Magazine article National Association of School Psychologists. Communique

What School Psychologists Need to Know about Analysis of Variance

Magazine article National Association of School Psychologists. Communique

What School Psychologists Need to Know about Analysis of Variance

Article excerpt

One of the most common questions asked within school psychology research concerns the relationship between an intervention (independent variable) and an outcome (dependent variable). For example, say a researcher is interested in the effect of a cognitive- behavioral intervention (independent variable) on student depression (dependent variable). They evaluate this relationship by comparing depression levels across three groups, including those who received the intervention in a one-on-one format, those who received it in a small-group format, and those who received "treatment-as-usual" with no counseling. A second researcher wants to determine whether a Tier 2 small-group reading intervention (independent variable) demonstrates a positive effect on student reading comprehension (dependent variable). They evaluate this relationship with a single group of students by comparing preintervention comprehension levels to postintervention comprehension levels.

For several decades, the most common approach to evaluating these causal questions has been through analysis of variance (ANOVA). ANOVA represents a family of analyses that might be used to determine if mean scores on some dependent variable differ across multiple time points and/or multiple groups. Different ANOVA-related statistics can be calculated in evaluating the extent of this difference in terms of both statistical and practical significance.

WHEN ANOVA SHOULD BE USED

ANOVA is rather flexible in that it can accommodate a variety of research designs and corresponding research questions. Below is a brief description of six common ANOVA models, and Table 1 includes a summary of those models.

One-way ANOVA. One-way ANOVA is used to compare the dependent variable of three or more levels, usually groups, of a single independent variable at a single time point. A one-way ANOVA can also be used to compare only two levels, with results being equivalent to an independent samples t-test.

Multi-way ANOVA. A multi-way ANOVA (e.g., two-way ANOVA or three-way ANOVA) is used to evaluate the effect of two or more independent variables, each of which has two or more levels, on a dependent variable measured at a single time point.

Repeated measures ANOVA. Repeated measures ANOVA is used to evaluate the effect of an independent variable with a single level on a dependent variable measured at two or more time points. In this scenario, time (also referred to as a within-group factor) is of particular interest, with results indicating the extent to which scores have changed across time points.

Mixed factorial ANOVA. Mixed factorial ANOVA is used to examine the effect of one or more independent variables, each of which has two or more levels, on a dependent variable measured at two or more time points.

Multivariate ANOVA. A multivariate ANOVA, which is commonly abbreviated as MANOVA, is used when evaluating two or more dependent variables at the same time. MANOVA can include any of four prior models, thus permitting simultaneous examination of multiple independent and dependent variables.

Analysis of covariance. Analysis of covariance, or ANCOVA, is an extension of any of the above ANOVA models that also includes additional covariates to remove bias in the dependent variable(s). Including covariates in the model increases the accuracy of conclusions regarding the independent variable. Covariates are those variables that predict the dependent variable despite not being related to the independent variable. By accounting for covariates, one is able to reduce within-group variance (also referred to as error variance), thus enhancing the ability to identify meaningful differences between groups on the dependent variable(s). When ANCOVA includes multiple dependent variables, it is normally referred to as MANCOVA.

KEY TERMS

There are several terms commonly used when describing ANOVA procedures. Some of the most important terms are described below, but readers are referred to Thompson (2006) for a more thorough review of ANOVA and the terms below. …

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