Estimation and Confidence Interval Based on the Wilcoxon

Suppose we have two populations which we assume differ by at most a shift in locations. Let $\Delta$ be the difference in locations: Population Y - Population X. We draw random samples from each population. Let X₁, X₂, ... ,X_m denote the sample of size m from the first population. Let Y₁, Y₂, ... ,Y_n denote the sample of size n from the second population. Think of $\Delta$ as positive for a moment. Then typical Y's are shifted up from a typical X's by $\Delta$. If we knew $\Delta$, we could unshift the Y's by subtracting $\Delta$ from each Y. This leads to a point estimate. Confused? Here's an example for the Wilcoxon point estimate:

Suppose the samples are:

• X: 8 12 16

• Y: 14 19 22

Lets estimate $\Delta$ by the median of the differences Y_j - X_i. Here are the 9 differences.

14-8=6,  14-12=2, 14-16=-2, 19-8=11, 19-12=7,
19-16=6, 22-8=14, 22-12=10, 22-16=6.

Here are the sorted differences:

-2  2   3   6   6  7   10   11   14

As our point estimate we will take the median of the differences, i.e., 6. Here are the X's and the unshifted Y's; i.e., Y-6:

X:       8   12  16
Y - 6:   8   13  16

Now compute the Wilcoxon test statistic on the X's and the unshifted Y's. You will get T=4.5 which is $\frac{m \times n}{2} = 9/2$. This is what you expect T to be if there are no differences. Hence, the median of the differences has unshifted the Y's.

In general, the estimate of the shift in locations based on the Wilcoxon is the median of the differences  Y_j - X_i.

Consider the battery example  of the last chapter. Recall that we had two types of batteries XX and YY and we wanted to see if a typical YY lasts longer that a typical XX. Lets estimate the difference in lifetimes of typical YY and XX batteries. Here are the samples (lifetime in hours):

XX    49     53     74    111    113    335
YY    62    101    167    174    190

Here is the comparison dotplot :

              ..  .      :                                    .
  XX       -----+---------+---------+---------+---------+---------+-

  YY            .      .          ..  .
           -----+---------+---------+---------+---------+---------+-
               60       120       180       240       300       360

It seems though YY's are beating XX's. To estimate the shift we need to get all 30 differences of the form YY_j-XX_i. When we get this estimate by hand calculation, the table of differences discussed in the last chapter really helps. Sort the samples. Then the columns of the table are the sorted Y's and the rows of the table are the sorted X's. Then obtain the differences Y_j - X_i. As you will see the median is easy to get.

	62	101	167	174	190
49	13	52	118	125	141
53	9	48	114	121	137
74	-12	27	93	100	116
111	-49	-10	56	63	79
113	-51	-12	54	61	77
335	-273	-234	-168	-161	-145

Our point estimate is the median which is 53 (do a quick stem-leaf then compute the median). Could you guess it from the plot? (Take the YY's shift them back 53 units. Do these "aligned" samples seem about the same?). So a typical YY battery lasts 53 hours longer than a typical XX battery. Takes care of that problem. What's that? Oh right, it could just be sampling error. We need a confidence interval!

We will use percentile confidence intervals based on resampling. So its old stuff! The steps for a general situation are:

1.

Resample m X's with replacement.

2.

Resample n Y's with replacement.

3.

Obtain the median of the differences of the resampled Y's minus the resampled X's.

4.

Record this median.

5.

Repeat steps (1) through (4) 1000 times.

6.

Sort the 1000 medians,

7.

Pick off the 25th and 976th sorted medians. This is our 95% confidence interval.

It looks great until you see step (5). To get an idea of what's going on, I did step (5) 100 times. Here are the resorted 100 resampled medians of the differences:

