In this chapter we create and use the variables

GndC_verbwhich is equal toiq_verbcentered around the grand mean;GrpMC_verbwhich contains the group means ofGndC_verb, so it contains the group means ofiq_verbcentered around the grand mean.

Table 4.1, p. 47.

The random intercept only model.

proc mixed data=schools covtest noclprint noitprint method=ml; class schoolnr; model langpost = / solution; random intercept / subject=schoolnr; run; <output omitted>Covariance Parameter Estimates Standard Z Cov Parm Subject Estimate Error Value Pr Z Intercept schoolNR 19.4126 3.0962 6.27 <.0001 Residual 64.5704 1.9729 32.73 <.0001 Fit Statistics -2 Log Likelihood 16253.2 AIC (smaller is better) 16259.2 AICC (smaller is better) 16259.2 BIC (smaller is better) 16267.8 Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > |t| Intercept 40.3642 0.4262 130 94.70 <.0001

Creating the variable

GndC_verbwhich is equal toiq_verbcentered around the grand mean.

proc sql; create table schools1 as select *, iq_verb - mean(iq_verb) as GndC_verb from schools; quit;

Table 4.2, p. 49.

Random intercept model with effect for IQ centered around the grand mean.

proc mixed data=schools1 covtest noclprint noitprint; class schoolnr; model langpost = GndC_verb / solution; random intercept / subject=schoolnr; run; <output omitted>Covariance Parameter Estimates Standard Z Cov Parm Subject Estimate Error Value Pr Z Intercept schoolNR 9.6015 1.5927 6.03 <.0001 Residual 42.2446 1.2893 32.77 <.0001 Fit Statistics -2 Res Log Likelihood 15255.8 AIC (smaller is better) 15259.8 AICC (smaller is better) 15259.8 BIC (smaller is better) 15265.5 Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > |t| Intercept 40.6082 0.3082 130 131.77 <.0001 GndC_verb 2.4876 0.07008 2155 35.50 <.0001

Fig. 4.2, p. 49.

Randomly chosen regression lines according to the random intercept model of table 4.4.

proc mixed data=schools2 covtest noitprint noclprint method=ml; class schoolnr; model langpost = GndC_verb / outp=p ; random intercept / subject=schoolnr type=un; run; goptions reset=all; symbol i=j r=24; axis1 order=(-4 to 4 by 1) label=('IQ'); axis2 order=(20 to 60 by 5) label=(a=90 'Predicted'); proc gplot data=p; where schoolnr < 27; plot pred*GndC_verb = schoolnr / vaxis=axis2 haxis=axis1 href=0 ; run; quit;

Table 4.3, p. 51.

Ordinary least squares regression of the model in table 4.2.

proc reg data=schools1; model langpost = GndC_verb; run; quit;Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 1 68916 68916 1352.84 <.0001 Error 2285 116402 50.94159 Corrected Total 2286 185317 Root MSE 7.13734 R-Square 0.3719 Dependent Mean 40.93485 Adj R-Sq 0.3716 Coeff Var 17.43585 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 40.93485 0.14925 274.28 <.0001 GndC_verb 1 2.65390 0.07215 36.78 <.0001

Table 4.4, p. 55.

Random intercept model with different within and between group regressions.

Note: The “IQ here is the raw variable” means that we are using the grand mean centered variableGndC_verband notiq_verb. Furthermore, the group mean variable for IQ is actually the group mean variable ofGndC_verbwhich we here have calledGrpMC_verb.

*Group mean centering of GndC_verb. ;proc sql; create table schools2 as select *, mean(GndC_verb) as GrpMC_verb from schools1 group by schoolnr; quit; proc mixed data=schools2 covtest noitprint noclprint method=ml; class schoolnr; model langpost = GndC_verb GrpMC_verb / solution; random intercept / subject=schoolnr; run; <output omitted>Covariance Parameter Estimates Standard Z Cov Parm Subject Estimate Error Value Pr Z Intercept schoolNR 7.7275 1.3093 5.90 <.0001 Residual 42.1519 1.2842 32.82 <.0001 Fit Statistics -2 Log Likelihood 15227.5 AIC (smaller is better) 15237.5 AICC (smaller is better) 15237.6 BIC (smaller is better) 15251.9 Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > |t| Intercept 40.7423 0.2844 129 143.26 <.0001 GndC_verb 2.4148 0.07166 2155 33.70 <.0001 GrpMC_verb 1.5885 0.3127 2155 5.08 <.0001

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