analysis of variance and covariance

Chapter 16

In ANOVA and ANCOVA, the dependent variable is metric in the independent variables are all categorical, or combinations of categorical and metric variables. One way ANOVA involves a single independent categorical variable. Interest lies in testing the null hypothesis that the category means are equal in the population. The total variation and the dependent variable is decomposed into two components: variation related to the independent variable and variation related to error. The variation is measured in terms of the sum of squares corrected for the mean (SS). The mean square is obtained by dividing the SS by the corresponding degrees of freedom (df). The null hypothesis of equal means is tested by an F statistic, which is the ratio of the mean square related to the independent variable to the mean square related to error.

N-way analysis of variance involves a simultaneous examination of two or more categorical independent variables. A major advantage is that the interactions between the independent variables can be examined. The significance of the overall effect, interaction terms, and main effects of individual factors are examined by appropriate F tests. It is meaningful to test the significance of main effects only of the corresponding interaction terms are not significant.

ANCOVA includes at least one categorical independent variable and at least one animal or metric independent variable. The metric independent variable, or covariate, is commonly used remove extraneous variation from the dependent variable.

When the analysis of variance is conducted on two or more factors, interactions can arise. And interaction occurs when the effective one independent variable on the dependent variable is different for different categories or levels of another independent variable. If the interaction is significant, it may be ordinal or disordinal. Disordinal an action may be of a non-crossover or crossover type. In balanced designs, the relative importance of factors in explaining the variation in the dependent variable is measured by omega squared. Multiple comparisons in the form of a priori or a posteriori contrasts can be used for examining differences among specific means.

And repeated measures analysis of variance, observations on each subject are obtained under each treatment condition. This design is useful for controlling the differences in subjects that exist prior to the experiment. Not metric analysis of variance involves examining the differences in the central tendencies of two or more groups when the dependent variable is measured on a ordinal scale. Multivariate analysis of variance (MANOVA) involves two or more metric dependent variables.

Analysis of variance (ANOVA) -- a statistical technique for examining the differences among means for two more populations
factors -- categorical independent variables. The independent variables must be all categorical (nonmetric) to use ANOVA
treatment -- in ANOVA, a particular combination of factor levels or categories
one-way analysis of variance -- an ANOVA technique in which there is only one factor
n-way analysis of variance -- an ANOVA model where two or more factors are involved
analysis of covariance (ANCOVA) -- an advanced analysis of variance procedure in which the effects of one or more metric scaled extraneous variables are renewed from the dependent variable before conducting the ANOVA
covariate -- and metric independent variable used in ANOVA
decomposition of the total variation -- in one-way ANOVA, separation of the variation observed in the dependent variable into the variation due to the independent variables plus the variation due to error
interaction -- when assessing the relationship between two variables, and interaction occurs if the effects of X1 depends on the level of X2, and vice versa
significance of the overall effect -- a test that some differences exist between some of the treatment groups
significance of interaction effect -- a test of the significance of the interaction between two or more independent variables
significance of the main effect -- a test of the significance of the main effect for each individual factor
ordinal interaction -- and interaction where the rank order of the effects attributable to one factor does not change across the levels of the second factor
disordinal interaction -- a change in the rank order of the effects of one factor across the levels of another
omega squared -- a measure indicating the proportion of the variation in the dependent variable explained by a particular independent variable or factor
contrasts -- in ANOVA, a method of examining differences among two or more means of the treatment groups
a priori contrasts -- contrasts that are determined before conducting the analysis, based on the researcher's theoretical framework
a postpriori contrasts -- contrast made after the analysis. These are generally multiple comparison tests
multiple comparison test -- a postpriori contrast that enable the researcher to construct generalized confidence intervals they can be used to make pairwise comparisons of all treatment means
repeated measures ANOVA -- an ANOVA technique used when respondents are exposed to more than one treatment condition and repeated measurements are obtained
nonmetric ANOVA -- an ANOVA technique for examining the difference in the central tendencies of more than two groups when the dependent variable is measured on an ordinal scale
k-sample median test -- non-parametric test that is used to examine differences among groups when the dependent variable is measured on an ordinal scale
Kruskal-Wallis one-way analysis of variance -- a nonmetric ANOVA test that uses the rank value of each case, not merely its location relative to the median
multivariate analysis of variance (MANOVA) -- an ANOVA technique using two or more metric dependent variables