Graph the R Matrix in Linear Mixed Model

Transcription

1 Paper SP01 Graph the R Matrx n Lnear Mxed Model Jan Wu, Roche Products Australa, Dee Why, NSW, Australa Peter Button, Roche Products Australa, Dee Why, NSW, Australa ABSTRACT In the longtudnal studes, subjects are usually measured repeatedly over the tme. Accordngly, the ndependence can not be assumed between the data collected wthn the same subject. Based nference could result from the longtudnal data analyss wthout accountng for the wthn-subject correlaton. Lnear Mxed Model allows for the wthn-subject correlaton and s a proposed model to analyse longtudnal data. The elements n the R matrx for the Lnear Mxed Model reflect the wthn-subject data relatonshp (resdual covarance structure). One of the tasks for the model buldng s to select the most sutable resdual covarance structure for the R matrx. More than 30 types of structures for the R matrx are avalable as an opton n SAS Verson 8. In addton to numercal methods, vsual nspecton of the varous covarance structures has proved to be helpful n selectng the most sutable one for the R matrx. Our experence wth plottng the resdual covarance for the R matrx s dscussed n ths paper. PROC MIXED, PROC GPLOT, PROC G3GRID and PROC G3D are proposed to graph the resdual covarance for the R matrx. Some napproprate methods of plottng are also dscussed n the paper. LINEAR MIXED MODEL Lnear Mxed Model s proposed to analyse the longtudnal data when the repeated data wthn the same subject are correlated. Both fxed effects and random effects can be specfed n the mxed model. The coeffcents for fxed effects are assumed to be the same for all subjects. The coeffcents for random effects reflect the varaton from ndvdual to ndvdual. For subject, Lnear Mxed model can be expressed as [Verbeke and Molenberghs, 2000] Y = X β + Z b + ε Y s the p 1 response vector for subject, where p s the number of observatons for subject. X and Z are (p by q) and (p by r) matrces of known covarates, where q s the number of fxed effects and r s the number of random effects n the model. β s the (q by 1) vector contanng the fxed effects. b s the (r by 1) vector contanng the random effects. The dstrbuton of b follows (0, D), where D s a general (r by r) varance covarance matrx. ε s the (p by 1) vector contanng resdual components, n whch the elements are no longer requred to be uncorrelated. The dstrbuton of ε follows (0, ), where s a (p by p ) varance covarance matrx dependng on the number of observatons (p ) subject has. The mxed model assumes that the data from dfferent subjects are ndependent and that there s no covarance between b and ε. The general mxed model can be further expressed as the followng by stackng the data from all subjects. Y = Xβ + Zb + ε The covarance G for b s a block dagonal matrx wth block D on the man dagonal and zeros for other elements. The dmenson of G s (r N by r N), where N s the number of dstnct subjects n the model. The covarance R for ε s also a block dagonal matrx wth block on the man dagonal and zeros for other elements. The dmenson of R N N s ( p by = 1 p ). The form of R matrx s of the followng: = 1 R ~ N 1 N

2 where zeros are denoted by blank. The number of on the dagonal of the R matrx s N. The elements n the G and R matrces are constructed based on the estmated covarance parameters. Approprate covarance structures need to be selected for the G and R matrces whle estmatng the covarance parameters. Covarance parameters can be estmated by one of the three methods: restrcted maxmum lkelhood (REML), maxmum lkelhood (ML) and mnmum varance quadratc unbased estmaton (MIVQUE0) [Khattree and Nak, 1999]. GRAPH RESIDUAL COVARIANCE FOR R MATRIX As mentoned above, the R matrx for ε s a block dagonal matrx wth block on the man dagonal and zeros for other elements. The number of elements n depends on the number of observatons subject has. If two subjects have the same number of observatons, these two subjects should have the same. If two subjects have dfferent number of observatons, ther should have dfferent number of elements but wth the same values for the common elements. When we depct the R matrx, we actually refer to plottng the elements n. It needs to be emphaszed that the elements (varance-covarance) n are based on resdual components. Computng the varance-covarance based on the raw data s not approprate. Even f the resdual could be computed frst, t s not suggested to use PROC CORR to compute the varance-covarance snce ths procedure assumes that all observatons are ndependent. Incorrect degree of freedom and, accordngly, ncorrect varance-covarance could be produced by PROC CORR when the data are repeatedly measured [Lttell et al, 2002]. Lttell et al [2000] suggested to ft the raw data nto PROC MIXED by selectng an UNSTRUCTURED structure for. The estmated covarance parameters can be used to construct the elements of. Ths wll ensure that the elements of are based on resdual components and the wthn-subject correlaton has been accounted for. UNSTRUCTURED structure for s selected snce no specal pattern but unque correlaton s assumed for each par of wthn-subject resduals. Other structures can be treated as a specal form of the UNSTRUCTURED structure. Then, SAS graphng procedures can be used to plot the varance-covarance for. Fortunately, PROC MIXED provdes an R opton n the REPEATED statement to allow us to easly access the (also called R block) matrx for each subject. PROC MIXED constructs each R block by usng the estmated covarance parameters followng the formula for the selected covarance structure. R block could be saved as a data set by usng ODS. Cautons need to be taken that the R block for subject wth ncomplete data can not be used for plottng. For nstance, f a subject has a mssng value for the response varable for the second observaton, the second row/column values n the R block provded by PROC MIXED for ths subject s not the covarance between the second observaton and the other observatons. The values for the second row/column n the R block are actually for the covarance between the next (thrd) observaton and other observatons. Subjects wth complete data should have the same R blocks. Therefore only one R block s used for plottng. A data set reported n SAS OnlneDoc V8 s used for llustraton purpose. The growth data for 11 grls and 16 boys at ages 8, 10, 12, and 14 were collected. The second observaton for the Person 3 s deleted to demonstrate the mpact of mssng on the R block. Person Gender y1 y2 y3 y4 1 F F F F F F F F F F F M M M M

