## Hypothesis Testing - Analysis of Variance (ANOVA)

The fundamental strategy of ANOVA is to systematically examine variability within groups being compared and also examine variability among the groups being compared.

The shape of the F-distribution depends on two degrees of freedom, the degrees of freedom of the numerator (among-group variance) and degrees of freedom of the denominator (within-group variance). The among-group degrees of freedom is the number of groups minus one. The within-groups degrees of freedom is the total number of observations, minus the number of groups. Thus if there are *n* observations in *a* groups, numerator degrees of freedom is *a*-1 and denominator degrees of freedom is *n*-*a*. For the example data set, there are 5 groups and 39 observations, so the numerator degrees of freedom is 4 and the denominator degrees of freedom is 34. Whatever program you use for the anova will almost certainly calculate the degrees of freedom for you.

## The hypothesis test for analysis of variance for populations:

Nested analysis of variance is an extension of one-way anova in which each group is divided into subgroups. In theory, you choose these subgroups randomly from a larger set of possible subgroups. For example, a friend of mine was studying uptake of fluorescently labeled protein in rat kidneys. He wanted to know whether his two technicians, who I'll call Brad and Janet, were performing the procedure consistently. So Brad randomly chose three rats, and Janet randomly chose three rats of her own, and each technician measured protein uptake in each rat.

## Here is an example of a two-level nested anova using the rat data.

There are a number of different tests for pairwise comparisons after a one-way anova, and each has advantages and disadvantages. The differences among their results are fairly subtle, so I will describe only one, the Tukey-Kramer test. It is probably the most commonly used post-hoc test after a one-way anova, and it is fairly easy to understand.

## The test statistic is the F statistic for ANOVA, F=MSB/MSE.

It's possible for a variance component to be zero; the groups (Brad vs. Janet) in our rat example had 0% of the variance, for example. This just means that the variation among group means is smaller than you would expect, based on the amount of variation among subgroups. Because there's variation among rats in mean protein uptake, you would expect that two random samples of three rats each would have different means, and you could predict the average size of that difference. As it happens, the means of the three rats Brad studied and the three rats Janet studied happened to be closer than expected by chance, so they contribute 0% to the overall variance. Using zero, or a very small number, in the equation for allocation of resources may give you ridiculous numbers. If that happens, just use your common sense. So if V_{subgroup} in our rat example (the variation among rats within technicians) had turned out to be close to 0%, the equation would have told you that you would need hundreds or thousands of observations per rat; in that case, you would design your experiment to include one rat per group, and as many measurements per rat as you could afford.

## Stats: Two-Way ANOVA - Richland Community College

The decision of whether or not to reject the null hypothesisthat the sample means are similar to each other requires that thevalue for F be compared with a.

## 02/01/2018 · Stats: Two-Way ANOVA

When the sample sizes in a nested anova are unequal, the *P* values corresponding to the F-statistics may not be very good estimates of the actual probability. For this reason, you should try to design your experiments with a "balanced" design, meaning equal sample sizes in each subgroup. (This just means equal numbers at each level; the rat example, with three subgroups per group and 10 observations per subgroup, is balanced). Often this is impractical; if you do have unequal sample sizes, you may be able to get a better estimate of the correct *P* value by using modified mean squares at each level, found using a correction formula called the Satterthwaite approximation. Under some situations, however, the Satterthwaite approximation will make the *P* values *less* accurate. If you cannot use the Satterthwaite approximation, the *P* values will be conservative (less likely to be significant than they ought to be), so if you never use the Satterthwaite approximation, you're not fooling yourself with too many false positives. Note that the Satterthwaite approximation results in fractional degrees of freedom, such as 2.87; don't be alarmed by that (and be prepared to explain it to people if you use it). If you do a nested anova with an unbalanced design, be sure to specify whether you use the Satterthwaite approximation when you report your results.