## Quiz: Correlation of dummy variables

One-way ANOVA with a balanced K-group design is equivalent to a regression on the intercept and K-1 dummy variables. The quiz is on the correlation between any two dummy variable——positive, zero, or negative? One may think it is zero for any two dummy variables are orthogonal with one another. The answer will emerges after your click.

## Understanding Tukey’s q & p

For a planned SINGLE comparison, the CI of $\mu_2-\mu_1$ can be interpreted as $(M_2-M_1)\pm qt_{1-\frac{\alpha}{2},df}\times SE$, in which $SE=s\times \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}$. The $df$ and $s$ should be defined according to the denominator of $F$.

For post hoc multiple comparisons, the CI of $\mu_2-\mu_1$ can still be interpreted as $(M_2-M_1)\pm \frac{qtukey_{1-\alpha,nmeans,df}}{\sqrt{2}}\times SE$, wherein $nmeans$ indicates the number of sub-groups whose means are compared. Note that qtukey(...) and ptukey(...) is defined with a two-tailed probability while qt(...) and pt(...) with a single-tailed one.

To verify it with a single click

## Mann-Whitney-Wilcoxon Test vs. Z-test, for non-normal samples

Control group includes 150 Poisson ($\lambda$=2.3) cases;
Experiment group includes 200 cases, with fixed effect +1 which is unknown but interested in.

Researcher A chose Z test, considering the big sample sizes.
Researcher B chose Mann-Whitney-Wilcoxon Test, considering the non-normal distributions.
Whose confidence interval will beat?

Researcher A might complain: "Your CI width is actually from Poisson distribution. Most time it is actually zero. Let's try $\chi^2_{df=1}$. It would be fairer for Z-Test to use a continual distribution."

Mann-Whitney-Wilcoxon plays better than his expectation even in the continual case.