WEBVTT 1 00:00:00.720 --> 00:00:01.920 Hello students, and welcome 2 00:00:01.920 --> 00:00:06.720 to Biostat ER Chapter Eight, Example Eight. 3 00:00:06.720 --> 00:00:09.780 In this example, we will learn how to calculate sample size 4 00:00:09.780 --> 00:00:12.960 for Two-Independent Samples Dichotomous Outcome 5 00:00:12.960 --> 00:00:15.480 to perform Hypothesis Testing. 6 00:00:15.480 --> 00:00:17.520 This example is from our textbook 7 00:00:17.520 --> 00:00:19.230 and it is Problem Four. 8 00:00:19.230 --> 00:00:21.127 So first, I'm going to read the problem. 9 00:00:21.127 --> 00:00:23.520 "A clinical trial was conducted comparing 10 00:00:23.520 --> 00:00:26.700 a new compound designed to improve wound healing 11 00:00:26.700 --> 00:00:29.490 in trauma patients to a placebo. 12 00:00:29.490 --> 00:00:31.200 After treatment for five days, 13 00:00:31.200 --> 00:00:34.320 58% of the patients taking the new compound 14 00:00:34.320 --> 00:00:37.920 had a substantial reduction in the size of their wound 15 00:00:37.920 --> 00:00:41.520 as compared to 44% in the placebo group. 16 00:00:41.520 --> 00:00:44.130 The trial failed to show significance. 17 00:00:44.130 --> 00:00:46.080 So how many subjects would be required 18 00:00:46.080 --> 00:00:50.310 to detect the difference in proportion observed 19 00:00:50.310 --> 00:00:53.190 in the trial with 80% power? 20 00:00:53.190 --> 00:00:58.190 A two-sided test is planned at alpha equal to 0.05." 21 00:00:58.860 --> 00:01:02.400 So first here, I have inserted the formula for you, 22 00:01:02.400 --> 00:01:07.050 and the next thing we really need to do here 23 00:01:07.050 --> 00:01:09.300 is determine the Effect Size, 24 00:01:09.300 --> 00:01:13.620 which is the denominator for this formula. 25 00:01:13.620 --> 00:01:15.030 Now for the Effect Size, 26 00:01:15.030 --> 00:01:18.420 the numerator here is the absolute difference between 27 00:01:18.420 --> 00:01:19.620 the two proportions, 28 00:01:19.620 --> 00:01:23.610 which is provided to us, 58% and 44%. 29 00:01:23.610 --> 00:01:26.040 But the denominator is the square root 30 00:01:26.040 --> 00:01:29.370 of the pooled proportion multiplied by one minus 31 00:01:29.370 --> 00:01:30.990 that pooled proportion, 32 00:01:30.990 --> 00:01:33.480 and that is not provided to us. 33 00:01:33.480 --> 00:01:37.710 However, below, I have shown you the step-by-step process 34 00:01:37.710 --> 00:01:40.170 on how to determine the pooled proportion. 35 00:01:40.170 --> 00:01:42.840 Here, the formula is, 36 00:01:42.840 --> 00:01:47.040 p equal to x1 plus x2 divided by n1 plus n2. 37 00:01:47.040 --> 00:01:49.953 So the x1 and x2 is the success, 38 00:01:51.750 --> 00:01:53.520 our number of successes, 39 00:01:53.520 --> 00:01:56.280 and n1 and n2 is the sample size. 40 00:01:56.280 --> 00:01:57.540 But here's the issue. 41 00:01:57.540 --> 00:02:01.260 We are actually here trying to determine the sample size. 42 00:02:01.260 --> 00:02:02.370 So what we need to do 43 00:02:02.370 --> 00:02:05.010 is solve this a little bit algebraically. 44 00:02:05.010 --> 00:02:09.960 So we also know that p1 is equal to x1 divided by n1 45 00:02:09.960 --> 00:02:14.190 and p2 is equal to x2 divided by n2. 