3 on the Plausibility of Unitary Language Proficiency Factor

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On the plausibility of unitary language proficiency factor Hossein Farhady

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On the Plausibility of the UnitaryLanguage Proficiency Factor*

Hossein Farhady

University of California, Los Angeles

University for Teacher Education, Tehran, Iran

Editors Introduction

In Chapter 2, Hossein Farhady explains certain fundamental facts about the

most widely accepted approaches to exploratory factor analysis. He is

concerned to demonstrate the inappropriateness of principal components

analysis as applied in Chapter 1 (and elsewhere) to the problem of testing

for the existence of an exhaustive general factor of language proficiency. He

flatly rejects the strongest possible version of a general factor hypothesis,

but he does not deny the possibility of a nonexhaustive general factor.

Farhady does insist, however, on more standard factoring methods withcommunality estimates on the diagonal of the original correlation matrix

(rather than unities) followed by an orthogonal rotational procedure to

achieve a terminal solution.

Introduction

The complexities and intricacies of human intellectual capacity have been a subject of

discussion for several centuries. The diversity of functions that the human mind is

capable of performing has led scholars to formulate numerous theories. Many

synonymous and/or overlapping terms have been coined to represent the underlying

traits of human performance. All these theories, I believe, should be subjected to

scientific scrutiny in order for them to be judged empirically valid. Otherwise,confusion and uncertainty will continue to overshadow systematicity and reality.

Probably, one of the best ways to investigate the plausibility of a given theory is to

test the hypotheses generated from the theory. And one of the most defensible ways of

testing a hypothesis may be attempting to quantify the relationship between the

variables in the hypothesis. Thus, quantification, testing, and measurement constitute

a necessary part of experimental investigation of theories.

Language is one of the most unique characteristics of human beings and is involved in

almost all mental activities in one way or another. Various theories have been

developed to explain and/or account for numerous facets of language behavior.

Language structures, language use and functions, language acquisition and/or

learning, and language instruction are some of the interrelated domains of

investigation. To evaluate the extent and nature of learner competence in specific

areas, various types of tests have been developed. The construction of tests for various

skills, modes, and components of language, however, has led to a confusing situation

in language testing.


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One reason for confusion may be the complex interrelationships among language

processing tasks. For example, skills such as reading, writing, listening, and speaking,

or processes such as recognition, comprehension, and production are so closely

interrelated that separating them from each other (not to mention from other skills) is

an almost impossible task. For this reason, many tests may be required to obtain a

comprehensive picture of the degree of a learners competence in language behavior.

Another source of confusion for the would-be tester of abilities is the plethora of tests

with different names and supposed functions. But the tester must beware of the fact

that bearing a certain name does not guarantee that the test actually measures

whatever it is named. For example, calling an instrument a test of listening

comprehension does not guarantee that it is actually measuring listening

comprehension. More importantly, a listening comprehension test probably taps not

one and only one aspect of language behavior but rather a combination of many

elements. Language processing in any modality is probably a more integrated

phenomenon.

The diversity of the dimensions of language behaviors, as well as of the tests, and the

interrelationship among the dimensions have led to duplication of efforts as well asarbitrary categorization of hypothesized traits. Of course, it is not an easy task to

isolate and identify the traits. It is possible, however, to utilize statistical methods to

determine the degree of relationship and/or overlap among tests. This approach will

help eliminate redundancy, and it will aid in the development of tests that are

representative of groups of traits (Guilford, 1954). In this way, the task of assessing

human capabilities in general, and language abilities in particular, will be simplified.

Fortunately, research in language testing has been moving in the desired direction.

Recent investigations have been carried out to simplify the task of language testing by

determining representative tests. The most common statistical technique used to

examine the traits underlying language tests has been factor analysis. Factor analysis

is a whole array of interrelated statistical procedures which allow researchers to

investigate the intercorrelations among observed variables and to group them in

relation to one or more underlying hypothetical factors. However, because of its

versatility, factor analysis has been overused and some of its fundamental

assumptions have been overlooked.

