Advanced and Multivariate Statistical Methods Practical Application and Interpretation, 7e Craig Mer

Answer Key Chapter 2 1. Bivariate regression 2. One-way MANOVA 3. T-test of independent samples 4. Bivariate correlation 5. Discriminant Analysis 6. One-way ANCOVA 7. One-way MANCOVA 8. Multiple regression 9. One-way ANOVA 10. Two-way MANOVA 11. Logistic regression 12. Two-way ANOVA 13. Factor Analysis Chapter 3 Output for Exercise 3.1.

1a. There are no missing values for either variable.

1b. Peduc2 has a large split in that n=121 (77%) with a master’s degree and n=36 (23%) with a postmaster’s degree. However, it is not extreme (90/10).

1c. Boxplots identify two outliers (cases 38 and 40) for the post-master’s group in Humility.

1d. Normality may be assumed as tests of normality are not significant. No transformation is necessary.

1e. Homogeneity of variances is not assumed since Levene’s is significant (p = .021) 1f. Remove the outliers for the post-master’s group, where HumilityMean < 3.0. This may improve homoscedasticity. The authors did NOT conduct this transformation.

Output for Exercise 3.2.

2a. There are no missing data. The box plots do not identify an outliers. However, since normality is violated for HumilityMean and MLQMean, we will examine the histograms to identify extreme values. Histograms do not reveal extreme outliers. No transformations will be made at this time.

2b. Chi-square criteria for three quantitative variables (df = 3) at p = .001 is 2 = 16.266. The box plot of multivariate outliers for Mahalanobis distance identifies two values that exceed 16.266. Use select cases.

2c. Scatter plot matrix indicates no curvilineary. Linearity will be assumed. Plots are fairly elliptical.

Normality will be assumed.

2d. Residuals plot shows not extreme clustering. Homoscedasticity will be assumed.

Chapter 4 Output for Exercise 4.1

1a. RQ1: Does principal gender and level of education significantly interact on Self-Awareness?

RQ2: Is there a difference in Self-Awareness between female and male principals?

RQ3: Is there a difference in Self-Awareness between principals with a master’s degree and postmaster’s degree?

1b. The line plot shows no interaction.

1c. Males score higher than females, but it may not be significant.

1d. Post-masters score higher than Master’s. The difference is sizable and is likely significant.

Output for Exercise 4.2

2a. Homegeneity of variances is assumed.

2b. Interaction is not significant; F(1,153) = 0.35, p =.555, partial 2 = .002.

2c. The main effect of gender is not significant; F(1,153) = 3.12, p =.08, partial 2 = .020. The main effect of education level is significant; F(1,153) = 18.16, p < .001, partial 2 = .106. Principals with a post-master’s degree score significantly higher is Self-Awareness.

2d. Results are likely fairly similar to estimation.

3. Factorial ANOVA was conducted to examine the interaction between principal gender and level of education on Self-Awareness. Of the 157 principals evaluated, 63 (40%) were female, while 121 (77%) had a master’s degree. Descriptive statistics are presented in Table 1. ANOVA results (see Table 2) indicate that interaction between gender and education level on Self-awareness is not significant; F(1,153) = 0.35, p =.555, partial 2 = .002. Although males scored higher than females in Self-awareness, the main effect of gender was not significant; F(1,153) = 3.12, p =.08, partial 2 = .020. The main effect of education level is significant where principals with a post-master’s degree score significantly higher in Self-Awareness.; F(1,153) = 18.16, p < .001, partial 2 = .106.

Table 1. Descriptive Statistics for Principal Self-awareness by Gender and Education Level

Master’s Post-Masters

n 43 20

Female M 3.00 3.64

SD .99 .80

Male M 3.13 3.82

n 78 16

SD .94 .75

Table 2. Two-way ANOVA Summary Table Source

SS

df

MS

F

p

Partial 2

Between treatments

48.40

3

16.13

23.70

<.001

.320

Gender

4.40

1

4.40

6.46

.012

.041

Education

26.28

1

26.28

38.61

< .001

.204

Gender x Education

2.84

1

2.84

4.18

.043

.027

Within treatments

102.78

151

0.68

Total

2098.55

155

Chapter 5 Output for Exercises 1 -5

1. RQ1: Does gender and age group (early, mid, late) interact on income when controlling for hours worked per week?

RQ2: Is there a difference in income by gender when controlling for hours worked per week?

