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Factor Analysis with SPSS

What is a Common Factor? • It is an abstraction, a hypothetical construct that affects at least two of our measurement variables. • We want to estimate the common factors that contribute to the variance in our variables. • Is this an act of discovery or an act of invention?

What is a Unique Factor? • It is a factor that contributes to the variance in only one variable. • There is one unique factor for each variable. • The unique factors are unrelated to one another and unrelated to the common factors. • We want to exclude these unique factors from our solution.

Iterated Principal Factors Analysis • The most common type of FA. • Also known as principal axis FA. • We eliminate the unique variance by replacing, on the main diagonal of the correlation matrix, 1’s with estimates of communalities. • Initial estimate of communality = R2 between one variable and all others.

Lets Do It • Using the beer data, change the extraction method to principal axis.

Look at the Initial Communalities • They were all 1’s for our PCA. • They sum to 5.675. • We have eliminated 7 – 5.675 = 1.325 units of unique variance. Communalities

COST SIZE ALCOHOL REPUTAT COLOR AROMA TASTE

Initial .738 .912 .866 .499 .922 .857 .881

Extraction .745 .914 .866 .385 .892 .896 .902

Extraction Method: Principal Axis Factoring.

Iterate! • Using the estimated communalities, obtain a solution. • Take the communalities from the first solution and insert them into the main diagonal of the correlation matrix. • Solve again. • Take communalities from this second solution and insert into correlation matrix.

• Solve again. • Repeat this, over and over, until the changes in communalities from one iteration to the next are trivial. • Our final communalities sum to 5.6. • After excluding 1.4 units of unique variance, we have extracted 5.6 units of common variance. • That is 5.6 / 7 = 80% of the total variance in our seven variables.

• We have packaged those 5.6 units of common variance into two factors: Total Variance Explained

Factor 1 2

Extraction Sums of Squared Loadings Total % of Variance Cumulative % 3.123 44.620 44.620 2.478 35.396 80.016

Extraction Method: Principal Axis Factoring.

Rotation Sums of Squared Loadings Total % of Variance Cumulative % 2.879 41.131 41.131 2.722 38.885 80.016

Our Rotated Factor Loadings • Not much different from those for the PCA. a Rotated Factor Matrix

Factor 1 2 TASTE .950 -2.17E-02 AROMA .946 2.106E-02 COLOR .942 6.771E-02 SIZE 7.337E-02 .953 ALCOHOL 2.974E-02 .930 COST -4.64E-02 .862 REPUTAT -.431 -.447 Extraction Method: Principal Axis Factoring. Rotation Method: Varimax with Kaiser Normalization. a. Rotation converged in 3 iterations.

Reproduced and Residual Correlation Matrices • Correlations between variables result from their sharing common underlying factors. • Try to reproduce the original correlation matrix from the correlations between factors and variables (the loadings). • The difference between the reproduced correlation matrix and the original correlation matrix is the residual matrix.

• We want these residuals to be small. • Check “Reproduced” under “Descriptive” in the Factor Analysis dialogue box, to get both of these matrices: • Reproduced Correlations

Reproduced Correlation COST SIZE ALCOHOL REPUTAT COLOR AROMA TASTE a Residual COST SIZE ALCOHOL REPUTAT COLOR AROMA TASTE

COST SIZE .745 b .818 .818 .914b .800 .889 -.365 -.458 1.467E-02 .134 -2.57E-02 8.950E-02 -6.28E-02 4.899E-02 1.350E-02 1.350E-02 -3.29E-02 1.495E-02 -4.02E-02 6.527E-02 3.328E-03 4.528E-02 -2.05E-02 8.097E-03 -1.16E-03 -2.32E-02

ALCOHOL REPUTAT COLOR AROMA TASTE .800 -.365 1.467E-02 -2.57E-02 -6.28E-02 .889 -.458 .134 8.950E-02 4.899E-02 b .866 -.428 9.100E-02 4.773E-02 8.064E-03 -.428 .385b -.436 -.417 -.399 b 9.100E-02 -.436 .892 .893 .893 4.773E-02 -.417 .893 .896 b .898 8.064E-03 -.399 .893 .898 .902 b -3.295E-02 -4.02E-02 3.328E-03 -2.05E-02 -1.16E-03 1.495E-02 6.527E-02 4.528E-02 8.097E-03 -2.32E-02 -3.47E-02 -1.88E-02 -3.54E-03 3.726E-03 -3.471E-02 6.415E-02 -2.59E-02 -4.38E-02 -1.884E-02 6.415E-02 1.557E-02 1.003E-02 -3.545E-03 -2.59E-02 1.557E-02 -2.81E-02 3.726E-03 -4.38E-02 1.003E-02 -2.81E-02

