Chapter 12 – Analysis of Variance (ANOVA)

What is the primary goal of ANOVA?

To decide whether the mean differences observed among samples are large enough to conclude that the corresponding population means are different.

In ANOVA, what are the two possible interpretations of observed mean differences and between-treatments variance?

(1) The populations really do not differ and the sample differences are due to chance (sampling error).

(2) The populations have different means, producing real differences in the samples.

In ANOVA terminology, what is a "factor"?

The independent (or quasi-independent) variable that designates the groups being compared.

In ANOVA terminology, what are "levels"?

The individual conditions, treatments, or values that make up a factor.

Into which two components is total sample variability partitioned in ANOVA?

Between-treatments variance

Within-treatments variance.

State the general formula for the F-ratio.

(F = MSbetween / MSwithin).

What value of F is expected when the null hypothesis is true?

An F value close to 1.00.

How does increasing the differences between treatment means affect the F-ratio

(with within-treatment variance held constant)?

It increases the numerator, producing a larger F-ratio.

How does increasing within-treatment variance affect the F-ratio

(with between-treatment variance held constant)?

It increases the denominator, producing a smaller F-ratio.

Why does ANOVA use variances rather than raw mean differences?

Because variances accumulate the total squared deviation information from all scores, allowing comparisons among more than two means in a single calculation.

What does the symbol k represent in ANOVA notation?

The number of treatment conditions (levels).

What does the symbol n represent in ANOVA notation?

The number of scores in each treatment (sample) condition.

What does the symbol N represent in ANOVA notation?

The total number of scores in the entire study.

What does the symbol T stand for in ANOVA notation?

The total of the scores (ΣX) for an individual treatment condition.

What does the symbol G stand for in ANOVA notation?

The grand total of all scores in the study (Σ of all X).

Give the formula for the total degrees of freedom (df_total).

df_total = N – 1

Give the formula for the within-treatments degrees of freedom (df_within).

df_within = N – k

Denominator treatment

Give the formula for the between-treatments degrees of freedom (df_between).

df_between = k – 1

Numerator treatment

Formula of SS_total

\Sigma X^2 - \frac{G^2}{N} = SS_total

OR SS_between + SS_within = SS_total

Formula of SS_within

SS_within = ΣSSinside each treatment

Formula of SS_between

\Sigma \frac{T^2}{n} - \frac{G^2}{N} = SS_between

How is MS_between calculated in ANOVA?

MS_between = SS_between ÷ df_between (variance estimate).

How is MS_within calculated in ANOVA?

MS_within = SS_within ÷ df_within (variance estimate).

Why are F-ratios always positive?

Because they are ratios of variances, and variance cannot be negative.

Why do F-ratios tend to cluster around 1.00 when H₀ is true?

Because between- and within-treatment variances both estimate the same error variance when no treatment effect exists.

What measure of effect size is commonly reported with one-way ANOVA?

Eta-squared (η²) = SSbetween ÷ SStotal

List the three assumptions for an independent-measures one-way ANOVA.

(1) Independence of observations

(2) Normally distributed populations,

(3) Homogeneity of variance among populations.

Under what two conditions are post hoc tests performed after ANOVA?

(1) The overall null hypothesis has been rejected, and

(2) There are three or more treatment conditions (k ≥ 3).

Distinguish between planned and unplanned (post hoc) comparisons.

Planned comparisons are specified before data collection based on specific hypotheses

unplanned comparisons are exploratory pairwise tests conducted after viewing the data.

State Tukey’s Honestly Significant Difference (HSD) formula.

HSD = q × √(MS_within / n)

Which three pieces of information are needed to obtain the q value used in Tukey’s HSD?

The chosen α level, the number of treatments (k), and df_within.

Is a single-sample ANOVA meaningful in the same way as a single-sample t-test?

No; ANOVA requires at least two groups because it analyses variance among different treatment conditions.

Hypothesis of ANOVA

• H₀: μ₁ = μ₂ = μ₃ (All population means are equal)

• H₁: At least one population mean is different