RQ3: Is there a difference in income by age group (early, mid, late) when controlling for hours worked per week?

2. Outliers are not extreme. No transformation will be conducted. Linearity is assumed.

3a.Test of normality indicate nonnormality. However, histograms reveal fairly normal distributions.

Groups sizes are fairly large, so no transformations will be made.

3b.Covariate and factor interaction is not significant; homogeneity of regression slopes is assumed.

3c.Levene’s Test is not significant. Homogeneity of variances is assumed.

4. Line plot does not show factor interaction.

5a.Covariate significantly adjusted group means.

5b. Factor interaction is not significant.

5c. Gender and age category significantly effect income after controlling for hours worked per week.

5d. Effect size for gender is small; effect size for age category is medium.

6. Note: Since full ANCOVA results are presented throughout the text, an ANCOVA summary table was not created.

Analysis of covariance was conducted to examine gender and age category interaction on 2020 income while controlling for hours worked per week. Among the 700 participants, 392 (56%) were male.

Three age groups were considered: early (less than 35 years old), mid (35-54 years old), and late (55+ years old). Income was divided into 21 levels. Participants reporting within the first four levels (less than $20,000) were eliminated as outliers (n=21). Most groups averaged between level 13 and 16.

Dollars per respective income levels are: 13 = $60-64.9k, 14 = 65-69.9k, 15 =70-74.9k, 16 = 75-79.9k.

Although most group distributions violated normality, histograms revealed fairly normal distributions.

Since ANCOVA is quite tolerant of nonnormality when group sizes are relatively large, no further transformations were conducted. ANCOVA results revealed that the covariate of hours worked per week did significantly adjust group means; F(1, 693) = 51.44, p < .001, partial 2 = .069. Factor interaction was not significant; F(2, 693) = 1.18, p = .309, partial 2 = .003. The main effect of gender was significant on 2020 income, with males earning significantly more income than females; F(1, 693) = 26.87, p < .001, partial 2 = .037. Age group also created significant income difference; F(1, 693) = 30.44, p < .001, partial 2 = .081. Age group 1 (early career) earned significantly less income than both mid and late career respondents. Late age respondents earned the highest income. Table 1 presents adjusted and original group means of income.

Table 1 . Adjusted and Unadjusted Group Means for 2020 Income n

Adjusted M

Unadjusted M

Females

407

13.62

13.45

Males

323

15.16

15.25

(1) Early age

229

12.90

12.93

(2) Middle age

390

14.99

15.11

(3) Late age

111

15.24

15.37

Chapter 6 Output for Exercise 6.1

1a. Research Questions 1. Do job satisfaction and age category interact on the combined DV of hours worked per week and years of education?

2. Does the combined DV of hours worked per week and years of education differ by job satisfaction? a. Does hours worked per week differ by job satisfaction? b. Does years of education differ by job satisfaction?

3. Does the combined DV of hours worked per week and years of education differ by age category? a. Does hours worked per week differ by age category? b. Does years of education differ by age category?

1b.Normality is an issues for every distribution. However, group sizes are quite large. Both DVs have high norms of 40 hours per week and 12 years of education. Neither DV has extreme outliers. No transformation will be made.

1c. Yes, curvilinearity is not an issue.

1d. i. Homogeneity of variance and covariance are assumed. Wilk’s Lambda will be used as the test statistic. ii. Factor interaction is not significant. iii. Neither main effect is significant on the combined DV. iv. Univariate results indicate that the IVs do not interact on either DV. The only significant difference is in years of education by job satisfaction. Group means reveal that employees have more education when they are satisfied in their job.

1e. MANOVA was conducted to examine the interaction of job satisfaction and age category on the combined DV of hours worked per week and years of education among 725 U.S. employees.