3 16 M M M M M M M M M M M M SORTING THE DATA The data are read nto SAS by the followng codes (SAS OnlneDoc V8): /* Read the data */ data pr; nput Person Gender $ y1 y2 y3 y4; y=y1; Age=8; Ageclss=age; output; y=y2; Age=10; Ageclss=age; output; y=y3; Age=12; Ageclss=age; output; y=y4; Age=14; Ageclss=age; output; drop y1-y4; datalnes; Before fttng the data nto the PROC MIXED, t s better to ensure that the data are sorted by Person and by Age. If there s a mssng value for one observaton, t s also suggested to keep the mssng n the data set to hold the place for the observaton. These efforts wll ensure that the correct vst order s recognzed by PROC MIXED. Otherwse, a CLASS varable ndcatng the vst order could be created n a data step and then used n the REPEATED statement n PROC MIXED. However, t s stll recommend to sort the data before PROC MIXED snce ths wll also ensure the elements n R block n correct order (see below). The varable Ageclss above wll be used as a CLASS varable n PROC MIXED below only for llustraton purpose snce the data set has already been n correct order. EXTRACTING THE RESIDUAL COVARIANCE FOR THE R BLOCK The followng codes are used to provde the R blocks for Person 1 and 3. Ageclss s used to ndcate the order of the observatons wthn each subject. ods output r (match_all)=rblock; proc mxed data=pr method=reml covtest; class Person Gender Ageclss; model y = Gender Age Gender*Age / s; repeated Ageclss / type=un subject=person r=1,3; The followng s the estmated covarance parameters: Covarance Parameter Estmates Cov Parm Subject Estmate UN(1,1) Person UN(2,1) Person UN(2,2) Person UN(3,1) Person UN(3,2) Person UN(3,3) Person UN(4,1) Person UN(4,2) Person UN(4,3) Person UN(4,4) Person

4 These estmated covarance parameters are used to construct the R blocks for all subjects. The opton r=1,3 n the REPEATED statement requests the PROC MIXED prnt the R blocks for Person 1 and Person 3 as the followng. An ODS OUTPUT statement s used to save the two blocks n two data sets RBLOCK and RBLOCK1 separately. Estmated R Matrx for Person 1 Row Col1 Col2 Col3 Col Estmated R Matrx for Person 3 Row Col1 Col2 Col The R block for Person 1 s correspondng to the estmated covarance parameters. The R block for Person 3 only has 3 by 3 dmensons snce the second observaton s mssng. Therefore, t needs to be repeated that only one R block for subjects wth complete data needs to be used for plottng. The followng codes wll make the selected R block always from subjects wth complete data. /* select one subject wth complete data */ proc sql noprnt; create table count as select person, count (y) as count from pr group by person; select person nto :person from count where count n (select max(count) from count); qut; /* Extract R block for subject wth compelte data */ ods output r =rblock; proc mxed data=pr method=reml covtest; class Person Gender Ageclss; model y = Gender Age Gender*Age / s; repeated Ageclss/ type=un subject=person r=&person; GRAPHING THE RESIDUAL COVARIANCE FOR THE R BLOCK BY USING PROC GPLOT The method s suggested by Lttell et al [2000]. The followng codes are used to plot the resdual covarance for the R block. /* Only half of the matrx s needed for plottng. */ data plot1; set rblock; test1=row; array col (4) col1-col4; do =1 to 4; test2=; ds=test1-test2; f ds>=0; cov= col(); output;

