Linear Mixed Models: Methodology and Algorithms

Transcription

1 Linear Mixed Models: Methodology and Algorithms David M. Allen University of Kentucky April 10, 2017

2 18SAS Ddfm Options You have been encouraged to use the Satterthwaite option for degrees of freedom. SAS does provide other options that are discussed here. It possible that some of the options will be depreciated in subsequent versions, so it is important to check the documentation for the version of SAS you are using. There is also an introduction to SAS Macros.

3 Section 18.1 SAS Degrees of Freedom Options This discussion pertains to SAS proc mixed or proc glimmix when used for linear mixed models. The estimate statement has a df option to specify the denominator degrees of freedom for the approximate t-distribution. However, with the possible exception of simple tests with balanced data, most people will want the program to provide the degrees of freedom. Here five methods for determining denominator degrees of freedom are described. There is a sixth, NONE, which means infinite degrees of freedom

4 The Containment Method The containment method is the default when the RANDOM statement is used. Otherwise, the containment method is invoked with the DDFM = CONTAIN option on the model statement. Denote the fixed effect in question A, and search the RANDOM effect list for the effects that syntactically contain A. Among the random effects that contain A, compute their rank contribution to the [X Z] matrix. The denominator degrees of freedom assigned to A is the smallest of these rank contributions. If A is not found on the random statement, the containment method is not invoked, and the denominator degrees of freedom are the residual degrees of freedom

5 Note Note that for a nested model, specified by the direct method, the containment method will not be invoked

6 The Between-Within Method The DDFM = BETWITHIN option is the default for REPEATED statement specifications (with no RANDOM statements). It is computed by dividing the residual degrees of freedom into between-subject and within-subject portions. PROC MIXED then checks whether a fixed effect changes within any subject. If so, it assigns within-subject degrees of freedom to the effect; otherwise, it assigns the between-subject degrees of freedom to the effect. If there are multiple within-subject effects containing classification variables, the within-subject degrees of freedom is partitioned into components corresponding to the subject-by-effect interactions

7 The Residual Degrees of Freedom The denominator degrees of freedom are the residual degrees of freedom. This will give exact test for all effects that are orthogonal to the Z matrix; i.e. split-plot treatment and interaction with whole-plot treatment

8 The Satterthwaite Method The Satterthwaite method has been described earlier

9 The Kenward-Roger Method The Kenward-Roger method implements the method described in [1]. This method is in SAS starting with Version 8. From what I can tell, the Kenward-Roger method gives the same degrees of freedom as the Satterthwaite method. The difference is in the methodology for computing the estimated standard error

10 Section 18.2 Comparison of Degrees of Freedom Previously, three different estimators, using traditional methods, were considered in the context of the balanced Drug-Alcohol data. This Section demonstrates the SAS denominator degrees of freedom options for these estimates. Then some of the data is removed, and the exercise is repeated

11 Drug-Alcohol Data with Missing Values Drugs Alcohol Subject A B C Yes RST Yes JBM Yes DGH Yes JBH Yes WJT Yes EEA No DCJ No CJW No RDF No RLA No HEM No AMR

12 The Nature of Missing Values Seven observations, or 19.4%, are removed. Four are from the alcohol group, and three are from the no alcohol group. Three observations are removed from both the Drug A and Drug B groups, and one observation is removed from Drug C

13 The SAS Code The SAS code used for this demonstration is proc glimmix data = balanced noprofile ; classes Alcohol Subject SubWithin Drug; model y = Alcohol Drug Alcohol Drug / ddfm = contain ; random Subject ; estimate 1 i n t 1 Alcohol 1 0 Drug 1 Alcohol drug 1; estimate 2 Alcohol 1 1 ; estimate 3 Drug ; run ; The objects of the red commands are changed from run to run. data will be balanced and missing. ddfm will take on all five methods of computing the denominator degrees of freedom. random will be Subject and SubWithin to illustrate the direct and product method of specifying the random effect

