- interpret a one-way ANOVA table
- interpret a multi-way ANOVA table
- know the assumptions of ANOVA

Source SS DF MS F p-value =========================================================================== Between SSk is the number of populations._{b}k-1 SS_{b}/(k-1) MS_{b}/MS_{w}area to right of F Within SS_{w}N-k SS_{w}/(N-k) =========================================================================== Total SS_{t}N-1

N is the total number of observations in all samples.

The formula for SS_{b} and SS_{w} are in the textbook.
A simpler formula for SS_{w}, depending on the sample standard deviations, is

SSIf s is the sample standard deviation of all N observations, then_{w}= (n_{1}-1)s_{1}^{2}+ ... + (n_{k}-1)s_{k}^{2}

SSIt is also true that SS_{t}= (N-1)s^{2}

Finding these sums of squares is the computationally tedious part of the computation. The remainder of the computations are straightforward.

The F statistic from the ANOVA table is compared to an F distribution with these degrees of freedom. The p-value is the area to the right of this F statistic. This p-value is interpreted like any other p-value. It is the probability of observing a result at least as extreme as the actual test statistic, assuming the null hypothesis is true. Low p-values are indications of strong evidence against the null hypothesis.

- The individual populations are all normally distributed.
- The individual population standard deviations are all equal.

For the first assumption, it is really only important that the individual sample sizes are sufficiently large so that the sampling distribution of the sample mean is approximately normal in each case, as the central limit theorem implies. For small samples (say n < 10), outliers in the samples or extensive skewness may invalidate the F test. A side-by-side boxplot of the data grouped by the categories of the explanatory variable should show each sample is fairly symmetric. For larger samples, some skewness in the population is not a problem.

For the second assumption, as long as the population standard deviations are within a factor of 10 or so from one another, the lack of exact equality can be safely ignored.

You are given a partial ANOVA table for a problem in which there are three samples of size 5, 3, and 3.

Source SS DF MS F p-value ================================================================== Between 20 Within ================================================================== Total 50

Below is S-PLUS output from a launcher experiment in which the ball type was the only variable factor.

Df Sum of Sq Mean Sq F Value Pr(F) ball 2 183.267 91.63333 1.048481 0.3643071 Residuals 27 2359.700 87.39630

- How many different types of balls were used in the experiment?
- How many total measurements were made?
- Which statement is the most appropriate interpretation of the data?
- There is strong evidence that the mean distance each ball travels is not the same for all balls.
- The data is consistent with the hypothesis that the mean distance each ball travels is the same. However, we cannot claim with high confidence that the population means are exactly equal.
- There is strong evidence that the population means are all equal.

In a larger experiment with two factors (explanatory variables), the distance a ball travels is modeled to depend on ball type and angle of the launcher. The S-PLUS output is below.

Df Sum of Sq Mean Sq F Value Pr(F) ball 2 1.056 0.528 0.418 0.6618089 angle 2 3012.389 1506.194 1193.830 0.0000000 Residuals 31 39.111 1.262

- How many different types of balls were used in the experiment?
- How many different angles were used in the experiment?
- How many total measurements were made?
- Which statement(s) are the most appropriate interpretation of the data?
- There is strong evidence that the different balls travel different distances on average.
- The data is consistent with there being no difference in the balls regarding the average distance they travel.
- There is overwhelming evidence that changing the angle changes the average distance the balls travel.
- It seems likely that changing the angle changes the average distance the balls travel, but there is still room for reasonable doubt.
- The data is consistent with the angle having effect on the average distance the ball travels.

Last modified: April 13, 1999

Bret Larget, larget@mathcs.duq.edu