Introduction to Analysis of Variance (ANOVA)

These flashcards cover fundamental concepts, formulas, hypotheses, test statistics, variance components, degrees of freedom, F distribution, and post hoc procedures associated with one-way and factorial ANOVA.

What does ANOVA stand for and what is its primary purpose?

Analysis of Variance; a hypothesis-testing procedure used to evaluate mean differences among two or more treatments or populations.

How many independent variables (factors) are involved in a single-factor ANOVA?

One independent variable (factor) with two or more levels.

When an experiment includes more than one factor, what design is used instead of single-factor ANOVA?

A two-factor or factorial design.

Give an example of a 2×2 factorial design from the lecture.

Factor 1: Anxiety (Low vs. High) × Factor 2: Audience (With vs. Without), producing four treatment conditions.

State the null hypothesis (H0) for a study comparing learning across three temperatures (15 °C, 24 °C, 34 °C).

H0: μ1 = μ2 = μ3 (room temperature has no effect on learning performance).

What is the general alternative hypothesis (H1) in ANOVA?

At least one population mean differs from another.

What test statistic is used in ANOVA and on what is it based?

The F-ratio, which is based on variances rather than mean differences.

Write the formula for the F-ratio in words.

F = Variance between treatments / Variance within treatments (error).

If the treatment effect is zero, the expected value of F is approximately ____.

1.00

What does ‘between-treatments variance’ measure?

Differences among sample means, which may be due to treatment effects plus chance.

What does ‘within-treatments variance’ measure?

Variability inside each treatment condition, reflecting only chance (error) differences.

List the symbols k, n, N, T, G, SS, M, and MS used in ANOVA.

k = number of treatments; n = scores per treatment; N = total scores; T = total of a treatment; G = grand total; SS = sum of squares; M = mean; MS = mean square (variance).

Provide the formula for the total sum of squares (SS_Total).

SS_Total = ΣX² − (G² / N).

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

df_Bet = k − 1.

Give the degrees of freedom for within-treatments (df_W/I).

df_W/I = N − k.

How is the mean square between treatments (MS_Bet) calculated?

MSBet = SSBet / df_Bet.

How is the mean square within treatments (MS_W/I) calculated?

MSW/I = SSW/I / df_W/I.

Explain how critical F values are chosen.

Based on numerator df (dfBet), denominator df (dfW/I), and the chosen alpha level (e.g., 0.05).

Why are post hoc tests needed after a significant F-ratio?

Because F only indicates that at least one mean difference is significant but does not specify which means differ.

Name two common post hoc tests discussed.

Tukey’s Honestly Significant Difference (HSD) and the Scheffé test.

Give the formula for Tukey’s HSD (equal n).

HSD = q × √(MS_W/I / n), where q is from a table, n is sample size per group.

What are the requirements for using Tukey’s HSD?

Equal sample size in all treatment conditions and the value q determined by alpha, k, and df_W/I.

How does the Scheffé test determine significance?

By computing an F-ratio for any two treatment means using their own SSBet but keeping the original dfBet; if this F exceeds the critical value, the difference is significant.

Describe the shape of the F-distribution when H0 is true.

It piles up around 1.00 and only includes positive values.

What happens to the F-ratio if the variance between treatments greatly exceeds variance within treatments?

F becomes much larger than 1, leading to potential rejection of H0.