ST706: Spring One-Way Random effects example.

Transcription

1 ST706: Spring One-Way Random effects example. DATA IS FROM CONTROL ARM OF SKIN CANCER STUDY AT DARTMOUTH. DC0 TO DC5 ARE MEASURES OF DIETARY INTAKE OVER 6 YEARS. For illustration here we treat the six years as if they were replicates. Later we will allow year effects. The data has all of the reps on one line so we first create a file with one line for each rep (for each person). This analysis is for the 154 people all with six years of data. n = 6, m = 154. Since the data is balanced some exact confidence intervals are available. But, the analysis here assumes normality and constant within subject variance, which is clearly suspect here. See figure at end. Below is the analysis of variance table with MSA = and MSE = σ 2 = and σ a 2 = (MSA MSE)/6 = A 95% confidence interval for σ 2 based on the chi-square with 770 degrees of freedom is ( , ) An approximate 95% confidence interval for σa 2 is ( , ). This is based on Satterthwaite s method which uses here a chi-square with degrees of freedom. A Wald confidence interval for σa 2 is (2.6234, ) Estimate of population mean µ is Ȳ.. = with an estimated standard error of (MSA/nm) 1/2 = and a 95% confidence interval of ( , ). options linesize=90; /* DATA IS FROM CONTROL ARM OF SKIN CANCER STUDY AT DARTMOUTH. DC0 TO DC5 ARE MEASURES OF DIETARY INTAKE OVER 6 YEARS. HERE WE TREAT THE 6 YEARS AS REPLICATES FOR ILLUSTRATION. */ data a; infile f:\s506\dietrep.dat ; input id age sex smkestat quetelet labcar0 labcar1 labcar2 labcar3 labcar4 labcar5 dc0 dc1 dc2 dc3 dc4 dc5; nmiss = nmiss(of dc0-dc5); diet=dc0; rep=0;output; diet=dc1;rep=1;output; diet=dc2;rep=2;output; diet=dc3;rep=3;output; diet=dc4;rep=4;output; diet=dc5;rep=5;output; /* THIS CREATES A DATA SET WITH JUST THE ONES HAVING ALL 6 YEARS OF DATA*/ data balanced;set a; if nmiss=0; proc means data=balanced; var diet; The MEANS Procedure 1

2 Analysis Variable : diet N Mean Std Dev Minimum Maximum * title original data unbalanced ; * proc glm data=a; /* UNCOMMENT TWO LINES ABOVE AND COMMENT OUT THE NEXT TWO TO RUN WITH FULL UNBALANCED DATA */ title original data. Balanced ; proc glm data=balanced; class id; model diet=id; Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total id <.0001 USING VARCOMP proc varcomp; class id; model diet=id; proc varcomp method=reml; class id; model diet=id; Variance Components Estimation Procedure MIVQUE(0) Estimates Variance Component diet Var(id) Var(Error) Variance Component Estimate Var(id) Var(Error) Asymptotic Covariance Matrix of Estimates Var(id) Var(Error) Var(id) Var(Error) If you run varcomp with method=type1, it will display the expected mean squares for this example. Type 1 Analysis of Variance Sum of Source DF Squares Mean Square Expected Mean Square id Var(Error) + 6 Var(id) Error Var(Error) Corrected Total

3 /* USING PROC MIXED */ proc mixed asycov covtest cl nobound; /* default method in proc mixed is reml*/ class id; model diet=/ ddfm=satterth solution cl; random id/cl ; The Mixed Procedure Data Set WORK.BALANCED Dependent Variable diet Covariance Structure Variance Components Estimation Method REML Residual Variance Method Profile Fixed Effects SE Method Model-Based Degrees of Freedom Method Satterthwaite Covariance Parameter Estimates Standard Z Cov Parm Estimate Error Value Pr Z Alpha Lower Upper id < Residual < Asymptotic Covariance Matrix of Estimates Row Cov Parm CovP1 CovP2 1 id Residual Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > t Alpha Lower Upper Intercept < Solution for Random Effects Std Err Effect id Estimate Pred DF t Value Pr > t Alpha Lower Upper id id id Additional comments about proc mixed. The random statement indicates there is a random effect associated with id. Note that the model is diet = since there is just an overall mean (i.e, intercept) in the fixed effects part of the model. In the model statement the ddfm = satterth is referring to the use of a Satterthwaite method associated with getting a degrees of freedom to go into the confidence interval for the fixed effect µ. Here it gives 3

4 the exact interval based on 153 degrees of freedom. You get the same answer if you use no ddfm= option here. With only a random statement (and no use of a repeated statement which we will discuss later), the program automatically incorporates the ǫ and the variance for this is referred to as residual variability. The results for σ 2 are the ones associated with the label Residual. Those for σa 2 are associated with the label id (since id is the grouping variable). The confidence interval given for σ 2 is based on the chi-square with m (n 1) = = 770 degrees of freedom. The proc mixed above had a nobound in the proc mixed statement. This allows for a negative estimate of a variance component. This option doesn t come into play here with respect to point estimation since the estimate of σ 2 a is positive. However, since nobound is used, the confidence interval for σ2 a is based on a Wald interval. With nobound, all intervals for all variance components associated with the random statement are computed using the Wald method. To get the Satterthwaite based confidence interval for σ 2 a, which we computed directly above, eliminate the nobound option as below. The interval associated with id is now the Satterthwaite interval for σ 2 a. proc mixed asycov covtest cl; class id; model diet=/ solution cl; random id; Standard Z Cov Parm Estimate Error Value Pr Z Alpha Lower Upper id < Residual < Doing some direct computing to double check what is getting done in varcomp and mixed. proc iml; n=6; m=154; dfe=r*(n-1); alpha=.05; MSA = ; MSE = ; sigma2hat = mse; sigmaa2hat = (msa-mse)/n; cvalue1 = cinv(1-(alpha/2),dfe); cvalue2 = cinv(alpha/2,dfe); lowsigma2 = (m*(n-1)*mse)/cvalue1; upsigma2= (m*(n-1)*mse)/cvalue2; print Estimation of sigma2 sigma2hat dfe lowsigma2 upsigma2; /* Computing the chi-square interval for sigma2mu using satterthwaite approximate degrees of freedom */ den = (msa**2/(m-1)) + (mse**2/(m*(n-1))); dfmu = (n*sigmaa2hat)**2/den; cvalmu1 = cinv(1-(alpha/2),dfmu); cvalmu2 = cinv(alpha/2,dfmu); 4

5 lowsigmaa2 = (dfmu*sigmaa2hat)/cvalmu1; upsigmaa2 = (dfmu*sigmaa2hat)/cvalmu2; print Estimation of sigmaa2 sigmaa2hat dfmu lowsigmaa2 upsigmaa2; /* Estimation of mu */ ybar = ; seybar = sqrt(msa/(m*n)); tval = tinv(1 - (alpha/2),r-1); lowermu = ybar -tval*seybar; uppermu = ybar + tval*seybar; print Estimate of population mean ybar seybar lowermu uppermu; quit; SIGMA2HAT DFE LOWSIGMA2 UPSIGMA2 Estimation of sigma SIGMAA2HAT DFMU LOWSIGMAA2 UPSIGMAA2 Estimation of sigmaa YBAR SEYBAR LOWERMU UPPERMU Estimate of population mean Figure 1: Measure of diet betacarotene intake. 5

6 Figure 2: Plot of id st.deviation versus id mean. 6 replicates 6