Procedures for Ranking Data

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Chapter: Biostatistics for the Health Sciences: Nonparametric Methods

Ranking data becomes useful when we are dealing with inferences about two or more populations and believe that parametric assumptions such as the normality of their distributions do not apply.


PROCEDURES FOR RANKING DATA

Ranking data becomes useful when we are dealing with inferences about two or more populations and believe that parametric assumptions such as the normality of their distributions do not apply. Suppose, for example, that we have two samples from two distinct populations. Our null hypothesis is that the two populations are identical. You may think of this as stating that they have the same medians. We are not checking for differences in means because the mean may not even exist for these populations. Table 14.1 shows how to rank data from two populations.

Let us denote the sample from the first population with n1 observations x1, x2, x3, . . . , xn1. The second sample consists of n2 observations. For the purpose of the analysis, we will pool the data from the two samples. We will label the observations from the second sample xn1+1, xn1+2, xn1+3, . . . , xn1+n2. Now, to rank the data, we or-der the observations from smallest to largest and denote the ordered observations as y’s. If x5 is the smallest observation, x5 becomes y1, and if x3 is the next smallest, x3 becomes y2, and so forth. We continue in this way until all the x’s are assigned to all the y’s.

TABLE 14.1. Terminology for Ranking Data from Two Independent Samples


In Table 14.2, we present hypothetical data to illustrate ranking. The y’s refer to the ranked observations from the first and second samples. We have two groups, control and treatment, xc and xt, respectively.

To illustrate the procedures described in the previous paragraph, suppose a re-searcher conducted a study to determine whether physical therapy increased the weight lifting ability of elderly male patients. As the researcher believed that the data were not normally distributed, a nonparametric test was applied. The data un-der the unsorted scores column represent the values as they were collected directly from the subjects. Then the two data sets were combined and sorted in ascending order. Each score was then assigned a rank, which is shown in parentheses. (Refer to the columns labeled “sorted scores.”) The term ΣR means that we should sum the ranks in a particular column; the symbols T and T refer to the sum of the ranks in the control and treatment groups, respectively. In this example, T = 25 and T = 30. We do not need to keep track of both of these statistics because the sum of all the ranks is T + T and is known to be n(n + 1)/2, where n is the sum of the sample sizes in the two groups, in this case n = 2(5) = 10, and so the sum of the ranks is 10(11)/2 = 55. In summing all the ranks we are just adding up the integers from 1 to 10 in our example.

A possible ambiguity can occur when some data points share the same value. In that case, the ordering among the tied values can be done by any system (e.g., choose the lowest indexed x first). Rather than assigning them separate ranks in ar-bitrary order, sometimes we prefer to give all the tied observations the same rank. That rank would be the average rank among the tied observations. If, for example, the 3rd, 4th, 5th, and 6th smallest values were all tied, they would all get the rank of 4.5 [i.e., (3 + 4 + 5 + 6)/4]. 

TABLE 14.2. Left Leg Lifting Test Data among Elderly Male Patients Who Are Receiving Physical Therapy; Maximum Weight (Unsorted, Sorted, and Ranked) For Treatment and Control Groups


Now that the x’s have been rearranged from the smallest to the largest values (the arrangement is sometimes called the rank order), the rank transformation is made by replacing the value of the observation with its y sub-script. This subscript is called the rank of the observation. Refer to Table 14.1 for an example. You can see that the lowest rank is y1. If x5 is the smallest observation, its rank would be 1. If x3 and x9 are tied, they both would be assigned to y2 and y3 and have a rank of 2.5.

If the two distributions of the parent populations are the same, then the ranks will be well mixed among the populations (i.e., both groups should have a similar num-ber of high and low ranks in their respective samples). However, if the alternative is true (that the population distributions are different) and the median or center of one distribution is very different from the other, the group with the smaller median should tend to have more lower ranks than the group with the higher median. A test statistic based on the ranks of one group should be able to detect this difference. In Section 14.3, we will consider an example: the Wilcoxon rank-sum test.

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