46 00:02:14.190 --> 00:02:18.540 So when we do the cross-multiplication here, 47 00:02:18.540 --> 00:02:22.263 we find that x1 is actually equal to p1, 48 00:02:24.240 --> 00:02:29.220 multiplied by n1 and x2 is equal to p2 multiplied by n2. 49 00:02:29.220 --> 00:02:30.110 And given... 50 00:02:32.970 --> 00:02:34.320 The problem here, 51 00:02:34.320 --> 00:02:38.370 the n2 and the n1 is equal, 52 00:02:38.370 --> 00:02:40.573 and we can just call them n. 53 00:02:45.300 --> 00:02:50.300 So now, if we replace the n1 and n2 just by the n, 54 00:02:50.430 --> 00:02:52.320 which I have done here, 55 00:02:52.320 --> 00:02:55.530 then we can simplify everything 56 00:02:55.530 --> 00:02:57.960 by inserting these new values. 57 00:02:57.960 --> 00:02:59.730 And once we do that, 58 00:02:59.730 --> 00:03:02.040 the ns all cancel out, 59 00:03:02.040 --> 00:03:06.870 and all we are left with then is p1 plus p2 divided by two. 60 00:03:06.870 --> 00:03:09.660 And we definitely have the value of p1 and p2, 61 00:03:09.660 --> 00:03:11.640 because that is given to us, 62 00:03:11.640 --> 00:03:15.300 which is 0.58 and 0.44. 63 00:03:15.300 --> 00:03:18.670 And I have highlighted those values in yellow 64 00:03:21.090 --> 00:03:22.140 in the problem. 65 00:03:22.140 --> 00:03:24.600 So once we add those and divide it by two, 66 00:03:24.600 --> 00:03:27.120 we get 0.51, 67 00:03:27.120 --> 00:03:31.500 and now we will insert those values into the formula 68 00:03:31.500 --> 00:03:33.033 to determine the Effect Size. 69 00:03:33.960 --> 00:03:36.540 The numerator here is the absolute difference, 70 00:03:36.540 --> 00:03:38.580 and that's exactly what I've inserted here. 71 00:03:38.580 --> 00:03:39.840 And in the denominator, 72 00:03:39.840 --> 00:03:43.710 we have the pooled proportion, 73 00:03:43.710 --> 00:03:45.423 which is 0.51, 74 00:03:46.260 --> 00:03:48.570 and then one minus the pooled proportion, 75 00:03:48.570 --> 00:03:50.640 which is 0.51. 76 00:03:50.640 --> 00:03:53.700 And once we algebraically solve this, 77 00:03:53.700 --> 00:03:57.240 we get 0.28. 78 00:03:57.240 --> 00:04:00.430 Now we need to move forward in terms of inserting 79 00:04:01.770 --> 00:04:03.690 the other values into the formula, 80 00:04:03.690 --> 00:04:06.120 so that we can determine the sample size. 81 00:04:06.120 --> 00:04:09.480 So the value for Z1 minus alpha divided by two 82 00:04:09.480 --> 00:04:13.920 is the Z of the 97.5th percentile, 83 00:04:13.920 --> 00:04:15.720 and that is 1.96. 84 00:04:15.720 --> 00:04:18.750 And the value for Z1 minus beta 85 00:04:18.750 --> 00:04:20.910 is the Z of the 80th percentile, 86 00:04:20.910 --> 00:04:23.580 because one minus beta is the power of the study, 87 00:04:23.580 --> 00:04:26.460 and that is 0.84. 88 00:04:26.460 --> 00:04:28.440 So once we insert those values 89 00:04:28.440 --> 00:04:32.340 and perform the algebra, 90 00:04:32.340 --> 00:04:33.690 we get 200. 91 00:04:33.690 --> 00:04:36.930 So the sample size of n1 equal to 200, 92 00:04:36.930 --> 00:04:41.930 and the sample size of n2 is also equal to 200, 93 00:04:42.060 --> 00:04:46.320 will ensure that the test of hypothesis will have 80% power 94 00:04:46.320 --> 00:04:48.810 to detect a difference in proportions 95 00:04:48.810 --> 00:04:53.810 on the order of 0.58 versus 0.44 between groups. 96 00:04:57.050 --> 00:04:58.860 (computer mouse clicking)