WEBVTT Kind: captions; language: en-us NOTE Treffsikkerhet: 84% (HØY) 00:00:00.100 --> 00:00:08.400 In this video we will talk about nonparametric tests, we're not going to talk about all nonparametric 00:00:08.400 --> 00:00:14.500 tests there are very many of them, and many different variants in formulations. We will only talk 00:00:14.500 --> 00:00:23.450 about two, the simplest ones that can be used as substitutes for T tests. What are nonparametric tests ? 00:00:23.450 --> 00:00:30.799 Why do you use the word nonparametric ? Because these tests do not depend on parametric NOTE Treffsikkerhet: 86% (HØY) 00:00:30.799 --> 00:00:39.200 assumption, so assumptions about parameters in the populations that we are sampling. So these are tests 00:00:39.200 --> 00:00:45.250 that then can be used when the assumptions for the parametric tests do not hold. NOTE Treffsikkerhet: 90% (HØY) 00:00:45.250 --> 00:00:52.200 For example the most frequently encountered situation is when the assumption of normality is 00:00:52.200 --> 00:01:00.300 violated. Normality is a prerequisite for running t-tests, and so when our samples are not normally 00:01:00.300 --> 00:01:08.800 distributed that is when we obtain a significant result on the Shapiro Wilkes test, then we cannot 00:01:08.800 --> 00:01:14.500 use the t-test and have to resort to a nonparametric test instead. NOTE Treffsikkerhet: 81% (HØY) 00:01:14.600 --> 00:01:22.100 These nonparametric tests that we will talk about concern differences between two groups, just like 00:01:22.100 --> 00:01:31.100 the t-test. The dependent variable must be measured on at least an ordinal scale, but since we usually 00:01:31.100 --> 00:01:37.800 use these as replacement for the t-test the most frequently encountered situation is that the 00:01:37.800 --> 00:01:43.800 dependent variable is actually a quantitative variable, a numeric variable in other words, NOTE Treffsikkerhet: 71% (MEDIUM) 00:01:43.800 --> 00:01:51.700 therefore at least on an interval scale. However the nonparametric tests work with ordinal scale 00:01:51.700 --> 00:01:54.400 dependent variables anyway. NOTE Treffsikkerhet: 91% (HØY) 00:01:54.500 --> 00:02:02.700 The independent variable, like for the t-test, must be a categorical or quantitative variable with 00:02:02.700 --> 00:02:08.949 just two levels, that is our independent variable is it to group split. NOTE Treffsikkerhet: 91% (HØY) 00:02:08.949 --> 00:02:18.000 These tests do not test a hypothesis about means because there aren't any means, means require assumptions 00:02:18.000 --> 00:02:25.800 and we want to use these tests when assumptions do not hold instead, these nonparametric 00:02:25.800 --> 00:02:33.800 tests test the hypothesis that the two groups are sampled from different distributions. With no 00:02:33.800 --> 00:02:37.900 reference to population parameters such as means. NOTE Treffsikkerhet: 87% (HØY) 00:02:37.900 --> 00:02:45.700 And there are variants for independent samples and for paired samples, we will only see the simplest 00:02:45.700 --> 00:02:52.649 of those and indeed in their simplest possible formulation. You may see different variants in books 00:02:52.649 --> 00:02:54.600 or online. NOTE Treffsikkerhet: 83% (HØY) 00:02:55.600 --> 00:03:03.750 To understand nonparametric tests the important concept we need to remember is rank. NOTE Treffsikkerhet: 91% (HØY) 00:03:03.750 --> 00:03:12.200 Because nonparametric tests generally work by replacing the actual measured values with ranks. What 00:03:12.200 --> 00:03:18.650 is a rank? A rank is the position of a value in a sorted list. NOTE Treffsikkerhet: 81% (HØY) 00:03:18.650 --> 00:03:27.000 So the smallest value gets a rank of 1 and the largest gets a rank of n, where n is how many values 00:03:27.000 --> 00:03:28.300 you have. NOTE Treffsikkerhet: 75% (MEDIUM) 00:03:28.300 --> 00:03:37.500 Equal values lead to average ranks, so they remain equal but they're still ranks. let's see an 00:03:37.500 --> 00:03:45.100 example of how this can work. Imagine we have collected these data so there are six values, perhaps these 00:03:45.100 --> 00:03:48.000 are some sort of test scores- NOTE Treffsikkerhet: 