One of the major issues that has emerged from factor-analytic studies has been the

unitary language proficiency hypothesis the claim that there is a unitary factor of

language proficiency which accounts for almost all variations in almost all language

processing tasks. Oller has presented data from numerous sources in support of this

hypothesis (Oller, Chapter 1, 1978, 1979b; Oller & Perkins, 1978, 1980). Though

these reports have been based on generally well-designed and carefully conductedresearch, questions have been raised about the appropriateness of the interpretations

offered for the results obtained (Vollmer, 1979, 1980, 1981; Briere, 1980; Abu-Sayf,

et al. 1979; Farhady, 1979, 1980a, 1980b; and also see the other chapters of this

section and their references).

The purpose of this paper, then, is to critically examine the previous reports and

question the correctness of the statements made about the unitary factor hypothesis. In

order to provide an accurate perspective on the issue, some theoretical and technical


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clarifications are warranted. First, a brief explanation of some concepts such as

reliability, communality, and specificity, which are crucial to explaining the results of

factor analysis, will be provided. Second, a brief description of the theory of factor

analysis and various alternative factor-analytic methods will be given and data from

different sources will be compared in order to justify the most appropriate method of

data analysis. And finally, the implications of alternative interpretations of the

relevant findings will be discussed.

Definition of Terms

In order to clarify the process of factor analysis, it is necessary to explain the terms

reliability and communality, as well as the process of analyzing test variance into

common, unique, specific, and error components. Because each of these concepts has

often been the topic of technical papers and books, I will try to avoid theoretical

complexities and to explain the functions and relationships of the terms in a relatively

non-technical way.

Reliability

Reliability refers to the consistency of scores obtained from an instrument on its

repeated administrations to the same person or group of persons. Psychometrically,

reliability is the proportion of standardized variance which can be consistently and

systematically obtained. If the scores on a test are standardized, the total variance

produced by that test will be unity (i.e., total variance = 1). Thus, the reliability will

be that portion of unity which is consistently observed.

If a test is perfectly reliable, the reliability coefficient will equal unity. In most cases,

however, the reliability coefficient will be considerably greater than zero but less than

unity.

The difference between unity and the reliability coefficient is referred to as the error

variance. It is variance that cannot be attributed to nonrandom sources. So, we have:

Reliability (rel.) + error variance (Ve) = 1

or

rel. = 1 Ve

For example, if the reliability coefficient for a test is reported to be .81, the error

variance will be:

Ve = 1 - .81 = .19

Communality

Communality (h2) refers to the amount of variance which is shared by two or more

variables. Its magnitude for two tests is simply the square of the correlation

coefficient between the two tests. For example, if the correlation coefficient between

two tests is .80, the common variance between them will be:

h2

= (.80)2

= .64


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This means that 64 percent of the variance is common between the two tests and is

accounted for by either of these test. If more than two tests are involved, which is

usually the case, communality is computed by methods more complex than simple

correlation, though the concept remains unchanged.

Specific and Unique Variance

Depending on the degree of correlation between two or more measures and depending

on their respective reliabilities, there may be in each test a portion of reliable variance

over and above the communality. This portion of variance by definition is not shared

by any other test in the analysis. This component is referred to as the specific variance

(Vs) and its value is the difference between the reliability (the maximum amount of

explainable variance) and the communality (common variance among the tests) of the

test in question:

Vs = rel. h2

For example, if two highly reliable tests (.90 and .95, respectively) are moderately

correlated (.75), there will be a considerable amount of specific variance in each

Vs = rel. h2

Vs (for test 1) = .90 (.70)2

= .90 - .49 = .41

and

Vs (for test 2) = .95 - .49 = .46

Therefore, there is a close relationship between the components of reliable, common,

and specific variance. One of the uses of factor analysis is to decompose test variance

into factor components in a meaningful way. It should be noted that in addition to

reliable variance, there is always (in fallible tests) error variance as well. The

combination of the Vs, which is unique to a particular test, with its error variance (Ve)

gives the total unique variance Vu. This quantity, Vu, is to be differentiated from the

specific variance:

Unique variance (Vu) = Vs + Ve

Different Components of Variance

Thus, the variance in a test can be decomposed into three components, common

variance, specific variance, and error variance:

Total variance (Vt) = h2

+ Vs + Ve

To determine the value of h2, we simply compute the sum of squares of factor

loadings, which are in fact correlation coefficients among factors and variables. For

example, from a variable, its common variance (i.e., that shared with other variables)

will be:

h2

= a2

+ b2

+ c2


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Two tests can be said to provide the same information if their values on a, b, and c are

the same and if the sum of squared loadings on a, b, and c for each test is equal to the

reliability of that test. In such a case, the specific variance for each test will be zero,

and the two tests can be used interchangeably.

