WEBVTT 1 00:00:03.060 --> 00:00:04.050 Hello students. 2 00:00:04.050 --> 00:00:06.390 Welcome to biostat ER example nine, 3 00:00:06.390 --> 00:00:08.073 chapter seven continuation. 4 00:00:09.120 --> 00:00:11.820 In this example, we have already learned how 5 00:00:11.820 --> 00:00:14.160 to perform hypothesis testing 6 00:00:14.160 --> 00:00:17.940 for multi sample independent continuous outcome data. 7 00:00:17.940 --> 00:00:21.840 And for that purpose we used ANOVA, 8 00:00:21.840 --> 00:00:23.553 our analysis of variance. 9 00:00:25.260 --> 00:00:29.460 For this example, I used problem 14 from our textbook. 10 00:00:29.460 --> 00:00:32.430 And for your convenience I will read the problem again 11 00:00:32.430 --> 00:00:35.763 as we are continuing with post hoc analysis. 12 00:00:37.410 --> 00:00:39.540 Suppose a hypertension trial is mounted 13 00:00:39.540 --> 00:00:41.970 and 18 participants are randomly assigned 14 00:00:41.970 --> 00:00:44.400 to one of the comparison treatments. 15 00:00:44.400 --> 00:00:47.220 Each participant takes the assigned medication 16 00:00:47.220 --> 00:00:49.500 and his or her systolic blood pressure 17 00:00:49.500 --> 00:00:52.533 is recorded after six months on the assigned treatment. 18 00:00:54.000 --> 00:00:57.570 The data is shown in table 7-58. 19 00:00:57.570 --> 00:01:02.570 Is there a difference in mean systolic blood pressure 20 00:01:02.910 --> 00:01:04.380 among treatments? 21 00:01:04.380 --> 00:01:08.490 Run the appropriate test at alpha = 0.05. 22 00:01:08.490 --> 00:01:12.780 Now we have performed the initial analysis of variance 23 00:01:12.780 --> 00:01:15.750 and we have found that there is a significant difference 24 00:01:15.750 --> 00:01:18.660 in means systolic blood pressure among treatments, 25 00:01:18.660 --> 00:01:20.610 so now we are going to proceed 26 00:01:20.610 --> 00:01:23.913 with performing post hoc analysis. 27 00:01:31.020 --> 00:01:34.233 And for our post hoc analysis, first we will compare, 28 00:01:36.870 --> 00:01:39.360 the means between the group 29 00:01:39.360 --> 00:01:41.760 that received the standard therapy 30 00:01:41.760 --> 00:01:43.560 and the placebo group. 31 00:01:43.560 --> 00:01:45.960 So in step one we are going to set up the hypothesis 32 00:01:45.960 --> 00:01:47.760 and determine level of significance. 33 00:01:47.760 --> 00:01:50.460 The null here states that there is no difference 34 00:01:50.460 --> 00:01:52.830 between the two means and the alternative states 35 00:01:52.830 --> 00:01:54.750 that there is a difference. 36 00:01:54.750 --> 00:01:59.310 The alpha here is 0.05 as provided. 37 00:01:59.310 --> 00:02:02.400 In step two we will select the appropriate test statistics, 38 00:02:02.400 --> 00:02:04.953 which is again provided here. 39 00:02:08.640 --> 00:02:09.990 In step three we are going 40 00:02:09.990 --> 00:02:12.210 to determine the decision rule. 41 00:02:12.210 --> 00:02:13.920 So the degrees of freedom here 42 00:02:13.920 --> 00:02:17.523 is going to be still 18 - 3 = 15. 43 00:02:18.510 --> 00:02:21.480 And we will reject null if the T calculated 44 00:02:21.480 --> 00:02:24.930 is less than or equal to 2.131, 45 00:02:24.930 --> 00:02:29.930 or if T calculated is greater than or equal to 2.131. 46 00:02:30.060 --> 00:02:32.350 Now in step four we are going to perform 47 00:02:33.420 --> 00:02:36.090 the calculation for the test statistics. 