## Example 11.3. Hypotheses with One Sample of One Measurement Variable

### State Null and Alternative Hypotheses

**Example:** Three different subjects were taught by two different instructors to three different students with the following results. The responses are examination results as a percentage. The null hypothesis: instructor and subject means do not differ.

### Example 11.4. Hypotheses with Two Samples of One Categorical Variable

It will be seen that the two-way analysis procedure is an extension of the patterns described in the one-way analysis. Recall that a one-way ANOVA has two components of variance: Treatments and experimental error (may be referred to as columns and error or rows and error). In the two-way ANOVA there are three components of variance: Factor A treatments, Factor B treatments, and experimental error (may be referred to as columns, rows, and error).

This analysis indicates that rejection of the null hypothesis is appropriate because the p-value is lower than 0.05. The probability values for the test of homogeneity of variances indicates that there is not enough information to reject the null hypothesis of equality of variances. No pattern or outlier data are apparent in either the “residuals versus order of the data” or “residuals versus fitted values .” The normal probability plot and histogram indicate that the residuals may not be normally distributed. Perhaps a transformation of the data could improve this fit; however, it is doubtful that any difference would be large enough to be of practical importance .

## State Null and Alternative Hypotheses

These are used to examine the assumptions underlying the ANOVA, and will also show whether there are any serious outliers. An example of some residuals plots produced by MINITAB version 14 is given below. The data are the percent of liver cells staining positive following treatment of rats with a hormone.

## We then have three null hypotheses (and three alternatives).

A two-way anova without replication and only two values for the interesting nominal variable may be analyzed using a The results of a paired *t*–test are mathematically identical to those of a two-way anova, but the paired *t*–test is easier to do and is familiar to more people. Data sets with one measurement variable and two nominal variables, with one nominal variable nested under the other, are analyzed with a

## State Null and Alternative Hypotheses

Suppose it was decided to add two diets to this experiment. This would then be a 2x2x2x2 factorial design. The problem would be that there would be no degrees of freedom left to estimate error because there would only be one animal on each of the 16 treatment combinations. This may not be an insuperable problem. High order interactions (3-way and 4-way or higher) usually turn out to be non-significant (i.e. the interaction SS is not very different in magnitude from the error SS). So it is possible to poole these and use them as the error term. High level factorials of this sort are widely used in industrial research, but are not disusses in further detail here.

## State Null and Alternative Hypotheses

Suppose 16 animals were available, and the experimenter wanted to compare two strains (S), two sexes (X) and two treatments (T). This would be a 2x2x2 factorial with 8 different treatment combinations, and there could be two animals on each treatment combination. Such a design would be analysed in a very similar way to the one above, except that there would now be three "main effects" to be estimated (differences between the two strains, the two sexes and the two treatments in each case averaging across the other factors), there would be three two-way interactions (SxX, SxT and XxT), and one three-way interaction (SxXxT), and there would be eight degrees of freedom to estimate the error, which although a bit low according to the resource equation method of determining sample size (see Sample size button), is not too bad.

## State Null and Alternative Hypotheses

Some people plot the results of a two-way anova on a 3-D graph, with the measurement variable on the *Y* axis, one nominal variable on the X-axis, and the other nominal variable on the *Z* axis (going into the paper). This makes it difficult to visually compare the heights of the bars in the front and back rows, so I don't recommend this. Instead, I suggest you plot a bar graph with the bars clustered by one nominal variable, with the other nominal variable identified using the color or pattern of the bars.