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

More documents

Recommendations

Info

3986 ✦ Chapter 56: The MIXED Procedure The MIXED procedure computes DFFITS when the EFFECT= or SIZE= modifier of the INFLUENCE option is not in effect. In general, an external estimate of the estimated standard error is used. When ITER > 0, the estimate is ese.byi/ D q x 0 i .X0 V.b .u// X/ 1 xi When ITER=0 and 2 is profiled, then q ese.byi/ D b .u/ x 0 i .X0 V.b / 1 X/ xi When the EFFECT=, SIZE=, or KEEP= modifier is specified, the MIXED procedure computes a multivariate version suitable for the deletion of multiple data points. The statistic, termed MDFFITS after the MDFFIT statistic of Belsley, Kuh, and Welsch (1980, p. 32), is closely related to Cook’s D. Consider the case V D 2 V. / so that VarŒbˇ D 2 .X 0 V. / 1 X/ and let fVarŒbˇ .U / be an estimate of VarŒbˇ .U / that does not use the observations in U . The MDF- FITS statistic is then computed as MDFFITS.ˇ/ D ı 0 .U / fVarŒbˇ .U / ı .U /=rank.X/ If ITER=0 and 2 is profiled, then fVarŒbˇ .U / is obtained by sweeping b 2 .U / .X0 .U / V .U /.b / X .U // The underlying idea is that if were known, then .X 0 .U / V .U /. / 1 X .U // would be VarŒbˇ= 2 in a generalized least squares regression with all but the data in U . In the case of iterative influence analysis, fVarŒbˇ .U / is evaluated at b .U /. Furthermore, a MDFFITStype statistic is then computed for the covariance parameters: MDFFITS. / D .b b .U // 0 cVarŒb .U / 1 .b b .U // Covariance Ratio and Trace These statistics depend on the availability of an external estimate of V, or at least of 2 . Whereas Cook’s D and MDFFITS measure the impact of data points on a vector of parameter estimates, the covariance-based statistics measure impact on their precision. Following Christensen, Pearson, and Johnson (1992), the MIXED procedure computes CovTrace.ˇ/ D jtrace. cVarŒbˇ fVarŒbˇ .U // rank.X/j CovRatio.ˇ/ D detns. fVarŒbˇ .U // detns. cVarŒbˇ/ where detns.M/ denotes the determinant of the nonsingular part of matrix M.
Residuals and Influence Diagnostics ✦ 3987 In the case of iterative influence analysis these statistics are also computed for the covariance parameter estimates. If q denotes the rank of VarŒb, then CovTrace. / D jtrace. cVarŒb cVarŒb .U // qj CovRatio. / D detns. cVarŒb .U // detns. cVarŒb/ Likelihood Distances The log-likelihood function l and restricted log-likelihood function lR of the linear mixed model are given in the section “Estimating Covariance Parameters in the Mixed Model” on page 3968. Denote as the collection of all parameters, i.e., the fixed effects ˇ and the covariance parameters . Twice the difference between the (restricted) log-likelihood evaluated at the full-data estimates b and at the reduced-data estimates b .U / is known as the (restricted) likelihood distance: RLD .U / D 2flR.b / lR.b .U //g LD .U / D 2fl.b / l.b .U //g Cook and Weisberg (1982, Ch. 5.2) refer to these differences as likelihood distances, Beckman, Nachtsheim, and Cook (1987) call the measures likelihood displacements. If the number of elements in that are subject to updating following point removal is q, then likelihood displacements can be compared against cutoffs from a chi-square distribution with q degrees of freedom. Notice that this reference distribution does not depend on the number of observations removed from the analysis, but rather on the number of model parameters that are updated. The likelihood displacement gives twice the amount by which the log likelihood of the full data changes if one were to use an estimate based on fewer data points. It is thus a global, summary measure of the influence of the observations in U jointly on all parameters. Unless METHOD=ML, the MIXED procedure computes the likelihood displacement based on the residual (=restricted) log likelihood, even if METHOD=MIVQUE0 or METHOD=TYPE1, TYPE2, or TYPE3. Noniterative Update Formulas Update formulas that do not require refitting of the model are available for the cases where V D 2 I, V is known, or V is known. When ITER=0 and these update formulas can be invoked, the MIXED procedure uses the computational devices that are outlined in the following paragraphs. It is then assumed that the variance-covariance matrix of the fixed effects has the form .X 0 V 1 X/ . When DDFM=KENWARDROGER, this is not the case; the estimated variance-covariance matrix is then inflated to better represent the uncertainty in the estimated covariance parameters. Influence statistics when DDFM=KENWARDROGER should iteratively update the covariance parameters (ITER > 0). The dependence of V on is suppressed in the sequel for brevity. Updating the Fixed Effects Denote by U the .n k/ matrix that is assembled from k columns of the identity matrix. Each column of U corresponds to the removal of one data point. The point being targeted by the ith column of U corresponds to the row in which a 1 appears. Furthermore,