14 Estimate 1 Drug A with no alcohol Denominator degrees of freedom Method Balanced Missing Containment Between-within Residual Satterthwaite Kenward-Roger

15 Estimate 2 Alcohol versus no alcohol Denominator degrees of freedom Method Balanced Missing Containment 20(10) 13(10) Between-within Residual Satterthwaite Kenward-Roger For the containment method, the first number is for direct specification, and the number in parentheses is for product specification

16 Estimate 3 Drug A versus drug C Denominator degrees of freedom Method Balanced Missing Containment Between-within Residual Satterthwaite Kenward-Roger

17 Section 18.3 SAS Macros Production of the tables in the preceding section was tedious. It involved all combinations of two data sets, five ddfm methods, and two ways of specifying the random effect indicator variables. The tedium is reduced somewhat if macros, as illustrated in the following listing, are used. title "DDFM Demonstration " ; data balanced ; input Alcohol $ Subject $ SubWithin Drug $ y x ; cards ; YES RST 1 A YES RST 1 B YES RST 1 C YES JBM 2 A YES JBM 2 B YES JBM 2 C YES DGH 3 A

18 YES DGH 3 B YES DGH 3 C YES JBH 4 A YES JBH 4 B YES JBH 4 C YES WJT 5 A YES WJT 5 B YES WJT 5 C YES EEA 6 A YES EEA 6 B YES EEA 6 C NO DCJ 1 A NO DCJ 1 B NO DCJ 1 C NO CJW 2 A NO CJW 2 B NO CJW 2 C NO RDF 3 A NO RDF 3 B NO RDF 3 C

19 NO RLA 4 A NO RLA 4 B NO RLA 4 C NO HEM 5 A NO HEM 5 B NO HEM 5 C NO AMR 6 A NO AMR 6 B NO AMR 6 C ; proc print ; data=balanced ; run ; data missing ; set balanced ; if x=1 then delete ; run ; %MACRO lmm( dataset=,method=,subject=); / Linear mixed models / proc glimmix data=&dataset noprofile ; classes Alcohol Subject SubWithin Drug ;

20 model y = Alcohol Drug Alcohol Drug / ddfm=&method ; random &subject ; estimate 1 intercept 1 Alcohol 1 0 Drug 1 Alcohol drug 1 ; estimate 2 Alcohol 1 1 ; estimate 3 Drug ; run ; %MEND lmm ; contain ; %lmm( dataset=balanced, method=contain, subject=subject ) ; %lmm( dataset=missing, method=contain, subject=subject ) ; %lmm( dataset=balanced, method=contain, subject=subwithin( Alcohol ) ) ; %lmm( dataset=missing, method=contain, subject=subwithin( Alcohol ) ) ; betwithin ; %lmm( dataset=balanced, method=betwithin, subject=subject ) ; %lmm( dataset=missing, method=betwithin, subject=subject ) ; %lmm( dataset=balanced, method=betwithin, subject=subwithin( Alcohol ) ) %lmm( dataset=missing, method=betwithin, subject=subwithin( Alcohol ) ) ; residual ; %lmm( dataset=balanced, method=residual, subject=subject ) ; %lmm( dataset=missing, method=residual, subject=subject ) ;

21 %lmm( dataset=balanced, method=residual, subject=subwithin( Alcohol ) ) ; %lmm( dataset=missing, method=residual, subject=subwithin( Alcohol ) ) ; satterthwaite ; %lmm( dataset=balanced, method=satterthwaite, subject=subject ) ; %lmm( dataset=missing, method=satterthwaite, subject=subject ) ; %lmm( dataset=balanced, method=satterthwaite, subject=subwithin( Alcoh %lmm( dataset=missing, method=satterthwaite, subject=subwithin( Alcoho kenwardroger ; %lmm( dataset=balanced, method=kenwardroger, subject=subject ) ; %lmm( dataset=missing, method=kenwardroger, subject=subject ) ; %lmm( dataset=balanced, method=kenwardroger, subject=subwithin( Alcoho %lmm( dataset=missing, method=kenwardroger, subject=subwithin( Alcohol