SAS/STAT 922 User's Guide: The MIXED Procedure (Book Excerpt)

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4556 ✦ Chapter 56: The MIXED Procedure The DDFM=SATTERTHWAITE option performs a general Satterthwaite approximation for the denominator degrees of freedom, computed as follows. Suppose is the vector of unknown parameters in V, and suppose C D .X 0 V 1 X/ , where denotes a generalized inverse. Let bC and b be the corresponding estimates. Consider the one-dimensional case, and consider ` to be a vector defining an estimable linear combination of ˇ. The Satterthwaite degrees of freedom for the t statistic t D `bˇ p ` OC` 0 is computed as D 2.` OC` 0 / 2 g 0 Ag where g is the gradient of `C` 0 with respect to , evaluated at b, and A is the asymptotic variance-covariance matrix of b obtained from the second derivative matrix of the likelihood equations. For the multidimensional case, let L be an estimable contrast matrix and denote the rank of LbCL 0 as q > 1. The Satterthwaite denominator degrees of freedom for the F statistic F D bˇ 0 L 0 .LbCL 0 / 1 Lbˇ q are computed by first performing the spectral decomposition LbCL 0 D P 0 DP, where P is an orthogonal matrix of eigenvectors and D is a diagonal matrix of eigenvalues, both of dimension q q. Define `m to be the mth row of PL, and let m D 2.Dm/ 2 g 0 m Agm where Dm is the mth diagonal element of D and gm is the gradient of `mC` 0 m , evaluated at b. Then let E D qX mD1 m m 2 I. m > 2/ where the indicator function eliminates terms for which m F are then computed as D 2E E q provided E > q; otherwise is set to zero. with respect to 2. The degrees of freedom for This method is a generalization of the techniques described in Giesbrecht and Burns (1985), McLean and Sanders (1988), and Fai and Cornelius (1996). The method can also include estimated random effects. In this case, append b to bˇ and change bC to be the inverse of the coefficient matrix in the mixed model equations. The calculations require extra memory to hold
E E1 E2 E3 MODEL Statement ✦ 4557 c matrices that are the size of the mixed model equations, where c is the number of covariance parameters. In the notation of Table 56.25, this is approximately 8q.p C g/.p C g/=2 bytes. Extra computing time is also required to process these matrices. The Satterthwaite method implemented here is intended to produce an accurate F approximation; however, the results can differ from those produced by PROC GLM. Also, the small sample properties of this approximation have not been extensively investigated for the various models available with PROC MIXED. The DDFM=KENWARDROGER option performs the degrees of freedom calculations detailed by Kenward and Roger (1997). This approximation involves inflating the estimated variancecovariance matrix of the fixed and random effects by the method proposed by Prasad and Rao (1990) and Harville and Jeske (1992); see also Kackar and Harville (1984). Satterthwaite-type degrees of freedom are then computed based on this adjustment. By default, the observed information matrix of the covariance parameter estimates is used in the calculations. For covariance structures that have nonzero second derivatives with respect to the covariance parameters, the Kenward-Roger covariance matrix adjustment includes a second-order term. This term can result in standard error shrinkage. Also, the resulting adjusted covariance matrix can then be indefinite and is not invariant under reparameterization. The FIRSTORDER suboption of the DDFM=KENWARDROGER option eliminates the second derivatives from the calculation of the covariance matrix adjustment. For the case of scalar estimable functions, the resulting estimator is referred to as the Prasad-Rao estimator em @ in Harville and Jeske (1992). The following are examples of covariance structures that generally lead to nonzero second derivatives: TYPE=ANTE(1), TYPE=AR(1), TYPE=ARH(1), TYPE=ARMA(1,1), TYPE=CSH, TYPE=FA, TYPE=FA0(q), TYPE=TOEPH, TYPE=UNR, and all TYPE=SP() structures. When the asymptotic variance matrix of the covariance parameters is found to be singular, a generalized inverse is used. Covariance parameters with zero variance then do not contribute to the degrees-of-freedom adjustment for DDFM=SATTERTHWAITE and DDFM=KENWARDROGER, and a message is written to the log. This method changes output in the following tables (listed in Table 56.22): Contrast, CorrB, CovB, Diffs, Estimates, InvCovB, LSMeans, Slices, SolutionF, SolutionR, Tests1–Tests3. The OUTP= and OUTPM= data sets are also affected. requests that Type 1, Type 2, and Type 3 L matrix coefficients be displayed for all specified effects. For ODS purposes, the name of the table is “Coef.” requests that Type 1 L matrix coefficients be displayed for all specified effects. For ODS purposes, the name of the table is “Coef.” requests that Type 2 L matrix coefficients be displayed for all specified effects. For ODS purposes, the name of the table is “Coef.” requests that Type 3 L matrix coefficients be displayed for all specified effects. For ODS purposes, the name of the table is “Coef.”