91% (HØY) 00:03:48.000 --> 00:03:52.250 The first thing we do is to sort them NOTE Treffsikkerhet: 91% (HØY) 00:03:52.250 --> 00:04:00.150 in ascending order so the first one is the smallest, the last one is the largest. NOTE Treffsikkerhet: 90% (HØY) 00:04:00.150 --> 00:04:09.700 And then we use this order to replace each value with the corresponding rank, so the smallest value 00:04:09.700 --> 00:04:13.200 50 is replaced with the number one. NOTE Treffsikkerhet: 87% (HØY) 00:04:13.200 --> 00:04:22.250 The next largest is 53 and 53 and these are replaced with the next two ranks two and three, NOTE Treffsikkerhet: 80% (HØY) 00:04:22.250 --> 00:04:32.500 and still one up to the largest the sixth value which obtains the value the rank 6. So our ranks 00:04:32.500 --> 00:04:37.400 range from 1 to the number of values that we have. NOTE Treffsikkerhet: 91% (HØY) 00:04:37.400 --> 00:04:47.700 However these two values are equal, therefore their ranks are averaged and instead of one having the 00:04:47.700 --> 00:04:55.900 value of 2 and the other having the value of 3 they both take the value of 2.5 which is the average 00:04:55.900 --> 00:04:57.900 of two and three. NOTE Treffsikkerhet: 85% (HØY) 00:04:58.700 --> 00:05:10.700 What this does is that you start with a quantitative variable in which distances are important and 00:05:10.700 --> 00:05:18.700 you end up with a variable in terms of ranks, which essentially functions as an ordinal scale. 00:05:18.700 --> 00:05:26.900 Differences are no longer important here differences, I mean differences between the original values. NOTE Treffsikkerhet: 91% (HØY) 00:05:26.900 --> 00:05:37.400 So even if this had been five instead of 50, so a crazy outlier, it will still be just one here. If 00:05:37.400 --> 00:05:46.750 this weren't 66 but 600 it would still be ranked number 6. So the relative distances of these values 00:05:46.750 --> 00:05:48.550 are irrelevant, NOTE Treffsikkerhet: 89% (HØY) 00:05:48.550 --> 00:05:58.000 and they become treated as an ordinal scale measurement by this rank transformation. This ignores any 00:05:58.000 --> 00:06:06.100 distortions that may have been introduced in the data by outliers, and it also ignores any deviations 00:06:06.100 --> 00:06:12.200 from any distribution such as the normal distribution, these issues are no longer relevant. We just 00:06:12.200 --> 00:06:18.250 get ranks up to the number of data points, therefore the only thing that matters NOTE Treffsikkerhet: 78% (HØY) 00:06:18.250 --> 00:06:23.799 from the original data set is the relative position of each measurement. NOTE Treffsikkerhet: 88% (HØY) 00:06:23.799 --> 00:06:31.600 Let's see how that works in actual analysis. The first example will concern independent samples and 00:06:31.600 --> 00:06:39.900 we're going to use the familiar situation with vocabulary scores for 15 boys and 15 girls, so these 00:06:39.900 --> 00:06:47.900 are actual vocabulary raw scores from a receptive vocabulary test, and these are the values for the 00:06:47.900 --> 00:06:53.600 boys in blue and the values for the girls in purple. NOTE Treffsikkerhet: 91% (HØY) 00:06:53.800 --> 00:07:03.100 To run a nonparametric tests on these two groups and test whether the two groups come from different 00:07:03.100 --> 00:07:11.750 distributions we first need to convert them to ranks. We do not convert them to ranks by group, 00:07:11.750 --> 00:07:21.700 instead we rank them with the two groups considered together. This is how this works; the smallest of 00:07:21.700 --> 00:07:24.150 all of these values is 47 NOTE Treffsikkerhet: 78% (HØY) 00:07:24.150 --> 00:07:37.500 so this one becomes rank 1. The next largest is 50 and this gets the rank of 2, next is 51 which 00:07:37.500 --> 00:07:46.000 gets a rank of three, next is 52 which is actually in the other group it doesn't matter there's two 52s 00:07:46.000 --> 00:07:54.550 so 4 and 5 would go here, but since they're equal they both become 4.5, so the average NOTE Treffsikkerhet: 76% (HØY) 00:07:54.550 --> 00:07:57.000 age of four and five. NOTE Treffsikkerhet: 82% (HØY) 00:07:57.000 --> 00:08:06.500 The next rank rank 6 would go to the next largest value which