Hello everyone, this is Professor Britain. >> I am back with the two factor ANOVA hypothesis test. >> I am getting ready to start step two. >> Now. >> Remember with B2 factor hypothesis test to factor ANOVA hypothesis test, when you go into step two, you go straight into the calculations. Okay, so please see my previous video where we start the hypothesis test for more information. So I'm gonna go ahead and move forward. So step two is in two stages. Stage one, this is where we calculate the total between and within sums of squares and degrees of freedom. >> So just like last time, you have total within and between sums of squares. >> And let me just remind you of where these are coming from, where these numbers are coming from. >> So the sum, this sum of squares total here, okay? So this sum of X squared minus G squared over N. >> Okay? >> Let me enlarge that. >> Okay? >> Sum of Squares Total equals the sum of X squared minus G squared over N. That's filled in with 216 minus 66 squared divided by 24. >> And let me show you where these numbers are coming from. >> So go back up here. >> And here is a sum of squares. 216 here is G is 66 and n is 24 now. >> All right, so we're going to go back down and as you can see, some of x-squared 216 G 66 in 24. >> Okay? >> So we have to add 16 minus 66 squared divided by 24. So I'm going to pull up my calculator here and we're not gonna mess with this 216 it, we're going to follow the order of operations. So 66 squared, okay? 66 squared is 4,356, okay? >> And we're going to divide that by 24. >> Okay? >> So that gives us 216 minus 101.85. >> And so 216 minus 1.581 gives us 34.5. >> So that is the sum of squares, total sum of squares within. This is sums of squares inside each treatment. >> And so scroll back up and so >> As you can see, the sums of squares in each of the treatments. >> Okay? So senseless wheres one is 2.7 sites, 2.75 sums of squares for group two is 2.75 sums of squares shift Group three is 0.75. Sums of squares for the group four is 2.75. That subject squares for group five is 0.7546 is 0.75. >> So the way you find sums of squares within is you just add up all of the sums of squares value, okay? >> Yep, 2.75 plus 2.75 plus 0.75 plus 2.75 plus 0.75 plus 0.75, and that gives us 10.5. >> Okay? >> So we get 10.5 for sums of squares within. >> Ok. And then of course you have sums of squares between. >> Okay? >> The easiest way to get sums of squares between. I've already shown how to calculate sums of squares between with the long way before, so I'm not going to do it again to save time. >> You, So the way you find sums of squares between is you take sums of squares total and subtract sums of squares within. So sums of squares total minus sum of squares within. So 34.5 minus 10.5, that's 24. >> So break out the calculator and 34.5 minus 10.5 equals 24, okay? >> Plus equals to one to four. >> So we have the sums of squares total within and between. >> So let's go ahead and do the degrees of freedom for total within in-between. >> So total is total number of participants minus one. >> We had 24 participants. >> So 24 minus one is 23 sums of squares, or sorry, degrees of freedom within. >> Degrees of freedom within is n minus k. >> K is the total number of, of conditions. >> Okay? >> So n minus k, it's going to be 24 minus six. >> Because look at this, you've got six conditions, okay? >> 123456, we have six conditions, okay? >> So 24 minus six gives you 18. >> You could also calculated out this long way, but the easy way to do it is just N minus K. >> So 24 minus six, it's 18. >> Okay? >> And then degrees of freedom between is the number of cells minus one. So there's six cells, okay? So that's 123456. So you have six cells. >> Okay, so we have six minus 16 minus one equals five. >> So we've got our Sum of Squares Between she's 24 sums of squares within just 10.5. And total is 34.5 degrees of freedom between is five, degrees of freedom within is 18, and the total is 23. >> Okay? >> Now what we have to do is we're going to be breaking down that between treatments variance into the variance for factor a, factor B, and the interaction. >> Okay, so factor a, remember that was the type of therapy, factor B, that was the dosage of the drug. >> And the interaction is an interaction between those two. >> Ok, so let's look at the sums of squares for factor a. >> Now what you see here is you see this part of the formula, this argument in the formula. >> Okay? I'm going to zoom in on that. >> So you have the sum of T squared rho divided by n row. >> So what does this mean? >> Does this mean that you're going to add up all the rows, all the t's, all the tools for the rows. >> And then square it divided by the total number of participants in each row. >> That's not how it works. >> So let's back up what you have here. >> This is going to be broken down by the number of levels you have for factor a. >> So remember, factor means independent variable level means how the independent variable is broken down. >> How many conditions are within that independent variables? >> So if you recall, we scroll back up here to our data table, zoom out. >> Factor a is a type of therapy. >> So you have these two rows. These two rows. >> Okay? >> These two rows, these represent two conditions. >> This is the condition with the workbook, this, this condition without the workbook. >> So there are two conditions, okay? >> So when you do the sums of squares of a, what you're going to have to do is to break this argument down, first of all, into one fraction for every level of the independent variable represented by factor a. Okay? >> So we have two conditions. >> We have with workbook, without Workbook. >> Ok? >> So this is why we have these two fractions, this right here. >> This is the total for the rows