GDA Models Results - Matrices Tab

Select the Matrices tab of the GDA Models Results dialog to access options to review various matrices involved in the computations of the main results, as well as detailed collinearity statistics and partial and semi-partial (or part) correlations.

Between design

The options under Between design are used to review various matrices computed for the between design.

Design terms

Click the Design terms button to display a spreadsheet of all the labels for each column in the design matrix (see GLM Introductory Overview). This spreadsheet is useful in conjunction with the Coefficients  button (available on the Effects tab) 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. Since the categorical predictor variables were coded according to the sigma-restricted parameterization, this spreadsheet will show the two levels of the respective factors that were contrasted in each column of the design matrix.

Partial corrs (Partial correlations)

Click the Partial corrs button to display 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 corrs) and DV semi-parl (error corrs) in the Between effects group box (see below).

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 (Estimable functions).

Click the Est. functions button to display 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 option Sums of squares on the GDA Quick Specs Dialog - Advanced tab. The estimable functions are used for computing the sums of squares for the effects in the ANOVA (MANOVA) table, and are discussed in the GLM Introductory Overview (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 the Effects tab) 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: sigma-restricted parameterization and estimable functions. Since sigma-restricted coding is used for the categorical predictor variables in the design, 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, the (Coefficients - see the Effects tab) 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 predictor variables in the design.

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

General form (General form estim. functions).

Click the General form button to display 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).

Raw SSCP

Click the Raw SSCP button to display a spreadsheet with the sums of squares and cross-product matrix for all effects and dependent variable levels 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 (Inverse)

Click the Raw SSCP inv button to display a spreadsheet with the (partial) inverse of the raw sums of squares and cross-product matrix for all effects and dependent variable levels in the current model (see Raw SSCP); 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 (Deviation SSCP).

Click the Dev. SSCP button to display 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, the numbers in this matrix are computed as the sums of squares and cross-products from the respective column (variable) means.

Dev. SSCP inv (Inverse).

Click the Dev. SSCP inv button to display 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); 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 Deviation SSCP matrix.

Covariances

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

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

Covariances inv (Inverse)

Click the Covariances inv button to display a spreadsheet with the (partial) inverse of the covariances for all effects and dependent variable levels 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 display a spreadsheet with the standardized variances and covariances (i.e., correlations) for all effects and dependent variable levels in the current model.

Correlation inv (Inverse)

Click the Correlation inv button to display a spreadsheet with the (partial) inverse of the correlations for all effects and dependent variable levels 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.

Between effects

The options under Between effects are used to review the sums of squares and cross-product matrices and derived matrices for the between effects in the design.

Error SS

Click the Error SS {Error SSCP} button to display a spreadsheet with the residual sums of squares and cross-product matrix; this matrix is used as the error term in the ANOVA (MANOVA) tables.

Cov (Covariances)

Click the Cov {Error Covariances} button to display a spreadsheet with the between effects residual variances and covariances; the values in this matrix are computed by dividing the Error SSCP values (residual sums of squares and cross products) by the number of valid cases minus 1. See option Covariances (above) for a description of the computations when Weights are in use.

DV partial corr (DV partial correlations)

Click the DV partial corr button to display a spreadsheet with the between effects residual correlations; the values in this matrix are computed by standardizing the values in the Error SSCP matrix. This is also the matrix of partial correlation among the dependent (response) variables. Partial and semi-partial correlations of predictor variables (columns in the design matrix) with the dependent (response) variables can be computed via the Partial corrs option in the Between design group box (see above).

DV semi-partl (DV semi-partial corrs)

Click the DV semi-partl button to display a spreadsheet with the matrix of semi-partial correlations between the dependent (response) variables, controlling for all predictor effects in the current model. Partial and semi-partial correlations of predictor variables (columns in the design matrix) with the dependent (response) variables can be computed via the can be computed via the Partial corrs option in the Between design group box (see above).

Effect SSCPs

Click the Effect SSCPs button to display a spreadsheet with the sums of squares and cross-product matrices for the between effects. These matrices are used to evaluate the significance of the effects in the model (in the ANOVA/MANOVA tables).

Effect Covs (Covariances)

Click the Effect Covs button to display a spreadsheet with the covariances for the between effects (see also option Effect SSCPs above). See Covariances above for a description of the computations when Weights are in use.

Effect Corrs (Correlations)

Click the Effect Corrs button to display a spreadsheet with the correlations (standardized covariances) for the between effects (see also option Effect SSCPs).

Show intercept

Select the Show intercept check box to include the effect for the intercept in the effect matrices (Effect SSCPs, Effect covs, Effect corrs, see above); clear this check box if those matrices are not of interest.