Chapter 4The Two-Sample Location Problem

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Order the combined X-sample and Y-sample in an ascending order.

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S2 = the rank of Y2

. ….

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in the combined sample of m+n.

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Table 4.1 Tritiated Water Diffusion Across Human Chorioamnion

Pd (10-4 cm/s)

At Term 12-26 weeks Gestational Age

0.80 1.15

0.83 0.88

1.89 0.90

1.04 0.74

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1.38

1.91

1.64

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1.46

Example 4.1: Water Transfer in Placental Membrane. The data in Table 4.1are a portion of the data obtained by Lloyd et al. (1969). Among other things, these authors investigated whether there is a difference in the transfer of tritiated water (water containing tritium, a radioactive isotope of hydrogen) across the tissue layers in the term human chorioamnion (a placental membrane) and in the human chorioamnion between 3 to 6 months gestation age. The objective measure used was the permeability constant Pd of the human chorioamnion to water. The tissues used for the study were obtained within 5 min of delivery from the placentas of healthy, uncomplicated pregnancies in the following gestation age categories: (a) between 12 and 26 weeks following termination of pregnancy via abdominal hysterotomy (surgical incision of the uterus) for psychiatric indications; and (b) term, uncomplicated vaginal deliveries. Tissues from ten term pregnancies and five terminal pregnancies were used in the experiment. Table 4.1 gives the average permeability constant (in units of 10-4 cm/s) for six measurements on each of the 15 tissues in the study.

In this example, the alternative of interest is greater permeability of the human chorioamnion for the term pregnancy. Thus, if we let X correspond to the Pd values of tissues from term pregnancies and Y to the Pd values of tissues from terminated pregnancies, we perform test (4.5), which is designed to detect the alternative Δ < 0.

For purpose of illustration we choose α to be 0.082. From Table A.6 we find w0.082 = 52. Now we list the combined sample in increasing order to facilitate the joint ranking. The ranks are given in parentheses

X Y X X Y Y X Y

0.73 0.74 0.80 0.83 0.88 0.90 1.04 1.15

(1) (2) (3) (4) (5) (6) (7) (8)

Y X X X X X X

1.21 1.38 1.45 1.46 1.64 1.89 1.91

(9) (10) (11) (12) (13) (14) (15)

We see that the Y-ranks are 2,5,6,8,9 and thus

W = 2+5+6+8+9 = 30.

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Control SST

1042 (13) 874 (9)

1617 (23) 389 (2)

1180 (18) 612 (4)

973 (12) 798 (7)

1552 (22) 1152 (17)

1251 (19) 893 (10)

1151 (16) 541 (3)

1511 (21) 741 (6)

728 (5) 1064 (14)

1079 (15) 862 (8)

951 (11) 213 (1)

1319 (20)

Table 4.2 Alcohol Intake for 1 Year (cl of Pure Alcohol)

Example 4.2 : Alcohol Intakes. Eriksen, Bjornstad, and Gotestam (1986) studied a social skills training program for alcoholics. Twenty-four “alcohol-dependent” male inpatients at an alcohol treatment center were randomly assigned to two groups. The control group patients were given a traditional treatment program. The treatment group patients were given the traditional treatment program plus a class in social skills training (SST). After being discharged from the program, each patient reported – in 2-week intervals – the quantity of alcohol consumed, the number of days prior to his first drink, the number of sober days, the days worked, the times admitted to an institution, and the nights slept at home. Reports were verified by other sources (wives or family members). (Such data can be unreliable!) One patient in the SST group, discovered to be an opiate addict, disappeared after discharge and submitted no reports. The remaining 23 patients reported faithfully for a year. The results for alcohol intake are given in Table 4.2. The ranks in the joint ranking of the 23 observations are given in parentheses in Table 4.2.

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To test H0 vs the alternative that the SST group tends to have lower alcohol intakes, we need to test H0 : Δ = 0 vs H2 : Δ < 0. Suppose, for example, we choose α = 0.05. Then z0.05 = 1.645 and the normal approximation given by the display (4.11) is Reject H0 if W* -1.645; otherwise do not reject.

From Table 4.2, we find the sum of the SST ranks is W = 9+2+4+7+17+10+3+6+14+8+1=81

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Example 5.4: Effect of Feedback on Salivation Rate

The effect of enabling a subject to hear himself salivate while trying to increase or decrease his salivary rate has been studied by Delse and Feather (1968). Two groups of subjects were told to attempt to increase their salivary rates upon observing a light to the left, and decrease their salivary rates upon observing a light to the right. The apparatus for collecting and recording the amounts of saliva was described by Delse and Feather (1968) and also Feather and Wells (1966). Members of the feedback group received a 0.2-s, 1000-cps tone for each drop collected, whereas members of the no-feedback group did not receive any indication of their salivary rates. Table 5.7 gives differences of the form mean number of drops over 13 increase signals- mean number of drops over 13 decrease signals for the feedback group and the no-feedback group, each group consisting of 10 subjects.

Since the sample sizes are both equal to 10, we arbitrarily choose to label the feedback group data as the X-sample and the no-feedback group data as the Y-sample. Thus we have m = n = 10, N = (10+10)=20,

and d = 10. We simultaneously illustrate the calculation of the values of the empirical distribution functions F10(t) and G10(t) at the ordered combined sample values Z(1) ≤…≤ Z(20) from Table 5.7, as well as the absolute differences | F10(Z(i) ) - G10(Z(i) )|, in the following display.

0 2.071 20

8.60 19

0 8.38 18

7.74 17

5.00 16

4.29 15

3.71 14

3.25 13

2.69 12

2.55 11

2.48 10

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