is 53, but there are actually two fifty 00:08:06.500 --> 00:08:17.350 threes, so six and seven and both of these get a rank of 6.5, the average of 6 and 7. And so on NOTE Treffsikkerhet: 91% (HØY) 00:08:17.350 --> 00:08:27.500 up to the top rank of 30 which is taken by the largest value of 91, the important thing here is that 00:08:27.500 --> 00:08:32.600 ranking is done with the two groups considered together. NOTE Treffsikkerhet: 77% (HØY) 00:08:32.900 --> 00:08:39.000 But the values the ranks are still separated by group, NOTE Treffsikkerhet: 91% (HØY) 00:08:39.000 --> 00:08:48.000 so what we can do next is to add up the ranks for one group. So add all these up the sum of these 00:08:48.000 --> 00:08:52.450 numbers is 274 NOTE Treffsikkerhet: 91% (HØY) 00:08:52.450 --> 00:09:01.400 and then we subtract from this number, this Factor here, so this is the sample size for this group so 00:09:01.400 --> 00:09:05.600 the number of boys times the number of boys plus one NOTE Treffsikkerhet: 60% (MEDIUM) 00:09:05.600 --> 00:09:08.250 divided by 2 NOTE Treffsikkerhet: 91% (HØY) 00:09:08.250 --> 00:09:12.500 and this results in 154. NOTE Treffsikkerhet: 91% (HØY) 00:09:12.500 --> 00:09:20.100 Alternatively we could have done this for the girls group, in which case we would have gotten a sum 00:09:20.100 --> 00:09:29.000 of 191 subtracting this factor for the sample size of the girls, and we would get 71. These numbers 00:09:29.000 --> 00:09:34.750 are mann-whitney U, this is the U statistic. NOTE Treffsikkerhet: 91% (HØY) 00:09:34.750 --> 00:09:43.100 And it doesn't matter which group will work with because one can be derived from the other. Now what 00:09:43.100 --> 00:09:49.600 does this number mean, what is this 154? This seems a little arbitrary and a little complicated 00:09:49.600 --> 00:09:56.850 because of the extra Factor you subtract, but this value actually has a very straightforward meaning. NOTE Treffsikkerhet: 91% (HØY) 00:09:56.850 --> 00:10:07.100 You can obtain this value by counting how many times a boy score exceeds a girl score, so if you 00:10:07.100 --> 00:10:12.050 compare this score with all girls scores NOTE Treffsikkerhet: 91% (HØY) 00:10:12.050 --> 00:10:21.900 you will see that it exceeds this value, so 52 is greater than 47, is greater than 51 NOTE Treffsikkerhet: 86% (HØY) 00:10:22.500 --> 00:10:26.050 and it's greater than 50. NOTE Treffsikkerhet: 80% (HØY) 00:10:26.050 --> 00:10:30.900 So this exceeds three girl values, NOTE Treffsikkerhet: 91% (HØY) 00:10:30.900 --> 00:10:33.850 the same is true of this one. NOTE Treffsikkerhet: 84% (HØY) 00:10:33.850 --> 00:10:42.349 How many values does this one exceed, so how many girl values are less than 60, this is less than 60 00:10:42.349 --> 00:10:54.800 this is less than 60, so two. This is less than 60 so three, this is less than sixty so four, five, six, 00:10:54.800 --> 00:11:04.200 seven. So it exceeds seven values and is equal to 1, so it gets seven-and-a-half, equality's get half NOTE Treffsikkerhet: 77% (HØY) 00:11:04.200 --> 00:11:12.200 the point. And then you do this for every value and the total number of times NOTE Treffsikkerhet: 84% (HØY) 00:11:12.200 --> 00:11:24.200 each of these values exceeds one of these plus half of the times there are equality's is 154, 154 is 00:11:24.200 --> 00:11:31.700 therefore the number of times a boy value exceeds a girl value, and 71 is the number of times a girl 00:11:31.700 --> 00:11:38.500 value exceeds a boy value. And these are both corrected for equalities and now you understand why 00:11:38.500 --> 00:11:42.750 these two numbers are related and you can derive one from the other. NOTE Treffsikkerhet: 85% (HØY) 00:11:42.750 --> 00:11:51.600 Because they cannot be independent, if a value is not exceeded or equal then it must be greater so it 00:11:51.600 --> 00:11:58.600 doesn't matter which one of the two you compute. What is this number mean, how can it be evaluated 00:11:58.600 --> 00:12:06.000 statistically? Clearly if one group has values that are greater than the values in the other group 00:12:06.000 --> 00:12:12.650 then this is going to be a large number, because it would exceed the other