rtio

n of

Sam

ples

Agr

eein

g w

ith R

ule

App

lied

to P

opul

atio

nR

-Mat

rix

by N

per

Var

iabl

e (N

:p)

1.0

0.9

5

N P

er V

aria

ble

Fact

or R

eten

tion 59

Figu

re 1

6. P

ropo

rtio

n of

Sam

ples

Agr

eein

g w

ith R

ule

App

lied

to P

opul

atio

nR

-Mat

rix

by N

umbe

r of

Var

iabl

es P

er F

acto

r (p

:k)

a.,

..30.

6 -

E

1.0

0.9

-

0.8

-

=0=

CN

G°=

4:1S

E S

cree

==

t1=

MR

0t=

O=

PA m

h =

2PA

ci

='P

A 1

1M

AP

° 0

° K

SR°

0 °

M-K

SR

0.7

-O

°.

(:-,

s o 0

5z 1 0

0.4-

0.3

-

0.2

-

0.1

0.0

23

45

67

Num

ber

of V

aria

bles

Per

Fac

tor

89

1011

Fact

or R

eten

tion 60

Figu

re 1

7. P

ropo

rtio

n of

Sam

ples

Agr

eein

g w

ith R

ule

App

lied

to P

opul

atio

nR

-Mat

rix

by C

omm

unal

ity T

ype

1.0

0.9

0.8

0.7

cow t0.

6

4.6

0.5

0

cx)

ow 0

.441

4

0.3

0.2

0.1

0.0

Low

Wid

e

Hig

h

CN

GSE

Scr

eeM

Rt

PA m

hPA

cl

PA 1

1M

AP

KSR

M-K

SR

Rul

e of

Thu

mb

Figu

re 1

8. P

ropo

rtio

n of

Sam

ples

Agr

eein

gw

ith R

ule

App

lied

to P

opul

atio

n R

-Mat

rix

by P

hi

1.0

0.9

0.8

0.7

cu cu Sa.

§0.

6

c.n

0.5

0.4

CZ

)0.

3

0.2

0.1

0.0

Or

= .0

0

Or

= .3

0

Or

= .5

0

r =

Mix

ed

CN

GSE

Scr

eeM

RPA

mh

PA c

lPA

11

MA

PK

SRM

-KSR

Rul

e of

Thu

mb

Fact

or R

eten

tion 61

1.

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Title: Factor Retention in Exploratory Factor Analysis: A Comparison ofAlternative Methods

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