Quantitative MethodsSupplement Material for

Forecasting

A. Keramati2007

Agenda

• Canonical Correlation Analysis– Concept and applications– Example

• Regression Analysis with Moderator-Mediator Variables

Canonical Correlation Analysis (CCA)

• CCA was developed first by H. Hotelling.– H. Hotelling. Relations between two sets of variates.

Biometrika, 28:321-377, 1936.

• CCA measures the linear relationship between two multidimensional variables.

• CCA finds two bases, one for each variable, that are optimal with respect to correlations.

• Applications in economics, medical studies, bioinformatics and other areas.

• More than one canonical correlations will be found each corresponding to a different set of basis vectors/Canonical variates

• Correlations between successively extracted canonical variates are smaller and smaller

• Correlation coefficients : Proportion of correlation between the canonical variates accounted for by the particular variable.

Canonical Correlation Analysis

Canonical Correlation Analysis (CCA)• Two multidimensional variables

– Two different measurement on the same set of objects

• Web images and associated text• Protein (or gene) sequences and related literature

(text)• Protein sequence and corresponding gene expression • In classification: feature vector and class label

– Two measurements on the same object are likely to be correlated.

• May not be obvious on the original measurements.• Find the maximum correlation on transformed space.

Canonical Correlation Analysis (CCA)

TXXW

Correlatio

n

TYYW

measurement transformationTransformed data

Problem definition

•Find two sets of basis vectors, one for x and the other for y, such that the correlations between the projections of the variables onto these basis vectors are maximized.

: and yx ww

Given

Compute two basis vectors

ywy y ,

Problem definition

•Compute the two basis vectors so that the correlations of the projections onto these vectors

are maximized.

Algebraic derivation of CCA

•In general, the k-th basis vectors are given by the k–th eigenvector of

• The two transformations are given by ypyyY

xpxxX

wwwW

wwwW

,,

,,

21

21

Example 1

The aim of this problem is to empirically test whether the use of IT in companies leads to productivity improvement or not

1. Is there any positive and significant correlation between the use of IT and firm performance?

2. What is the effect of different aspects of IT usage on company performance?

Research Model

IT UsageIT in communicationIT in Production and OperationsIT in Decision SupportIT in Administration and pecuniary affairs

PerformanceCustomer resultsEmployee resultsOperational results

Figure 1. Research model

Canonical correlation analysis (1)

• Seven variables of the use of IT in companies are designed as the set of multiple independent variables or the predictor variables.

• Three measures of company performance are specified as the set of multiple dependent variables or the criterion variables.

• The benefit of the canonical correlation analysis, in contrast to the univariate analysis of scales is that it takes into account the simultaneous interaction between all scales.

Canonical correlation analysis (2)

• The statistical problem involves identifying any latent relationships between extent of use of IT in companies and the level of company performance.

Canonical correlation analysis (3)

• The canonical correlation analysis was restricted to drive three canonical functions, since the dependent variable set (company performance) contained three variables.

• To determine the number of canonical functions included in the interpretation stage, our analysis focused on the level of statistical significance, and the redundancy indices for each variate.

• To do statistical significance test, multivariate tests of three functions are performed simultaneously. The test statistics employed are Wilk’s lambda and Chi-Square tests.

• Table 3 shows the multivariate test statistics, which both indicate that the first canonical function is statistically significant at .001 levels. In addition to statistical significance, the canonical correlation of first function is (0.639) which is partially significant.

Step1Table 3) Multi variate test of significance

Canonical Function

Canonical Correlation

Wilk’sChi-SQDFSig.

1.639.513**59.026**21.000.000

2.429.86712.66312.000.394

3.292.9623.4295.000.634

Step 2• Table 4A shows that redundancy index for the

dependent variate is 0.15. Table 4b indicates that the redundancy index for independent variates is 0.323.

• These redundancy indices belong to the first canonical function.

• The variates for the second and third functions are too low to be of practical importance (Tables 4A and 4B). (These numbers indicate the amount of variance of one variate, explained by the other.)

• Explaining more than 25% of the variance in an organizational level of study can be fairly significant, considering all other factors that can contribute to performance measures.

• This, together with the results presented in Table 3, justifies the exclusion of the second and third functions.

Step 2Table 4a) Redundancy analysis of Dependent Variates for three functions

Canonical Function

Variance explained by own variablesVariance explained by opposite variables

PercentCumulative percentPercentCumulative percent

137.137.115.115.1

210.747.81.116.2

310.558.300.416.6

Table 4b) Redundancy analysis of Independent Variates for three functions

Canonical Function

Variance explained by own variables

Variance explained by opposite variables

PercentCumulative percent

PercentCumulative percent

179.379.332.332.3

213.292.51.333.6

37.51.00333.9

Step 3Interpretations

• In general, the researcher faces the choice of interpretation of the functions using: – canonical weights (standardised coefficients)– canonical loadings (structure correlations)– canonical cross loadings.

