Analyzes a one-way classification model with covariates.

The type double function is imsls_d_ancovar.

Pointer to an array of length 15 containing the one-way analysis of covariance assuming parallelism, organized as follows:

#include <imsls.h>

float *imsls_f_ancovar (int ngroup, int ncov, int ni[], float y[], float x[],

Function imsls_f_ancovar performs analyses for models that combine the features of a one-way analysis of variance model with that of a multiple linear regression model. The basic one-way analysis of covariance model is

In practice, sometimes the regression functions are not parallel. In addition to estimates for the model assuming parallelism, imsls_f_ancovar computes estimates and summary statistics for the separate regressions for each group. These estimates can be examined using the optional arguments IMSLS_COEF_TABLES and IMSLS_REG_ANOVA.

Estimates for the β0i’s and β1, β2, …, βm in the model assuming parallelism are returned using the optional argument IMSLS_COEF. Summary statistics are also computed for this model.

The adjusted group means, stored in the last column of xymean, are computed using the formula:

The estimated covariance between the i1-th and i2-th adjusted group mean is given by

where vpq is the entry in covb[(p - 1)(ngroup + ncov) + q -1] and is the estimated covariance between the p‑th and q‑th estimated coefficients in the regression function.

A discussion of formulas and interpretations for the one-way analysis of covariance problem appears in most elementary statistics texts, e.g., Snedecor and Cochran (1967, Chapter 14).

This example fits a one-way analysis of covariance model assuming parallelism using data discussed by Snedecor and Cochran (Table 14.6.1, pages 432−436). The responses are concentrations of cholesterol (in mg/100 ml) in the blood of two groups of women: women from Iowa and women from Nebraska. Age of a woman is the single covariate. The cholesterol concentrations and ages of the women according to state are shown in the following table. (There are 11 Iowa women and 19 Nebraska women in the study. Only the first 5 women from each state are shown here.)

There is no evidence from the data to indicate that the regression lines for cholesterol concentration as a function of age are not parallel for Iowa and Nebraska women (p-value is 0.5425). The parallel line model suggests that Nebraska women may have higher cholesterol concentrations than Iowa women. The cholesterol concentrations (adjusted for age) are 195.5 for Iowa women versus 224.2 for Nebraska women. The difference is 28.7 with an estimated standard error of

This example fits a one-way analysis of covariance model and performs a test for parallelism using data discussed by Snedecor and Cochran (1967, Table 14.8.1, pages 438-443). The responses are weight gains (in pounds per day) of 40 pigs for four groups of pigs under varying treatments. Two covariates-initial age (in days) and initial weight (in pounds) are used. For each treatment, there are 10 pigs. Only the first five pigs from each treatment are shown here.

For these data, the test for non-parallelism is not statistically significant (p = 0.901). The one-way analysis of covariance test for the treatment means adjusted for the covariates, assuming parallel slopes, is statistically significant at a stated significance level of α = 0.05, (p = 0.04931).

Multiple comparisons can be done using the least significant difference approach of comparing each pair of treatment groups with the two-sample student-t test. Since the adjusted means in the one-way analysis of covariance are correlated, the standard error for these comparisons must be computed using the variances and covariances in covm. The standard errors for these comparisons are fairly similar ranging from 0.0630 to 0.0638. The Student’s t comparisons identify differences between groups 1 and 2, and 1 and 4 as being statistically significant with p-values of 0.01225 and 0.03854 respectively.