Multiple samples
One-way ANOVA
STATS 191
2024-01-01
Outline
Case studies:
Does diet affect longevity?
the Spock consipracy trial
Sums of squares
F-tests
Case study A: does diet affect longevity?
One-way ANOVA model: generalization of two-sample
Model details
Data: \(Y_{ij}, 1 \leq i \leq n_j, j \in {\tt [lopro, N/N85, N/R40, N/R50, NP, R/R50]}\)
Model: \(Y_{ij} \sim N(\mu_j, \sigma^2)\) (Note: assumed equal variance here!)
Null: \(H_0\) no difference: \(\mu_{\tt lopro}=\dots=\mu_{\tt R/R50}\)
Alternative: \(H_a\) model holds for some values \(\mu_{\tt lopro},\dots,\mu_{\tt R/R50}\) but they are not all identical.
Fitting the model
Null model
Comparing the models
How much better a fit is model then null_model?
Extra sum of squares
\[ \begin{aligned} SSE_R-SSE_F &= \|\hat{Y}_F-\hat{Y}_R\|^2_2 \\ &= \sum_{j=1}^6 \sum_{i=1}^{n_j}(\bar{Y}_j-\bar{Y}_{\cdot})^2 \end{aligned} \]
\(F\)-statistic
Convert “extra” sum of squares to unitless quantity
\[ S^2_P = \frac{\sum_{j=1}^6 (n_j-1) \cdot S^2_j}{\sum_{j=1}^6 (n_j-1)} \]
Using anova
Comparing two groups: N/N85 vs N/R50
Confidence interval
Comparing two groups: R/R50 vs N/R50
Confidence interval
Case study B: is the jury pool representative?
Any differences between judges?
How about Spock’s judge vs. others?
Using anova
How about variation only among others?
Summarizing all 3 models
Some diagnostic plots
longevity study
Residual vs. fitted
