# One-Way ANOVA

One-Way ANOVA: Theory And Logic

INTRODUCTION In Unit 9, we will study the theory and logic of analysis of variance (ANOVA). Recall that a t-test requires a predictor variable that is dichotomous (it has only two levels or groups). The advantage of ANOVA over a t-test is that the categorical predictor variable can have two or more groups. Just like a t-test, the outcome variable in ANOVA is continuous and requires the calculation of group means.

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Logic of a ‘One-Way’ ANOVAThe ANOVA, or F-test, relies on predictor variables referred to as factors. A factor is a categorical (nominal) predictor variable. The term ‘one-way’ is applied to an ANOVA with only one factor that is defined by two or more mutually exclusive groups. Technically, an ANOVA can be calculated with only two groups, but the t-test is usually used instead. Instead, the one-way ANOVA is usually calculated with three or more groups, which are often referred to as levels of the factor.If the ANOVA includes multiple factors, it is referred to as a factorial ANOVA. An ANOVA with two factors is referred to as a ‘two-way’ ANOVA; an ANOVA with three factors is referred to as a ‘three-way’ ANOVA, and so on.

Factorial ANOVA is studied in advanced inferential statistics. In this course, we will focus on the theory and logic of one-way ANOVA.ANOVA is one of the most popular statistics used in social sciences research. In non-experimental designs, the one-way ANOVA compares group means between naturally existing groups, such as political affiliation (Democrat, Independent, Republican). In experimental designs, the one-way ANOVA compares group means for participants randomly assigned to different treatment conditions (for example, high caffeine dose; low caffeine dose; control group).

Avoiding Inflated Type I ErrorYou may wonder why a one-way ANOVA is necessary. For example, if a factor has four groups ( k = 4), why not just run independent sample t-tests for all pairwise comparisons (for example, Group A versus Group B, Group A versus Group C, Group B versus Group C, et cetera)? Warner (2013) points out that a factor with four groups involves six pairwise comparisons. The issue is that conducting multiple pairwise comparisons with the same data leads to the inflated risk of a Type I error (incorrectly rejecting a true null hypothesis—getting a false positive).

The ANOVA protects the researcher from inflated Type I error by calculating a single omnibus test that assumes all k population means are equal. Although the advantage of the omnibus test is that it helps protect researchers from inflated Type I error, the limitation is that a significant omnibus test does not specify exactly which group means differ, just that there is a difference ‘somewhere’ among the group means. A researcher, therefore, relies on either (a) planned contrasts of specific pairwise comparisons determined prior to running the F test or (b) follow-up tests of pairwise comparisons, also referred to as post-hoc tests, to determine exactly which pairwise comparisons are significant. Hypothesis Testing in a One-Way ANOVA

The null hypothesis of the omnibus test is that all k (group) population means are equal, or H0: µ1 = µ2 = … µk. By contrast, the alternative hypothesis is usually articulated by stipulating that ‘at least one’ pairwise comparison of population means is unequal. Keep in mind that this prediction does not imply that all groups must significantly differ from one another on the outcome variable. Assumptions of One-Way ANOVAThe assumptions of ANOVA reflect the assumptions of the t-test. ANOVA assumes independence of observations. ANOVA assumes that outcome variable Y is normally distributed.

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