group very often. NOTE Treffsikkerhet: 84% (HØY) 00:12:12.650 --> 00:12:22.200 And this will be a small number because it will be often exceeded. In contrast if the two groups 00:12:22.200 --> 00:12:30.500 have about the same values then they will not exceed each other very often, and they will exceed each 00:12:30.500 --> 00:12:36.700 other about equally often, so this is going to be smaller, this is going to be larger and they're 00:12:36.700 --> 00:12:41.000 going to be closer to one another if the groups do not differ. NOTE Treffsikkerhet: 91% (HØY) 00:12:41.000 --> 00:12:47.700 What we need to do to evaluate probabilistically these numbers is to obtain the sampling 00:12:47.700 --> 00:12:50.100 distribution of U. NOTE Treffsikkerhet: 89% (HØY) 00:12:50.100 --> 00:12:57.850 Of course there are formulas to do that mathematically and the statistics programs do that for us, 00:12:57.850 --> 00:13:05.400 but the most straightforward way conceptually is to obtain a sampling distribution by randomly 00:13:05.400 --> 00:13:14.500 sampling from groups that aren't different and then calculating U for those groups. Let us try 00:13:14.500 --> 00:13:20.900 100,000 random shuffles of these 30 scores which amounts to random NOTE Treffsikkerhet: 86% (HØY) 00:13:20.900 --> 00:13:28.300 ranking essentially and this is the sampling distribution of U for two groups of 15. NOTE Treffsikkerhet: 79% (HØY) 00:13:28.400 --> 00:13:39.000 These are U values calculated for sets of two groups, independent groups, that have 15 values each 00:13:39.000 --> 00:13:47.800 and are randomly ranked so they're not related. This is the null distribution for U for these group 00:13:47.800 --> 00:13:56.349 sizes. As you can see the values 154 and 71 NOTE Treffsikkerhet: 71% (MEDIUM) 00:13:56.349 --> 00:14:05.600 lie symmetrically around the center of this distribution, of course because they are related. NOTE Treffsikkerhet: 91% (HØY) 00:14:05.600 --> 00:14:14.450 When one sample exceeds the other more, the other sample will exceed less, so these move in tandem 00:14:14.450 --> 00:14:23.200 outwards or inwards. And it turns out that on this sampling distribution there are about 4.4 00:14:23.200 --> 00:14:33.750 of values that are higher than 154, and 4.4 percent approximately that a 71 or lower, NOTE Treffsikkerhet: 78% (HØY) 00:14:33.750 --> 00:14:44.000 and so in total the probability that a U value would be as observed or more extreme with groups 00:14:44.000 --> 00:14:50.300 of the size is approximately 0.088. NOTE Treffsikkerhet: 79% (HØY) 00:14:50.800 --> 00:15:00.100 So that's how you use a sampling distribution to locate the observed value and count the frequency 00:15:00.100 --> 00:15:06.900 of occurrence of this or more extreme values, which is what the P value means. How frequently The 00:15:06.900 --> 00:15:15.200 observed or more extreme indices will be obtained by random sampling, this P value does not allow us 00:15:15.200 --> 00:15:21.400 to reject the null hypothesis, which was that the two groups did not differ NOTE Treffsikkerhet: 91% (HØY) 00:15:21.400 --> 00:15:29.500 in vocabulary. Therefore we cannot conclude that boys and girls have different vocabulary scores at 00:15:29.500 --> 00:15:37.800 this age we, conclude that there is no difference in vocabulary by sex, or that the variable sex and 00:15:37.800 --> 00:15:46.900 vocabulary are not associated. Let us now turn to a second example, this time with paired samples. We 00:15:46.900 --> 00:15:51.349 will use our familiar situation with the 20 children that were NOTE Treffsikkerhet: 77% (HØY) 00:15:51.349 --> 00:15:53.750 administered an intervention NOTE Treffsikkerhet: 81% (HØY) 00:15:53.750 --> 00:16:00.900 when tested before and after the intervention, and we asked if there is a difference between the 00:16:00.900 --> 00:16:09.000 before and after scores. Sow can this be approached nonparametricly, so not with a paired samples T 00:16:09.000 --> 00:16:15.800 Test, but with a nonparametric test for paired samples. Again we will begin NOTE Treffsikkerhet: 77% (HØY) 00:16:15.800 --> 00:16:24.300 with the before-and-after scores, so each child here is a row and the second set