• Given a choice, it is suggested that cross loadings are superior to loadings, which are in turn superior to weights (Hair et al., 1998). Hence, the interpretation presented here is based on cross loadings.

Step 3 (Cont’)Interpretations

• Table 5 includes the cross-loadings for the three canonical functions.

• Considering the first function, canonical cross-loadings for the independent variate range from 0.300 to 0.481.

• The canonical cross loadings for the dependent variate are more than 0.523 for the first function.

• Both of the canonical cross-loadings for the dependent and independent variates are acceptable for interpretation.

• All cross-loadings are positive. This gives one more indication of a valid relationship between two variates.

Step 3 (Cont’)Interpretations

• The results for the dependent variables indicate that the strongest correlations in descending order of importance, are associated with the extent to which IT is used in the planning, administration affaires, pecuniary affaires, production and operations, communication, decision support and quality control.

• The overall correlation between dependent and independent variables indicates that high ratings on the dependent variables are associated with higher levels of company performance.

Table 5) Canonical cross-loadings of the three functions

Function 1Function 2 Function 3

Independent variate

IT in planning0.4810.065.-060

IT in administration affaires0.480.-041.-014

IT in pecuniary affaires0.422.-1180.089

IT in production and operations

0.3410.1760.098

IT in communication0.3380.008.-025

IT in decision support0.3140.131.-035

IT in Quality control0.3000.079.-068

Dependent variate

Customer results0.558.-083.-080

Staff results0.5230.180.-009

Operation results0.621.-0060.046

Conclusions (1)• Bivariate correlation analysis reveals that

there are significant correlations between three company performance variables including:– “customer results”– “employee results” – “operational performance results”

and the seven scales of the IT usage including:– IT in communication, decision support, planning,

operation, quality control, administration and pecuniary affairs.

• The effects of IT in quality control, IT in planning and IT in pecuniary affairs on firm performance have not been reported in the literature.

Conclusions (2)

• This study strongly showed positive significant associations between three out of seven scales of IT usage (IT in planning, IT in administration, IT in pecuniary affaires) and the company performance indicators.

• This shows the importance of use of IT in planning, administration and pecuniary affaires to realize the potential of IT.

• According to canonical correlation findings (table 5), the effect of IT usage on operational results is more than the effects of IT usage on the two other company performance measures.

What Is Mediation?

•    Consider a variable X that is assumed to affect another variable Y.  The variable X is called the initial variable and the variable that it causes or Y is called the outcome.  In diagrammatic form, the unmediated model is

What Is Mediation?

•    The effect of X on Y may be mediated by a process or mediating variable M, and the variable X may still affect Y. Path c is called the total effect.  The mediated model is

• Path c' is called the direct effect.• The mediator has been called an intervening

or process variable.• Complete mediation is the case in which

variable X no longer affects Y after M has been controlled and so path c' is zero.

• Partial mediation is the case in which the path from X to Y is reduced in absolute size but is still different from zero when the mediator is controlled.

•     Note that a mediational model is a causal model.  For example, the mediator is presumed to cause the outcome and not vice versa.  If the presumed model is not correct, the results from the mediational analysis are of little value.

Baron and Kenny (1986) and Judd and Kenny (1981) have discussed four steps in establishing mediation: – Step 1:  Show that the initial variable is

correlated with the outcome.  Use Y as the criterion variable in a regression equation and X as a predictor (estimate and test path c). This step establishes that there is an effect that may be mediated.

Y = f(X)

– Step 2: Show that the initial variable is correlated with the mediator.  Use M as the criterion variable in the regression equation and X as a predictor (estimate and test path a).  This step essentially involves treating the mediator as if it were an outcome variable.

M = f(X)

– Step 3:  Show that the mediator affects the outcome variable.  Use Y as the criterion variable in a regression equation and X and M as predictors (estimate and test path b).  It is not sufficient just to correlate the mediator with the outcome; the mediator and the outcome may be correlated because they are both caused by the initial variable X.  Thus, the initial variable must be controlled in establishing the effect of the mediator on the outcome.

Y = f (X, M)

– Step 4:  To establish that M completely mediates the X-Y relationship, the effect of X on Y controlling for M (path c')should be zero .  The effects in both Steps 3 and 4 are estimated in the same equation.

• If all four of these steps are met, then the data are consistent with the hypothesis that variable M completely mediates the X-Y relationship, and if the first three steps are met but the Step 4 is not, then partial mediation is indicated.

Example 2

• Hypothesis :

The relationship between IT and firm performance will be mediated by the extent of BPR associated with the IT.

