1. ABOUT TWO DEPENDENT POPULATIONS
HYPOTHESIS TESTING:
ABOUT TWO DEPENDENT POPULATIONS

2. Two Dependent Populations
The significance test for mean difference in paired (two dependent) populations
Wilcoxon Signed Ranks Test
The Significance Test for The Difference Between Two Dependent Population Proportions
2*2 Chi Square Tests for Paired Populations

3. mean difference in paired populations
Significance test for
mean difference in paired populations
A method frequently used for assessing the effectiveness of a treatment or experimental procedure is one that makes use of related observations resulting from non-independent samples. A hypothesis test based on this type of data is known as a paired comparisons test.
Related or paired observations may be obtained in a number of ways. The same subjects may be measured before and after receiving some treatment or same subjects can be evaluated at two different conditions. Instead of performing the analysis with individual observations, we use the difference between individual pairs of observations as the variable of interest.

4. H0: d = 0 Ha: d  0 (1) H0: d = 0 Ha: d> 0 (2) H0: d = 0
(3)
When the sample differences constitute a simple random sample from a normally distributed population of differences, the test statistic for testing hypotheses about the population mean differences d is
Population variance of the differences is known
Where is the sample mean difference
Population variance of the differences is unknown

5. Example Twelve subjects participated in an experiment to study the effectiveness of a certain diet, combined with a program of exercise, in reducing serum cholesterol levels.The serum cholestrol levels for 12 subjects at the beginning of the program (Before) and at the end of the program (After) are shown in the table. Is the diet program effective?

6. H0: d = 0 Ha: d < 0 t(11,0.05)=1.7959 Since, reject H0.
Subject
Serum Cholestrol
Before
After 1
201
200 2
231
236 3
221
216 4
260
233 5
228
224 6
237 7
326
296 8
235
195 9
240
207 10
267
247 11
284
210 12
209
Difference
(after-before) di -1 +5 -5
-27 -4
-21
-30
-40
-33
-20
-74 +8
t(11,0.05)=1.7959
Since, reject H0.
We conclude that diet-exercise program is effective.

7. Wilcoxon Signed Ranks Test
Wilcoxon Signed Ranks Test is a nonparametric alternative for the significance test for mean difference in paired samples. It is used with paired data that are measured on at least the ordinal scale and is especially effective when the sample size is small and distribution of the data to be examined does not meet the assumptions of normality, as is required in paired t test.

8. n: The number of positive and negative ranks, excluding ties.
T: The sum of positive or negative ranks, depending on the proposed alternative hypothesis
n: The number of positive and negative ranks, excluding ties.
Calculation of T statistic:
Take the difference between two measurements for each subject.
New values are ranked by taking absolute values.
A. When sample size is smaller than 25, T statistic is used
B. When sample size is larger than or equal to 25, z statistic is used.

9. Example Activity scores of 12 subjects before and after the treatment are given:62 27 38 54 33 41 46 30 44 61
After 68 45 56 36 26 40
Is there any difference between before treatment and after treatment activity scores of 12 subjects?

10. Before
After
Difference (di) 62 68 -6 27 33 38 45 -7 54 56 -2 -5 41 46 -8 30 36 26 4 40 1 44 -1 61
Sorted di values
Rank
Signed Rank
1,5
-1,5 1
3,5 -1
-3,5 -2 5 -5 4 6 7 -7 -6 9 -9 11
-11 -8 12
-12

11. Assigning signes to ranks:
For zero differences:
If there is one zero difference, exclude this observation from the analysis.
If the number of zero differences is even, assign plus signs to half of the ranks and minus signs to the other half.
If the number of zero differences is odd, exclude one observation from the analysis and follow the procuderes defined in the above step.
For others:
Assign the sign of the difference to the rank.

12. Sum of negative ranks (T-)=62
Sum of positive ranks (T+)=11
T = 1,5+3,5+6=11
T =11 < Ttable = 14 , reject H0, p<0.05

13. The Significance Test for The Difference Between Two Dependent Population Proportions
Instead of measurements, if number of subjects possessing a certain characteristic are recorded on two dependent samples, we test the hypothesis whether the two dependent proportions are equal or not.

14. p1 = (a+b) / n p2 = (a+c) / n OR After Before + - Total a b a+b c d
b+d
a+b+c+d=n
p1 = (a+b) / n p2 = (a+c) / n OR
n is small
n is large

15. Example Two radiologists interpreted magnetic resonance images (MRI) of shoulders of 15 symptomatic patients. They independently evaluated the patients. The results of their evaluations are given in the table. Is there any significant difference between the propotions of positive evaluations of two radiologists?

16. Rad. A Rad. B Total + - 3 6 9 15 3 6 3 H0:p1=p2 Ha: p1 p2
Patient
Radiologist A
Radiologist B 1 + 2 3 - 4 5 6 7 8 9 10 11 12 13 14 15 3 6 9 15 3 6 3
H0:p1=p2
Ha: p1 p2
p>0.05, accept H0.

17. Chi Square Test for paired Samples
(McNemar Test)
After
Before + -
Total a b
a+b c d
c+d
a+c
b+d
a+b+c+d=n

18. Rad. A
Rad. B
Total + - 3 6 9 15
2(1,0.05)=3.841>0.444, p>0.05; accept H0.