Produces summary statistics for a regression model given the information from the fit.

The type double function is imsls_d_regression_summary.

IMSLS_ANOVA_TABLE, float **anova_table (Output)

Function imsls_f_regression_summary computes summary statistics from a fitted general linear model. The model is y = Xβ + ɛ, where y is the n × 1 vector of responses, X is the n × p matrix of regressors, β is the p × 1 vector of regression coefficients, and ɛ is the n × 1 vector of errors whose elements are each independently distributed with mean 0 and variance σ2. Function regression can be used to compute the fit of the model. Next, imsls_f_regression_summary uses the results of this fit to compute summary statistics, including analysis of variance, sequential sum of squares, t tests, and an estimated variance-covariance matrix of the estimated regression coefficients.

Some generalizations of the general linear model are allowed. If the i‑th element of ɛ has variance of

and the weights wi are used in the fit of the model, imsls_f_regression_summary produces summary statistics from the weighted least-squares fit. More generally, if the variance-covariance matrix of ɛ is σ2V, imsls_f_regression_summary can be used to produce summary statistics from the generalized least-squares fit. Function regression can be used to perform a generalized least-squares fit, by regressing y* on X* where y* = (T -1)Ty, X* = (T-1)TX and T satisfies TTT = V.

The sequential sum of squares for the i‑th regression parameter is given by

The regression sum of squares is given by the sum of the sequential sums of squares. If an intercept is in the model, the regression sum of squares is adjusted for the mean, i.e.,

is not included in the sum.

The estimate of σ2 is s2 (stored in anova_table[7]) that is computed as SSE/DFE.

If R is nonsingular, the estimated variance-covariance matrix of

(stored in coef_covariances) is computed by s2R-1(R-1)T.

If R is singular, corresponding to rank(X) < p, a generalized inverse is used. For a matrix G to be a gi (i = 1, 2, 3, or 4) inverse of a matrix A, G must satisfy conditions j (for j ≤ i) for the Moore-Penrose inverse but generally must fail conditions k (for k > i). The four conditions for G to be a Moore-Penrose inverse of A are as follows:

In the case where R is singular, the method for obtaining coef_covariances follows the discussion of Maindonald (1984, pp. 101–103). Let Z be the diagonal matrix with diagonal elements defined by the following:

Let G be the solution to RG = Z obtained by setting the i‑th ({i : rii = 0}) row of G to 0. Argument coef_covariances is set to s2GGT. (G is a g3 inverse of R, represented by,

the result

is a symmetric g2 inverse of RTR = XTX. See Sallas and Lionti 1988.)

Note that argument coef_covariances can be used only to get variances and covariances of estimable functions of the regression coefficients, i.e., nonestimable functions (linear combinations of the regression coefficients not in the space spanned by the nonzero rows of R) must not be used. See, for example, Maindonald (1984, pp. 166–168) for a discussion of estimable functions.

The estimated standard errors of the estimated regression coefficients (stored in Column 1 of coef_t_tests) are computed as square roots of the corresponding diagonal entries in coef_covariances.

When R is nonsingular and comes from an unrestricted regression fit, coef_covariances is the estimated variance-covariance matrix of the estimated regression coefficients, and coef_covariances = (SSE/DFE) (RTR). Otherwise, variances and covariances of estimable functions of the regression coefficients can be obtained using coef_covariances, and coef_covariances = (SSE/DFE) (GDGT). Here, D is the diagonal matrix with diagonal elements equal to 0 if the corresponding rows of R are restrictions and with diagonal elements equal to 1 otherwise. Also, G is a particular generalized inverse of R.