DSC6008 Chap.10 One-Way ANOVA and Model Diagnostics
One-Way ANOVA and Model Diagnostics
Partition one source of variation at a time
One-way ANOVA compares variation among fitted group means with variation remaining within groups. Each observation's deviation from the grand mean is separated into a treatment-mean component and a residual component.
Under the fitted model those components are orthogonal, which is why total sum of squares equals treatment sum of squares plus error sum of squares.
Attach degrees of freedom to each component
With k groups and n independent units, treatment degrees of freedom are k minus one and error degrees of freedom are n minus k. Their sum is n minus one, the total degrees of freedom.
Dividing each sum of squares by its degrees of freedom creates a mean square. These identities provide an immediate audit of an output table before any p-value is interpreted.
Read the F statistic as a variance ratio
Under the global null and common-variance model, treatment and error mean squares estimate the same error scale, so their ratio tends to be near one.
A large F means variation among fitted group means is large relative to within-group noise. The reference distribution uses both degrees of freedom. Its global null states that all population group means are equal.
Connect ANOVA to regression coding
A one-way ANOVA is a regression model with a categorical predictor. Treatment coding uses one group as the reference and expresses other means as differences from it.
Sum coding uses deviations from a grand-mean parameter. The coefficient labels change, but an equivalent model produces the same fitted group means, residuals and global F test.
Diagnose the error model before chasing pairs
Residual-versus-fitted plots examine changing spread and omitted structure, while normal Q-Q plots examine the distributional shape used for reference inference.
Independence follows from design and execution, not from a residual plot.
A funnel pattern, run-order drift or an influential observation has a different implication, so describe the feature and its consequence rather than declaring assumptions passed or failed.
Earn the follow-up conclusion
A large F statistic provides evidence that at least one mean differs, but it does not identify a pair or imply that every pair differs.
Planned contrasts answer questions fixed before the outcomes were examined. When all pairwise comparisons matter, a procedure such as Tukey control addresses that family. Selecting unadjusted tests only for the most separated sample means inflates false claims.
Write contrasts as scientific questions
Contrast coefficients sum to zero and express a comparison among treatment means.
Control versus the average of active settings is different from a linear dose trend, even if both use the same four groups. Record coefficients and scaling so the estimate can be reproduced. Report the contrast in response units with uncertainty and the chosen multiplicity rule.
Return the table to the experiment
Name the response, treatment factor, independent unit and global decision.
Then report diagnostics and any justified follow-up contrast. Subsamples do not enlarge the error degrees of freedom, and blocked or repeated-measures designs need additional terms. The final conclusion is credible only when the ANOVA decomposition matches the assignment that generated the data.
What this chapter covers
- 01
Group means
- 02
Sums of squares
- 03
Degrees of freedom
- 04
F statistics
- 05
Multiple comparisons
- 06
Residual diagnostics
Worked example · free
Complete an ANOVA ratio from mean squares
- 2Compute the between-group mean square as 96/3 = 32.
- 2Compute the within-group mean square as 160/20 = 8.
- 2Compute F = 32/8 = 4 and state that it tests the global equality of means.
Key terms
- Between-group Variation
- Variation explained by differences among fitted group means.
- Within-group Variation
- Variation of observations around their fitted group means.
- F Statistic
- The ratio of a model mean square to an error mean square.
One-Way ANOVA and Model Diagnostics FAQ
What does the ANOVA F test compare?
It compares variation attributed to differences among group means with variation remaining among units within groups. Under the global null and model assumptions, both mean squares estimate the same error variance, so an unusually large ratio counts against equality of all means.
Why not run every pairwise t test after ANOVA?
A collection of unadjusted tests raises the chance of at least one false rejection across the family. Planned contrasts can target questions chosen before seeing the data, while procedures such as Tukey comparisons control a familywise error criterion for broad pairwise exploration.
Which residual pattern suggests unequal variance?
A residual-versus-fitted plot whose vertical spread grows or shrinks systematically across fitted values suggests nonconstant variance. The response may need a different variance model, transformation or robust analysis. The remedy depends on the measurement scale and inferential target, not on appearance alone.
Assessment move
Reconstruct every ANOVA table from sources, sums of squares and degrees of freedom before reading software output. Check that degrees of freedom add correctly, compute both mean squares and state the global null in words. After a significant result, name the exact follow-up contrast and its multiplicity rule.
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