Extraction Method: Principal Axis Factoring. a. Residuals are computed between observed and reproduced correlations. There are 2 (9.0%) nonredundant residuals with absolute values greater than 0.05. b. Reproduced communalities

Nonorthogonal (Oblique) Rotation • The axes will not be perpendicular, the factors will be correlated with one another. • the factor loadings (in the pattern matrix) will no longer be equal to the correlation between each factor and each variable. • They will still equal the beta weights, the A’s in X j = A1 j F1 + A2 j F2 +  + Amj Fm + U j

• Oblique solutions make me uncomfortable. • but I did one just for you – • a Promax rotation. • First a Varimax rotation is performed. • Then the axes are rotated obliquely. • Here are the beta weights, in the “Pattern Matrix,” the correlations in the “Structure Matrix,” and the correlation between factors:

Beta Weights

Correlations Structure Matrix

Pattern Matrixa

Factor

Factor

1 .947 .946 .945 .123 .078 -.002 -.453

2 .030 .072 .118 .956 .930 .858 -.469

1 2 TASTE .955 -7.14E-02 AROMA .949 -2.83E-02 COLOR .943 1.877E-02 SIZE 2.200E-02 .953 ALCOHOL -2.05E-02 .932 COST -9.33E-02 .868 REPUTAT -.408 -.426

TASTE AROMA COLOR SIZE ALCOHOL COST REPUTAT

Extraction Method: Principal Axis Factoring. Rotation Method: Promax with Kaiser Normalization. a. Rotation converged in 3 iterations.

Extraction Method: Principal Axis Factoring. Rotation Method: Promax with Kaiser Normalization.

Factor Correlation Matrix Factor 1 2

1 1.000 .106

2 .106 1.000

Extraction Method: Principal Axis Factoring. Rotation Method: Promax with Kaiser Normalization.

Exact Factor Scores • You can compute, for each subject, estimated factor scores. • Multiply each standardized variable score by the corresponding standardized scoring coefficient. • For our first subject, Factor 1 = (-.294)(.41) + (.955)(.40) + (-.036)(.22) + (1.057)(-.07) + (.712)(.04) + (1.219)(.03) + (-1.14)(.01) = 0.23.

• SPSS will not only give you the scoring coefficients, but also compute the estimated factor scores for you. • In the Factor Analysis window, click Scores and select Save As Variables, Regression, Display Factor Score Coefficient Matrix.

• Here are the scoring coefficents: Factor Score Coefficient Matrix Factor COST SIZE ALCOHOL REPUTAT COLOR AROMA TASTE

1 .026 -.066 .036 .011 .225 .398 .409

2 .157 .610 .251 -.042 -.201 .026 .110

Extraction Method: Principal Axis Factoring. Rotation Method: Varimax with Kaiser Normalization. Factor Scores Method: Regression.

• Look back at the data sheet and you will see the estimated factor scores.

Use the Factor Scores • Let us see how the factor scores are related to the SES and Group variables. • Use multiple regression to predict SES from the factor scores. Model Summary Model 1

R R Square .988a .976

Adjusted R Square .976

a. Predictors: (Constant), FAC2_1, FAC1_1

Std. Error of the Estimate .385

ANOVAb Model 1

Regression Residual Total

Sum of Squares 1320.821 32.179 1353.000

df 2 217 219

Mean Square 660.410 .148

F 4453.479

Sig. .000 a

a. Predictors: (Constant), FAC2_1, FAC1_1 b. Dependent Variable: SES

Coefficientsa

Model 1

Standardized Coefficients Beta (Constant) FAC1_1 FAC2_1

.681 -.718

a. Dependent Variable: SES

t 134.810 65.027 -68.581

Sig. .000 .000 .000

Correlations Zero-order Part .679 -.716

.681 -.718

• Also, use independent t to compare groups on mean factor scores. Group Statistics