in one. >> Ro Okay. >> So what let me explain that. >> Let me explain what I just said. >> Okay. >> So here you see the totals in this row, and you have the totals in this row, what's going on is you add up all the totals for with workbook. You add up all the totals for without workbook. >> So let's see what that looks like. >> So you have nine plus 13 plus 19 equals 41. >> Okay? And let me do that again. >> Nine plus 13 plus 19 gives you 41. Okay? So let's go back down. Okay, so that gives you 41. Ok? So that is the sum of the first row, that's the sum of the totals in the first row squared divided by 12, where this is 12 come from. >> This 12 comes from adding up all of the number of participants in each of these conditions in this row. >> Okay? >> So we have four people here for people here, for people here. >> So four plus four plus four is 12. >> Ok? So that's where that 12 came from. >> And so that's how we get 12. >> Okay? >> Okay, so where does this 25 come from? Let's go look the second row. >> You have these totals here. >> Okay? >> You have these totals. >> So you have seven plus nine plus nine gives each one you five, okay? Gives you 25. >> Maybe a shutdown Chrome because it's using up too much space season, it too much memory right now. >> So we take that 25, we go back down. >> These were the 25 goes and again the 12 here. >> This 12 here comes from four plus four plus four. >> All right. I'm sorry. >> I apologize. >> It's a little bit noisy in the background right here, but I'm here to teach you. >> So we have 41 squared divided by 12 plus 25 squared divided by 12. Okay? >> And of course, G squared over N, we've already calculated G squared over N right here. >> So that's 181.80.5. Okay. >> 181.5. We don't need to calculate that all over again. Okay, there's no need to reinvent the wheel. >> Let's work smarter, not harder. >> Okay, so 41 squared is one thousand six hundred and eighty one twenty five squared is 625. >> Okay? >> So 1681, or 1681 divided by 12 gives you 140.8.08, excuse me, 140.08333 repeating. >> And then 625 divided by 12 gives you 52.0833. >> Okay? So I cut off the decimal two to the fourth decimal place for guard digits. >> Okay, to make the final calculation more accurate. So 140.0833 plus 52.0833. >> That gives us 192.1666, okay? Minus 181.5 equals 10.6666. >> Okay? >> Then the degrees of freedom for a is a number of rows minus one. >> So there are two rows, so two minus one is one. >> Okay? >> And now for factor B, now we're going to do the same principle, is the same principle as we did for the sums of squares for factor a. >> But now we're going to go by the columns. >> Ok, so this 16 times, let me zoom in there. This 16 is the total of those C values for the first column. For the placebo column, this is 22 is the total of the total values for the 50 milligram column and for the 100 milligram column, this is the sum of the totals for that column. >> So let's look back at the data set here. >> Okay, so we have nine plus seven. So let's see what that does. >> Nine plus seven is 16, and then 13 plus nine is 22, and then 19 plus nine gives us 28. >> So what I was doing is for each one of these, I was adding the totals for each of these columns here, adding the totals for each of those columns. >> And these 8's >> These come from the number of participants in each column. >> So four plus four is 84, plus four is 84 plus four is eight. >> So that's where the eighth come from. Alright, and G squared over N, And we already have that. It's 101.5181.5. >> Sees me my allergies are really messing me up right now and it's a little bit hard for me to think clearly, but I'm ok. >> So 181.5111.5. >> So we have 16 squared, that's 256, and then 22 squared is 484, and then 28 squared, that's 784. Okay, so let's go ahead and divide 256 divided by 83244 divided by eight is 60.5 in 784 divided by eight is 98. >> Okay? So 32 plus 60.5 plus 98 gives us 190.5. >> Then 190.5 minus 181.5 gives us nine. Ok, so here's nine. Now, the degrees of freedom for factor B is the number of columns minus one. So we're three columns minus one, that gives us two. >> And for the sons of square root of a times b is just sums of squares between minus census. Whereas a minus census wears a B says sums of squares between is 24, since this wedge of a was 10.6666 and sums of squares to be was nine. >> So 24 minus 10.6666 minus nine, this 4.3334. >> And degrees of freedom interaction is disagreed. >> And degrees of freedom between minus degrees of freedom as a minus degrees of freedom of b. >> So that's five minus one minus two gives us two. >> And so here are the answers placed in the ANOVA summary table. >> Notice here. >> Ok. >> Notice here, all of these are going to add up to 24. >> Okay? >> All of these add up to 24. >> So they might not add up to 24 perfectly because these are repeating decimals and I've rounded them. >> But let's take a look. So 10.6666 plus nine plus 4.3334. Ok, so it adds up to 24. >> Okay? >> So it does add up to 24 because this between treatments variance is being broken down into variance from factor a, factor B, Indians for action, Same thing with the degrees of freedom. >> One plus two plus two is five. >> Okay? >> We have one plus two plus two equals five, okay? >> In between, in, within out, up to the total. >> Ok, so this video is already 21 minutes long. >> I'm going to stop it here and I will continue with the hypothesis test in the next video. So thank you for your time and attention and patience. >> I apologize that I'm a little bit under the weather at the moment, so but I'm okay. >> Don't worry, I don't have covert 19. Alright, so hang tight. >> I'll be, I'll be making another video to finish out the two-way ANOVA. >> Thank you for your time and attention. >> See you next time. >> Bye.