Answer Key for Advanced and Multivariate Statistical Methods Practical Application and Interpretatio

Chapter 1 There is no exercise for Chapter 1.

Chapter 2 1. Bivariate regression.

2. One-way MANOVA. 3. Path analysis.

4. t test of independent samples.

5. Bivariate correlation.

6. Discriminant analysis or logistic regression.

7. One-way ANCOVA. 8. One-way MANCOVA. 9. Multiple regression.

10. One-way ANOVA. 11. Factorial MANOVA. 12. Logistic regression.

13. Factorial MANCOVA. 14. Factor analysis.

Chapter 3 1. (a) No missing values. (b) No. Frequencies in groups are equivalent. (c) Below median: outliers would be those ≥ 22.5; above median: outliers would be those ≥ 17. (d) Below median: 64; above median: 26 and 57. (e) Distributions display moderate positive skewness and would be transformed by taking the square root. (f) No. Levene’s test is significant, which indicates unequal variance between groups.

2. Taking the inverse produced the most normal distribution.

3. (a) All variables should be transformed: grad94 is moderately positive (SQRT); loinc93 is moderately negative (SQRT[K-X]); read94me is substantially positive (LG10[X]); math94me is substantially positive (LG10[X]). (b) Zero subjects exceeded the chi-square critical value of 18.467. (c) Some variables display slight curvilinear tendencies. (d) Homoscedasticity is questionable because residuals do not cluster around the center; however, there is a nice horizontal spread throughout the plot.

Chapter 4 1. (a) Are there significant mean differences for hours worked per week between those satisfied and dissatisfied with job? Are there significant mean differences for hours worked per week among general 1

happiness categories? Is there significant interaction of hours worked per week between job satisfaction and general happiness? (b) Factors intersect slightly. (c) Most likely because lines differ so much. Not too happy may be significantly different from very happy and pretty happy. (d) Probably not.

2. (a) Factor interaction is not significant [F(2, 891) = 1.682, p = .187]. (b) Main effect for satjob2 [F(1, 891) = .147, p = .701] and happy [F(2, 891) = 1.597, p = .203] are not significant. (c) Student responses may vary depending on the accuracy of the estimates they made in response to Questions 1b, 1c, and 1d.

3. (a) Are there significant mean differences in current salaries between males and females? Are there significant mean differences in current salaries between minority status? Is there significant interaction in current salaries between gender and minority status? (b) Outliers equal to or greater than 26,000 were eliminated based on minority status. Because group distributions of salnow are substantially positively skewed, it was transformed by computing its logarithm. Levene’s test indicates that homogeneity of variances cannot be assumed. Line graph reveals no factor interaction. (c) & (d) Factor interaction is not significant [F(1, 431) = 1.05, p = .306]. Gender significantly affects current salary [F(1, 431) = 92.69, p < .001, partial η2 = .177]. Minority status significantly affects current salary [F(1, 431) = 27.56, p < .001, partial η2 = .060]. Results reveal that gender accounts for 17.7% of variance in current salary.

Chapter 5 1. Does income differ by gender among employees when controlling for differences in hours worked per week? Does income differ by age category when controlling for differences in hours worked per week?

Does the relationship between income and gender differ by age category among employees when controlling for differences in hours worked per week?

2. Recode all values in rincom91 that are greater than or equal to 22 as sysmis.

3. (a) Tests for normality indicate nonnormal distributions. Histograms reveal that income distributions for all groups are moderately negatively skewed. The reader may select to transform income by taking its reflection and square root in order to create more normal distributions; however, the authors have selected not to transform it. (b) No. Interaction between the factors and covariate is not significant [F(7, 691) = 1.11, p = .353]. Yes. Tests for homogeneity of regression slopes indicate fulfillment of assumption. (c) Using Levene’s test of equal variances, homogeneity of variance can be assumed [F(7, 696) = 1.43, p = .189].

4. Line plot reveals slight factor interaction.

5. (a) No. Interaction between agecat4 and sex is not significant [F(3, 695) = .827, p = .479, partial η2 = .004]. (b) Yes. Main effects for age [F(3, 695) = 21.73, p < .001, η2 = .086] and gender [F(1, 695) = 28.09, p < .001, partial η2 = .039] are significant. (c) Yes. The covariate of hrs1 significantly influences the DV [F(1, 695) = 25.71, p < .001, partial η2 = .036]. (d) Although main effects for each factor are significant, effect sizes indicate that each factor accounts for a small percentage of variability in the DV (age: 8.6%, gender: 3.9%).