5 end; keep test1 test2 ds cov; goptons reset=all ftext=swssb; symbol1 =j value=1 c=black; symbol2 =j value=2 c=black; symbol3 =j value=3 c=black; symbol4 =j value=4 c=black; axs1 label=(angle=90 'Wthn-Subject Resdual Covarance' ); axs2 label=('dstance Between Observatons'); legend1 label=('from Test'); proc gplot data=plot1; plot cov*ds=test2/vaxs=axs1 haxs=axs2 legend=legend1 noframe; qut; The graph s shown n Graph 1. The horzontal axs ndcates the dstance between the observatons. The dstance between the frst test and tself s 0, the dstance between the frst test and the second test s 1 and so on. The lne ndcates the varance for the frst test when the Dstance Between Observatons s 0, the covarance between the frst test and the second test when the Dstance Between Observatons s 1, the covarance between the frst test and the thrd test when the Dstance Between Observatons s 2, and so on. So t s the same for the other lnes. The Graph 1 shows that there s no obvous trend between the varances for the 4 tests (Dstance Between Observatons=0). However, the covarance between the tests s nearly stable (or flat) as the dstance between the tests ncreases (Dstance Between Observatons=1, 2, 3). Ths would suggest a CS (Compound Symmetry) pattern. Graph 1. The resdual covarance for the R block by usng PROC GPLOT

6 GRAPHING THE RESIDUAL COVARIANCE FOR THE R BLOCK BY USING PROC G3GRID AND PROC G3D The followng codes are used to plot a 3 D graph for the resdual covarance for the R block. The graph contans the same nformaton as n the Graph 1, but vsually much more ntutve. The PROC G3GRID s used for smoothng the data before the PROC G3D. /* sort the data */ data plot2 (keep=vsta vstb estmate); set rblock; array col (4) col1-col4; vsta=row; do =1 to 4; vstb=nput(substr(vname(col()),4),3.); estmate=col(); output; end; /* smoothng the data*/ proc g3grd data=plot2 out=smooth_s; grd vsta*vstb=estmate/splne smooth=0.001 naxs1=99 naxs2=99 jon; proc g3d data=smooth_s; plot vsta*vstb=estmate/grd caxs=green ctext=red ctop=purple sde zmn=0 ztcknum=7 xtcknum=4 ytcknum=4 rotate=20 tlt=80; label vsta='test' vstb='test' estmate='cov'; The graph s shown n Graph 2. The covarance s nearly stable as the tme (or duraton) between the tests ncreases. The R matrx based on UNSTRUCTURED s regarded as beng close to realty, but a lnear mxed model based on UNSTRUCTURED covarance structure mght not be the best choce snce more covarance parameters have to be estmated. The objectve s to try to select a covarance structure whch s as close as possble to UNSTRUCTURED, but wth the most parsmonous covarance parameters to be estmated. Numercal ft statstcs such as AIC and BIC can be used for model selecton more formally, but plottng the covarance structure enables the analyst to vsualse and to compare these structures. Other covarance structures can also be plotted n the smlar way. In the PROC MIXED, you can change the opton type=un n the REPEATED statement to type=cs, and then re-run the plottng procedures agan. Ths wll produce a graph wth a resdual covarance structure of CS. The graph s shown n Graph 3. The two graphs (Graph 2 and 3) roughly have smlar pattern, though the varance for the second test s smaller n UNSTRUCTURED. So, the CS mght be a better choce than UNSTRUCTURED snce less number of covarance parameters needs to be estmated, and the wth-subject correlaton has been accounted for. A plot for covarance structure of AR(1) s also shown n Graph 4 for comparson purpose. The pattern s slghtly dfferent from that for UNSTRUCTURED. The BIC for UNSTRUCTURED, CS and AR (1) s 454.5, and respectvely, reflectng that CS fts the model better snce CS corresponds to the smallest BIC value (CS s the approprate covarance structure to ft n ths case).

7 Graph 2. The resdual covarance for the R block by usng PROC G3GRID and PROC G3D UNSTRUCTURED Graph 3. The resdual covarance for the R block by usng PROC G3GRID and PROC G3D CS

8 Graph 4. The resdual covarance for the R block by usng PROC G3GRID and PROC G3D AR(1) REFERENCES Verbeke G, Molenberghs G. (2000). Lnear mxed models for longtudnal data. New York: Sprnger-Verlag. Khattree R, Nak DN. (1999). Multvarate Data Reducton and Dscrmnaton wth SAS Software. 2 nd edton. Cary: SAS Publshng. Lttell RC, Stroup WW, Freund RJ. (2002). SAS for lnear models. 4 th edton. Cary: SAS Publshng. SAS OnlneDoc V8 CONTACT INFORMATION Your comments and questons are valued and encouraged. Contact the author at: Jan Wu Roche Products Australa 4-10 Inman Road, Dee Why, NSW 2099, Australa Work Phone: Fax: Emal: jan.wu@roche.com SAS and all other SAS Insttute Inc. product or servce names are regstered trademarks or trademarks of SAS Insttute Inc. n the USA and other countres. ndcates USA regstraton. Other brand and product names are trademarks of ther respectve companes.