of 10 children is 00:16:24.300 --> 00:16:31.100 here and for each child we calculate the difference, what I've done here is I have separated the 00:16:31.100 --> 00:16:39.200 absolute difference from its sign. So whenever the difference is positive that is whenever the after 00:16:39.200 --> 00:16:44.700 score exceeds the before score, that's a positive difference 4 minus 3 is 1. NOTE Treffsikkerhet: 89% (HØY) 00:16:44.700 --> 00:16:54.000 So this would be a plus, I don't write anything for a plus, but when this is less, the after score is 00:16:54.000 --> 00:17:01.100 less than that before score that is a negative difference. So it's a difference of 2 in absolute 00:17:01.100 --> 00:17:08.500 terms and I enter the negative sign here. I'm going to use it but first I need the absolute value, so 00:17:08.500 --> 00:17:14.750 how big this difference is. The next step is to rank the NOTE Treffsikkerhet: 80% (HØY) 00:17:14.750 --> 00:17:22.300 differences without signs, so rank the absolute differences. First of all we throw out all the 0 00:17:22.300 --> 00:17:31.700 differences, 0 differences do not contribute to a difference between groups. In fact if all the pairs 00:17:31.700 --> 00:17:37.000 would be equal before and after there would be no difference to talk about, well then we wouldn't even 00:17:37.000 --> 00:17:44.400 need to run a test. So zero differences don't count, they're thrown out, and then we rank the rest of 00:17:44.400 --> 00:17:44.900 the absolute NOTE Treffsikkerhet: 84% (HØY) 00:17:44.900 --> 00:17:53.100 differences. There are actually nine instances of a difference of an absolute difference equal to 00:17:53.100 --> 00:18:02.500 1 in this case. So nine times nine children had an after score that was one scaled Point higher than 00:18:02.500 --> 00:18:11.050 their before score. These nine instances of one all receive the average rank, so the average the mean 00:18:11.050 --> 00:18:14.800 of 1 through 9 is 5, so NOTE Treffsikkerhet: 87% (HØY) 00:18:14.800 --> 00:18:18.300 all of these receive a rank of 5. NOTE Treffsikkerhet: 80% (HØY) 00:18:18.300 --> 00:18:27.600 Enter up to rank 9 so the next rank should be 10, the 10th value and two more values are equal to 2, 00:18:27.600 --> 00:18:33.900 so there are three occurrences of an absolute difference of two points and note that this doesn't 00:18:33.900 --> 00:18:41.900 count we're only considering the absolute values. So we have three instances of 2-point difference NOTE Treffsikkerhet: 74% (MEDIUM) 00:18:41.900 --> 00:18:47.900 ranks 10 through 12 are replaced by their mean of 11 NOTE Treffsikkerhet: 89% (HØY) 00:18:47.900 --> 00:18:50.750 and we're up to rank 12 NOTE Treffsikkerhet: 91% (HØY) 00:18:50.750 --> 00:19:01.200 so rank 13 and 14 concern to kids with a 3-point difference and since these are equal they both 00:19:01.200 --> 00:19:09.000 receive the average rank of 13.5. So these are our ranks NOTE Treffsikkerhet: 91% (HØY) 00:19:09.200 --> 00:19:13.400 of the absolute differences NOTE Treffsikkerhet: 91% (HØY) 00:19:13.400 --> 00:19:25.150 corresponding to ranks from 1 to 14, there are 14 nonzero differences here. And there is a minus here 00:19:25.150 --> 00:19:33.300 there could have been more in other situations, all we need to do now is take these signs, apply them to 00:19:33.300 --> 00:19:34.800 the ranks NOTE Treffsikkerhet: 88% (HØY) 00:19:34.800 --> 00:19:39.650 so all of these ranks will be positive NOTE Treffsikkerhet: 91% (HØY) 00:19:39.650 --> 00:19:46.200 reflecting the fact that these were positive differences. This rank will be negative, so this will 00:19:46.200 --> 00:19:50.650 become minus 11, so take on this sign. NOTE Treffsikkerhet: 86% (HØY) 00:19:50.650 --> 00:20:00.200 And then we just add them up with their signs, and it turns out the sum of all of these is 83 and 00:20:00.200 --> 00:20:07.900 this is the simplest variant for what is called Wilcoxons W statistic, there are other variants but 00:20:07.900 --> 00:20:15.550 this is the simplest one. So our Wilcoxons W is equal to 83, NOTE Treffsikkerhet: 91% (HØY) 00:20:15.550 --> 00:20:23.550 and then to evaluate that probabilistically we need to consider the sampling