MATLAB Examples

# Multiple Comparisons Using One-Way ANOVA

```load carsmall ```

MPG represents the miles per gallon for each car, and Cylinders represents the number of cylinders in each car, either 4, 6, or 8 cylinders.

Test if the mean miles per gallon (mpg) is different across cars that have different numbers of cylinders. Also compute the statistics needed for multiple comparison tests.

```[p,~,stats] = anova1(MPG,Cylinders,'off'); p ```
```p = 4.4902e-24 ```

The small p-value of about 0 is a strong indication that mean miles per gallon is significantly different across cars with different numbers of cylinders.

Perform a multiple comparison test, using the Bonferroni method, to determine which numbers of cylinders make a difference in the performance of the cars.

```[results,means] = multcompare(stats,'CType','bonferroni') ```
```results = 1.0000 2.0000 4.8605 7.9418 11.0230 0.0000 1.0000 3.0000 12.6127 15.2337 17.8548 0.0000 2.0000 3.0000 3.8940 7.2919 10.6899 0.0000 means = 29.5300 0.6363 21.5882 1.0913 14.2963 0.8660 ```

In the results matrix, 1, 2, and 3 correspond to cars with 4, 6, and 8 cylinders, respectively. The first two columns show which groups are compared. For example, the first row compares the cars with 4 and 6 cylinders. The fourth column shows the mean mpg difference for the compared groups. The third and fifth columns show the lower and upper limits for a 95% confidence interval for the difference in the group means. The last column shows the p-values for the tests. All p-values are zero, which indicates that the mean mpg for all groups differ across all groups.

In the figure the blue bar represents the group of cars with 4 cylinders. The red bars represent the other groups. None of the red comparison intervals for the mean mpg of cars overlap, which means that the mean mpg is significantly different for cars having 4, 6, or 8 cylinders.

The first column of the means matrix has the mean mpg estimates for each group of cars. The second column contains the standard errors of the estimates.