Analyzes a balanced complete experimental design for a fixed, random, or mixed model.

The type double function is imsls_d_anova_balanced.

The p-value for the F-statistic.

Function imsls_f_anova_balanced analyzes a balanced complete experimental design for a fixed, random, or mixed model. The analysis includes an analysis of variance table, and computation of subgroup means and variance component estimates. A choice of two parameterizations of the variance components for the model can be made.

Scheffé (1959, pages 274-289) discusses the parameterization for model = 1. For example, consider the following model equation with fixed factor A and random factor B:

The fixed effects αi’s are subject to the restriction

the bj’s are random effects identically and independently distributed

cij are interaction effects each distributed

and are subject to the restrictions

and the eijk’s are errors identically and independently distributed N(0, σ2). In general, interactions of fixed and random factors have sums over subscripts corresponding to fixed factors equal to zero. Also in general, the variance of a random interaction effect is the associated variance component times a product of ratios for each fixed factor in the random interaction term. Each ratio depends on the number of levels in the fixed factor. In the earlier example, the random interaction AB has the ratio (a-1)/a as a multiplier of

and

In a three-way crossed classification model, an ABC interaction effect with A fixed, B random, and C fixed would have variance

Searle (1971, pages 400−401) discusses the parameterization for model = 0. This parameterization does not have the summation restrictions on the effects corresponding to interactions of fixed and random factors. Also, the variance of each random interaction term is the associated variance component, i.e., without the multiplier. This parameterization is also used with unbalanced data, which is one reason for its popularity with balanced data also. In the earlier example,

Searle (1971, pages 400−404) compares these two parameterizations. Hocking (1973) considers these different parameterizations and concludes they are equivalent because they yield the same variance-covariance structure for the responses. Differences in covariances for individual terms, differences in expected mean square coefficients and differences in F tests are just a consequence of the definition of the individual terms in the model and are not caused by any fundamental differences in the models. For the earlier two-way model, Hocking states that the relations between the two parameterizations of the variance components are

where

are the variance components in the parameterization with model = 0.

The computations for degrees of freedom and sums of squares are the same regardless of the option specified by model. imsls_f_anova_balanced first computes degrees of freedom and sum of squares for a full factorial design. Degrees of freedom for effects in the factorial design that are missing from the specified model are pooled into the model effect containing the fewest subscripts but still containing the factorial effect. If no such model effect exists, the factorial effect is pooled into error. If more than one such effect exists, a terminal error message is issued indicating a misspecified model.

The analysis of variance method is used for estimating the variance components. This method solves a linear system in which the mean squares are set to the expected mean squares. A problem that Hocking (1985, pages 324−330) discusses is that this method can yield a negative variance component estimate. Hocking suggests a diagnostic procedure for locating the cause of the negative estimate. It may be necessary to re-examine the assumptions of the model.

The percentage of variation explained by each random effect is computed (output in variance_components column 6) as the variance of the associated random effect divided by the variance of y. The two parameterizations can lead to different values because of the different definitions of the individual terms in the model. For example, the percentage associated with the AB interaction term in the earlier two-way mixed model is computed for model = 1 using the formula

while for the parameterization model = 0, the percentage is computed using the formula

In each case, the variance components are replaced by their estimates (stored in variance_components column 5).

Confidence intervals on the variance components are computed using the method discussed by Graybill (1976, Theorem 15.3.5, page 624, and Note 4, page 620).

An analysis of a generalized randomized block design is performed using data discussed by Kirk (1982, Table 6.10-1, pages 293−297). The model is

where yijk is the response for the k-th experimental unit in block j with treatment i; the αi’s are the treatment effects and are subject to the restriction

the bj’s are block effects identically and independently distributed

cij are interaction effects each distributed

and are subject to the restrictions

and the eijk’s are errors, identically and independently distributed N(0, σ2). The interaction effects are assumed to be distributed independently of the errors.

The data are given in the following table: