Module 6 - General Linear Models: ANOVA & ANCOVA The two exercises below utilize the data sets career-a.sav and career-f.sav, which can be downloaded from this Web site: www.Pyrczak.com/data 1. You are interested in evaluating the effect of job satisfaction ( satjob2) and age category (agecat4) on the combined DV of hours worked per week ( hrsl) and years of education (educ). Use career-a.sav for steps a and b. a. Develop the appropriate research questions and/or hypotheses for main effects and interaction. b. Screen data for missing data and outliers. What steps, if any, are necessary for reducing missing data and outliers? For all subsequent analyses in Question 1, use career-f.sav and the transformed variables of hrs2 and educ 2. c. Test the assumptions of normality and linearity of DVs. i. What steps, if any, are necessary for increasing normality? ii. Are DVs linearly related? d. Conduct MANOVA with post hoc (be sure to test for homogeneity of variancecovariance). a. Can you conclude homogeneity of variance-covariance? Which test statistic is most appropriate for interpretation of multivariate results? b. Is factor interaction significant? Explain. c. Are main effects significant? Explain. 1 d. What can you conclude from univariate ANOVA and post hoc results? e. Write a results statement. 2. Building on the previous problem, in which you investigated the effects of job satisfaction (satjobl) and age category (agecat4) on the combined dependent variable of hours worked per week (hrsl) and years of education (educ), you are now interested in controlling for respondents' income such that rin- com91 will be used as a covariate. Complete the following using career-a.sav. i. Develop the appropriate research questions and/or hypotheses for main effects and interaction. ii. Screen data for missing data and outliers. What steps, if any, are necessary for reducing missing data and outliers? For all subsequent analyses in Question 2, use career-f.sav and the transformed variables of hrs2, educ2, and rincoml. iii. Test the assumptions of normality and linearity of DVs and covariate. i. What steps, if any, are necessary for increasing normality? ii. Are DVs and covariate linearly related? c. Conduct a preliminary MANCOVA to test the assumptions of homogeneity of variance- covariance and homogeneity of regression slopes/planes. i. Can you conclude homogeneity of variance-covariance? Which test statistic is most appropriate for interpretation of multivariate results? ii. Do factors and covariate significantly interact? Explain. d. Conduct MANCOVA. 2 i. Is factor interaction significant? Explain. ii. Are main effects significant? Explain. iii. What can you conclude from univariate ANOVA results? e. Write a results statement. 3. Compare the results from Question 1 and Question 2. Explain the differences in main effects. The following output was generated from conducting a forward multiple regression to identify which IVs {urban, birthrat, Inphone, and Inradio) predict Ingdp. The data analyzed were from the SPSS country-a.sav data file. Variables Entered/Removed Variables Variables Model 1 Entered 1 Removed Method Forward (Criterion: Probability LNPHONE y-of-F-to-e nter <= .050) Forward (Criterion: Probability y-of-F-to-e nter <= 050) 2 BIRTHRAT a Dependent Variable: LNGDP Model Summary Std. Error Model 1 2 R ,941a .943" R Square .886 .890 Adjusted of the R Square R Square .885 888 Estimate 5180 .5109 Change 886 .004 Change Statistics Sig. F F Change 862 968 4.095 df1 df2 1 1 111 110 Change ,000 .045 a Predictors: (Constant), LNPHONE b Predictors: (Constant), LNPHONE, BIRTHRAT 3 B Std Error 110 Collinearity Statistics Zero-order Partial Part Tolerance VIF .941 941 941 1.000 1.000 941 -811 .824 - 189 482 -064 .322 322 3 104 3 662.000 000 29.376 6 878 663 -1 .248 .044 006 849 - 113 29E-02 2 (Constant) LNPHONE BIRTHRAT Correlations Beta 941 6 389 .736 058 .025 1 (Constant) LNPHONE Coefficients" Sifl. t Standardi zed Coefficien ts