SAS/STAT 9.2 User's Guide: The MIXED Procedure (Book Excerpt)

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3892 ✦ Chapter 56: The MIXED Procedure Figure 56.3 Dimensions Dimensions Covariance Parameters 1 Columns in X 15 Columns in Z 0 Subjects 1 Max Obs Per Subject 18 The “Number of Observations” table in Figure 56.4 displays information about the sample size being processed. Figure 56.4 Number of Observations Number of Observations Number of Observations Read 18 Number of Observations Used 18 Number of Observations Not Used 0 The “Covariance Parameter Estimates” table in Figure 56.5 displays the estimate of 2 for the model. Figure 56.5 Covariance Parameter Estimates Covariance Parameter Estimates Cov Parm Estimate Residual 2.1000 The “Fit Statistics” table in Figure 56.6 lists several pieces of information about the fitted mixed model, including values derived from the computed value of the restricted/residual likelihood. Figure 56.6 Fit Statistics Fit Statistics -2 Res Log Likelihood 41.6 AIC (smaller is better) 43.6 AICC (smaller is better) 44.1 BIC (smaller is better) 43.9 The “Type 3 Tests of Fixed Effects” table in Figure 56.7 displays significance tests for the three effects listed in the MODEL statement. The Type 3 F statistics and p-values are the same as those produced by the GLM procedure. However, because PROC MIXED uses a likelihood-based esti-
Clustered Data Example ✦ 3893 mation scheme, it does not directly compute or display sums of squares for this analysis. Figure 56.7 Tests of Fixed Effects Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F Gender 1 10 17.63 0.0018 Family 3 10 5.90 0.0139 Family*Gender 3 10 2.89 0.0889 The Type 3 test for Family*Gender effect is not significant at the 5% level, but the tests for both main effects are significant. The important assumptions behind this analysis are that the data are normally distributed and that they are independent with constant variance. For these data, the normality assumption is probably realistic since the data are observed heights. However, since the data occur in clusters (families), it is very likely that observations from the same family are statistically correlated—that is, not independent. The methods implemented in PROC MIXED are still based on the assumption of normally distributed data, but you can drop the assumption of independence by modeling statistical correlation in a variety of ways. You can also model variances that are heterogeneous—that is, nonconstant. For the height data, one of the simplest ways of modeling correlation is through the use of random effects. Here the family effect is assumed to be normally distributed with zero mean and some unknown variance. This is in contrast to the previous model in which the family effects are just constants, or fixed effects. Declaring Family as a random effect sets up a common correlation among all observations having the same level of Family. Declaring Family*Gender as a random effect models an additional correlation between all observations that have the same level of both Family and Gender. One interpretation of this effect is that a female in a certain family exhibits more correlation with the other females in that family than with the other males, and likewise for a male. With the height data, this model seems reasonable. The statements to fit this correlation model in PROC MIXED are as follows: proc mixed; class Family Gender; model Height = Gender; random Family Family*Gender; run; Note that Family and Family*Gender are now listed in the RANDOM statement. The dummy variables associated with them are used to construct the Z matrix in the mixed model. The X matrix now consists of a column of 1s and the dummy variables for Gender. The G matrix for this model is diagonal, and it contains the variance components for both Family and Family*Gender. The R matrix is still assumed to equal 2 I, where I is an identity matrix.
