 Each version of the analysis of variance that we've looked at so far into MOVI has something in common, and that is that they all rely on population parameters or assumptions about population parameters, like the population mean or the population variance. So the one factor or two factor analysis of variance, the repeated measures analysis of variance, the ANCOVA, the analysis of covariance, even the multivariate analysis of covariance, all of them have this parametric assumption in common. Very, really, it's typical among analyses. But there are other options. These are called non parametric tests, and they're ones that don't make assumptions about population parameters. And traditionally, they're based on ranks. And so for the analysis of variance, the one way version, the non parametric or ranked version of it is called the Kruskal-Wallis test. And it's actually really easy to set up if you're concerned about non normal distributions, this might be a good choice. Now I'm going to use the iris data, we've looked at it lots of times. And I'm going to look at the sepal width and break it down by the three different species. But let's take a quick look at the sepal width and break it down in a couple of different ways. I'm going to go over here to exploration and go to descriptives. And what I'm going to choose is sepal width as the only variable that I'm looking at. And I'm going to split it by species. I'm not really concerned about this great big table. But what I do want is the plots, there's two kinds of plots that I want to get, I want to get the density plots that will split it by species. And I also want to get not the box plot, but this time I want to get the data plot, it's a dot plot. And I have a couple of options on that one I can do jittered which randomly shuffles them from left to right. Let's see if I come down here for showing up yet. But I'm going to actually get the nice and orderly stacked version because it makes it a little easier to tell what we're dealing with that was the jitter that popped up there for just a second version. Now, let's take a quick look at these analyses. What you can see from this is that in terms of sepal width, the irisatosa, their sepals are a little wider than the sepals on the iris versa color or virginica, which are pretty similar to each other. If we were doing a parametric test, we'd be looking at the means of each of these and we've done that previously. But the Kruskal Wallace looks at ranks. And in that case, you're sort of looking at where is the highest score, this is the number one, this is the second highest, the third highest, the fourth highest. And if you go through and as you come down to here, you have to start saying, Well, this might be the 17th highest and this one here might be the 34th highest and this one is 125th highest, you take every single data point across all three groups and you rank them. And so this one, I know there's 150 total, this will be the 150th right here. So we go from rank number one to rank 150. And the question is whether the ranks are evenly distributed across the three versions. That's the question that the Kruskal Wallace says is trying to answer. So let's go and see how to do the Kruskal Wallace. It's actually really simple. Just come to ANOVA. And then under non parametric, go to one way ANOVA where it says Kruskal Wallace. And we click on that. And it's actually a very small dialogue box. It only asked me for two things. It says, what's your dependent variable? What's the thing you're looking at? I'm doing just sepal width. And what's your grouping variable? And I am using the species of the iris. And I got this tiny little table that just has three numbers in it. It gives me the test statistic, which is actually a chi squared, I know that looks like an x, but it's a Greek, see it's a capital, chi, with two degrees of freedom. And this is a highly significant result, meaning that there's these aren't even remotely close to identical distributions. Now normally, when you get a significant analysis of variance, you want to do some sort of follow up with post talks. And because we're using ranked non parametric data, we need to do something a little bit different. And Jamal, because it's the option of the DSCF pairwise comparisons, which stands for Duas, Steel, Critchlow, and Fligner or Flignier, pairwise comparisons, you can think of them as similar to the Sheffa or Bonferroni or Tukey. Because what it does is it gets every possible comparison. And with three groups, there's three possible comparisons, it calculates a test statistic. And then this part here at the end that's important is the p value that tells us whether it is statistically significant. Now you can see this one's taking a while. And that's one of the things about when you're dealing with ranks, and especially if you have a lot of data is an iterative procedure, and it takes a long time to get through all of it. But here we have it. And you can see that in all three cases, the p value, the probability value of this happening at random, if the null hypothesis is true is way less than the standard 5%. This is 0.5%. And here they're even lower. And so these all serve to reject the null hypothesis that everything is similar or that their ranks are sort of randomly distributed across the three groups. And it gives us the same general impression that we got from doing the one way analysis of variance earlier that the three groups are not different. Obviously, this one's really high, these two are here. But even then the rank shows us that there are still differences between these two groups because of the way the ranks are divvied up between them. And so the Kruskal Wallace is a nonparametric or rank based analog to the one factor or one way analysis of variance that you can use, especially when you're concerned about non normality in your data.