The next section describes the principles of factor analysis and how different methods

operate to deal with different components of variance.

Principles of Factor Analysis

Factor analysis, in oversimplified terms, is a body of statistical procedures, based on

correlation coefficients, used to investigate underlying patterns of interrelationships

among observed variables. The main purpose of factor-analytic methods is to

associate the variables with a smaller number of traits in order to define the variables

in a more precise way. Factor analysis can be used to test a certain theory, in which

case, it is referred to as confirmatory, or it may be used to seek out a convenient

model for the structuring of variables, in which case, it is referred to as exploratory

factor analysis.

As with other statistical methods, factor analysis depends heavily both on rigorous

mathematical foundations and on theoretical interpretations. The computational side

of factor analysis is determined with mathematical precision, but how to interpret the

findings falls within the scope of theory formation. An example may help clarify the

point. Obtaining a certain correlation coefficient relates to the mathematical

dimension. For instance, a given correlation coefficient may be determined

mathematically to be statistically significant at a particular probability level.

However, just what that significant correlation coefficient means cannot be

determined by statistics alone. We must formulate some theoretical explanation based

on sound, but not mathematically rigorous, reasoning.

It is usually on the theoretical side that controversies arise because theory formation is

influenced by many extraneous factors such as the predispositions of the investigator,

the purpose of the analysis, the statistical hypotheses, the expected and hoped-for

implications of results, and so forth. In simple terms, the theory dimension is partly a

matter of taste rather than mathematical rigor.

Unfortunately, divergence between mathematical and interpretive dimensions of

factor analysis may be more marked than in other statistical procedures. Controversies

among scholars in interpreting the results of factor analysis have led to the

development of various factor-analytic methods and auxiliary techniques. Although

all the common methods can be mathematically justified, their outcomes may vary

significantly in regard to theoretical interpretations. Thus, numerous conflicts arise.

Some of the common disagreements on the interpretive side of factor analysis concern

(1) how to do the initial factor extraction, (2) how to decide on the number of factors

to be extracted, and (3) how to arrive at a final solution by applying rotational

techniques to the extracted factor structures. In spite of disagreements on these issues,

there are procedures upon which most scholars agree though it is true that their

agreement consists of suggested preferences rather than mathematical necessities.


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Extracting Initial Factors

Until recently, the centroid method was the most commonly used method for

extracting initial factors. However, owing to the increased availability of computers,

the principal-axes method, which is mathematically more sophisticated (though

computationally more complex) than the centroid method, has become the most often

used method of initial factor extraction. The principal-axes method in actual practiceconsists of two different techniques: principal component analysis (PCA) and

principal factor analysis (PFA).

There are two major differences between PCA and PFA. In PCA, the values in the

diagonal entries of the correlation matrix are somewhat arbitrarily set at unity. This

means that all of the variance generated by the tests used to obtain the correlations is

entered into the analysis. Thus, common, specific, and error variance will be used by

PCA to define the factors (in this case, they should be called components). In

PFA, on the other hand, estimated communalities are assigned to the diagonal cells of

the original correlation matrix. Thus, specific and error variance components are not

included in the analysis. A PCA matrix is illustrated in Table 1 and a PFA matrix in

Table 2.

The second major difference between PCA and PFA is the process of factor

extraction. In PCA, the factor loadings are extracted form the 1s in the diagonals. In

PFA, by contrast, an iterative (successive approximation) method is used to refine

estimates of the communalities to some predefined level of accuracy, and then, these

values are placed in the diagonal of the correlation matrix. The iterative approach is

designed to obtain the best possible estimates of the communalities for various steps

of factor extraction. There are actually several acceptable ways of accomplishing this

(Harman, 1976).