Although normality was violated for every distribution, group sizes are fairly large, so no adjustments were made. MANOVA results indicate that job satisfaction and age category do not interact on the combined DV of hours worked per week and years of education; Wilks’  = .991, F(6, 1442) = 1.00, p = .426, multivariate 2 = .004. In addition, neither job satisfaction [Wilks’  = .994, F(2, 721) = 2.12, p = .121, multivariate 2 = .006] nor age category [Wilks’  = .892, F(6, 1442) = 0.89, p = .500, multivariate 2 = .004 significantly affected the combined DV. Univariate results indicated that neither DV had significant differences by job satisfaction or age category.

Output for Exercise 6.2

Preliminary MANCOVA

Final MANCOVA

2a. Develop the appropriate research questions and/or hypotheses for main effects and interaction.

1. Does the combined DV of transformational leadership (idealized influence, inspirational motivation, intellectual stimulation, and individualized consideration) differ by gender when controlling for education level?

2. Does idealized influence differ by gender when controlling for education level?

3. Does inspirational motivation differ by gender when controlling for education level?

4. Does intellectual stimulation differ by gender when controlling for education level?

5. Does individualized consideration differ by gender when controlling for education level?

2b. Screen data for missing data and outliers. What steps, if any, are necessary for reducing missing data and outliers? Multivariate outliers were examined using Mahalanobis’ distance with df=4, p<.001, 2 = 18.467. Two outliers were eliminated using Select Cases, IF (MAH_3<=18.467).

2c. Since the covariate (Peduc2) is categorical, it is not examined for normality or linearity. Normality tests indicate most distributions violate normality. Histograms are somewhat skewed, especially for female inspirational motivation. No transformations for normality were made. Linearity is fulfilled as demonstrated by the scatter plot matrix.

2d. Preliminary MANCOVA indicates that univariate homogeneity of variances is fulfilled. Box’s test for homogeneity of covariance reveals test violation. Therefore, the more conservative Pillai’s Trace will be used a the test statistic. Factor and covariate interaction is not significant, therefore, homogeneity of regression slopes is fulfilled.

2e. The covariate significantly adjusts group means for the combined DV. Gender significantly affects the combined DV. Univariate results indicate that the covariate significantly adjusts group means for intellectual stimulation and individual consideration. In addition, gender differences are significant for only two of the four subscales (idealized influence and individual consideration). Adjusted group means reveal that females score higher in all leadership subscales.

2f. Gender differences were examined in the combined dependent variable of transformational leadership while controlling for education level among 155 K-12 principals. Transformational leadership consists of four subscales: idealized influence, inspirational motivation, intellectual stimulation, and individualized consideration. Education level was measured categorically as having a master’s degree or post master’s degree. Multivariate outliers were examined using Mahalanobis’ distance with df=4, p<.001, 2 = 18.467. Two outliers were eliminated that exceed the chi-square criteria. Normality was an issue, however, group sizes are relatively large. Box’s M test was significant indicating that homogeneity of covariance was violated; therefore, the more conservative test statistic of Pillai’s Trace was utilized in interpreting the multivariate results. MANCOVA results reveal that the covariate of education level did significantly adjust group means of the combined transformational leadership; Pillai’s Trace = .098, F(4, 149) = 4.04, p = .004, multivariate 2 = .098. Adjusted group means of transformational leadership significantly differed by gender; Pillai’s Trace = .156, F(4, 149) = 6.87, p = .004, multivariate 2 = .156. Univariate ANOVA results reveal that the covariate significantly adjusts groups means for intellectual stimulation and individual consideration (see Table 1). In addition, gender differences are significant for only two of the four subscales (idealized influence and individual consideration). Adjusted group means reveal that females score higher in all leadership subscales (see Table 2).

