Variance Components and Mixed Model ANOVA/ANCOVA - Summary: Components of Variance Spreadsheets

The option selected in the Method group box of the Variance Components and Mixed Model ANOVA/ANCOVA Results dialog - Advanced tab will affect the list of spreadsheets that are available below the Summary: Components of Variance button (also on the Advanced tab).

Expected MS. If Expected MS (mean squares) is selected in the Method group box, three additional spreadsheets can be displayed when the Summary: Components of variance button is clicked.

Expected mean squares.  Select the Expected mean squares check box to display a spreadsheet that contains the elements of the expected mean squares coefficient matrix, which are used to estimate the variation for each effect in the model.

Coefficients for denominator synthesis. Select the Coefficients for denominator synthesis check box to display a spreadsheet that contains the elements of the denominator synthesis coefficient matrix, which are used to estimate the error term for testing the significance of each random effect in the model.

Denominator synthesis ANOVA. Select the Denominator synthesis ANOVA check box to display a spreadsheet that contains the results of the denominator synthesis ANOVA, which shows the tests of significance for the effects in the model, using synthesized denominators for testing the significance of random effects. Satterthwaite's method of denominator synthesis (Satterthwaite, 1946) is used when Expected MS has been selected as the Method for estimating variance components.

MIVQUE0. If MIVQUE0 has been selected in the Method group box, the additional spreadsheets listed under the Summary: Components of variance button will not be available. If you click the Summary: Components of variance button a spreadsheet containing the Quadratic sums of squares (SSQ) matrix will automatically be displayed. The elements of the SSQ matrix are the sums of squares of the sums of squares and cross products for each random effect in the model. To estimate the variance components, the partition of the SSQ matrix for the random effects is inverted and post-multiplied by the dependent variable column vector. This amounts to solving the system of equations that relates the dependent variable to the random independent variables, taking into account the covariation among the independent variables.

REML and ML. If REML or ML is selected in the Method group box, three additional spreadsheets can be displayed when the Summary: Components of variance button is clicked.

Show iteration history. Select the Show iteration history check box to display a spreadsheet that contains the iteration history, which includes the REML or ML log likelihood and the variance component estimates at each iteration of the solution.

Show parameter covariance matrix. Select the Show parameter covariance matrix check box to display a spreadsheet that contains the parameter covariance matrix, which is the asymptotic (large sample) variance/covariance matrix of the nonzero variance components at convergence of the solution.

Show parameter correlation matrix. Select the Show parameter correlation matrix check box to display a spreadsheet that contains the parameter correlation matrix, which is the asymptotic (large sample) correlation matrix of the nonzero variance components at convergence of the solution.

Note: Singular Hessian matrix at point of conversion in maximum likelihood estimation. In some cases you may see this message at the top of the spreadsheet reporting the estimates of the variance components; usually, you will also see at least one component with a reported asymptotic standard error of 0 (zero). This message indicates that during the iterative computations, at least two components were found to be entirely redundant, and that no unique estimates can be derived for those components. This may, for example, happen when there are 0 (zero) degrees of freedom for the error term, and hence, when the error variance is not defined. The computational algorithm used in STATISTICA will always attempt to resolve any redundancies so as to retain a positive (>0) estimate for the Error variance component (otherwise, if during the estimation an intermediate solution makes the Error component equal to 0 [zero], the algorithm may prematurely terminate with a non-maximum likelihood solution). However, when this message appears in the header of a results spreadsheet in your analysis, you should carefully review the current model (e.g., use option SS and MS Effect on the Estimation tab), and interpret the estimates of the variance components with caution (as they are not unique!).

Note that there are several modules in STATISTICA that will perform Analysis of Variance for factorial or specialized designs. For a discussion of these modules and the types of designs for which they are best suited refer to Methods for Analysis of Variance. Note also that the General Linear Model (GLM) module can analyze designs with any number and type of between effects and compute ANOVA-based variance component estimates for any effect in a mixed-model analysis using any of the six types of sums of squares.