H1 H1

The influence of IT on business processes (BPRM)

Order flow (BPOF)Strategic processes (BPST)Product (BPPR)Marketing and sales (BPMS)Services (BPSE)Accounting (BPAC)Personnel (BPPE)Technology (BPTE)

IT application (ITU)Performance upgrading (PER)

H1 H1

BPRM

ITUPER

1. The independent variable must affect the mediator [equation (1)].

2. The independent variable must affect the dependent variable [equation (2)].

3. The mediator must affect the dependent variable equation (3).

4. If these conditions hold, then the effect of the independent variable on the dependent variable must be less in equation (3) than in equation (2).

H1 H1

BPRM

ITU PER

Results of regression analysis: Testing mediating effects of BPRM

No.Measurement scaleB

t-test

R2Adj .R2SE

F-test

StatisticsSig.StatisticsSig.

PER1=f(ITU).5407.114.000.348.34

1.635850.607.000

Business process of order flow: BPOF

1PER1=f(BPOF).3244.726.000.190.18

2.7082

922.335.000

2BPOF =f(ITU).4373.691.000.125.11

6.9912

213.623.000

3PER1=f(ITU, BPOF).4565.856.000

.407.39

4.6095

132.224.000

.1933.062.003

Business process of strategy: BPST

1PER1=f(BPST).1753.632.000.123.11

4.7381

013.190.000

2BPST =f(ITU).4952.713.008.073.06

31.5173

7.359.008

3PER1=f(ITU, BPST).4866.256.000

.383.37

0.6225

228.842.000

.1042.461.016

In the case of BPOF, the beta-coefficient of equation PER*=f(ITU) suggests that Performance is a function IT Usage (b=0.0.540, p<0.000). The coefficient of row 2 suggests that changes in BPOF is a function of IT Usage (b=0.437, p50.000) and the coefficients of row 3 suggest that performance improvement is a function of IT Usage (b=0.456, p50.000) and changes in BPOF (b=0.193, p<0.003).Since the coefficient associated with IT Usage is less in row 3 than in equation PER*=f(ITU), and rows 1 and 2 are both significant, a mediating effect is implied.

Moderator

A variable that changes the impact of one variable on

another.

Predictor Outcome

Moderator

Testing a Moderator Hypothesis

X*MInteraction (OIS* ITU)

XPredictor (ITU)

YOutcome

(PER)M

Moderator (OIS)

Hierarchical regression• Hierarchical regression analysis conducts in three steps:• Step 1: Enter moderator into the equation (1).

Y = f (M)

• Step 2: Enter the predictor variable into the equation (2).

Y=f (M, X)

• Step 3: Finally, enter the interaction between moderator and predictor into the regression equation (3).

Y=f (M, X, X*M)

Hierarchical regression

• When these interaction terms account for a significant amount of incremental variance in the dependent variable, as measured by the t-tests for each interaction or by significance tests for the incremental F-statistic, then there is evidence to support that there is a moderating effect of M on relationship between X and Y.

Example 3

OIS* ITU

ITU

PEROrganizational

Infrastructure (OIS):•Delegation of power (reducing hierarchy) (INEM)•Decentralization (INDE)•Training (INTR)•Group work (INGR)•Process management (INPM)•Relationship with customers and suppliers (INRM)

• For each of the seven organizational infrastructure scales, this analysis is conducted in three steps:

• 1. In each equation, one of the organizational infrastructure scales is entered into the equation [for example INEM is entered in equation (1)].

• 2. Total mean of the ITU is entered into the equation.

• 3. Finally, the interaction between respective organizational infrastructure and ITU are entered into the regression equation.

Results of hierarchical regression analysis: Testing moderating effects of OIS

StepMeasurement

scaleB

t-test

R2∆R2

F-test

StatisticsSig.StatisticsSig.

Empowerment: INEM

1INEM.3935.498.00

0.241.24130.229.000

2ITU.4194.903.00

0.396.15424.038.000

3ITU*INEM.-155-3.026.00

3.450.0549.155.003

Decentralization: INDE

1INDE.3154.354.00

0.168.16818.953.000

2ITU.4635.239.00

0.357.19027.452.000

3ITU*INDE.-147-2.687.00

9.404.0477.220.009

Overall

20.806.000

• Table 9 shows the results of a hierarchical regression with PER* as the dependent

• variable, the organizational infrastructure scales (for example INEM in first part of

• the table 9), ITU, and their interactions entered in the sequential manner described

• above. Results of hierarchical regression with, for example, INEM as the moderating

• variable will be discussed in the following paragraph. The discussion of the other

• moderating variables is the same as INEM.

Results• The organizational infrastructure scale, which is

entered into the model in the first step, accounts for a significant amount of variance (an R2 of 0.241, P<0.000).

• The inclusion of ITU in the second step provides a significant improvement (an incremental R2 of 0.154, P<0.000), and also the interaction terms of step 3 result in an incremental R2 of 0.054, which is significant (P<0.01). The overall effect of the model is the explanation of 39.6% of the variance in PER*, which the associated F test indicates is significant at P<0.000.

Structural Equation Modeling (SEM)