Comparing Multiple Categories
Previously, you examined the procedures for analyzing univariate data that had one or two values of their categorization variable. It is a logical extension to now look at cases where there are more than two values of this categorization variable.
An Appropriate Data Matrix
Two variables are used when comparing multiple categories. One variable has measurement values and the other has the categorization values. A typical data matrix is:
OBS WEIGHT BAIT
- - - - - - - - - -
1 112.3 LIVE
2 144.2 LIVE
3 98.1 FLY
4 139.8 LIVE
5 106.4 FLY
6 87.2 LURE
7 103.7 LURE
8 105.2 NET
...
This form of data matrix is the same as the one used when comparing two categories (page 145). This example shows the important difference: there are now more than two values of BAIT, the categorization variable. In preparation for the statistical test that compares the distribution means for differences, you will ordinarily perform a series of procedures that provide both descriptive and diagnostic information. Specifically, it is a good idea to:
Check the measurement values of each category for normality of its distribution. In the example above, you would check the normality of the WEIGHT for each of the different values of BAIT (e.g., LIVE, FLY, LURE, and NET).
View the distribution of each category's values. This is done, most often, with PROC CHART. The purpose of showing each of these frequency distributions is so that you will see the approximate distribution of the values (thereby giving you a cross-check on the assumption of normality) and get a preliminary idea of how the distributions compare (looking at the "peak" of each distribution as well as its "width").
If your data fulfill the assumption regarding normality of each distribution then you can proceed with the statistical tests in PROC ANOV A. If the data for your categories are not normally distributed, then you will have to consider either performing some data transformation (beyond the scope of this discussion) or using a non-parametric statistical test. Generally, it is also considered safer if you have approximately the same number of observations for each value of the categorization variable. In the fishing example, this would mean that approximately the same numbers of fish caught with each type of bait would be used in the analysis. Needless to say, observations should be based on random sampling, not by the subjective selection of individuals to be included in the analysis.
The Duncan-Waller Tests
There are a number of statistical tests that can be used to identify distributions that are not significantly different from each other. The Duncan-Waller tests (called DUNCAN and WALLER in SAS) are two of those that are used commonly. Others include the LSD and Tukey tests.
The purpose of the Duncan-Waller tests is to show the distributions that are not significantly different from each other. Actually, the distributions are likely to form sets that overlap so that clear-cut groupings in not identified.
You can see how this works by two examples.
If you have four values for the categorization variable (such as shown in the example data matrix). The mean measurement values for two of these categories (for example, the mean weights for LURE and NET) are similar to each other and relatively large. The mean measurement values for the other two categories (for example, weights for LIVE and FLY) are are similar and relatively small. In this case you would expect that the weights for LURE and NET would not be significantly different from each other and the weights for LIVE and FLY would not be significantly different from each other. There would be two groups and the weights of these two groups would be significantly different from each other.
WALLER GROUPING MEAN N BAIT
A 122.07 50 LURE
A
A 119.05 50 NET
B 99.33 50 LIVE
B
B 97.29 50 FLY
In contrast to this are distributions whose means form a series. In this case, the NET might have the largest mean weight, the mean FLY and LURE similar to each other and in the middle of the range, and LIVE the smallest mean weight. The grouping below shows that some of the BAITS whose means are adjacent to each other are not significantly different from each other, hence they share the same group. However, those BAITS that are more widely separated from each other are significantly different. This results in overlapping of groups (as in B and C below for the LURE bait).
WALLER GROUPING MEAN N BAIT
A 129.41 50 NET
B 109.17 50 FLY
B
C B 106.64 50 LURE
C
C 98.27 50 LIVE