Between Design Matrices in GLM, GRM, and ANOVA

The Between design matrices are available on the GLM, GRM, and ANOVA Results - Matrix tab and the GLM, GRM, and ANOVA More Results - Matrix tab.

Between design

The options in the Between design group box allow you to review various matrices computed for the between design.

Design terms

Click the Design terms button to create a spreadsheet of all the labels for each column in the design matrix (see Introductory Overview, note that this same option is available on the Summary tab, in the Between effects group box, see Summary results for between effects in GLM and ANOVA). This spreadsheet is useful in conjunction with the Coefficients button (available on the Summary tab, see Summary results for between effects in GLM and ANOVA) to unambiguously identify how the categorical predictors in the design were coded, that is, how the model was parameterized, and how, consequently, the parameter estimates can be interpreted. The Introductory Overview discusses in detail the overparameterized and sigma-restricted parameterization for categorical predictor variables and effects, and how each parameterization can yield completely different parameter estimates (even though the overall model fit, and ANOVA tables are usually invariant to the method of parameterization). If in the current analysis the categorical predictor variables were coded according to the sigma-restricted parameterization, then this spreadsheet will show the two levels of the respective factors that were contrasted in each column of the design matrix; if the overparameterized model was used, then the spreadsheet will show the relationship of each level of the categorical predictors to the columns in the design matrix (and, hence, the respective parameter estimates).

Partial corrs

Click the Partial corrs button to create a spreadsheet with various collinearity statistics, as well as the partial and semi-partial correlations (and related statistics) between the predictor variables (columns in the design matrix) and the dependent (response) variables. Note that matrices of partial and semi-partial correlations among dependent variables (controlling for the effects currently in the model) can be reviewed via options DV partial corr (error correlations) and DV semi-partl (error correlations) in the Between effects group box.

Note: Collinearity statistics may possibly be omitted from this spreadsheet if the matrix inversion routine detects numerical round-off problems according to the precision specified by the SDELTA parameter (Sweep Delta and Inverse Delta for matrix inversion and determining estimable functions) on the analysis dialog. To attempt to display the statistics when omitted, try inputting larger values for the DELTA parameters and rerun the analysis. Be aware, though, that you will need to carefully review your results for consistency, as they may be subject to round-off errors.

Est. functions.

Click the Est. functions button to create a spreadsheet with the estimable functions for all between effects in the model, and for the current method of computing sums of squares (as specified via the Sums of squares group box on the Quick Specs Dialog - Options tab, or via the SSTYPE keyword in the GLM Analysis Syntax Editor). The estimable functions are used for computing the sums of squares for the effects in the ANOVA (MANOVA) table, and are discussed in the Introductory Overview section (for detailed discussion see also Milliken and Johnson, 1992, and Searle, 1987). In short, the estimable functions specify the linear combinations of the parameter estimates (Coefficients, see Summary results for between effects in GLM and ANOVA) that are tested (against zero; i.e., the hypothesis being tested is Lb = 0, where L is a matrix of estimable function parameters for a particular effect, and b is the respective solution matrix of Coefficients, for the current model).

Note: overparameterized model and estimable functions. The estimable functions are of particular interest in the context of the overparameterized model (clear the Sigma-restricted check box on the Quick Specs Dialog - Options tab, or use keyword PARAM = OVERP in the GLM Analysis Syntax Editor), when the design includes categorical predictor variables and missing cells. In that case, the tests of the main effects and interactions may not be unique, i.e., different orderings of the levels of the factors or the factors themselves may produce different results (sums of squares for the respective effects). In that case, the estimable functions identify unambiguously the specific hypotheses (linear combinations of parameter estimates) that are being tested for each effect in the ANOVA (MANOVA) table.

Note: sigma-restricted parameterization and estimable functions. If sigma-restricted coding is used for the categorical factors in the design (select the Sigma-restricted check box on the Quick Specs Dialog - Options tab, or use keyword PARAM = SIGMA in the GLM Analysis Syntax Editor), the X'X matrix (where X is the design matrix) will usually be of full rank, and hence, the matrix of estimable functions will be a diagonal matrix. In other words, if sigma-restricted parameterization is used, the Coefficients (see Summary results for between effects in GLM and ANOVA) pertain to differences between factor levels, and thus, testing the respective Coefficients themselves against zero usually provides appropriate tests of main effects and interactions for the categorical factors in the design.