48 00:02:36.090 --> 00:02:38.760 So for the denominator, I'm going to go through 49 00:02:38.760 --> 00:02:40.953 that calculation first step by step. 50 00:02:41.850 --> 00:02:45.900 So here we have to take the square root of MSW, 51 00:02:45.900 --> 00:02:48.060 which is something we are going to get 52 00:02:48.060 --> 00:02:50.580 from the ANOVA table and the MSW stands 53 00:02:50.580 --> 00:02:52.200 for means square within. 54 00:02:52.200 --> 00:02:56.130 And then we are going to multiply that 55 00:02:56.130 --> 00:02:59.340 after we divide the sample size by one, 56 00:02:59.340 --> 00:03:01.350 and add them for those two groups. 57 00:03:01.350 --> 00:03:03.120 So the sample size for the groups 58 00:03:03.120 --> 00:03:04.710 that we are comparing here, 59 00:03:04.710 --> 00:03:07.320 which is the group that received the standard therapy 60 00:03:07.320 --> 00:03:09.240 and the group that received the placebo, 61 00:03:09.240 --> 00:03:11.640 and that is equal to six. 62 00:03:11.640 --> 00:03:13.530 So that is exactly what I've done here. 63 00:03:13.530 --> 00:03:16.080 The MSW is 56.4. 64 00:03:16.080 --> 00:03:17.850 I have inserted that here. 65 00:03:17.850 --> 00:03:22.080 The sample size, the small n here is six 66 00:03:22.080 --> 00:03:25.050 and it's also six here, I've inserted that. 67 00:03:25.050 --> 00:03:26.490 Then in the next step, again, 68 00:03:26.490 --> 00:03:29.700 I'm doing it step by step algebraically. 69 00:03:29.700 --> 00:03:32.823 So we are going to divide one by six, 70 00:03:33.659 --> 00:03:36.180 and we get 0.17. 71 00:03:36.180 --> 00:03:40.470 We divide one by six again we get 0.17. 72 00:03:40.470 --> 00:03:44.460 Then what we do is we add them and we get 0.34. 73 00:03:44.460 --> 00:03:49.460 We multiply that by 56.4, and then we get 19.2. 74 00:03:50.640 --> 00:03:54.630 We take the square root of 19.2 and we get 4.4. 75 00:03:54.630 --> 00:03:56.310 So now this is the denominator 76 00:03:56.310 --> 00:04:01.310 which we have to insert into our actual formula. 77 00:04:01.890 --> 00:04:05.220 So the numerator here is the difference 78 00:04:05.220 --> 00:04:10.220 between the two means, and that is 146.2 - 121, 79 00:04:10.440 --> 00:04:12.667 and then we are going to divide that by 4.4. 80 00:04:13.590 --> 00:04:16.950 So once we do the subtraction, we get 25.2, 81 00:04:16.950 --> 00:04:20.880 we divide that by 4.4, we get 5.7. 82 00:04:20.880 --> 00:04:22.320 So what is the conclusion here? 83 00:04:22.320 --> 00:04:23.580 The conclusion here is 84 00:04:23.580 --> 00:04:27.870 that our T calculated is greater than 2.131. 85 00:04:27.870 --> 00:04:30.420 So we have statistically significant evidence 86 00:04:30.420 --> 00:04:32.280 to reject null hypothesis, 87 00:04:32.280 --> 00:04:34.980 and conclude that there is a significant difference 88 00:04:34.980 --> 00:04:37.600 in the mean systolic blood pressure 89 00:04:38.520 --> 00:04:41.640 between the standard and the placebo groups. 90 00:04:41.640 --> 00:04:44.220 So we have compared two groups right now. 91 00:04:44.220 --> 00:04:47.670 Now we are going to proceed with the other comparisons. 92 00:04:47.670 --> 00:04:50.010 Next we are going to compare the placebo 93 00:04:50.010 --> 00:04:51.570 and the new treatment group. 94 00:04:51.570 --> 00:04:54.690 So again, for step one we will set up the hypothesis 95 00:04:54.690 --> 00:04:56.673 and determine level of significance. 