6. A 2 × 4 analysis of covariance was conducted on income. Independent variables consisted of gender and age category. The covariate was hours worked per week. Initial data screening led to the transformation of income by eliminating all values greater than or equal to 22. After significant adjustment by the covariate 2

of hours worked per week, income varied significantly with gender [F(1, 695) = 28.09, p < .001, partial η2 = .039] and with age category [F(3, 695) = 21.73, p < .001, η2 = .086]. Interaction between gender and age category was not significant [F(3,695) = .827, p = .479, partial η2 = .004]. The covariate of hours worked per week significantly influenced income [F(1,695) = 25.71, p < .001, partial η2 = .036].

Comparison of adjusted group means reveals that males (M = 14.43) earned significantly more in income than females (M = 12.56) and that the age category of 40–49 earned more than all other age categories.

Chapter 6 1. (a) Does the combined DV of hours worked per week and years of education differ by job satisfaction?

Does the combined DV of hours worked per week and years of education differ by age category? Is there a significant interaction between job satisfaction and age category on the combined DV of hours worked per week and years of education? (b) The following variables have been transformed to eliminate outliers. Please note that the reader may have selected different transformations that are also appropriate. hrs1 was transformed to hrs2, in which those less than or equal to 16 were recoded 17 and those greater than or equal to 80 were recoded 79. educ was transformed to educ2 in order to eliminate cases with 6 or fewer years. (c-i) Tests for normality indicate nonnormal distributions for both hrs2 and educ2. However, histograms reveal fairly normal distributions. Therefore, normality will be assumed; no further transformations will be made. (c-ii) Yes. Scatterplot and correlation coefficient indicate linear relationship. (d-i) Yes. Box’s test is not significant [F(21, 801322) = 1.06, p = .384]. Wilks’ Lambda will be used. (d-ii) Factor interaction is not significant [Wilks’ Λ = .989, F(6, 1460) = 1.338, p = .237]. (d-iii) Job satisfaction is significant [Wilks’ Λ = .992, F(2, 730) = 3.009, p = .050]. Age category is not significant [Wilks’ Λ = .994, F(6,1460) = .758, p = .603]. (d-iv) Job satisfaction significantly affects years of education [F(1, 731) = 4.64, p = .032], at α = 0.05 but not at the conservative alpha level of α = 0.025. (e) A two-way MANOVA was conducted to determine the effect of age category and job satisfaction on the combined dependent variable of hours worked per week and years of education. Prior to the test, variables were transformed to eliminate outliers. Hours worked per week was transformed; those less than or equal to 16 were recoded 17 and those greater than or equal to 80 were recoded 79.

Years of education was also transformed to eliminate cases with 6 or fewer years. MANOVA results revealed significant differences in job satisfaction on the combined dependent variable [Wilks’ Λ = .992, F(2, 730) = 3.009, p = .050, multivariate η2 = .008]. The main effect of age category was not significant.

Univariate ANOVA was conducted as a follow-up test to MANOVA. Job satisfaction significantly affected years of education [F (1, 731) = 4.64, p = .032, η2 = .006, at α = .05], but not at the new conservative alpha level of α = .025. Hours worked per week does not significantly differ for job satisfaction [F (1, 731) = 2.09, p =.149, η2 = .003].

2. (a) Does the combined DV of hours worked per week and years of education differ by job satisfaction when controlling for income? Does the combined DV of hours worked per week and years of education differ by age category when controlling for income? Is there a significant interaction between job satisfaction and age category on the combined DV of hours worked per week and years of education when controlling for income? (b) The following variables have been transformed to eliminate outliers. Please note that the reader may have selected different transformations that are also appropriate. hrs1 was transformed to hrs2, in which those less than or equal to 16 were recoded 17 and those greater than or 3

equal to 80 were recoded 79. educ was transformed to educ2 in order to eliminate cases with 6 or fewer years. rincom91 was transformed to rincom2 in order to eliminate cases with income of zero and equal to or exceeding 22. (c-i) Tests for normality indicate nonnormal distributions for both hrs2 and educ2.

