In this chapter, the data set is

popular.Table 4.1

Part 1: The variable

sexis a fixed effect, not centered.

proc mixed data = pop ; model popular = sex/solution; random intercept / subject = school type = un; run;

The Mixed Procedure Covariance Parameter Estimates Cov Parm Subject Estimate UN(1,1) SCHOOL 0.8622 Residual 0.4599 Fit Statistics -2 Res Log Likelihood 4492.9 AIC (smaller is better) 4496.9 AICC (smaller is better) 4496.9 BIC (smaller is better) 4502.1 Null Model Likelihood Ratio Test DF Chi-Square Pr > ChiSq 1 1728.72 <.0001 Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > |t| Intercept 4.8972 0.09530 99 51.39 <.0001 SEX 0.8437 0.03096 1899 27.25 <.0001

Part 2: The variable

sexis a fixed effect, raw centered. We first created a centered variablecsexforsex.

proc means data = pop mean ; var sex; run;Analysis Variable : SEX pupil sex Mean ------------ 0.4870000 ------------

data popc; set pop; csex = sex - .487; run; proc mixed data = popc ; model popular = csex/solution; random intercept / subject = school type = un; run;

The Mixed Procedure Covariance Parameter Estimates Cov Parm Subject Estimate UN(1,1) SCHOOL 0.8622 Residual 0.4599 Fit Statistics -2 Res Log Likelihood 4492.9 AIC (smaller is better) 4496.9 AICC (smaller is better) 4496.9 BIC (smaller is better) 4502.1 Null Model Likelihood Ratio Test DF Chi-Square Pr > ChiSq 1 1728.72 <.0001 Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > |t| Intercept 5.3081 0.09410 99 56.41 <.0001 csex 0.8437 0.03096 1899 27.25 <.0001

Part 3: The variable

sexis included as a random effect.

proc mixed data = pop ; model popular = sex/solution; random intercept sex/ subject = school type = un; run;

Covariance Parameter Estimates Cov Parm Subject Estimate UN(1,1) SCHOOL 0.9402 UN(2,1) SCHOOL -0.1410 UN(2,2) SCHOOL 0.2725 Residual 0.3924 Fit Statistics -2 Res Log Likelihood 4336.3 AIC (smaller is better) 4344.3 AICC (smaller is better) 4344.3 BIC (smaller is better) 4354.7 Null Model Likelihood Ratio Test DF Chi-Square Pr > ChiSq 3 1885.30 <.0001 Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > |t| Intercept 4.8901 0.09901 99 49.39 <.0001 SEX 0.8431 0.05963 99 14.14 <.0001

Part 4: The variable

sexis centered and is a random effect.

proc mixed data = popc ; model popular = csex/solution; random intercept csex/ subject = school type = un; run;

Covariance Parameter Estimates Cov Parm Subject Estimate UN(1,1) SCHOOL 0.8675 UN(2,1) SCHOOL -0.00825 UN(2,2) SCHOOL 0.2725 Residual 0.3924 Fit Statistics -2 Res Log Likelihood 4336.3 AIC (smaller is better) 4344.3 AICC (smaller is better) 4344.3 BIC (smaller is better) 4354.7 Null Model Likelihood Ratio Test DF Chi-Square Pr > ChiSq 3 1885.30 <.0001 Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > |t| Intercept 5.3007 0.09424 99 56.25 <.0001 csex 0.8431 0.05963 99 14.14 <.0001

Table 4.2 on page 60.

Part 1: No interaction, no centering.

proc mixed data = pop ; model popular = sex texp /solution; random intercept sex/ subject = school type = un; run;

Covariance Parameter Estimates Cov Parm Subject Estimate UN(1,1) SCHOOL 0.4116 UN(2,1) SCHOOL 0.02089 UN(2,2) SCHOOL 0.2733 Residual 0.3925 Fit Statistics -2 Res Log Likelihood 4275.9 AIC (smaller is better) 4283.9 AICC (smaller is better) 4283.9 BIC (smaller is better) 4294.3 Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > |t| Intercept 3.3400 0.1608 98 20.77 <.0001 SEX 0.8431 0.05969 99 14.13 <.0001 TEXP 0.1084 0.01022 1800 10.61 <.0001

Part 2: With interaction, but no centering.

data pop1; set pop; genxexp = sex*texp; run; proc mixed data = pop1 ; model popular = sex texp genxexp/solution; random intercept sex /subject = school type=un ; run;