Unstandardized Coefficients 27 744 15238000 .000 045 104 -2 024 a Dependent Variable LNGDP a - Predictors: (Constant), LNPHONE b Predictors: (Constant), LNPHONE, BIRTHRAT c. Dependent Variable: LNGDP Excluded Variables Collinearitv Statistics Partial Model 1 2 Beta In 095a -.113a URBAN BIRTHRAT t 1.901 -2.024 Sip. .060 045 Minimum Correlation .178 -.189 Tolerance .404 .322 VIF 2.475 3.104 Tolerance 404 .322 LN RADIO .026a .557 .579 .053 .461 2.171 .461 URBAN LNRADIO ,091b 021b 1.848 .455 .067 .650 .174 044 .403 .459 2.479 2.178 .225 .243 a Predictors in the Model: (Constant), LNPHONE b- Predictors in the Model: (Constant), LNPHONE. BIRTHRAT c. Dependent Variable: LNGDP Model Sum of Squares 1 Regression Residual Total 2 Regression Residual Total 231.539 Mean Square 1 231.539 .268 29.782 261.321 232.608 ANOVAc df F Siq. 862.968 ,000a 445.561 .000" 111 112 28.713 2 110 112 116.304 .261 261.321 a. Predictors: (Constant), Inphone b. Predictors: (Constant), Inphone, birthrat c. Dependent Variable: Ingdp 4 i. Evaluate the tolerance statistics. Is multicollinearity a problem .' ii. What variables create the model to predict Ingdp? What statistics support your response? iii. Is the model significant in predicting Ingdp? Explain. iv. What percentage of variance in Ingdp is explained by the model? v. Write the regression equation for Ingdp. This question utilizes the data sets profile-a.sav and profile-b.sav, which can be downloaded from this Web site: www.Pvrczak.com/data You are interested in examining whether the variables shown here in brackets [years of age (age), hours worked per week (hrsl), years of education (educ), years of education for mother (maeduc), and years of education for father (paeduc)] are predictors of individual income (rincmdol). Complete the following steps to conduct this analysis. a. Using profile-a.sav, conduct a preliminary regression to calculate Mahalanobis distance. Identify the critical value for chi-square. Conduct Explore to identify outliers. Which cases should be removed from further analysis? For all subsequent analyses, use profile-b.saw Make sure that only cases where MAH l < 22.458 are selected. b. Create a scatterplot matrix. Can you assume linearity and normality? c. Conduct a preliminary regression to create a residual plot. Can you assume normality and ho- moscedasticity? d. Conduct multiple regression using the Enter method. Evaluate the tolerance statistics. Is multicollinearity a problem? e. Does the model significantly predict rincmdol? Explain. 5 f. Which variables significantly predict rincmdol? Which variable is the best predictor of the DV? g. What percentage of variance in rincmdol is explained by the model? h. Write the regression equation for the standardized variables. i. Explain why the variables of mother's and father's education are not significant predictors of rincmdol. The following exercises seek to determine what underlying structure exists among the following variables in profile-a.sav: highest degree earned (degree), hours worked per week (hrsl), job satisfaction (satjob), years of education (educ), hours per day watching TV (tvhours), general happiness {happy), degree to which life is exciting(life), and degree to which the lot of the average person is getti ng worse (anomiaS). 1. The following output was generated for the initial analysis. Varimax rotation wa$ utilized. Communalities Initial Extraction degree hrs1 1 000 1 000 933 602 satjob 1 000 447 educ 1 000 939 tvhours 1 000 556 happy 1 000 .576 life 1 000 500 anomia5 1 000 317 Extraction Method Principal Component Analysis Component Total Variance Explained Extraction Sums of Squared loadings Initial Eigenvalues Total % of Variance Cumulative % Total % of Variance Rotation Sums of Squared Loadings Cumulative % Total % of Variance Cumulative % 1 2 2423 1 426 30 293 17.822 30 293 48 115 2 423 1 426 30 293 17 822 30 293 48 115 1 879 1 734 23 488 21.676 23 488 45 165 3 1 021 12 760 60 875 1 021 12 760 60 875 1 257 