However, histograms reveal fairly normal distributions. Therefore, normality will be assumed; no further transformations will be made, however. (c-ii) Yes. Scatterplots present linear relationships. Pearson correlation coefficients are statistically significant. (d-i) Yes. Box’s test is not significant [F(21, 666972) = 1.02, p = .433]. Wilks’ Lambda will be used to interpret remaining tests. (d-ii) Factor-covariate interaction is not significant [Wilks’ Λ = .975, F(14, 1332) = 1.19, p = .276]. (e-i) Factor interaction is not significant [Wilks’ Λ = .990, F(6, 1340) = 1.16, p = .327]. (e-ii) The main effect of age category on the combined DV is significant [Wilks’ Λ = .981, F(6, 1340) = 2.11, p = .049]. The main effect of job satisfaction is not significant. (e-iii) Age category significantly affects the DV of years of education and not hours worked per week. (f) A two-way MANCOVA was conducted to determine the effect of age category and job satisfaction on the combined dependent variable of hours worked per week and years of education while controlling for income. Prior to the test, variables were transformed to eliminate outliers.

Cases with income equal to zero and equal to or exceeding 22 were eliminated. Hours worked per week was transformed; those less than or equal to 16 were recoded 17 and those greater than or equal to 80 were recoded 79. Years of education was also transformed to eliminate cases with 6 or fewer years. MANOVA results revealed significant differences among the age categories on the combined dependent variable [Wilks’ Λ = .981, F(6, 1340) = 2.11, p = .049, multivariate η2 = .009]. The main effect of job satisfaction was not significant. The covariate (income) significantly influenced the combined dependent variable [Wilks’ Λ = .840, F(2, 670) = 63.70, p < .001, multivariate η2 = .160]. Analysis of covariance (ANCOVA) was conducted on each dependent variable as a follow-up test to MANCOVA. Age category differences were significant for years of education [F(3, 671) = 3.22, p = .022, partial η2 = .014] but not hours worked per week [F(3, 671) = 1.192, p = .312, partial η2 = .005].

3. In Question 1, a MANOVA was conducted in which job satisfaction was significantly different for the combined DV. In Question 2, the covariate of income was added to the analysis, which adjusted group means of the combined DV. This adjustment changed the test results in that job satisfaction was no longer significant; instead, the variable of age category was significant.

Chapter 7 1. (a) Multicollinearity is not a problem; all tolerance statistics are greater than .1. (b) Variables entered into the model are lnphone (R2chg = .886) and birthrat (R2chg = .004). (c) The model significantly predicts lngdp [R = .943, R2 = .890, R2adj = .888, F(2, 110) = 445.56, p < .001]. (d) The model accounts for 89% of the variance in lngdp. (e) lngdp = .663Xlnphone – .013Xbirthrate + 6.878.

2. (a) The 2 critical value is 22.458. Any cases with mah_1 > 22.458 should be eliminated from the regression analysis. Thus, cases 406, 18, 508, 1129, 351, 750, and 466 are eliminated. (b) Scatterplots display fairly elliptical shapes. Linearity and normality are assumed. (c) Residual plot shows some scattering in the upper-right quadrant; however, it is not too severe. Assumptions are fulfilled. (d) Tolerance for all variables is greater than .1. Multicollinearity is not a problem. (e) The model significantly predicts rincmdol [R2 = .398, F(5, 603) = 79.57, p < .001]. (f) The variables of age (B = 546.02,  = .296, t = 8.78, p < .001), educ (B = 2595.49, = .316, t = 9.04, p < .001), and hrs1 (B = 4

605.49, = .400, t = 12.51, p < .001) significantly predict the DV. The variable of hrs1 is the best predictor of rincmdol as indicated by the beta weight and respective t and p values. (g) The model accounts for 39.8% of variance in the DV. (h) Zrespondent’s income = .296 Zage + .316 Zeduc + .400 Zhrs1 + .036 Zmaeduc + .006 Zpaeduc. (i) Bivariate and partial correlation coefficients of these two variables with the DV are very low. Therefore, these variables are not significantly related to the DV.

Chapter 8 1.

Reproduced Correlation r̂13

Path Decomposition p31 (D)

r̂14

r12p42 (U)

r̂15

p31p53 + r12p52 + r12p42 p54 (I) (U) (U)

r̂23

r12 p31 (U)

r̂24

p42 (D)

r̂25

p52 + p42p54 + r12p31p53 (D) (I) (U)

r̂34

p31r12p42 (S)

r̂35

p53 + p31 r12 p52 + p31r12p42p54 (D) (S) (S)

r̂45

p54 + p42p52 + p42r12p31p53 (D) (S) (S)

2. Regression analyses for initial model: Analysis 1 2 3

Endogenous Variable deathrat birthrat lifeexpf

Exogenous Variables lndocs lngdp lngdp, deathrat, birthrat

3. (a) r12 = .824. (b) p31 = –.643. (c) p42 = – .803. (d) p52 = .313. (e) p53 = –.425. (f) p54 = –.377.