Covariance Parameter Estimates Cov Parm Subject Estimate UN(1,1) SCHOOL 0.4120 UN(2,1) SCHOOL 0.02343 UN(2,2) SCHOOL 0.2264 Residual 0.3924 Fit Statistics -2 Res Log Likelihood 4268.4 AIC (smaller is better) 4276.4 AICC (smaller is better) 4276.5 BIC (smaller is better) 4286.9 Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > |t| Intercept 3.3135 0.1610 98 20.58 <.0001 SEX 1.3296 0.1330 98 9.99 <.0001 TEXP 0.1102 0.01023 1800 10.77 <.0001 genxexp -0.03403 0.008457 1800 -4.02 <.0001

Part 3: Centering, but no interaction. We first created new variables

csexandctexpand their interaction term.

proc means data = pop mean; var sex texp; run;The MEANS Procedure Variable Label Mean ------------------------------------------------------- SEX pupil sex 0.4870000 TEXP teacher experience in years 14.2630000 -------------------------------------------------------data pop2; set pop; csex = sex - 0.487; ctexp = texp - 14.263; cx = csex*ctexp; run; proc mixed data = pop2 ; model popular = csex ctexp /solution; random intercept csex /subject = school type=un ; run;

Covariance Parameter Estimates Cov Parm Subject Estimate UN(1,1) SCHOOL 0.4967 UN(2,1) SCHOOL 0.1540 UN(2,2) SCHOOL 0.2733 Residual 0.3925 Fit Statistics -2 Res Log Likelihood 4275.9 AIC (smaller is better) 4283.9 AICC (smaller is better) 4283.9 BIC (smaller is better) 4294.3 Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > |t| Intercept 5.2960 0.07192 98 73.63 <.0001 csex 0.8431 0.05969 99 14.13 <.0001 ctexp 0.1084 0.01022 1800 10.61 <.0001

Part 4: Centering and with interaction.

proc mixed data = pop2 ; model popular = csex ctexp cx/solution; random intercept csex /subject = school type=un ; run;

Covariance Parameter Estimates Cov Parm Subject Estimate UN(1,1) SCHOOL 0.4885 UN(2,1) SCHOOL 0.1337 UN(2,2) SCHOOL 0.2264 Residual 0.3924 Fit Statistics -2 Res Log Likelihood 4268.4 AIC (smaller is better) 4276.4 AICC (smaller is better) 4276.5 BIC (smaller is better) 4286.9 Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > |t| Intercept 5.2969 0.07135 98 74.24 <.0001 csex 0.8442 0.05562 98 15.18 <.0001 ctexp 0.09366 0.01085 1800 8.63 <.0001 cx -0.03403 0.008457 1800 -4.02 <.0001

Figure 4.3. Regression lines for popularity of girls and boys, predicted by teacher experience,

texp.

This uses model in Part 2 of Table 4.2. We will have to manually create the predicted values by the fixed part of the model.

data fig4_3; set pop1; pred = 3.3135 + 1.3296* sex + 0.1102*texp -0.03403*genxexp; run; axis1 order = (0 to 30 by 10) minor = none label = (" "); axis2 order = (3.5 to 7 by .5) minor = none label = (" "); symbol i = join; proc gplot data = fig4_3; plot pred*texp = sex /vaxis =axis2 haxis=axis1; run; quit;

Table 4.3

Part 1: Intercept only.

This has been done in Chapter 2, Table 2.1.

Part 2: The variable

sexis included as a fixed effect.

This has been done in Table 4.1.

Part 3: The variable

texpis also included.

proc mixed data = pop1 ; model popular = sex texp / outp = test solution; random intercept /subject = school type=un ; run;

Covariance Parameter Estimates Cov Parm Subject Estimate UN(1,1) SCHOOL 0.4860 Residual 0.4599 Fit Statistics -2 Res Log Likelihood 4444.4 AIC (smaller is better) 4448.4 AICC (smaller is better) 4448.4 BIC (smaller is better) 4453.6 Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > |t| Intercept 3.5607 0.1715 98 20.76 <.0001 SEX 0.8447 0.03095 1899 27.29 <.0001 TEXP 0.09345 0.01085 1899 8.61 <.0001

Part 4: This has been done in Table 4.2.

Part 5: This is done in Table 4.2.

Table 4.4 can be produced manually based on the equations provided in this section. We omit the calculation here.