15.710 60 875 4 886 11.077 71 952 5 796 9 955 81 907 6 728 9 094 91 001 7 .607 7 589 98 590 8 113 1 410 100 000 Extraction Method: Principal Component Analysis 6 "n Metho Scree Plgf 9 plot Component Number degree Reproduced Correlation Residual?? hrs1 Reproduced Correlations satjob educ tvhours happy life anomia5 degree hrs1 .933" .176 .176 ,602b -.039 -239 .935 .194 -.239 -.576 -.119 -.077 .230 .141 .118 -.049 satjob -.039 -.239 447b -.062 .214 .469 -.436 -.297 educ .935 194 -062 939b -.255 -.142 .252 .131 tvhours -.239 -.576 .214 -.255 .556" .066 -.136 .047 happy -.119 -.077 .469 -.142 .066 576b -.526 -.412 life .230 .141 -.436 .252 -.136 -.526 500b .371 anomia5 .118 -.049 -.297 .131 .047 -.412 .371 ,317b .004 -068 -.050 .032 -.004 -.034 -.037 .104 .011 .361 -.031 -.046 .112 -.037 -.105 -.197 .151 .158 .026 -.002 -029 -.012 .014 -.012 -.099 .159 .177 degree hrs1 .004 satjob -.068 .104 educ -.050 .011 -.037 tvhours 032 .361 -.105 .026 happy -.004 -.031 -.197 -.002 .014 life -.034 -.046 .151 -.029 -.012 .159 anomia5 -.037 .112 .158 -.012 -099 .177 -.217 -.217 Extraction Method: Principal Component Analysis. a Residuals are computed between observed and reproduced correlations. There are 12 (42.0%) nonredundant residuals with absolute values greater than 0.05. b. Reproduced communalities 7 a. Assess the eigenvalue criterion. How many components were retained? Is the eigenvalue appropriate, considering the number of factors and the communalities? b. Assess the variance explained by the retained components. What is the total variability explained by the model? Is this amount adequate? c. Assess the scree plot. At which component does the plot begin to level off? d. Assess the residuals. How many residuals exceed the .05 criterion? e. Having applied the four criteria, do you believe the number of components retained in this analysis is appropriate? If not, what is your recommendation? 2. Assume that you believe four components should be retained from the analysis in the previous exercise. Conduct a factor analysis with varimax rotation (be sure to retain four components). a. Evaluate each of the four criteria. Has the model fit improved? Explain. b. Provide two alternatives for improving the model. 8 Prediction and Association Practice Exercise Use Practice Data Set 2 in Appendix B. If we want to predict salary from years of education, what salary would you predict for someone with 12 years of education? What salary would you predict for someone with a college education (16 years)? Use Practice Data Set 2 in Appendix B. Determine the prediction equation for pre dicting salary based on education, years of service, and sex. Which variables are significant predictors? If you believe that men were paid more than women were, what would you conclude after conducting this analysis? Data Set 2 Appendix B A survey of employees is conducted. Each employee provides the following information: Salary (SALARY), Years of Service (YOS), Sex (SEX), Job Classification (CLASSIFY), and Education Level (EDUC). Note that you will have to code SEX (Male = 1, Female = 2) and CLASSIFY (Clerical = 1, Technical = 2, Professional = 3), and indicate that they are measured on a nominal scale. Name A Jj?> "{"Decimals] 8 2 None None 8 9 Right f Scale yos Numeric 8 2 None None 8 m Right # Scale 3 sex Numeric 8 2 (1 00. Male) None 8 m Right 4 classify Numeric 8 2 {1 00, Cleric None 8 9 Right A Nominal \\ Input 5 educ Numeric 8 2 None None 8 9 Right f Scale \\ Input salary 2 SALARY 35,000 18,000 20,000 50,000 38,000 20,000 75,000 40,000 30,000 22,000 23,000 45,000 YOS 8 4 1 20 6 6 17 4 8 15 16 2 Label Values Missin g ] Columns SEX Male Female Male Female Male Female Male Female Male Female Male Female CLASSIFY Technical Clerical Professional Professional Professional Clerical Professional Technical Technical Clerical Clerical Professional E Align Measure T~ Role [Jwidth Numeric 1 Nominal S Input \\ Input \\ Input EDUC 14 10 16 16 20 12 20 12 14 12 12 16 9