this will be the third and final video on analysis of variants um please make sure you've gone and watched the first two videos um if you haven't done so yet um this video will not make sense to you so watch the first two Inova videos before you've watched this one so in the event that we reject the null hypothesis for an NOA and that the means are not equal to one another um we've got a variety of different alternative hypothesis which may be true um and an NOA doesn't initially indicate to us which of these alternative hypotheses are most appropriate now we can use multiple comparisons which are going to be discussing in this particular lecture in order to determine which of those means are different from one another now the key is that the null hypothesis needs to be rejected before we can follow this step so you cannot do multiple comparisons if the null hypothesis is not rejected so this is a key so let's go to our example from our fish in our different base systems um based on our Anova we found that the F value was greater than our critical value so we can reject our n hypothesis because we have a significant P value so there's a statistical difference in the size of these fish but then the question is well which fish are which bays are different from the others so we can use the least significant difference in order to determine which each of those Bay is different from one other another using pairwise comparisons again this is only if a NOA is significant if we don't reject the null hypothesis for Anova we wouldn't take this next step so the least significant difference is the smallest difference between two of the means that we're comparing that indicate that they're statistically different meaning that those groups are different and this method uh approaches it as if we are treating these two means as if they'd been the only ones that had been compared so any difference between these two groups that's equal to or larger than the least significant difference suggests and indicates to us that they're statistically different so in our particular case we have our three different bays and we know that there's a there's at least one group that's different than the others so now our trick is to find out which groups are different from each other statistically and we're going to use both our raw data as well as our Anova table to do this and so again the first step will be to make sure that we have a significant uh difference between at least one of our groups using an NOA and then we're going to use the following formula here in which our least significant difference is going to equal Q times the square root of the mean sum of square of the mean sum of squares within groups divided by sample size so these first first two components here on the left are things that we're familiar with so the mean sum of squares within groups that's going to come from our Anova table and the group sample size is going to come from our data now this Q value is a little bit different so we can go ahead and we can fill in that aspect of the formula but we're going to need to uh use a different table in which to determine what this Q value is and this is our student eyes T value so so we can use the following table in which to find that Q value and here we're going to use the number of groups and the error degree of freedom in which to do so so these values in the interior of the table are those Q values this is what we're after now the area in red here the number of groups um this is relatively easy to for us to discern because this is just the number of different groups that we have in our test in our case um we have three different Bays so that would be the number of groups now this error degree of freedom is something new um but if you've been doing the reading in the book you know that there's uh synonymous terms of this within groups and the error so the sum of squares within groups is synonymous to the sum of squares the error we're going to be using the sum of squares of error moving forward so this is a good time to become familiar with it and the sum of squares of error uh refers to to again this portion of the variance um when we're doing the Inova so the degrees of freedom of the error are equivalent to the degrees of freedom within groups so we can take these this information that we have and we can find where these values intersect just like if we were doing an F test a t test or an anova or a Ki Square test we then find our Q value which is 3.77 in this particular example we can then take that Q value and put it into our formula and we get our least significant difference so this is the minimum difference between the means of our groups that would indicate that they're statistically different from one another again an NOA is telling us that at least one of those groups is different but it's not telling us which group is different so now we're going to take our data with this new information and we can then discern which of those groups are different so the first step is going to be to take the mean of our respective data and then we need to calculate the difference between these means so we need to take the mean the difference in the means of the galvas bay and the Matagorda Bay fish we need to take the difference between the galison bay and Corpus chrisy fish and the Matagorda and the Corpus fish and then we need to compare these to the least significant difference and anything that is less than this least significant difference will not be different from one another statistically the only difference that we observe here is the fish between galison Bay and Corpus Christie and that the the difference in the means is 13.8 8 cm and our least significant difference meaning that minimum threshold that we need to exceed is 11.69 so it does indeed exceed that threshold so we can say statistically that the fish in Galvis Bay are different from the fish in Corpus Christie Bay so if we were to represent this visually say on a box and whisker plot we would plot our data and we've got our mean our median our cor tiles and our whiskers and we would indicate that galason bay um is one particular group group statistically we can indicate this by a and we know that it's different from Corpus Christie in the size of the fish and so we would indicate that Corpus Christie is a different group statistically now Matagorda Bay is not different from galison Bay or Corpus chrisy Bay so we labeled this with both A and B indicating to the viewer that galison bay is different from Corpus chrisy Bay but Matagorda Bay is not different from either of these two now we went through this example pretty quick um but we'll be doing so in class more and I'd also encourage you to check out section 11.3.2 of your book so we're at the end of the road for an NOA and we're going to move forward after this lecture um but some important things that you should know is when you should be using a Nova and why we're not using multiple T tests um extra calculations increased likelihood of rejecting the null hypothesis when it's two type one error you should know how to calculate the total sum of squares among group sum of squares and within group sum of squares if you're given the other two and how to determine the degrees of freedom for each of these um and then how to calculate the F value based on the mean sum of squares and determine if the null hypothesis should be rejected and then lastly as we covered in this video how to determine which means are significantly different from one another based on the least significant difference if we have uh significant uh difference in these means based on a Nova so moving forward we're going to start talking about correlation and we're going to move from data that are in particular categories to continuous numerical V variables that are compared to one another