One-Way Between-Subjects ANOVA

When is a one-way between-subjects ANOVA used?

When comparing the means of three or more independent groups.

Why not run multiple t-tests instead of an ANOVA?

Multiple t-tests increase the risk of Type I error.

What does “one-way” mean in ANOVA?

There is one independent variable.

What does “between-subjects” mean?

Each participant is in only one condition.

What is the dependent variable in a one-way ANOVA?

A continuous outcome variable.

What is the central question of a one-way ANOVA?

Is between-group variability larger than within-group variability?

What is within-group variability?

Differences between individuals within the same group due to individual differences, measurement error, or noise.

What is between-group variability?

Differences between group means potentially caused by the independent variable.

If the independent variable has no effect, what should happen?

Between-group variability should be similar to within-group variability.

If the independent variable has an effect, what should happen?

Between-group variability should be greater than within-group variability.

What is the F-statistic?

A ratio comparing variability explained by the model to unexplained variability.

What does a large F-value indicate?

The model explains much more variance than error, suggesting a significant effect.

What is the grand mean?

The overall mean across all participants and groups.

What is Total Sum of Squares (SST)?

The total variability in the data around the grand mean.

How is SST calculated?

Square the difference between each score and the grand mean, then sum them.

What is Model Sum of Squares (SSM)?

The variability explained by group differences.

How is SSM calculated?

Square the difference between each group mean and the grand mean, multiply by group size, then sum.

What is Residual Sum of Squares (SSR)?

The variability not explained by the model (error).

How is SSR calculated?

Square the difference between each score and its group mean, then sum.

What is the relationship between SST, SSM, and SSR?

SST = SSM + SSR.

Why can’t we compare sums of squares directly?

They depend on the number of observations.

What is a mean square (MS)?

The average sum of squares (SS divided by degrees of freedom).

How is mean square calculated?

MS = SS / df.

What is MSM?

Model sum of squares divided by model degrees of freedom.

What is MSR?

Residual sum of squares divided by residual degrees of freedom.

What is the formula for the F-statistic?

F = MSM / MSR.

What does the F-ratio represent conceptually?

The proportion of explained variance to unexplained variance.

What are degrees of freedom?

The number of independent pieces of information used to estimate variability.

What is the degrees of freedom for the model?

dfM = number of groups − 1.

What is the degrees of freedom for residual/error?

dfR = total sample size − number of groups.

What does a significant F-test tell us?

At least one group mean differs from another.

What does a significant F-test NOT tell us?

Which specific groups differ.

Why are post-hoc tests needed?

To determine which specific group comparisons are significant.

What are planned comparisons?

Hypothesis-driven comparisons specified before collecting data.

What are post-hoc tests?

Exploratory comparisons decided after a significant ANOVA.

When should planned comparisons be used?

When there are specific theoretical predictions.

What is the risk of using LSD (Least Significant Difference)?

Higher risk of Type I error.

When is Bonferroni correction preferred?

When making multiple comparisons and wanting to reduce Type I error.

What is the trade-off of Bonferroni correction?

Reduced statistical power.

What does Levene’s test assess?

Homogeneity of variances across groups.

What does homogeneity of variance mean?

Groups have similar spreads of scores.

What does p > .05 in Levene’s test indicate?

Variances are equal (assumption met).

What happens if variances are unequal?

Some post-hoc tests may not be appropriate.

Why does unequal group size matter?

Large differences in group size can affect robustness of ANOVA.

Is ANOVA robust to small differences in group size if variances are equal?

Yes.

How should a one-way ANOVA be reported?

F(df between, df within) = value, p = value, effect size (η²).

What does η² (eta squared) represent?

The proportion of total variance explained by the independent variable.

If η² = .40, what does this mean?

40% of the variance in the dependent variable is explained by group differences.

What type of independent variable is required in one-way ANOVA?

A categorical variable with three or more levels.

What type of dependent variable is required?

A continuous variable.

What increases the F-statistic?

Large differences between group means and/or small within-group variability.

What decreases the F-statistic?

High within-group variability relative to between-group variability.

Why does running multiple t-tests inflate Type I error?

Each test increases the probability of falsely rejecting the null hypothesis.

What does it mean if F is close to 1?

Between-group and within-group variability are similar.

What is the null hypothesis in one-way ANOVA?

All group means are equal.

What is the alternative hypothesis in one-way ANOVA?

At least one group mean differs.