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Master ANOVA for Effective Statistical Analysis

By Isla Nguyen

When comparing the means of two or more samples regarding a variable of interest, statistical tests determine whether significant differences exist. Analysis of Variance (ANOVA) is one such parametric test. This article details ANOVA’s definition, function, types, and the critical assumptions required for its proper use.

What is Analysis of Variance (ANOVA)?

In statistics, Analysis of Variance (ANOVA) is a collection of statistical models and associated procedures where the observed variance in a particular variable is partitioned into components due to various explanatory variables. The acronym ANOVA stands for ANalysis Of VAriance.

ANOVA is classified as a parametric test. This means specific assumptions must be met to apply it, and the dependent variable must be quantitative (at least interval-level, such as IQ scores).

History and Origins of ANOVA

The initial techniques for analysis of variance were developed in the 1920s and 1930s by R.A. Fisher, a renowned statistician and geneticist. Consequently, ANOVA is sometimes referred to as “Fisher’s ANOVA” or “Fisher’s analysis of variance,” partly due to its use of Fisher’s F-distribution in hypothesis testing.

ANOVA originates from the concepts of linear regression. In statistics, linear regression is a mathematical model used to approximate the dependent relationship between a dependent variable (Y), independent variables (Xi), and a random error term.

Function of This Parametric Test

An ANOVA test determines whether different treatments (e.g., psychological interventions) exhibit significant differences in their effects, or if their population means are effectively equal, indicating no significant difference.

Specifically, ANOVA is used to test hypotheses about differences between means when comparing more than two groups. It involves analyzing and decomposing the total variability, which is primarily attributed to two sources of variation:

  • Inter-group variability: Differences in means between the groups being compared.
  • Intra-group variability (or error): Variability within each group, not explained by the independent variable.

Types of ANOVA

There are two main types of Analysis of Variance:

ANOVA I (One-Way ANOVA)

This type is applied when there is only one classification criterion or independent variable (e.g., different types of therapeutic techniques). It can be:

  • Between-subjects: Involves multiple distinct experimental groups.
  • Within-subjects (Repeated Measures): Involves a single experimental group measured multiple times.

ANOVA II (Factorial ANOVA)

This type is used when there is more than one classification criterion or independent variable. Similar to ANOVA I, it can be:

  • Between-subjects: Involves multiple distinct experimental groups for each factor combination.
  • Within-subjects (Repeated Measures): Involves a single group measured across multiple conditions for each factor.

Characteristics and Assumptions

In experimental studies utilizing ANOVA, each group may consist of a differing number of subjects. When the number of subjects is equal across all groups, the design is referred to as a balanced model.

To validly apply ANOVA, several statistical assumptions must be met:

1. Normality

The scores on the dependent variable (e.g., anxiety levels) must follow a normal distribution within each group. This assumption is typically assessed using goodness-of-fit tests.

2. Independence

This implies no autocorrelation between the scores, meaning each observation is independent of the others. To ensure this, a simple random sampling (SRS) method should be employed when selecting the study sample.

3. Homoscedasticity

This term signifies the equality of variances across the sub-populations (groups). Variance is a measure of variability and dispersion, increasing with greater spread in scores. The assumption of homoscedasticity is tested using Levene’s Test or Bartlett’s Test. If violated, a logarithmic transformation of the scores may be considered as an alternative.

Additional Assumptions for Within-Subjects ANOVA

While the above assumptions apply to between-subjects ANOVA, an ANOVA with repeated measures (within-subjects design) requires two additional assumptions:

1. Sphericity

This assumption implies that the variances of the differences between all possible pairs of within-subject conditions are equal. Violation of sphericity suggests that different sources of error correlate with each other. A potential solution if this assumption is not met is to use Multivariate Analysis of Variance (MANOVA).

2. Additivity

This assumption posits no interaction between the subject and the treatment. A violation of additivity would inflate the error variance, potentially leading to incorrect conclusions.