One-Way ANOVA

Why do we use One-way ANOVA?

A one-way ANOVA (between-subjects analysis of variance) is used in research situations where the researcher wants to compare means on a quantitative Y outcome variable across two or more groups. Group membership is identified by each participant’s score on a categorical X predictor variable. ANOVA is a generalization of the t test; a t test provides information about the distance between the means on a quantitative outcome variable for just two groups, whereas a one-way ANOVA compares means on a quantitative variable across any number of groups. The categorical predictor variable in an ANOVA may represent either naturally occurring groups or groups formed by a researcher and then exposed to different interventions. When the means of naturally occurring groups are compared (e.g., a one-way ANOVA to compare mean scores on a self-report measure of political conservatism across groups based on religious affiliation), the design is nonexperimental. When the groups are formed by the researcher and the researcher administers a different type or amount of treatment to each group while controlling extraneous variables, the design is experimental.

What is the null hypothesis for ANOVA?

In ANOVA, the focus is on different types of variance inherent in a multigroup design, yet one-way ANOVA is very much an extension of the t test for independent groups. A null hypothesis will be tested that states there is no difference among a number of group means on a response variable. We will still obtain a single derived value and compare it with a distribution of values, the F distribution (named in honor of Fisher). If the derived F value exceeds the tabled critical value of F, then the null hypothesis will be rejected. The null hypothesis in

ANOVA is as follows:

H0 = μ1 = μ2 = ··· = μk

where

μ1 = the mean for Group 1 on the response variable,

μk = the last group’s mean.

Although we will always analyze sample means, notice that the null hypothesis states that the population means are not different from each other. Thus, if there is no real effect of the independent variable on the dependent variable, then it is as if each of these samples was taken from the same population. If the samples were all taken from the same population, then the sample means should not vary from each other based on reasons stated earlier in the book about the sampling distribution of means and the standard error of the mean. Only chance or random differences will differentiate among the group means.

What are the assumptions for ANOVA?

The assumptions of an ANOVA are similar to the independent t test. First, it is assumed that the dependent variable is drawn from a population of values that is normally distributed (also called the normality assumption). Second, it is assumed that the participants have been randomly assigned to each group and the scores in each group are independent of each other. Third, it is assumed that the variances about each group’s means are not substantially different from each other (also called the assumption of homogeneity of variance).

– Independent Groups

– Normality of the Dependent Variable

– Homogeneity of Variance

It is important to note, however, that ANOVA is a robust statistical test, and violations of the assumptions may still result in a correct statistical decision (to reject or retain the null hypothesis). Also, large samples (minimum of 10–15 participants per group) help minimize violations of the assumption of normality, and using an equal number of participants in each group helps minimize violations of the assumption of homogeneity of variance.