FAC1_1 FAC2_1

GROUP 1 2 1 2

N 121 99 121 99

Mean -.4198775 .5131836 .5620465 -.6869457

Std. Deviation .97383364 .71714232 .88340921 .55529938

Std. Error Mean .08853033 .07207552 .08030993 .05580969

Independent Samples Test Levene's Test for Equality of Variances

F FAC1_1

FAC2_1

Equal variances assumed Equal variances not assumed Equal variances assumed Equal variances not assumed

19.264

25.883

Sig. .000

.000

t

t-test for Equality of Means 95% Confidence Interval of the Difference df Sig. (2-tailed) Lower Upper

-7.933

218

.000

-1.16487

-.701253

-8.173

215.738

.000

-1.15807

-.708049

12.227

218

.000

1.047657

1.450327

12.771

205.269

.000

1.056175

1.441809

Unit-Weighted Factor Scores • Define subscale 1 as simple sum or mean of scores on all items loading well (> .4) on Factor 1. • Likewise for Factor 2, etc. • Suzie Cue’s answers are • Color, Taste, Aroma, Size, Alcohol, Cost, Reputation

• 80, 100, 40, 30, 75, 60, 10 • Aesthetic Quality = 80+100+40-10 = 210 • Cheap Drunk = 30+75+60-10 = 155

• It may be better to use factor scoring coefficients (rather than loadings) to determine unit weights. • Grice (2001) evaluated several techniques and found the best to be assigning a unit weight of ± 1 to each variable that has a scoring coefficient at least 1/3 as large as the largest for that factor. • Using this rule, we would not include Reputation on either subscale and would drop Cost from the second subscale.

Item Analysis and Cronbach’s Alpha • Are our subscales reliable? • Test-Retest reliability • Cronbach’s Alpha – internal consistency – Mean split-half reliability – With correction for attenuation – Is a conservative estimate of reliability

• AQ = Color + Taste + Aroma – Reputation • Must negatively weight Reputation prior to item analysis. • Transform, Compute, NegRep = -1∗Reputat.

• Analyze, Scale, Reliability Analysis

• Statistics • Scale if item deleted.

• Continue, OK

• Shoot for an alpha of at least .70 for research instruments.

• Note that deletion of the Reputation item would increase alpha to .96.

Comparing Two Groups’ Factor Structure • Eyeball Test – Same number of well defined factors in both groups? – Same variables load well on same factors in both groups?

• Catell’s Salient Similarity Index – Factors(one from one group, one from the other group) are compared in terms of similarity of loadings. – Summary statistic, s, can be transformed to p value testing null that the factors are not related to one another. – See the handout for details.

• Pearson r – Just correlate the loadings for one factor in one group with those for the corresponding factor in the other group. – If there are many small loadings, r may be large due to the factors being similar on small loadings despite lack of similarity on the larger loadings.

• Cross-Scoring – Obtain scoring coefficients for each group. – For each group, compute factor scores using coefficients obtained from the analysis for that same group (SG) and using coefficients obtained from the analysis for the other group (OG). – Correlate SG factor scores with OG factor scores.

Required Number of Subjects and Variables • Rules of Thumb (not very useful) – 100 or more subjects. – at least 10 times as many subjects as you have variables. – as many subjects as you can, the more the better.

• It depends – see the references in the handout.

• Start out with at least 6 variables per expected factor. • Each factor should have at least 3 variables that load well. • If loadings are low, need at least 10 variables per factor. • Need at least as many subjects as variables. The more of each, the better. • When there are overlapping factors (variables loading well on more than one factor), need more subjects than when structure is simple.

• If communalities are low, need more subjects. • If communalities are high (> .6), you can get by with fewer than 100 subjects. • With moderate communalities (.5), need 100-200 subjects. • With low communalities and only 3-4 high loadings per factor, need over 300 subjects. • With low communalities and poorly defined factors, need over 500 subjects.

What I Have Not Covered Today

• LOTS. • For a general introduction to measurement (reliability and validity), see http://core.ecu.edu/psyc/wuenschk/docs2210/R

Practice Exercises

• Animal Rights, Ethical Ideology, and Misanthro • Rating Characteristics of Criminal Defendants