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 and 10:
Clustered Data Example ✦ 3891 The
Page 11 and 12:
Clustered Data Example ✦ 3893 mat
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
Page 61 and 62: RANDOM Statement ✦ 3943 TDATA= en
Page 63 and 64: RANDOM Statement ✦ 3945 GCORR dis
Page 65 and 66: RANDOM Statement ✦ 3947 available
Page 67 and 68: Table 56.12 continued Option Descri
Page 69 and 70: proc mixed; class a; model y = a x;
Page 71 and 72: REPEATED Statement ✦ 3953 TYPE=co
Page 73 and 74: Table 56.15 Covariance Structure Ex
Page 75 and 76: REPEATED Statement ✦ 3957 TYPE=AR
Page 77 and 78: TYPE=SP(EXPGA)(c1 c2) TYPE=SP(GAUGA
Page 79 and 80: TYPE=UN@AR(1) TYPE=UN@CS REPEATED S
Page 81 and 82: Mixed Models Theory ✦ 3963 where
Page 83 and 84: proc mixed; class indiv; model y =
Page 85 and 86: 2 6 Z D 6 4 2 6 G D 6 4 1 1 1 1 1 1
Page 87 and 88: Mixed Models Theory ✦ 3969 In man
Page 89 and 90: Mixed Models Theory ✦ 3971 Finall
Page 91 and 92: Mixed Models Theory ✦ 3973 When L
Page 93 and 94: Parameterization of Mixed Models
Page 95 and 96: Interaction Effects Parameterizatio
Page 97 and 98: Table 56.20 Example of Continuous-b
Page 99 and 100: as ei ei p D pvi VarŒei and studen
Page 101 and 102: Residuals and Influence Diagnostics
Page 103: Residuals and Influence Diagnostics
Page 107 and 108: Default Output Default Output ✦ 3
Page 109 and 110: Default Output ✦ 3991 batch proce
Page 111 and 112: ODS Table Names ✦ 3993 You can us
Page 113 and 114: Table 56.22 continued ODS Table Nam
Page 115 and 116: Table 56.23 continued Table Name Va
Page 117 and 118: ODS Graphics ✦ 3999 The graphical
Page 119 and 120: ODS Graphics ✦ 4001 precision is
Page 121 and 122: Table 56.24 continued ODS Graph Nam
Page 123 and 124: Computational Issues ✦ 4005 For f
Page 125 and 126: Memory Computational Issues ✦ 400
Page 127 and 128: The following statements fit the sp
Page 129 and 130: Output 56.1.6 Split-Plot Analysis (
Page 131 and 132: proc mixed; class A B Block; model
Page 133 and 134: Example 56.2: Repeated Measures ✦
Page 135 and 136: Output 56.2.6 Repeated Measures Ana
Page 137 and 138: Output 56.2.9 Repeated Measures Ana
Page 139 and 140: Output 56.2.11 continued Dimensions
Page 141 and 142: Output 56.2.17 Analysis with Compou
Page 143 and 144: Example 56.2: Repeated Measures ✦
Page 145 and 146: The results from this analysis are
Page 147 and 148: Output 56.3.6 Estimated Covariance
Page 149 and 150: Output 56.3.10 Solutions of the Mix
Page 151 and 152: Output 56.3.14 Plot of Likelihood S
Page 153 and 154: Example 56.4: Known G and R ✦ 403
Page 155 and 156:
Example 56.4: Known G and R ✦ 403
Page 157 and 158:
Output 56.4.9 Solutions of the Mixe
Page 159 and 160:
Example 56.5: Random Coefficients E
Page 161 and 162:
Output 56.5.3 continued Number of O
Page 163 and 164:
Output 56.5.9 Random Coefficients A
Page 165 and 166:
Example 56.5: Random Coefficients
Page 167 and 168:
Example 56.6: Line-Source Sprinkler
Page 169 and 170:
Output 56.6.3 Model Dimensions and
Page 171 and 172:
Example 56.6: Line-Source Sprinkler
Page 173 and 174:
Example 56.7: Influence in Heteroge
Page 175 and 176:
Example 56.7: Influence in Heteroge
Page 177 and 178:
Output 56.7.5 Covariance Parameter
Page 179 and 180:
Output 56.7.7 Restricted Likelihood
Page 181 and 182:
Output 56.7.9 Covariance Parameter
Page 183 and 184:
Example 56.8: Influence Analysis fo
Page 185 and 186:
Output 56.8.2 Restricted Likelihood
Page 187 and 188:
Page 189 and 190:
Page 191 and 192:
Output 56.8.8 Distribution of Condi
Page 193 and 194:
Output 56.9.2 Type 3 Tests in Split
Page 195 and 196:
Output 56.9.5 Parameter Estimates i
Page 197 and 198:
References ✦ 4079 Carroll, R. J.
Page 199 and 200:
References ✦ 4081 Hanks, R.J., Si
Page 201 and 202:
References ✦ 4083 Littell, R. C.,
Page 203 and 204:
References ✦ 4085 Smith, A. F. M.
Page 205 and 206:
Subject Index 2D geometric anisotro
Page 207 and 208:
graphics, residual panel, 3998 grap
Page 209 and 210:
maximum likelihood estimation mixed
Page 211 and 212:
parameterization, 3975 Pearson resi
Page 213:
subject effect MIXED procedure, 391
Page 216 and 217:
INFLUENCE option, MODEL statement (
Page 218 and 219:
OUTG= option, 3942 OUTGT= option, 3
Page 220 and 221:
INFLUENCE option, MODEL statement (
Page 223:
SAS® Publishing Delivers! Whether
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?