Table 1. ANCOVA Results for Transformational Leadership Subscales by Covariate (Educational Level) and Gender Variable Idealized Influence Inspirational Motivation Intellectual Stimulation Individual Consideration

Covariate F(1,152) = 2.12, p = .148, partial 2 = .014 F(1,152) = 3.48, p = .062, partial 2 = .022 F(1,152) = 5.07, p < .001, partial 2 = .032 F(1,152) = 9.50, p = .002, partial 2 = .059

Gender F(1,152) = 8.63, p = .004, partial 2 = .054 F(1,152) = 3.13, p = .079, partial 2 = .020 F(1,152) = 0.83, p = .364, partial 2 = .005 F(1,152) = 7.19, p = .008, partial 2 = .045

Table 2. Unadjusted and Adjusted Group Means by Gender

Idealized Influence Inspirational Motivation Intellectual Stimulation Individual Consideration

Unadjusted M Female Male 3.88 3.38 3.85 3.53 3.27 3.08 3.56 3.07

Adjusted M Female Male 3.86 3.39 3.82 3.55 3.24 3.10 3.52 3.10

Chapter 7 1a. Tolerance statistics exceed .1. Multicollinearity is not a problem.

1b. Three variables (fertility rate, doctors, and GDP) predict birth rate. The model summary indicates these variables were selected. Coefficients indicate variable significance.

1c. The model is significant; F(3, 98) = 573.85, p < .001.

1d. The three variable model accounts for 93.8% of the variance in birth rate.

1e. Birth rate = 5.06Xfertrate – 1.463Xlndocs + 0.0XGDP + 13.04 Output for 7.2

2a. 2 critical = 22.458. Only one value exceeds 22.458.

2b. Scatter plots are elliptical and support linearity and normality.

2c. Scatter plot of residuals are spread out. Normality and homoscasdicity are assumed.

2d. Tolerance statistics exceed .1. Multicollinearity is not a problem.

2e. Model is significant; F(5, 139) = 41.40, p < .001.

2f. An examination of the beta coefficients indicate that task confidence is the best predictor followed by appreciation.

2g. Model explains 59.8% of the variability in MLQ. 2h. MLQ = .541Xtaskcon + .311Xappreciation + 0.243Xleaderefficacy + 0.02Xselfaware - 0.118Xteachability - .199 2i. Self-awareness (p = .814) and teachability (p = .274) do not significantly contribute to the model.

Chapter 8 1a. One component has been retained, which has an eigenvalue of 7.76. Although sample size is low (n = 137), 11 items are being analyzed. Bartlett’s test of sphericity is significant, indicating that our sample is adequate for dimension reduction. Although several communalities are below 0.7, the average is .705. Eigenvalue criterion is adequate.

1b. Component 1 accounts for 70.5% of the total variance, which meets the 70% criterion.

1c. The screeplot levels off at Component 2.

1d. 43% of the residuals exceed .05, which is not great but it is less than 50%.

1e. The one component model seems appropriate.

2. Factor analysis was conducted to determine what, if any, underlying structure exists for measures on the following 11 items measuring aspects of humility (Q15-Q25) from the Perceptions of School Leadership Survey (Oyer, 2011) to measure perceived confidence of K-12 principals as reported by their respective teachers. These 11 items measured three aspects of humility: teachability, appreciation, and self-awareness. The sample consisted of 137 K-12 teachers from the Midwest. Principal components analysis was conducted utilizing a varimax rotation with all 11 items entered into the analysis. Due to the small sample size, Bartlett’s Test of Sphericity was conducted and generated a significant result, indicating that the sample was adequate for dimension reduction. The analysis generated one component, which had an eigenvalue of 7.76 and accounted for 70.53% of the total variability of the original items. The scree plot indicates that the eigenvalues after the first component drop off drastically. An evaluation of residuals (differences between the reproduced correlations and the original correlations) indicate that 43% exceed the .05 criterion, below 50%. Factor loadings ranged from .783 to .912. Reliability analysis generated a Cronbach’s alpha of .957. Analysis supports the overall factor of humility, but not the three subscales of teachability, appreciation, and selfawareness.

