The TwoSample Location Problem

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Chapter 4 The Two-Sample Location Problem
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4.1 The Wilcoxon-Mann-Whitney Rank Sum Test
for the Location-Shift Model Order
the combined X-sample and Y-sample in an
ascending order. Let S1 the rank of Y1
S2 the rank of Y2 .
. Sn the rank of Yn in the combined
sample of mn.
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Sampling from a finite Popualtion Suppose a
population consists of N numbers
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Large Sample Tests
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Table 4.1 Tritiated Water Diffusion Across
Human Chorioamnion
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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.


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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 ? lt 0. For purpose of illustration
we choose a 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
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• X Y X X Y Y X Y
• 0.73 0.74 0.80 0.83 0.88
  0.90 1.04 1.15
• (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 25689 30.

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Table 4.2 Alcohol Intake for 1 Year (cl of
Pure Alcohol)
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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 ? 0 vs H2 ? lt 0. Suppose, for
example, we choose a 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
9247171036148181 Then from equation
(4.9) we obtain
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4.2 An Estimator for ? - the Location Shift
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4.3 CIs for ? based on the Mann-Whitney
Statistic
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Large Sample CIs For large m n, approximate Ca
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4.4 Relative Efficiencies
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4.5 Kolmogorov-Smirnov test for general
differences Will abandon the location-shift
model Y X ? Assume no particular
relationship between F, the df of X-population,
and G, the df of Y-population. Still assume X and
Y both are continuous r.v.s, and independence
both within a sample and between the samples.
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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 (1010)20,
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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.
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Large Sample Approximation
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4.6 One Sample Goodness-of-Fit test
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Confidence Bands for F (unknown d.f.)