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It may come as a surprise to learn that statisticians cannot agree on exactly what a quartile is and that there are many different ways to compute a quartile.
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Details about this approach will be covered in Chapter 15. Yet another strategy is to estimate the probability that a randomly sampled observation from the first group is less than a randomly sampled observation from the second. For example, the command boxplot (x,y) will create a boxplot for the data in both x and y. Another useful approach is to create a boxplot for both groups on the same scale. An area of zero corresponds to no overlap, and an area of 1 occurs when the distributions are identical and the groups do not differ in any manner whatsoever. You estimate the distributions associated with both groups and then compute the area under the intersection of these two curves see Clemons and Bradley (2000) for recent results on how this might be done. Another approach to measuring effect size is the so-called overlapping coefficient. If the groups have identical quantiles, a plot of the quantiles should be close to a line having slope 1 and intercept zero. Another approach is to examine a so-called quantile−quantile plot. The methods listed in this section are not exhaustive. This is because for x 17, the upper confidence band extends up to ∞. Notice that the left end of the lower dashed line in Figure 8.5 begins at approximately x = 22. The + along the x-axis indicates the location of the median of the first group and the o's indicate the lower and upper quartiles. The hypothesis of equal quantiles is rejected if the lower (upper) dashed line is above (below) zero.įIGURE 8.5. The dashed lines in Figure 8.5 mark the confidence band for the difference between the quantiles. If there are no differences between the quantiles, the shift function should be a straight horizontal line at 0. Looking at the graph as a whole suggests that the effect of ozone becomes more pronounced as we move along the x-axis, up to about 19, but then the trend reverses, and in fact in the upper end we see more weight gain in the ozone group. Now there is more weight gain among rats in the ozone group. However, the difference between the upper quartiles is 26.95 − 17.35 = 9.6. So based on the medians it is estimated that the typical rat in the ozone group gains less weight than the typical rat in the control group.
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For example, the median of the control group is M = 22.7, as indicated by the +, and the difference between the median of the ozone group (which is 11.1) and the control group is given by the solid line and is equal to −11.6.
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The + indicates the location of the median for the control group (the data stored in the first argument, x), and the lower and upper quartiles are marked with an o to the left and right of the +. Storing the data for the control group in the S-PLUS variable x, and storing the data for the ozone group in y, sband produces the graph shown in Figure 8.5. A control group of 23 rats of the same age was kept in an ozone-free environment. (These data were taken from Doksum & Sievers, 1976.) The experimental group consisted of 22 70-day-old rats kept in an ozone environment for 7 days. Table 8.6 contains data from a study designed to assess the effects of ozone on weight gain in rats.