Analyzes a one-way classification model with covariates.

Routine AONEC 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

where the observed value of yij constitutes the j-th response in the i-th group, β0 ι denotes the y intercept for the regression function for the i-th group, β1, β2, …, βm are the regression coefficients for the covariates, and the ɛ ij’s are independently distributed normal errors with mean zero and variance σ2. This model allows the regression function for each group to have different intercepts. However, the remaining m regression coefficients are the same for each group, i.e., the regression functions are parallel. Often in practice, the regression functions are not parallel. In addition to estimates for the model assuming parallelism, AONEC computes estimates and summary statistics for the separate regressions for each group. With IPRINT = 2, the estimates and summary statistics for each group are printed. If ITEST = 1, a test for parallelism is performed.

AONEC requires (xij1, xij2, …, xijk, yij) as input into a single data matrix XY with the data for each group occupying contiguous rows of XY.

Estimates for the β0i’s and β1, β2, …, βm in the model assuming parallelism are computed and stored in COEF. Summary statistics are also computed for this model. The adjusted group means (stored in column m + 3 of XYMEAN) are given by

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

where vpq is the pq-th entry in COVB and is the estimated covariance between the p-th and q-th estimated coefficients in the regression function.

The design of AONEC can be used with routines described in Chapter 2, “Regression.” For example, confidence intervals and diagnostics for the individual cases can be computed by using the output matrices R and COEF as input into regression routines for case analysis.

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 4 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 5 pigs from each treatment are shown here.