Example
Example 79.1. Using the Four Estimation Methods
In this example, a and b are classification variables and y is the dependent variable. a is declared fixed, and b and a * b are random. Note that this design is unbalanced because the cell sizes are not all the same. PROC VARCOMP is invoked four times, once for each of the estimation methods. The data are from Hemmerle and Hartley (1973). The following statements produce Output 79.1.1.
data a; input a b y @@; datalines; 1 1 237 1 1 254 1 1 246 1 2 178 1 2 179 2 1 208 2 1 178 2 1 187 2 2 146 2 2 145 2 2 141 3 1 186 3 1 183 3 2 142 3 2 125 3 2 136 ; proc varcomp method=type1; class a b; model y=ab / fixed=1; run; proc varcomp method=mivque0; class a b; model y=ab / fixed=1; run; proc varcomp method=ml; class a b; model y=ab / fixed=1; run; proc varcomp method=reml; class a b; model y=ab / fixed=1; run;
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Variance Components Estimation Procedure Class Level Information Class Levels Values a 3 1 2 3 b 2 1 2 Number of Observations Read 16 Number of Observations Used 16 Dependent Variable: y
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The 'Class Level Information' table displays the levels of each variable specified in the CLASS statement. You can check this table to make sure the data are input correctly.
Variance Components Estimation Procedure Type 1 Analysis of Variance Sum of Source DF Squares Mean Square Expected Mean Square a 2 11736 5868.218750 Var(Error) + 2.725 Var(a*b) + 0.1 Var(b) + Q(a) b 1 11448 11448 Var(Error) + 2.6308 Var(a*b)+ 7.8 Var(b) a*b 2 299.041026 149.520513 Var(Error) + 2.5846 Var(a*b) Error 10 786.333333 78.633333 Var(Error) Corrected Total 15 24270 . .
The Type I analysis of variance consists of sequentially partitioning the total sum of squares. The mean square is the sum of squares divided by the degrees of freedom, and the expected mean square is the expected value of the mean square under the mixed model. The 'Q' notation in the expected mean squares refers to a quadratic form in parameters of the parenthesized effect.
Variance Components Estimation Procedure Type 1 Estimates Variance Component Estimate Var(b) 1448.4 Var(a*b) 27.42659 Var(Error) 78.63333
The Type I estimates of the variance components result from solving the linear system of equations established by equating the observed mean squares to their expected values.
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Variance Components Estimation Procedure Class Level Information Class Levels Values a 3 1 2 3 b 2 1 2 Number of Observations Read 16 Number of Observations Used 16
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The 'Class Level Information' is the same as before.
Variance Components Estimation Procedure MIVQUE(0) SSQ Matrix Source b a*b Error y b 60.84000 20.52000 7.80000 89295.4 a*b 20.52000 20.52000 7.80000 30181.3 Error 7.80000 7.80000 13.00000 12533.5
The MIVQUE0 sums-of-squares matrix is displayed in the previous table.
Variance Components Estimation Procedure MIVQUE(0) Estimates Variance Component y Var(b) 1466.1 Var(a*b) -35.49170 Var(Error) 105.73660
The MIVQUE0 estimates result from solving the equations established by the MIVQUE0 SSQ matrix. Note that the estimate of the variance component for the interaction effect, Var(a*b), is negative for this example.
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Variance Components Estimation Procedure Class Level Information Class Levels Values a 3 1 2 3 b 2 1 2 Number of Observations Read 16 Number of Observations Used 16 Dependent Variable: y
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The 'Class Level Information' is the same as before.
Variance Components Estimation Procedure Maximum Likelihood Iterations Iteration Objective Var(b) Var(a*b) Var(Error) 0 78.3850371200 1031.49070 0 74.3909717935 1 78.2637043807 732.3606453636 0 77.4011688154 2 78.2635471161 723.6867470850 0 77.5301774839 3 78.2635471152 723.6658365289 0 77.5304926877 Convergence criteria met.
The Newton-Raphson algorithm used by PROC VARCOMP requires three iterations to converge to the maximum likelihood estimates.
Variance Components Estimation Procedure Maximum Likelihood Estimates Variance Component Estimate Var(b) 723.66584 Var(a*b) 0 Var(Error) 77.53049
The ML estimate of Var(a*b) is zero for this example, and the other two estimates are smaller than their Type I and MIVQUE0 counterparts.
Variance Components Estimation Procedure Asymptotic Covariance Matrix of Estimates Var(b) Var(a*b) Var(Error) Var(b) 537826.1 0 107.33905 Var(a*b) 0 0 0 Var(Error) 107.33905 0 858.71104
One benefit of using likelihood-based methods is that an approximate covariance matrix is available from the matrix of second derivatives evaluated at the ML solution. This covariance matrix is valid asymptotically and can be unreliable in small samples.
Here the variance component estimates for B and the Error are negatively correlated and the elements for Var(a*b) are set to zero because the estimate equals zero. Also, the very large variance for Var(b) indicates a lot of uncertainty about the estimate for Var(b), and one contributing explanation is that B has only two levels in this data set.
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Variance Components Estimation Procedure Class Level Information Class Levels Values a 3 1 2 3 b 2 1 2 Number of Observations Read 16 Number of Observations Used 16 Dependent Variable: y
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The 'Class Level Information' is the same as before.
Variance Components Estimation Procedure REML Iterations Iteration Objective Var(b) Var(a*b) Var(Error) 0 63.4134144942 1269.52701 0 91.5581191305 1 63.0446869787 1601.84199 32.7632417174 76.9355562461 2 63.0311530508 1468.82932 27.2258186561 78.7548276319 3 63.0311265148 1464.33646 26.9564053003 78.8431476502 4 63.0311265127 1464.36727 26.9588525177 78.8423898761 Convergence criteria met.
The REML optimization requires four iterations to converge.
Variance Components Estimation Procedure REML Estimates Variance Component Estimate Var(b) 1464.4 Var(a*b) 26.95885 Var(Error) 78.84239
The REML estimates are all larger than the corresponding ML estimates (adjusting for potential downward bias) and are fairly similar to the Type I estimates.
Variance Components Estimation Procedure Asymptotic Covariance Matrix of Estimates Var(b) Var(a*b) Var(Error) Var(b) 4401703.8 1.29359 273.39651 Var(a*b) 1.29359 3559.1 502.85157 Var(Error) 273.39651 502.85157 1249.7
The Error variance component estimate is negatively correlated with the other two variance component estimates, and the estimated variances are all larger than their ML counterparts.
EAN: 2147483647
Pages: 132