  -161.0 -105.0  -78.5  -78.5  -76.0  -49.0  -49.0  -45.5  -44.5  -12.0
   -11.0  -11.0  -11.0  -10.0  -10.0   -0.5    9.0    9.0    9.0    9.0
    11.0   11.0   13.0   13.0   13.0   13.0   22.0   27.0   27.0   27.0
    27.0   27.0   27.0   30.5   34.5   37.5   37.5   48.0   48.0   48.0
    48.0   48.0   51.0   51.0   51.0   52.0   52.0   52.0   52.0   53.0
    53.0   53.0   54.0   54.0   54.5   55.0   56.0   56.0   56.0   56.0
    56.0   56.0   56.5   57.5   58.5   61.0   61.0   61.0   61.0   63.0
    63.0   64.5   70.0   72.5   77.0   77.0   77.0   77.0   77.0   79.0
    79.0   85.0   86.0   93.0   93.0   93.0   93.0   93.0   93.0   93.0
    93.0   96.5  100.0  100.0  107.0  114.0  116.0  117.0  121.0  137.0

The confidence interval is (-78.5, 117). It contains 0, hence, the results are inconclusive. Remember we took differences of the form YY minus XX, so positive values in the CI means YY beats XX, but negative values mean XX beats YY. Our conclusion would be: a typical YY battery has a shorter lifetime than a typical XX by 78.5 hours to a typical YY battery has a longer lifetime than a typical XX by 117 hours. Though right, this sounds a bit odd. It is better to say the results were inconclusive . The value of 53 did not overcome the noise level. Note that on this data set, this is the same conclusion which we came to in Chapter 8.

A picture is worth a 1000 words, so here is a histogram of the 100 resampled medians of the differences. I have located the CI on it with [ ]'s.

                                                 .
                                                 :
                                                 :
                                                 :
                                                 :
                                                 :
                                          :      :      .
                                          :  .   :   .  :
                                      :   :  :  .::  :  :
                           .    .     :   :  :. :::. :  :
             .         .   :    :.    : . : .:: ::::::: ::.::  .
          +---------+------[--+---------+---------+--------]+-------C1
       -180      -120       -60         0        60       120

Using 1000 resamples, I got the confidence interval (-105,115). So the conclusion remains the same.

Using the class code (Two-Sample hypothesis test and confidence interval for the location parameter based on the Wilcoxon) you try it. Simply bring up class code in a second window, drop the XX sample in the first box (data set 1), drop the YY sample in the second box (data set 2), and submit.

Exercise 10.2.1  

1.

To set ideas work on this simple data set.

      X   12 15 18
      Y   16 19 25 28

(a)

Obtain all 12 differences (Y minus X).

(b)

Next obtain the point estimate, the median of the differences.

(c)

Subtract this estimate from the Y's and obtain the value of the Wilcoxon test statistic.

2.

For the last problem, use the following list of random numbers to obtain 2 resampled median of differences.

   2  9  2  2  7  2  2  3  0  8  8  1  9  8  8 
   2  3  3  4  0  9  2  1  0  7  9  3  6  6  2 
   3  7  6  8  8  7  0  5  0  3  4  3  5  7  7 
   3  4  5  0  1

3.

Consider the batting averages of the switch hitters and the left-handed hitters from the baseball data set. Using the class code (Two-Sample Hypothesis and CI (Wilcoxon)), obtain the estimate of the difference (Left minus switch) of batting averages and determine a 95% confidence interval for the difference. What does the interval mean in terms of the problem?

    Switch  .212  .218  .236  .242  .251  .251  .254  .261  .270  .282
     Left   .238  .271  .279  .283  .284  .290  .300  .303

4.

Consider the following samples of Italian and Etruscan skull sizes. Use the class code (Two-Sample Hypothesis and CI (Wilcoxon)) to obtain the estimate of difference between a typical Etruscan skull and an Italian skull. Obtain a 95% confidence interval and interpret it in terms of the problem.

   Ital.  134  132  126  134  131  130  130  125  132  126
   Etru.  141  145  145  146  142  126  144  146  154  149  143  131

5.

Let $\Delta$ be the difference in weight between a typical pitcher and hitter, professional baseball players. Using the class code (Two-Sample Hypothesis and CI (Wilcoxon)) estimate $\Delta$ and determine a 95% confidence interval for it based on the following data. What does the interval mean in terms of the problem?

  Hitters:
   155   155   160   160   160   166   170   175   175   175   180
   185   185   185   185   185   185   185   190   190   190   190
   190   195   195   195   195   200   205   207   210   211   230

  Pitchers:
   160   175   180   185   185   185   190   190   195   195   195
   200   200   200   200   205   205   210   210   218   219   220
   222   225   225   232