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4. Calculation of reproduced correlations. r̂13 = p31 = –.643

r̂14 = r12p42 = (.824)(–.803) = –.662 r̂15 = p31p53 + r12p52 + r12p42 p54 = (–.643)(–.425) + (.824)(.313) + (.824)(–.803)(–.377) = .781 r̂23 = r12p31 = (.824)(–.643) = –.530

r̂24 = p42 = –.803 r̂25 = p52 + p42p54 + r12p31p53 = (.313) + (–.803)(–.377) + (.824)(–.643)(–.425) = .841 r̂34 = p31r12p42 = (–.643)(.824)(–.803) = .425 r̂35 = p53 + p31r12p52 + p31r12p42p54 = (–.425) + (–.643)(.824)(.313) + (–.643)(.824)(–.803)(–.377) = –.751 r̂45 = p54 + p42p52 + p42r12p31p53 = (–.377) + (–.803)(.313) + (–.803)(.824)(–.643)(–.425) = –.809

5. The following reproduced correlations exceed the .05 criterion: rˆ14 , rˆ15 , rˆ34 , rˆ45 .

6. The model is not consistent with empirical data.

The following missing paths should be analyzed: Analysis 1 2 3

Endogenous Variable deathrat birthrat lifeexpf

Exogenous Variables lndocs, lngdp lngdp, lndocs, deathrat lndocs, lngdp, deathrat, birthrat

Chapter 9 1. (a) Three components were retained. The eigenvalue criterion is questionable because several of the communalities are below .70. (b) The model does not meet the variance criterion as it only accounts for 60.88% of the total variance. (c) The scree plot begins to level off after four components. (d) Twelve residuals exceed the .05 criterion. (e) No, four components should be investigated.

2. (a) The eigenvalue criterion is in question because several communalities are less than .70. Variance is improved as the new model accounts for 71.95% of total variance. Results from the scree plot indicate that four components should be retained. Eleven residuals exceed the .05 criterion, indicating minimal model improvement. The four-component model is better than the three-component model; however, it still is not a solid model. (b) An option for improving the model would be to remove the variable of anomia5 because this variable is the only one loaded into the fourth component. Another option would be to add or delete other variables to the model.

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Chapter 10 1. Can a school district’s income level be predicted by knowledge of graduation rate ’94, average ACT scores ’94, percent taking ACT ’94, math scores in ’93, % meeting/exceeding math standards in ’94, reading scores in ’93, % meeting/exceeding reading standards in ’94, science scores in ’93, % meeting/exceeding science standards in ’94?

2. All variables show significant group differences at p < .01.

3. The Box’s M test is significant at p < .01 and therefore shows equality of variance.

4. math94me and read94me.

5. One function was generated due to the DV having only two levels.

6. The function is significant [Λ = .669, χ2(2, N = 62) = 23.69, p < .001], indicating that the function of predictors significantly differentiated between school income levels.

7. η2 = .5752 = .331.

8. Exceeding math/reading standards.

9. Only 74.2% of the below-median schools were correctly classified, while 75% of above-median schools were correctly classified. Of the total group, 74.6% were correctly classified. Cross-validation results were lower as only 69.8% of the total group were correctly grouped.

10. Schools with high percentage of students meeting/exceeding math and reading standards will likely be classified as above the median income.

Chapter 11 1. Which independent variables (age, years of education, hours worked per week, number of siblings, life perspective, and income) are predictors of job satisfaction?

2. (a) 2(5) = 20.515. All cases that exceed this value should be eliminated. Therefore, the following cases are eliminated: 50, 406, 121, 689, 1129, 192, 268, and 632. (b) No, tolerance for all variables exceeds .1.

3. (a) life2 and rincom91. (b) Model fit is questionable because index is so large, –2 Log Likelihood = 753.9. (c) Yes, Model Chi-Square = 19.96. (d) The model correctly classified only 59.90% of subjects. (e) Odds ratios for the model variables are rincom91 (eB = .942) and life2 (eB = 4.58) (because it was designated as a categorical covariate with the last category as the reference, routine/exciting [2] is now the reference category compared to dull [1]). Odds ratios indicate that those whose life perspective is dull are 4.6 times more likely to be very dissatisfied with their job than those whose life perspective is routine/exciting, the reference category.

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