Table 1. Correlation Matrix for Principal-Component Method

Variable 1 2 3 4 5 6 7

1 1 r12 r13 r14 r15 r16 r17

2 r21 1 r23 r24 r25 r26 r27

3 r31 r32 1 r34 r35 r36 r37

4 r41 r42 r43 1 r45 r46 r47

5 r51 r52 r53 r54 1 r56 r57

6 r61 r62 r63 r64 r65 1 r67

7 r71 r72 r73 r74 r75 r76 1

Table 2. Correlation Matrix for Principal-Factor Method

Variable 1 2 3 4 5 6 7

1 * r12 r13 R14 r15 r16 r172 r21 * r23 R24 r25 r26 r27

3 r31 r32 * R34 r35 r36 r37

4 r41 r42 r43 * r45 r46 r47

5 r51 r52 r53 R54 * r56 r57

6 r61 r62 r63 R64 r65 * r67

7 r71 r72 r73 R74 r75 r76 *

*Communalities are assigned to these cells.


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Thus, factors with loadings of .30 or less can be eliminated, and this criterion can be

used to terminate factor extraction.

When a judgment must be made concerning a larger or smaller number of initial

factors, it seems reasonable to start with the larger number because it is better to

examine all the meaningful factors at the start and then eliminate unnecessary ones

rather than to exclude meaningful factors from the beginning without carefulexamination. Eliminating any factor without sufficient care may well result in loss of

information and distortion of the final outcome.

Rotating the Initial Factors

Probably, the most important step in factor analysis is the rotation of the initial factor

structures. Regardless of the method of factor extraction and the number of factors to

be extracted, almost all factor analysts unanimously agree that in order to obtain

psychologically meaningful factor patterns, the initial factor structures should be

rotated.iii

Although unrotated factors are mathematically as accurate as rotated factors,

they are hardly as useful for scientific purposes (Comrey, 1973; Nunnally, 1967;

Guilford, 1954; Guilford & Fruchter, 1973; Harman, 1976, and also see theirreferences).

The major reason for using rotation is to achieve a simpler factor structure, preferably

with each variable loading primarily on only one factor, and each factor accounting

for a maximum of the variance generated by the variables that load on it. The PFA

technique extracts the first factor in such a way as to account for the maximum

amount of variance in each and all of the variables. A given variable may actually be

better explained by two factors other than the first, but the first factor may still

account for a substantial portion of variance in that variable which may cause it to

look uncorrelated with the other factors. The procedure of getting the initial factors is

such that the second factor will necessarily account for less total variance (i.e., have a

smaller eigenvalue) than the first factor, and the third will explain less than the

second, and so forth. The procedure of factor extraction (by PFA or PCA) is similar to

a stepwise regression. The first factor variance is extracted, and so on. The algorithm

will cause each successive factor to account for less total variance than its

predecessor.

This means that the initial unrotated factors may not give the best picture of the factor

structure. At each step, variance from many different common factor sources is being

extracted because the factor vector is placed in such a way that as many of the

variables as possible have substantial projections on it (Comrey, 1973, p.103). Figure

1 illustrates how the first factor may in effect usurp variance that would otherwise fall

to two other uncorrelated factors, namely, X and Y in the figure which fall on thehorizontal and vertical axes at 45-degree angles from factor 1 which would be

extracted by PFA.

It is an unfortunate possibility that in the first step of PFA, the first factor can be, and

usually is, a composite of variance components usurped in just this way from clearly

distinct factors. In such cases, taking the first factor as a general factor is a mistake.

There is no easy solution for the problem, but the one approach that most factor

analysts recommend is the rotation of the initial factor structures.iv Though there is no


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mathematical justification for rotating the factors, rotating the initial factors is

expected to lead to psychologically more interpretable factor structures. For example,

if we rotate the factors illustrated in Figure 1 (approximately 45 degrees), we obtain

two distinct and probably more meaningful factors at positions X and Y. Figure 2

shows the result of such a rotation. The language variables in Figure 2 load heavily on

factor 1 and the body-measurement variables load heavily on factor 2. This seems to

make more sense theoretically than having a general factor as shown in Figure 1.


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Of course, determining the kind of rotation technique to use will depend on many

considerations such as the nature of the variables included in the analysis and the

kinds of interpretations that the researcher might want to make. However, for many

reasons, orthogonal solutions which arrive at uncorrelated rotated factors are preferred

(see Hinofotis, Chapter 8, for discussion of orthogonal versus oblique rotations).

To examine the influence of rotation, it will be helpful to investigate differences in the

outcome of rotated and unrotated solutions. Data are presented below from various

versions of the UCLA English as a Second Language Placement Examination

(ESLPE) administered in the fall quarters of 1977, 1978, and 1979. A principal factor

solution with iterations was used to extract the initial factors. Then, the number of

factors was set at four after eliminating non-significant factors. The results of rotated

and unrotated solutions are presented in pairs of tables for each of the three data sets

to demonstrate as clearly as possible the difference in the factor structures obtained.