96 00:04:58.050 --> 00:05:00.630 The null hypothesis here again is going to state 97 00:05:00.630 --> 00:05:01.920 that the means are equal. 98 00:05:01.920 --> 00:05:05.310 The alternative is going to state that they are not equal, 99 00:05:05.310 --> 00:05:08.523 And the alpha is still 0.05 as provided. 100 00:05:12.720 --> 00:05:15.240 The test statistics here will remain the same. 101 00:05:15.240 --> 00:05:16.860 It is going to be the T calculated, 102 00:05:16.860 --> 00:05:18.420 the formula is exactly the same 103 00:05:18.420 --> 00:05:22.263 and I have provided you all the information here. 104 00:05:23.970 --> 00:05:26.370 The step three will be to set up the decision rule. 105 00:05:26.370 --> 00:05:28.020 Again, that will remain the same. 106 00:05:28.020 --> 00:05:30.600 The degrees of freedom will be 15, 107 00:05:30.600 --> 00:05:33.810 and we will reject if our T calculated again 108 00:05:33.810 --> 00:05:36.990 is less than or equal to 2.131, 109 00:05:36.990 --> 00:05:41.313 or if our T calculated is greater than or equal to 2.131. 110 00:05:42.570 --> 00:05:47.570 Now in step four we have to calculate our test statistics. 111 00:05:47.850 --> 00:05:52.850 Now given that our sample size was equal 112 00:05:52.920 --> 00:05:55.350 for each three groups, we do not have 113 00:05:55.350 --> 00:05:57.480 to recalculate the denominator, 114 00:05:57.480 --> 00:06:01.170 because the MSW is still the same, which is 56.4 115 00:06:01.170 --> 00:06:04.950 as given here, and the sample size is six 116 00:06:04.950 --> 00:06:07.800 again for each group. 117 00:06:07.800 --> 00:06:10.170 So we don't have to do that calculation again, 118.;l,// 00:06:10.170 --> 00:06:12.933 we can just basically use 4.4 here. 119 00:06:13.770 --> 00:06:15.660 And what we have to do right now 120 00:06:15.660 --> 00:06:19.920 is insert that into our T statistics calculation. 121 00:06:19.920 --> 00:06:22.500 However, the numerator here will be different, 122 00:06:22.500 --> 00:06:25.440 it's going to be 146.2, 123 00:06:25.440 --> 00:06:28.920 and from that we are going to deduct 122.7 124 00:06:28.920 --> 00:06:31.327 and then we are going to divide it by 4.4. 125 00:06:32.250 --> 00:06:36.000 So once we do the algebra, we get 5.3. 126 00:06:36.000 --> 00:06:38.600 So in step five we are going to draw the conclusion. 127 00:06:39.480 --> 00:06:41.760 So we have found that our T calculated 128 00:06:41.760 --> 00:06:44.940 is actually greater than 2.131, 129 00:06:44.940 --> 00:06:47.310 so we have statistically significant evidence 130 00:06:47.310 --> 00:06:49.530 to reject the null hypothesis 131 00:06:49.530 --> 00:06:52.250 and conclude that there is a significant difference 132 00:06:52.250 --> 00:06:55.980 in the mean systolic blood pressure between the placebo 133 00:06:55.980 --> 00:06:57.813 and the new treatment groups. 134 00:07:02.070 --> 00:07:04.380 Now given we have three groups, 135 00:07:04.380 --> 00:07:06.270 we will do the last comparison, 136 00:07:06.270 --> 00:07:08.340 and that is going to be between the group 137 00:07:08.340 --> 00:07:09.900 that received the standard treatment 138 00:07:09.900 --> 00:07:12.930 and the group that received the new treatment. 139 00:07:12.930 --> 00:07:16.050 So again, in step one we are going to set up the hypothesis 140 00:07:16.050 --> 00:07:18.540 and determine the level of significance. 