MATH 5010 Sections 18

Big idea

ANOVA asks whether several population means can reasonably be treated as equal. Instead of doing many two-sample tests, ANOVA compares two sources of variability in one model.

Model: Y_ij = μ_i + ε_ij, ε_ij independent N(0, σ²)

F = MS_between / MS_within

Interpretation

If the group means are equal, the between-group mean square and within-group mean square estimate the same noise level. A large F statistic suggests the group means are separated more than random noise would explain.

Learning goals

Compute a one-way ANOVA table from raw data.
Explain SST = SSB + SSW.
Use the F distribution for the global test.
Check assumptions: independence, normal errors, equal variance.
Use planned contrasts and pairwise comparisons carefully.

Notation map

Groups
i = 1, …, k

Observations
Y_ij, j = 1, …, n_i

Group means
Ȳ_i.

Grand mean
Ȳ_..

1. Interactive one-way ANOVA table

Enter one group per line. The default trout-toxin data match the ANOVA treatment example from the course lab.

Group data

F–

p-value–

η²–

Decision–

Hypotheses for one-way ANOVA

H0: μ1 = μ2 = ⋯ = μk. HA: at least one μi differs.

The ANOVA F-test is global. Rejecting H0 tells us that not all means are equal, but it does not by itself identify which means differ.

2. Decomposing variation

The identity behind one-way ANOVA is:

SST = SSB + SSW

Use the current data and choose a group to visualize deviations from the grand mean, group mean, and observation.

Group to display

3. Simulate F statistics

When δ = 0, all group means are equal and the rejection rate should be close to α = 0.05. Increase δ to see power rise.

Groups k: 4

Sample size n: 8

Within-group σ: 4

Mean separation δ: 0

4. Assumption explorer

ANOVA is most reliable when errors are independent, roughly normal within groups, and groups have similar variances. Explore how unequal variance changes group spread.

Group 1 σ: 2

Group 2 σ: 2

Group 3 σ: 2

Mean separation: 3

5. Planned linear contrast

A planned contrast tests a focused comparison, such as one treatment versus the average of two controls.

L = a1 μ1 + ⋯ + ak μk, with a1 + ⋯ + ak = 0

Sample means

Sample sizes

Contrast coefficients

Pooled MSE from ANOVA

Confidence level

Test statistic

T = L̂ / sqrt(MSE Σ a_i²/n_i), with df = N − k.

Reject a two-sided contrast test when |T| is larger than the corresponding t critical value.

6. Pairwise comparisons from current data

After a significant global ANOVA, pairwise comparisons help locate differences. This table uses the pooled MSE from the current ANOVA and reports unadjusted and Bonferroni-adjusted p-values.

Teaching warning

Do not replace ANOVA by many unplanned t-tests.
As the number of groups grows, the number of pairwise tests grows quickly and the chance of at least one false positive increases. Use planned contrasts or a multiplicity adjustment when needed.

Number of pairs = k(k − 1)/2

Number of groups k: 4

Quick self-checks

1. Global null

What is H0 in one-way ANOVA?

2. Large F

A large F statistic means:

3. After rejection

If ANOVA rejects H0, what do we know?