Examining these pairs of tables, it is clear that in unrotated factor structures (Tables

3a, 4a, and 5a), as we would expect, the first factor in each case, accounts for a large

proportion of the total variance. We might therefore be tempted to say that no otherfactors are needed to account for a large proportion of the total variance. We might

therefore be tempted to say that no other factors are needed to account for the data.

After all, the additional factors do not account for a significant amount of variance if

the factor structure is left in its initial form. But as we saw in Figure 1, this may be an

artifactual result. Indeed, when the initial factors are rotated, entirely different factor

patterns appear.v

The rotated factor structures (Tables 3b, 4b, and 5b) show that the

first unrotated factor was highly inflated and that the subsequent factors were

correspondingly deflated, giving us a distorted view. Factor 1 is not so powerful as it


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seems in Tables 3a, 4a, and 5a, and factors 2, 3, and 4 are not so weak as they seem.

This comes clear in the rotated solution given in Tables 3b, 4b, and 5b. It should be

noted that the total amount of variance accounted for in either type of solution is

unchanged from the initial to the rotated solution. They are explaining in fact the same

total variance but distributing it in different ways. The crucial point is that the first

factor can pull out variance from several unrelated variables and may therefore lead to

misinterpretations. In an unrotated factor structure, the first factor by the very natureof the extraction procedure, will account for the greatest amount of variance possible

and thus will be a composite of multiple unrelated factors.

Table 3a. Unrotated Factor Matrix Using Principal Factor

with Iteration for the Fall 77 ESLPE Subtests

Subtests F1 F2 F3 F4

Spelling .69 .59 * *Punctuation .40 .63 * *

Dictation .86 * * *

Cloze .79 * * *

Listening comprehension .71 * .35 *

Verbs .80 * * *Prepositions .80 * * *

Articles .72 * * *Vocabulary romance .77 * .37 *

Vocabulary germanic .81 * * *

Reading comprehension .88 * * *

Table 3b. Varimax Rotated Factor Matrix Using Principal Factor



Spelling * .83 * .36Punctuation * .74 * *

Dictation .45 .52 .55 *

Cloze .50 .58 .35 *Listening comprehension .37 * .78 *

Verbs .49 * .45 .49

Prepositions .67 * .39 *

Articles .46 * .52 .34

Vocabulary romance .77 * * *Vocabulary germanic .52 * .42 .62

Reading comprehension .59 * .38 .34

*Loadings less than .30 are deleted.




Cloze .42 .83 * *Dictation .46 .51 * *

Listening 1 .71 * .35 *

Listening 2 .80 * * *

Reading 1 .80 * * *Verbal 1 .72 * * *

Prepositions .77 * * *

Articles .81 * * *Verbal 2 .88 * * *

Reading 2 .63 * * .36



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Cloze * * .90 *

Dictation * * .69 *


Listening 2 .75 .35 * *Reading 1 .33 .49 * .54

Verbal 1 * .73 * *Prepositions .50 .54 * *

Articles .36 .63 * .37

Verbal 2 .38 .74 * .33

Reading 2 .30 * * .61Listening 3 .67 .31 * .31

Loadings less than .30 are deleted.



Subtests F1 F2 F3Cloze .80 * -.30

Dictation .81 * *

LCV .64 .45 *

LCW .72 .36 *RC .74 * *

Grammar (verbs) .81 * *

Grammar (prepositions) .83 * *Grammar (others) .79 * *

Functional .80 * *

LCV = listening comprehension: visualLCW = listening comprehension: written

RC = reading comprehension



Subtests F1 F2 F3

Cloze .51 * .65

Dictation .35 .58 .55

LCV * .74 *LCW .35 .71 *

RC .53 .31 .46

Grammar (verbs) .78 * .30Grammar (prepositions) .75 .39 *

Grammar (others) .70 * .35

Functional .66 .38 .30

*Loadings less than .30 are deleted.LCV = listening comprehension: visual

LCW = listening comprehension: writtenRC = reading comprehension

This might raise controversies about the utility of rotation. One might argue that as

long as the amount of explained variance remains the same and/or increases

insignificantly, there is no need to rotate the factors. However, how the variance is


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explained logically outranks the question of the magnitude of the explainable

variance.vi The reason is that decomposing the common variance into appropriate

factor loadings is precisely the purpose of factor analysis.