Page 1 and 2: SAS/STAT ® 9.22 User’s Guide The
Page 3 and 4: Chapter 56 The MIXED Procedure Cont
Page 5 and 6: Basic Features ✦ 4515 Repeated me
Page 7 and 8: PROC MIXED Contrasted with Other SA
Page 9 and 10: proc mixed data=heights; class Fami
Page 11 and 12: Figure 56.6 Fit Statistics Fit Stat
Page 13 and 14: Figure 56.10 Dimensions and Number
Page 15 and 16: Syntax: MIXED Procedure The followi
Page 17 and 18: Table 56.1 continued Statement Desc
Page 19 and 20: PROC MIXED Statement ✦ 4529 CL< =
Page 21 and 22: INFO PROC MIXED Statement ✦ 4531
Page 23 and 24: ORD PROC MIXED Statement ✦ 4533 d
Page 25 and 26: PROC MIXED Statement ✦ 4535 RESID
Page 27 and 28: PROC MIXED Statement ✦ 4537 RANDO
Page 29 and 30: CLASS Statement ✦ 4539 Because so
Page 31 and 32: contrast 'A narrow' A 1 -1 0 A*B .5
Page 33 and 34: ESTIMATE Statement ESTIMATE ’labe
Page 35 and 36: ID Statement ID variables ; ID Stat
Page 37 and 38: ADJUST=BON proc mixed; class A; mod
Page 39 and 40: LSMEANS Statement ✦ 4549 BYLEVEL
Page 41 and 42: SLICE= fixed-effect LSMESTIMATE Sta
Page 43 and 44: Table 56.6 Summary of Important MOD
Page 45: MODEL Statement ✦ 4555 DDFM=KENWA
Page 49 and 50: Table 56.8 continued Description Su
Page 51 and 52: SIZE=n MODEL Statement ✦ 4561 If
Page 53 and 54: MODEL Statement ✦ 4563 Table 56.1
Page 55 and 56: The estimated prediction variance i
Page 57 and 58: PARMS Statement PARMS (value-list)
Page 59 and 60: OLS PRIOR Statement ✦ 4569 By def
Page 61 and 62: PRIOR Statement ✦ 4571 JEFFREYS s
Page 63 and 64: RANDOM Statement ✦ 4573 TDATA= en
Page 65 and 66: RANDOM Statement ✦ 4575 GCORR dis
Page 67 and 68: RANDOM Statement ✦ 4577 component
Page 69 and 70: Table 56.12 continued Option Descri
Page 71 and 72: proc mixed; class a; model y = a x;
Page 73 and 74: REPEATED Statement ✦ 4583 assumed
Page 75 and 76: REPEATED Statement ✦ 4585 functio
Page 77 and 78: REPEATED Statement ✦ 4587 The fol
Page 79 and 80: TYPE=SP(EXPGA)(c1 c2) TYPE=SP(GAUGA
Page 81 and 82: TYPE=UN@AR(1) TYPE=UN@CS REPEATED S
Page 83 and 84: WEIGHT Statement WEIGHT variable ;
Page 85 and 86: Example: Growth Curve with Compound
Page 87 and 88: Mixed Models Theory ✦ 4597 Such a
Page 89 and 90: 2 6 Z D 6 4 2 6 G D 6 4 1 1 1 1 1 1
Page 91 and 92: Estimating Fixed and Random Effects
Page 93 and 94: Inference and Test Statistics Mixed
Page 95 and 96: Mixed Models Theory ✦ 4605 (2002,
Page 97 and 98:
Intercept Parameterization of Mixed
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Table 56.18 continued Data I A B(A)
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Residuals and Influence Diagnostics
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Scaled Residuals Residuals and Infl
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Predicted Values, PRESS Residual, a
Page 107 and 108:
When ITER=0 and 2 is profiled, then
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Using results in Cook and Weisberg
Page 111 and 112:
Default Output ✦ 4621 The Evaluat
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Null Model Likelihood Ratio Test De
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Table 56.22 continued ODS Table Nam
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ODS Table Names ✦ 4627 CAUTION: T
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ODS Graphics ✦ 4629 Some of the v
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Figure 56.16 Panel of the Condition
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Figure 56.18 Deletion Estimates ODS
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Computational Issues Computational
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Memory Computational Issues ✦ 463
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Examples: MIXED Procedure Examples:
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Example 56.1: Split-Plot Design ✦
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Example 56.1: Split-Plot Design ✦
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4 F 23.5 24.5 25.0 26.5 5 F 21.5 23
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Output 56.2.4 Repeated Measures Ana
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Output 56.2.7 Repeated Measures Ana
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Example 56.2: Repeated Measures ✦
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Output 56.2.14 Analysis with Compou
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Example 56.2: Repeated Measures ✦
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Example 56.3: Plotting the Likeliho
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Output 56.3.3 continued Number of O
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Output 56.3.8 Fit Statistics and Li
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Output 56.3.12 Least Squares Means
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Both G and R are known. 2 2 1 1 2 1
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Output 56.4.2 Model Dimensions and
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Example 56.4: Known G and R ✦ 466
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Example 56.5: Random Coefficients
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Example 56.5: Random Coefficients
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Output 56.5.7 Random Coefficients A
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Output 56.5.10 continued Covariance
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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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Page 191 and 192:
Page 193 and 194:
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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Output 56.10.1 Analysis of LS-Means
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References ✦ 4711 Cook, R. D. (19
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References ✦ 4713 Harville, D. A.
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References ✦ 4715 Matérn, B. (19
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Bulletin No. 343, Louisiana Agricul
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Subject Index 2D geometric anisotro
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graphics, influence diagnostics, 46
Page 213 and 214:
matrix notation, theory (MIXED), 45
Page 215 and 216:
parameter constraints, 4568, 4636 p
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subject effect MIXED procedure, 454
Page 220 and 221:
INFLUENCE option, MODEL statement (
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NSAMPLE= option, 4572 NSEARCH= opti
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ESTIMATE statement (MIXED), 4544 LS
Page 227:
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SAS/STAT 922 User's Guide: The MIXED Procedure (Book Excerpt)

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