Note: continuous predictor variables. Like in the sigma-restricted case (see previous paragraphs), for continuous predictor variables, testing the respective Coefficients themselves against zero provides appropriate tests for the respective continuous predictors. Therefore, the estimable functions for continuous predictor variables usually only contain a 1 for the respective predictor columns, and 0s otherwise.

General form

Click the General form button to create a spreadsheet of the general form of the estimable functions; these are computed as the product of the g2 inverse of the X'X matrix and X'X (where X is the design matrix); in the sigma-restricted case, when the X'X matrix is usually of full rank, the resultant matrix (general form estimable functions) will be an identity matrix.

Note: estimable functions. For details concerning estimable functions, see option Est. functions above or the Introductory Overview; see also Milliken and Johnson (1992), or Searle (1987).

Raw SSCP

Click the Raw SSCP button to create a spreadsheet with the sums of squares and cross-product matrix for all effects and dependent variables in the current model. Unlike in option Dev. SSCP (see below), the numbers in this matrix are computed as the sums of squares and cross-products from 0 (zero), and not from the respective means.

Raw SSCP inv

Click the Raw SSCP inv button to create a spreadsheet with the (partial) inverse of the raw sums of squares and cross-product matrix for all effects and dependent variables in the current model (see Raw SSCP, above); specifically, the spreadsheet will show the inverse for the X'X portion of the matrix (where X is the design matrix), the solution vectors for the X'Y portion of the matrix (where Y is the matrix of observations for the dependent variables), and the residual sums of squares and products matrix for the Y'Y portion of the complete Raw SSCP matrix.

Dev. SSCP.

Click the Dev. SSCP button to create a spreadsheet with the deviation sums of squares and cross-product matrix for all effects and dependent variables in the current model; unlike in option Raw SSCP (see above), the numbers in this matrix are computed as the sums of squares and cross-products from the respective column (variable) means.

Dev. SSCP inverse.

Click the Dev. SSCP inv button to create a spreadsheet with the (partial) inverse of the deviation sums of squares and cross-product matrix for all effects and dependent variables in the current model (see Dev. SSCP, above); specifically, the spreadsheet will show the inverse for the X'X portion of the matrix (where X is the design matrix), the solution vectors for the X'Y portion of the matrix (where Y is the matrix of observations for the dependent variables), and the residual sums of squares and products matrix for the Y'Y portion of the complete Dev. SSCP matrix.

Covariances

Click the Covariances button to create a spreadsheet with the variances and covariances for all effects and dependent variables in the current model; these values are the deviation sums of squares and cross-products (see option Dev. SSCP, above), divided by the number of valid observations minus 1.

Note: weights. If weights were specified for the current analysis, the divisor that is used to compute the variances and covariances from the deviation sums of squares and cross-products (see option Dev. SSCP, above) depends on the setting of the Weighted moments check box available on the GLM Startup Panel. If Weighted moments is not selected, the values found in the weight variable will be treated as case multipliers, and the denominator will be computed accordingly. If Weighted moments is selected and the DF = W-1 option button is selected, the denominator is computed as the sum of the weights minus 1. If the DF = N-1 option button is selected, the denominator is computed as the number of valid cases minus 1.

Covariances inverse

Click the Covariances inverse button to create a spreadsheet with the (partial) inverse of the Covariances (see above) for all effects and dependent variables in the current model; specifically, the spreadsheet will show the inverse for the X'X portion of the matrix (where X is the design matrix), the solution vectors for the X'Y portion of the matrix (where Y is the matrix of observations for the dependent variables), and the residual variances and covariances for the Y'Y portion of the complete matrix of Covariances.

Correlations

Click the Correlations button to create a spreadsheet with the standardized variances and covariances (i.e., correlations) for all effects and dependent variables in the current model.

Correlations inv

Click the Correlations inv button to create a spreadsheet with the (partial) inverse of the Correlations (see above) for all effects and dependent variables in the current model; specifically, the spreadsheet will show the inverse for the X'X portion of the matrix (where X is the design matrix), the solution vectors for the X'Y portion of the matrix (where Y is the matrix of observations for the dependent variables), and the residual correlations for the Y'Y portion of the complete matrix of Correlations.

See also GLM - Index.