An analogy may help clarify the point. If someone sees a green area in the mountains

from a distance, the most likely initial reaction is that the green area is a bunch of

trees. Even if the person knew how many trees were there, this would not allow himto conclude that all the trees were of the same kind. The green area might be

composed of trees of different kinds. Without a closer look, it might be quite incorrect

to claim that they are all of the same kind. The same thing may happen in factor

analysis. The first factor may be the most obvious sign of interrelationship among the

variables, but without further and more detailed examination, to claim that there is

only one factor would also be misleading.

The data presented here are consistent with almost all previous reports on the results

of factor analyses. Whenever the investigator(s) did not rotate the initial factor matrix,

the first factor appeared to be so strong that the researcher inferred, mistakenly, the

existence of one and only one general factor accounting for essentially all the

explainable variance in the data. However, when the same researchers rotated theinitial factors, various factor patterns appeared from the same data (see Oller &

Hinofotis, 1980; Scholz, et al. 1980; and other entries in the same volume).

Thus, the hypothesis of a unitary language proficiency factor will not be supported if

one follows the appropriate steps in conducting the relevant factor analyses. If, on the

other hand, one uses incomplete methods, it will appear, in study after study, that the

first factor, whatever it may be called, is the only factor underlying the variables.

Therefore, previous interpretations of unrotated factor matrices are called into

question and further investigation is required to determine the actual composition of

language proficiency.

If the appropriate steps are followed, one will be drawn to conclude that the unitary

language proficiency hypothesis is not plausible. However, it is still possible that

owing to the nature of language skills and their interrelationships, a general factor

may exist that accounts for a considerable amount of variance in a large variety of

language processing tasks.

But this is not the same as arguing for a unitary factor. A general factor would not

exhaust all of the reliable variance. Only that portion that all language skills share will

be manifested in the common factor. This does not mean, however, that a specific

factor related to individual language skills does not exist. These specific factors can

be expected to account for a portion of variance over and above the common variance.

Another way to test for the existence of specific variance and thus to examine the

plausibility of an exhaustive unitary language proficiency factor is to compare the

magnitude of the reliability coefficients with the values of communalities. The

difference between the two, if substantial, will directly reject this hypothesis.

A close examination of the data, presented either here or in the literature, indicates

that there is specific variance for almost all the tests included in the analyses.

Regardless of the accuracy of the type of factor analysis being used, and regardless of


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the application of rotation to the initial factor structures, the differences between the

reliability coefficients and the reported communalities are evidence for the existence

of specific variances, which refutes the unitary hypothesis.

For the purposes of illustration, the communalities and the reliability coefficients for

the data discussed earlier in this paper are reported in Table 6, 7, and 8. Reliability

coefficients are calculated using KR-21 for cloze and dictation, and Cronbachs alphafor other subtests. It should be noted that internal consistency reliability coefficients

for cloze and dictation are not strictly appropriate, as I have argued elsewhere

(Farhady, 1979, 1980a, and see also Chapters 14 and 17 of this volume). However,

since such estimates would err on the high side for cloze and dictation, it would only

strengthen the argument at stake here to use methods giving lower reliabilities and

thus higher specificities.

It can be observed from the data presented in Tables 6, 7, and 8 that most of the tests

have specific variances. In some of them, the specificity is substantial. Of course,

these results should be compared with those from other studies, but it seems to me

that looking at language tests from the unitary perspective is not defensible.

Table 6. Communalities and Reliability Coefficients for the Fall 77 ESLPE Subtests


Subtests Communality Reliability Specificity

Spelling .88 .88* .00

Punctuation .57 .60* .03Dictation .81 .93* .12

Cloze .72 .83* .11

Listening comprehension .81 .84 .03Verbs .74 .86 .12

Prepositions .69 .77 .08

Articles .62 .70 .08

Vocabulary romance .68 .88 .20

Vocabulary germanic .87 .89 .02Reading comprehension .67 .89 .22

* KR-21. : Alpha.