Table 1. Descriptive Statistics and Factor Loadings for Humility

Humility ( = .955) 21. open to advice 23. open to ideas 16. appreciates contributions 17. seeks feedback 19. notices others’ strengths 15. willing to learn 25. shows self-awareness

Loading

M

SD

.912 .885 .865 .864 .847 .832 .827

3.45 3.64 3.64 3.20 3.72 3.55 3.49

1.16 1.08 1.21 1.19 1.04 1.19 1.08

20. compliments others 22. admits mistakes 18. acknowledges others abilities 24. admits when lacks knowledge

Chapter 9 Output for Exercises 9.1-9.11

.820 .797 .796 .783

3.48 3.41 3.35 3.36

1.23 1.12 1.22 1.12

1. RQ--Which factors (age, income, years of education, and hour worked) best predict job satisfaction (not satisfied, moderately satisfied, very satisfied)?

2. Income (Wilks’ Λ = .980, p = .001) is the only significant variable that creates group differences.

3. Homogeneity of covariance is assumed since Box’s test is not significant.

4. Only income was entered into the function.

5. One function was generated.

6. The one function is significant; Wilks’ Λ = .980, χ2(2, N = 745) = 14.92, p < .001.

7. Effect size is (.141)2 = . or 1.98% of variance in DV is explained by income.

8. Only one variable is entered into the function, thus Income is an appropriate name.

9. Discriminant function equation is DF1 = .258(income) – 3.832 10. Function is not very accurate in predicting job satisfaction level in that only 49.5% are accurately classified. In addition, the function of income only explains approximately 2% of variance in job satisfaction.

11. Since only one variable was entered into one function, correlation coefficients were not included in Table 1. In addition, a table of unstandardized and standardized coefficients was not generated since these could easily be reported within the text.

Stepwise discriminant analysis was conducted to determine the ability of the four variables—age, hours worked per week, years of education, and income—to predict one’s level of job satisfaction (dissatisfied, moderately satisfied, and very satisfied). Group means each predictor by job satisfaction are presented in Table 1. Only the variable of income was entered into the one significant function [Wilks’ Λ = .980, χ2(2, N = 745) = 14.92, p < .001.], which accounts for only 1.98% of the variance in job satisfaction. This function of Income originally and cross-validated classified cases with 49.5% overall accuracy. The function was not accurate in predicting unsatisfied employees (0%). Group function means indicate that those who are very satisfied (.156) with their job have higher income that those who are moderately satisfied (-.091) and dissatisfied (-.205). These results suggest that as income increases so does job satisfaction. In summary, although a significant function was generated, it only accounts for 1.98% of the variance in job satisfaction and correctly classifies 49.5% of the cases with a discriminant function equation of: DF1 = .258(income) – 3.832.

Table 1. Group Means of Predictors by Job Satisfaction Level Dissatisfied (n = 109) Age Income Years of Education Hours Worked

41.94 14.85 14.06 45.99

Moderately Satisfied (n = 312) 40.19 14.50 13.89 45.65

Very Satisfied (n = 324) 41.72 15.46 14.24 46.92

Chapter 10 Output for Exercises 10.1 – 10.3

1. RQ: Which factors (teachability, self-awareness, appreciation, leader efficacy, and task outcomes) best predict the likelihood of being an effective leader among K-12 principals? a. 2 criterion is 20.515. Extreme values indicate one outlier based upon Mahalanobis distance. Prior to conducting logistic regression, the authors used Select Cases to eliminate outliers: IF MAH_1<=20.515.

2b.Multicollinearity is not an issue since tolerance exceeds .1.

3a. Task confidence, leader efficacy, and appreciation.

3b. Model is signficant -2 Log likelihood = 29.93, 2 (3, N = 87) = 91.09, p < .001; Negelkerke R2 indicates that the model accounts for 86.3% of the variance in leader effectivness.

3c. Model correctly classifies 94.3% of principals.

3d. Leader efficacy is the strongest predictor of leader effectiveness with an odd ratio of 19.84. The odds ratio indicates that as task confidence increases by 1, the likelihood of being classified as an effective leader increases by 19.84. The next predictor, leader efficacy has an odds ratio of 17.81, followed by appreciation (eB = 4.61).