Page 1 and 2: SAS/STAT ® 9.2 User’s Guide The
Page 3 and 4: Chapter 56 The MIXED Procedure Cont
Page 5 and 6: Basic Features ✦ 3887 clustered i
Page 7 and 8: variance components compound symmet
Page 9: Clustered Data Example ✦ 3891 The
Page 13 and 14: criterion of 1E 8. Figure 56.11 REM
Page 15 and 16: Syntax: MIXED Procedure ✦ 3897 Th
Page 17 and 18: Table 56.2 continued Option Descrip
Page 19 and 20: If H k is singular, then PROC MIXED
Page 21 and 22: MMEQ PROC MIXED Statement ✦ 3903
Page 23 and 24: PLOTS < (global-plot-options ) > <
Page 25 and 26: Residual Plot Options PROC MIXED St
Page 27 and 28: RATIO Multiple Plot Request PROC MI
Page 29 and 30: CONTRAST Statement ✦ 3911 TRUNCAT
Page 31 and 32: CONTRAST Statement ✦ 3913 differe
Page 33 and 34: ESTIMATE Statement ✦ 3915 ALPHA=n
Page 35 and 36: Table 56.5 Summary of Important LSM
Page 37 and 38: LSMEANS Statement ✦ 3919 The numb
Page 39 and 40: E LSMEANS Statement ✦ 3921 are di
Page 41 and 42: Table 56.6 continued Option Descrip
Page 43 and 44: Table 56.7 Aliases for DDFM= Option
Page 45 and 46: E E1 E2 E3 FULLX MODEL Statement
Page 47 and 48: MODEL Statement ✦ 3929 The modifi
Page 49 and 50: SIZE=n MODEL Statement ✦ 3931 ins
Page 51 and 52: Table 56.10 continued Suboption Sta
Page 53 and 54: MODEL Statement ✦ 3935 where Vm i
Page 55 and 56: PARMS Statement PARMS (value-list)
Page 57 and 58: OLS PRIOR Statement ✦ 3939 approp
Page 59 and 60: PRIOR Statement ✦ 3941 JEFFREYS s
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RANDOM Statement ✦ 3943 TDATA= en
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RANDOM Statement ✦ 3945 GCORR dis
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RANDOM Statement ✦ 3947 available
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Table 56.12 continued Option Descri
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proc mixed; class a; model y = a x;
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REPEATED Statement ✦ 3953 TYPE=co
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Table 56.15 Covariance Structure Ex
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REPEATED Statement ✦ 3957 TYPE=AR
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TYPE=SP(EXPGA)(c1 c2) TYPE=SP(GAUGA
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TYPE=UN@AR(1) TYPE=UN@CS REPEATED S
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Mixed Models Theory ✦ 3963 where
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proc mixed; class indiv; model y =
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2 6 Z D 6 4 2 6 G D 6 4 1 1 1 1 1 1
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Mixed Models Theory ✦ 3969 In man
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Mixed Models Theory ✦ 3971 Finall
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Mixed Models Theory ✦ 3973 When L
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Parameterization of Mixed Models
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Interaction Effects Parameterizatio
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Table 56.20 Example of Continuous-b
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as ei ei p D pvi VarŒei and studen
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Residuals and Influence Diagnostics
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Default Output Default Output ✦ 3
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Default Output ✦ 3991 batch proce
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ODS Table Names ✦ 3993 You can us
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Table 56.22 continued ODS Table Nam
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Table 56.23 continued Table Name Va
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ODS Graphics ✦ 3999 The graphical
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ODS Graphics ✦ 4001 precision is
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Table 56.24 continued ODS Graph Nam
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Computational Issues ✦ 4005 For f
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Memory Computational Issues ✦ 400
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The following statements fit the sp
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Output 56.1.6 Split-Plot Analysis (
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proc mixed; class A B Block; model
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Example 56.2: Repeated Measures ✦
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Output 56.2.6 Repeated Measures Ana
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Output 56.2.9 Repeated Measures Ana
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Output 56.2.11 continued Dimensions
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Output 56.2.17 Analysis with Compou
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Example 56.2: Repeated Measures ✦
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The results from this analysis are
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Output 56.3.6 Estimated Covariance
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Output 56.3.10 Solutions of the Mix
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Output 56.3.14 Plot of Likelihood S
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Example 56.4: Known G and R ✦ 403
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Example 56.4: Known G and R ✦ 403
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Output 56.4.9 Solutions of the Mixe
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Example 56.5: Random Coefficients E
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Output 56.5.3 continued Number of O
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Output 56.5.9 Random Coefficients A
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Example 56.5: Random Coefficients
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Example 56.6: Line-Source Sprinkler
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Output 56.6.3 Model Dimensions and
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Example 56.6: Line-Source Sprinkler
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Example 56.7: Influence in Heteroge
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Example 56.7: Influence in Heteroge
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Output 56.7.5 Covariance Parameter
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Output 56.7.7 Restricted Likelihood
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Output 56.7.9 Covariance Parameter
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Example 56.8: Influence Analysis fo
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Output 56.8.2 Restricted Likelihood
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Output 56.8.8 Distribution of Condi
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Output 56.9.2 Type 3 Tests in Split
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Output 56.9.5 Parameter Estimates i
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References ✦ 4079 Carroll, R. J.
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References ✦ 4081 Hanks, R.J., Si
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References ✦ 4083 Littell, R. C.,
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References ✦ 4085 Smith, A. F. M.
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Subject Index 2D geometric anisotro
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graphics, residual panel, 3998 grap
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maximum likelihood estimation mixed
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parameterization, 3975 Pearson resi
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subject effect MIXED procedure, 391
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INFLUENCE option, MODEL statement (
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OUTG= option, 3942 OUTGT= option, 3
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INFLUENCE option, MODEL statement (
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SAS/STAT 9.2 User's Guide: The MIXED Procedure (Book Excerpt)

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