Cloze .87 .97* .10

Dictation .56 .98* .42Listening 1 .69 .74 .05

Listening 2 .76 .77 .01

Reading 1 .71 .85 .14

Verbal 1 .66 .71 .05Prepositions .62 .68 .06

Articles .70 .72 .02

Verbal 2 .84 .89 .05Reading 2 .56 .67 .11

Listening 3 .70 .73 .03

* KR-21. : Alpha.


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Conclusions

Various conclusions can be drawn. First, factor analysis is complex and should be

applied with great care. Second, the most often recommended technique for factor

extraction is the principal-factor solution, which uses the communalities, estimated

through iteration, in the diagonals of the correlation matrix. Third, initial factor

structures should be rotated in order to obtain maximally meaningful and interpretable

factor patterns. And finally, the results should be interpreted only after the completion

of the sequence of required steps. This implies that the results obtained from

incomplete factor analyses are questionable and need to be reanalyzed and

reinterpreted in terms of the steps recommended in this paper and in many standard

resource books on factor analysis (see Harman, 1976). Therefore, alternatives other

than the unitary-factor hypothesis, some of which were discussed in this paper, and

perhaps others not touched on here, should be pursued to improve our knowledge of

the nature of language proficiency.

I wish to thank Frances Hinofotis, Andrew Cohen, and Ebrahim Maddahian for their comments on an

earlier draft of this paper.

* This is the revised version of the paper printed in W.J. Oller, Jr. (ed.) (1983). Issues in language

testing research. Rowley, Mass.: Newbury House Publishers, Inc.

Editors Notes

iIn Chapter 9, below, Upshur & Homburg discuss the results of applying a statistical correction for the

inflation of PCA loadings. Their method comes form Kazelskis (1978). It yields corrected estimateswhich are quite comparable with those obtained from PFA. However, it should be noted that the

adjustments do not by any means remove the general factor that is in dispute. What they do is reduce its

apparent strength, but only slightly.

iiThis stems from the logic that an eigenvalue which accounts for less variance than that which is

found in any single input variable is of little or no interest. This is in keeping with the general aim of

factor analysis to reduce the number of constructs to be taken into account. In PCA, with unities on the


Cloze .70 .71* .01

Dictation .75 .93* .18

LCV .62 .67 .05

LCW .66 .74 .08

RC .57 .78 .21Grammar (verbs) .75 .77 .02

Grammar (prepositions) .76 .77 .01

Grammar (others) .67 .78 .11Functional .66 .78 .12

* KR-21. : Alpha.

LCV = listening comprehension: visual

LCW = listening comprehension: written

RC = reading comprehension


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diagonal of the correlation matrix, if the allowed eigenvalues were much less than unity (or in PFA

where communalities are placed on the diagonal, if they were much less than the mean communalityfor the input variables), the number of factors extracted would exceed the number of variables to be

taken into account.

iiiBut what if only one factor appears? In such a case, rotation makes no sense. Rotation is sensible

only if there are two or more factors over which the variance from the several contributing variablesmay be distributed in a meaningful way. However, one of the problems which the experts readily

acknowledge is that there are always innumerable ways of doing this (Harman, 1976, p.19). Not all ofthem will be equally appealing in theory, but all will be equally defensible mathematically. The

potential escape from this aspect of the rotational dilemma was thought to be a virtue of the singlefactor solutions in Chapter 1. However, it is acknowledged that factoring methods without rotation are

applicable only when in fact only one factor can be discerned even then, PFA is to be preferred over

PCA, as Farhady argues. The confirmatory techniques discussed in Chapters 6 and 7 below are also

promising methods which offer more appropriate model testing capabilities.

ivThe use of any rotational procedure, however, presupposes a multiple-factor solution. However, the

assumption underlying the factoring done in Chapter 1 was to attempt to assess the strength of that

hypothesized factor. Must we reject as unconscionable any basis for looking for just such a generalfactor? Was not even the strongest version of this possibility worth examining if only to rule it out?

vHowever, the appearance of the different patterns in the rotated solutions is also a function of the

statistical procedure to some extent, is it not, just as the large first factor is in the unrotated solutions?

Therefore, isnt it possible to prefer rotated solutions without necessarily rejecting the possibility of a

general factor?

viBut isnt the magnitude of explainable variance always an important issue? Is any theorist interested

in elegant explanations of negligible factors?

Documents

3 on the Plausibility of Unitary Language Proficiency Factor