Welcome back to our course on experimental design. Today is the first class from Module 3. Module 3 is all about experiments with single factor. That's like what we talked about in Module 2. But in Module 2, all of those experiments had only two levels. The factor only had two levels. Here the factor could have multiple levels, more than two. What you're seeing on the slide is the outline for this chapter. We're going to cover a lot of the material from this chapter in the book in this module, in particular, this first group of sections here, 3.1 through 3.5, which is basically how the basic analysis procedure for these types of experiments works, a technique called the analysis of variance. We're also going to talk a little bit about determining sample size. We'll look at a couple of other examples of single factor experiments and then a couple of topics from the latter part of the chapter that I think are important. What do we do if we have more than two factor levels? Well, the t-test that we studied before, it just doesn't work. It does a really nice job of comparing the means of two factor levels, but it doesn't really work nicely for more than two factor levels. There's no really easy way to make it work. Of course, there are lots of practical situations where there are either more than two factor levels of interest. Of course, there are lots of experiments where we have several factors, more than just a single factor, irrespective of the number of levels. We need a technique that we can generalize to deal with these situations. The technique that we use is something called the Analysis of Variance or the ANOVA. The ANOVA was developed by RA Fisher back in the early 1920s. Of course, it was originally applied in agricultural experiments, but today it is the primary analysis engine that we use for all experimental work. Here's an example. I want to illustrate how the Analysis of Variance works with a very practical example and this is a very real example. We've got an engineer who is interested in investigating the relationship between RF power setting and the etch rate for a tool that is used in semiconductor manufacturing. This tool is a plasma etch and what it does is it removes material from the surface of the wafer. We want to do that at a particular desired etch rate. If you etch too slowly, it takes too long to get wafers processed through the system. If you etch too fast, then a lot of times the etch is uneven. That can lead to some non-uniformity on the surface of the wafer and that can lead to defects. So the response variable in this experiment is etch rate. She's going to use a particular etching gas, C2F6, and a particular gap between the anode and cathode in this tool. So the only factor she really wants to study in this experiment is the RF power plat. She has four levels of power that she's interested in testing, 160 watts, 180 watts, 200 watts, and 220 watts. So this is a single factor with four levels. She's decided to test five wafers at each of these factor levels. So this is a four level single factor experiment with five replicates. So there are a total of 20 trials, 20 wafers need to be used. This is a single wafer tool, and so it's fairly straightforward to do all of these runs in random order because the only variable that has to be changed between runs is the RF power setting. Here's a schematic that shows you what the tool looks like. The wafer is in etching chamber, and it sits between the anode and the cathode. There is a vacuum pump that pumps the chamber down to a certain vacuum level, and then the gas mixture is introduced into the chamber. Then, once the gas is in the chamber, the RF power generator is started up, and that applies energy to the anode and that's excites the electrons in the gas and that's what causes the etching to actually occur. So she's run this experiment, she's tested her 20 wafers. This has all been done in random order. This table, which is Table 3-1 from the book, shows you what the experimental results look like. This data table layout is very typical for a single factor experiment. You notice that the factor levels or the treatments are in the rows. So there's a row for 160, 180, 200 and 220 watts. Then the observations are displayed across that row. The etch rate here is an angstroms per minute. Then at the end of each row, we've got the totals and we've got the averages for those five wafers that were tested at each of those sets of conditions. Now, one of the things that you learn in many basic statistics courses, and I hope you've seen or heard this idea yourself before, is the first thing you should do is plot the data. So that's what I've done down at the bottom of the slide. On the left-hand side is comparative box plots, and on the right-hand side are scattered diagrams showing you the etch rate at each power setting. Well, what's the first thing you notice when you look at either of those graphical displays? Well, the first thing I noticed is that there appears to be a pretty strong upward movement in iterate as the RF power increases. In other words, more material is getting removed, or material is getting removed more quickly as we increase the power. That makes good sense from a physics viewpoint. The other thing that I noticed, and this was pretty clear in both plots, is if the inherent wafer to wafer of variability in etch rate seems to be about the same at each of these power settings. The way I'm determining that is I'm looking at the spread in these little clusters of observations that you see here and similarly you see them over here on the box plot. So the question that we'd like to answer initially is, does change in the power setting change the mean etch rate? Based on the graphics that we've seen, you would probably say yes, it looks like they're changing the etch rate, changing the power setting does affect the mean etch rate. Now, is there an optimum level for power? Well, that's another related question. Probably our engineer has a target etch rate in mind and she'd like to be able to use this data to help her once she's decided on target etch rate to decide what power setting to use. We'd like to have an objective way to do this. We'd like to have a reliable, objective way to go at addressing these two questions. As I've said earlier, the t-test really doesn't work here because we've got more than two levels. The appropriate technique to use is something called the Analysis of Variance. As I said, the Analysis of Variance is an old technique. It's been around since the early 1920s, but it is the primary analysis tool that we use for looking at data and analyzing data from Designed Experiments. Now, the table that you see on this slide, Table 3.2, is a typical data layout for a single factor multi-level experiment. The treatments or factor levels are in the rows and then the observations are the columns. Yij is the jth observation from treatment or factor level i. I've assumed that there are little a levels of the factor and that there are little n replicates of the total experiment. Of course, this is a completely randomized design. All of the runs are made in random order. Now, look at the last two columns. The next to the last column is a column of treatment totals. Y1. is the total of all the observations from factor Level 1. Y2. is the total of all the observations from factor Level 2 and so on. This dot subscript notation is used extensively in the analysis of variance. What it means is that when you replace a subscript with a dot, that implies that you are summing up all of the observations represented by that subscript. So Y1. says take all of the observations for treatment one, Y11, Y12 on to Yn and add them up. At the bottom of that column is the grand total, Y.. that's the total of all of the observations. The last column are treatment averages. The average is represented by taking the total Y_i. and put in a bar over the Y. So Y-bar 1. is the total or is the average of all the observations in treatment one. Y-bar 2. is the average of all the observations in treatment two and so on. Y-bar.. would be the grand average. Capital N is the symbol usually used to represent the total number of runs. Capital N in this case would be little a, the total number of treatments times N, the number of replicates. Now, we're going to be considering initially what we call the fixed effects case or the fixed effects model. The fixed effects model basically assumes that the particular factor levels that are being investigated in this day were chosen by the experimenter because he or she has specific interest in those. Our interest is in seeing whether the means of those factor levels are the same. So our hypothesis testing problem will be about the equality of treatment means. The technique that we're using, the Analysis of Variance, widely used. The name Analysis of Variance stems from a partitioning of the total variability of your response variable into component parts that are consistent with an underlying model for your experiment. What's the underlying model? Well, for a single factor Analysis of Variance, the model that we typically use is the one that you see in the middle of the slide. Yij, that is the jth observation in the ith treatment is assumed to be equal to an overall mean Mu, which is constant to all of the observations in the experiment, plus a parameter toss of i, which is the effect of the individual ith treatment. So if the treatment means differ, it's because these parameters toss of i are different. Then Epsilon ij is a random error term. I can go from one to a, j can go from one to n and our experimental error, these error terms are assumed to be normally and independently distributed random variables with mean zero and a constant variance sigma square. So this is the basic setup for a single factor experiment. It's an introduction to the basic single factor ANOVA model. Now, what we will need to do is to really dig into the mechanics of this Analysis of Variance and see how it works.