 Estimation of Variance Components - Testing the Significance of Variance Components

When maximum likelihood estimation techniques are used, standard linear model significance testing techniques may not be applicable. ANOVA techniques such as decomposing sums of squares and testing the significance of effects by taking ratios of mean squares are appropriate for linear methods of estimation, but generally are not appropriate for quadratic methods of estimation. When ANOVA methods are used for estimation, standard significance testing techniques can be employed, with the exception that any confounding among random effects must be taken into account.

To test the significance of effects in mixed or random models, error terms must be constructed that contain all the same sources of random variation except for the variation of the respective effect of interest. This is done using Satterthwaite's method of denominator synthesis (Satterthwaite, 1946), which finds the linear combinations of sources of random variation that serve as appropriate error terms for testing the significance of the respective effect of interest. The spreadsheet below shows the coefficients used to construct these linear combinations for testing the Variety and Plot effects.

 Denominator Synthesis: Coefficients (MS Type: 1) (wheat.sta) The synthesized MS Errors are linear combinations of the resp. MS effects Effect (F/R) VARIETY PLOT Error {1}VARIETY Random 1.000000 {2}PLOT Random 1.000000

The coefficients show that the Mean square for Variety should be tested against the Mean square for Plot, and that the Mean square for Plot should be tested against the Mean square for Error. Referring back to the Expected mean squares spreadsheet, it is clear that the denominator synthesis has identified appropriate error terms for testing the Variety and Plot effects. Although this is a simple example, in more complex analyses with various degrees of confounding among the random effects, the denominator synthesis can identify appropriate error terms for testing the random effects that would not be readily apparent.

To perform the tests of significance of the random effects, ratios of appropriate Mean squares are formed to compute F statistics and p-values for each effect. Note that in complex analyses the degrees of freedom for random effects can be fractional rather than integer values, indicating that fractions of sources of variation were used in synthesizing appropriate error terms for testing the random effects. The spreadsheet displaying the results of the ANOVA for the Variety and Plot random effects is shown below. Note that for this simple design the results are identical to the results presented earlier in the spreadsheet for the ANOVA treating Plot as a random effect nested within Variety.

 ANOVA Results for Synthesized Errors: DAMAGE (wheat.sta) df error computed using Satterthwaite method Effect Effect (F/R) df Effect MS Effect df Error MS Error F p {1}VARIETY Fixed 3 .270053 9 .056435 4.785196 .029275 {2}PLOT Random 9 .056435 ----- ----- ----- -----

As shown in the spreadsheet, the Variety effect is found to be significant at p < .05, but as would be expected, the Plot effect cannot be tested for significance because plots served as the basic unit of analysis. If data on samples of plants taken within plots were available, a test of the significance of the Plot effect could be constructed.

Appropriate tests of significance for MIVQUE(0) variance component estimates generally cannot be constructed, except in special cases (see Searle, Casella, & McCulloch, 1992). Asymptotic (large sample) tests of significance of REML and ML variance component estimates, however, can be constructed for the parameter estimates from the final iteration of the solution. The spreadsheet below shows the asymptotic (large sample) tests of significance for the REML estimates for the Wheat.sta data.

 Restricted Maximum Likelihood Estimates (wheat.sta) Variable: DAMAGE -2*Log(Likelihood)=4.50162399 Effect Variance Comp. Asympt. Std.Err. Asympt. z Asympt. p {1}VARIETY .073155 .078019 .937656 .348421 Error .057003 .027132 2.100914 .035648

The spreadsheet below shows the asymptotic (large sample) tests of significance for the ML estimates for the Wheat.sta data.

 Maximum Likelihood Estimates (wheat.sta) Variable: DAMAGE -2*Log(Likelihood)=4.96761616 Effect Variance Comp. Asympt. Std.Err. Asympt. z Asympt. p {1}VARIETY .048552 .050747 .956748 .338694 Error .057492 .027598 2.083213 .037232

It should be emphasized that the asymptotic tests of significance for REML and ML variance component estimates are based on large sample sizes, which certainly is not the case for the Wheat.sta data. For this data set, the tests of significance from both analyses agree in suggesting that the Variety variance component does not differ significantly from zero.