ABOUT TWO DEPENDENT POPULATIONS HYPOTHESIS TESTING: ABOUT TWO DEPENDENT POPULATIONS

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

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.

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

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?

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.

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.

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.

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?

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

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.

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

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.

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

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?

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.

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

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