Two-way ANOVA Method Research Articles

Testing non-additivity (interaction) in two-way ANOVA tables with no replication

Testing for any significant interaction between two variables depends on the number of replicates in each cell of the two-way table and structure of the interaction. If there is interaction between two factors model of observations include interaction term and is called 'non-additive model' which makes interaction and non-additivity equivalent in terms of meaning. When there are several observations taken at each level combination of two variables, testing non-additivity can easily be done by usual two-way ANOVA method which cannot be used when there is only one observation per cell. For the cases with only one observation per cell, some methods have been developed starting with Tukey's one-degree-of-freedom test in which interaction is supposed to be the product of two factor's effects. There are other methods which are used for different structures of interaction when there is only one observation. In this paper, we review some of these tests. After presenting general methodology for the two-factor linear model with interaction effect and the general two-way ANOVA method when there are n > 1 observations per cell, we present some methods for testing non-additivity when there is only one observation per cell. Finally, we illustrate these methods on examples.

Postweaning growth of unselected Hereford and Angus cattle fed two different diets.

Two unselected herds of purebred Hereford and Angus cattle were created and their progeny evaluated during a 4-yr period (1964 to 1967) for 168-d postweaning gain when they were fed either a high- or medium-energy diet. Birth weight and 200-d adjusted weaning weight also were measured and the importance of sire x diet interactions for postweaning gain examined. Year effects were significant (P less than .001) for all traits in Herefords and for postweaning gain in Angus. Postweaning gain of both breeds increased in successive years, but no trend was observed for birth and 200-d weights. Bulls were heavier than heifers (P less than .05) for all three traits in both breeds. Hereford and Angus calves receiving the high-energy diet gained more (P less than .001) than their contemporaries fed the medium-energy diet. Sire differences were significant for birth weight in Herefords and for all three traits in Angus. Sire x diet interactions were not significant for postweaning gain in either breed. Genetic correlations were calculated by two methods: the two-way ANOVA approach using sire and sire x diet interaction variance components and the one-way ANOVA approach in which gains by progeny of each sire on each diet were considered to be two distinct traits. The genetic correlations for gain in Herefords could not be estimated by either method because of negative sire variance component estimates. The genetic correlations for gain in Angus were 1.08 for the two-way ANOVA method and 1.43 +/- .64 for the one-way ANOVA method. These results indicate that sires ranked the same based on progeny performance when fed either diet.

Enlargement of the Anterior Horn of the Lateral Ventricle in Schizophrenic Patients: Chronological and Morphometrical Studies

The anterior horn and lateral ventricular sizes of the brain CT were selected for measurement and comparison between 47 schizophrenic patients and 48 neurotic cases, which constituted the control subjects. The ventricular brain ratio (VBR) and the linear ratio (LR 1-6) in multiple age groups were calculated, analyzed and compared using the Student's t test, the two-way ANOVA and Bonferroni's methods. It was found that the VBR of the anterior horn and modified bicaudate cerebroventricular index of the teenage schizophrenics were significantly greater than those of the teenage controls (p less than 0.01) and the ventricular sizes were not associated with the different stages of age except for the cases of the teenage group. These results support the hypothesis of previous investigators that the ventricular enlargement is present early in the course of schizophrenia and provide additional evidence that in teenage schizophrenic patients there is a tendency of the enlargement of the anterior horn of the lateral ventricle which may be related with a morphological vulnerability in the prefrontal cortex.