Think about if you have 2 friends who are both arguing over which one lives farther from work/school. You offer to settle the debate and ask them to measure how far they have to travel between home and work. They both report back to you, but one reports in miles and the other reports in kilometers, so you cannot compare the 2 numbers directly. You can convert the miles to kilometers or the kilometers to miles and make the comparison, which conversion you make does not matter, you will come to the same decision either way.

It is similar with test statistics, you cannot compare your alpha value to the F-statistic you need to either convert alpha to a critical value and compare the F-statistic to the critical value or you need to convert your F-statistic to a p-value and compare the p-value to alpha.

Alpha is chosen ahead of time (computers often default to 0.05 if you don't set it otherwise) and represents your willingness to falsely reject the null hypothesis if it is true (type I error). The F-statistic is computed from the data and represents how much the variability among the means exceeds that expected due to chance. An F-statistic greater than the critical value is equivalent to a p-value less than alpha and both mean that you reject the null hypothesis.

We don't compare the F-statistic to 1 because it can be greater than 1 due only to chance, it is only when it is greater than the critical value that we say it is unlikely to be due to chance and would rather reject the null hypothesis.

In the classes that I teach I have found that the students who are not quite as young as the others and are returning to school after working for a while often ask the best questions and are more interested in what they can actually do with the answers (rather than just worrying if it is on the test), so don't be afraid to ask.

Single-step and step-wise relate to the view of the procedures as dynamic.

A single-step procedure implies there is no dynamics: without looking at the data, the procedure offers some rejection threshold.

A step-wise procedure implies there is dynamics: rejection boundaries are data driven and are updated along the sequence of p-values/test statistics in the data.

In reality, there is no real dynamics, as even the step-wise procedures take all test statistics, and returns a rejection boundary. The name stems mainly from the motivation to the procedure.

## Best Answer

The Studentized Range Distribution is a function of q, k, and df, where k is the number of groups of means, and df is the degrees of freedom. If $\phi(z)$ is the standard normal PDF, and $\Phi(z)$ is the standard normal CDF:

$$RangeCDF(q,k,\infty) = k \int_{-\infty}^\infty\phi(z)[\Phi(z+q)-\Phi(z)]^{k-1}dz$$

This seems to agree with the tables of critical q values when $df=\infty$. However, if I replace the standard normal with Student's T, the calculated value does not match the table, except when $df \to \infty$. $$Studentized RangeCDF(q,k,df) = k \int_{-\infty}^\infty t(t,df)[T(t+q,df)-T(t,df)]^{k-1}dt$$

Edit: I now have correct values when k=2, or df=$\infty$: $$2* \int_{-\infty}^\infty\phi(z)[\Phi(z+q)-\Phi(z)]^{2-1}dz = 2 \int_{-\infty}^\infty\phi(z)*\Phi(z+q)-\phi(z)\Phi(z)dz$$ $$=2*[\int_{-\infty}^\infty\phi(z)\Phi(z+q)dz-\int_{-\infty}^\infty\phi(z)\Phi(z)dz]$$ if $u=\Phi(z); du=\phi(z)dz$ $$=2*[\int_{-\infty}^\infty\phi(z)\Phi(z+q)dz-\int_{-\infty}^\infty udu]$$ $$=2*[\int_{-\infty}^\infty\phi(z)*\Phi(z+q)dz-\frac{1}{2}(\Phi^2(\infty)-\Phi^2(-\infty)]$$ $$=2*\left[\int_{-\infty}^\infty\phi(z)*\Phi(z+q)dz-\frac{1}{2}\right]$$ $$=2*\left[\Phi\left(\frac{q}{\sqrt2}\right)-\frac{1}{2}\right]$$ Using Student's T distribution with the specified degrees of freedom yields the same results as the reference table. However, my numerical integration of the second (StudentizedRange) equation does not match. This must mean that the process used to integrate $\phi(z)\Phi(z+q)dz$ does not work for $t$ and $T$