141 00:07:18.540 --> 00:07:21.120 So here again the null will be stating 142 00:07:21.120 --> 00:07:23.760 that the means are equal, 143 00:07:23.760 --> 00:07:25.290 and the alternative will state 144 00:07:25.290 --> 00:07:26.700 that the means are not equal, 145 00:07:26.700 --> 00:07:29.163 the alpha is still going to be 0.05. 146 00:07:30.900 --> 00:07:32.340 So in step two we are going 147 00:07:32.340 --> 00:07:34.380 to select the appropriate test statistic. 148 00:07:34.380 --> 00:07:38.610 And again, it is the same formula that we used previously, 149 00:07:38.610 --> 00:07:40.260 I have copied and pasted it here. 150 00:07:40.260 --> 00:07:43.320 I have also provided you with the information 151 00:07:43.320 --> 00:07:45.540 regarding the degrees of freedom. 152 00:07:45.540 --> 00:07:49.740 Now step three, again, it will be identical, 153 00:07:49.740 --> 00:07:52.620 the degrees of freedom again is going to be 15 154 00:07:52.620 --> 00:07:55.440 and we will reject the null if our T calculated 155 00:07:55.440 --> 00:07:58.590 is less than or equal to 2.131, 156 00:07:58.590 --> 00:08:03.590 or if our T calculated is greater than or equal to 2.131. 157 00:08:04.620 --> 00:08:08.760 Now, in step four we have to compute the test statistic. 158 00:08:08.760 --> 00:08:10.680 Now, as I mentioned previously, 159 00:08:10.680 --> 00:08:13.530 again the denominator here will remain the same 160 00:08:13.530 --> 00:08:16.890 because our sample size is equal, six per group, 161 00:08:16.890 --> 00:08:18.810 and the MSW will not change, 162 00:08:18.810 --> 00:08:22.080 which we already obtained from the ANOVA table. 163 00:08:22.080 --> 00:08:25.080 So the denominator will still be 4.4, 164 00:08:25.080 --> 00:08:27.000 however, the numerator will change. 165 00:08:27.000 --> 00:08:30.600 So now it will be 122.7, 166 00:08:30.600 --> 00:08:33.780 and from that we are going to deduct 121, 167 00:08:33.780 --> 00:08:36.990 and then we are going to divide it by 4.4. 168 00:08:36.990 --> 00:08:39.780 So once we do the subtraction 169 00:08:39.780 --> 00:08:42.183 and we do the division, we get 0.39. 170 00:08:43.500 --> 00:08:44.463 And from this, 171 00:08:45.750 --> 00:08:48.150 we are going to draw the conclusion 172 00:08:48.150 --> 00:08:50.670 and in step five, when we draw the conclusion, 173 00:08:50.670 --> 00:08:54.630 we realize that now we are failing to reject the null, 174 00:08:54.630 --> 00:08:57.780 because our T calculated, which is 0.39, 175 00:08:57.780 --> 00:09:01.020 is actually less than 2.131. 176 00:09:01.020 --> 00:09:04.110 So we do not have statistically significant evidence 177 00:09:04.110 --> 00:09:06.690 to reject null hypothesis and conclude 178 00:09:06.690 --> 00:09:08.880 that there is a significant difference 179 00:09:08.880 --> 00:09:10.980 in the mean systolic blood pressure 180 00:09:10.980 --> 00:09:13.743 between the standard and new treatment groups. 181 00:09:14.880 --> 00:09:16.470 So thank you for your attention. 182 00:09:16.470 --> 00:09:18.870 I know this was a long problem, 183 00:09:18.870 --> 00:09:22.050 and as such I divided into two videos 184 00:09:22.050 --> 00:09:24.600 and I hope that was helpful, 185 00:09:24.600 --> 00:09:26.430 and please feel free to reach out 186 00:09:26.430 --> 00:09:28.470 if you have any questions. 187 00:09:28.470 --> 00:09:30.663 Thank you, again.