distribution. NOTE Treffsikkerhet: 84% (HØY) 00:20:23.550 --> 00:20:33.600 And the sampling distribution of w, for 14 nonzero differences remember we threw out the six zeros, so 00:20:33.600 --> 00:20:43.200 14 nonzero differences randomly sampled can result in this distribution of W statistics. This was a 00:20:43.200 --> 00:20:53.250 simulation with 100,000 samples, simulated samples, randomly drawn to have no difference. NOTE Treffsikkerhet: 90% (HØY) 00:20:53.250 --> 00:21:02.900 As you see this distribution is symmetric around zero, because depending on which sample you consider 00:21:02.900 --> 00:21:09.500 to be first and which you consider to be second. In other words depending on the order in which you 00:21:09.500 --> 00:21:18.100 subtract them you may end up with positive or negative differences, and since you take on the signs 00:21:18.100 --> 00:21:22.950 of the differences into the ranks after you rank the absolute differences NOTE Treffsikkerhet: 70% (MEDIUM) 00:21:22.950 --> 00:21:31.000 The same amount of difference between the groups could be either a positive or an equal 00:21:31.000 --> 00:21:38.500 negative number depending on the order of the subtraction, so this ends up being symmetric and our 00:21:38.500 --> 00:21:49.400 value of W for 83 which is there, is out at the Tails of the distribution. So 83 on the one side minus 83 on 00:21:49.400 --> 00:21:53.150 the other side and values of 83 or NOTE Treffsikkerhet: 81% (HØY) 00:21:53.150 --> 00:22:05.150 farther away from zero on either side add up to almost 0.008. So they're approximately 0.4 percent on 00:22:05.150 --> 00:22:11.800 each side with this or a larger sum of sign ranks. NOTE Treffsikkerhet: 79% (HØY) 00:22:11.800 --> 00:22:19.000 This means that we can reject the null hypothesis that the before and after scores come from the 00:22:19.000 --> 00:22:25.400 same distribution and we can conclude that there is a statistically significant difference between 00:22:25.400 --> 00:22:32.200 the before and after scores. Not between their means, between their distribution because this is a 00:22:32.200 --> 00:22:34.850 nonparametric test. NOTE Treffsikkerhet: 73% (MEDIUM) 00:22:34.850 --> 00:22:37.600 In sum NOTE Treffsikkerhet: 82% (HØY) 00:22:38.400 --> 00:22:44.800 we use rank based test when normality is violated. NOTE Treffsikkerhet: 90% (HØY) 00:22:44.800 --> 00:22:52.600 When we have two independent samples and cannot run an independent samples t-test because of the 00:22:52.600 --> 00:22:59.800 violation of the normality assumption, we then run a what is called the mann-whitney rank-sum test, 00:22:59.800 --> 00:23:09.300 which is the first one that we showed here, which provides an index named U this is statistic U 00:23:09.300 --> 00:23:14.800 which is equal to the count of times values in NOTE Treffsikkerhet: 73% (MEDIUM) 00:23:14.800 --> 00:23:17.750 one group exceed values in the other group, NOTE Treffsikkerhet: 69% (MEDIUM) 00:23:17.750 --> 00:23:25.100 and we can obtain a P value based on its sampling distribution accordingly. When we have paired 00:23:25.100 --> 00:23:32.400 samples and cannot use a paired samples t-test because the normality assumption is violated then we 00:23:32.400 --> 00:23:39.300 use what is called a wilcoxons signed rank test, the simplest variant of which I showed you in the 00:23:39.300 --> 00:23:48.000 example with the Intervention, which provides a W statistic which is equal to the sum of the NOTE Treffsikkerhet: 86% (HØY) 00:23:48.000 --> 00:23:51.700 sign ranks of the absolute differences, NOTE Treffsikkerhet: 82% (HØY) 00:23:51.700 --> 00:24:00.750 and can look up a p-value on its sampling distribution for the corresponding effective sample size. 00:24:00.750 --> 00:24:04.700 That is after discarding zero differences. NOTE Treffsikkerhet: 88% (HØY) 00:24:04.700 --> 00:24:13.600 Both of these tests test the hypothesis that the two samples or the two groups come from different 00:24:13.600 --> 00:24:22.600 distributions, and we know we have to use one of these when our test of normality is violated, that is 00:24:22.600 --> 00:24:27.600 when we get a statistically significant Shapiro Wilks test.