WEBVTT 00:00:00.000 --> 00:00:02.430 The following content is provided under a Creative 00:00:02.430 --> 00:00:03.730 Commons license. 00:00:03.730 --> 00:00:06.030 Your support will help MIT OpenCourseWare 00:00:06.030 --> 00:00:10.060 continue to offer high-quality educational resources for free. 00:00:10.060 --> 00:00:12.690 To make a donation or to view additional materials 00:00:12.690 --> 00:00:16.560 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.560 --> 00:00:17.904 at ocw.mit.edu. 00:00:21.730 --> 00:00:26.680 DUANE BONING: OK, so on Tuesday, Hayden 00:00:26.680 --> 00:00:29.290 brought us up through analysis of variance, 00:00:29.290 --> 00:00:34.270 which had an implicit model of the process hidden underneath. 00:00:34.270 --> 00:00:36.940 It was mostly focused on the idea of, 00:00:36.940 --> 00:00:41.960 are there effects if you make different kinds of conditions? 00:00:41.960 --> 00:00:43.630 What we want to do is pick up on that, 00:00:43.630 --> 00:00:45.700 but get back to the model a little bit more, 00:00:45.700 --> 00:00:49.180 talk a little bit more explicitly about ways 00:00:49.180 --> 00:00:53.140 to express, from an empirical point of view, 00:00:53.140 --> 00:00:56.830 experimental data or other kinds of run data that we might have. 00:00:56.830 --> 00:00:58.330 So I'm going to pick up a little bit 00:00:58.330 --> 00:01:04.000 on this idea of modeling effects when we've got multiple inputs. 00:01:04.000 --> 00:01:07.960 Much of the ANOVA analysis was described for a single input. 00:01:07.960 --> 00:01:12.550 And then what we will do is look at linear and quadratic models, 00:01:12.550 --> 00:01:17.420 and other kinds of models for fitting of that kind of data. 00:01:17.420 --> 00:01:22.570 We'll talk a little bit about calculation or fitting of model 00:01:22.570 --> 00:01:28.480 coefficients, both from a fairly powerful and general regression 00:01:28.480 --> 00:01:34.300 idea, but also, we'll start to build up the lingo for design 00:01:34.300 --> 00:01:35.680 of experiments. 00:01:35.680 --> 00:01:39.280 One advantage of a design of experiments approach 00:01:39.280 --> 00:01:43.450 is that there are shorthand ways for rapidly assessing 00:01:43.450 --> 00:01:48.100 and, in fact, building models based on experimental data 00:01:48.100 --> 00:01:50.590 points in the selection-- careful selection 00:01:50.590 --> 00:01:55.660 of where you sample from your process or your conditions. 00:01:55.660 --> 00:01:59.920 So things like contrast will be perhaps a new terminology 00:01:59.920 --> 00:02:03.363 for you that summarize a little bit what 00:02:03.363 --> 00:02:04.780 it is we're trying to do with some 00:02:04.780 --> 00:02:08.650 of these design of experiments, or DOE. 00:02:08.650 --> 00:02:11.410 So to pick up a little bit, here's, again, 00:02:11.410 --> 00:02:12.950 the injection molding data. 00:02:12.950 --> 00:02:14.620 We've looked at this from a number 00:02:14.620 --> 00:02:16.930 of different perspectives, and we'll look at it 00:02:16.930 --> 00:02:19.187 a little bit more today. 00:02:19.187 --> 00:02:20.770 So we've got some dimension that we're 00:02:20.770 --> 00:02:23.260 measuring with an injection molding, where 00:02:23.260 --> 00:02:25.760 we have a couple of different parameters-- 00:02:25.760 --> 00:02:27.670 one of which might be the velocity 00:02:27.670 --> 00:02:31.140 that we're using for the injection. 00:02:31.140 --> 00:02:32.590 Another may be the whole time. 00:02:32.590 --> 00:02:34.660 Perhaps these are minutes-- 00:02:34.660 --> 00:02:37.180 one minute, two minute. 00:02:37.180 --> 00:02:40.840 And one would ask, is there an effect from a point of view 00:02:40.840 --> 00:02:41.650 of ANOVA? 00:02:41.650 --> 00:02:47.170 And then we can look and compare things like the spread 00:02:47.170 --> 00:02:51.280 within that particular condition, say, at one minute-- 00:02:51.280 --> 00:02:56.560 compare this spread and its mean to this spread and its mean, 00:02:56.560 --> 00:02:58.960 and make some assessment of whether there 00:02:58.960 --> 00:03:02.740 is a statistically significant effect. 00:03:02.740 --> 00:03:05.860 But we might also want to build up 00:03:05.860 --> 00:03:09.370 a causal relationship, and actually 00:03:09.370 --> 00:03:13.480 a functional relationship between hold time 00:03:13.480 --> 00:03:14.590 and dimension. 00:03:14.590 --> 00:03:16.160 And then the question becomes, well, 00:03:16.160 --> 00:03:20.080 what do we use to fit such a set of data with? 00:03:20.080 --> 00:03:22.840 Would we fit it with a line? 00:03:22.840 --> 00:03:26.900 Would we fit it with a curve that looks something like that, 00:03:26.900 --> 00:03:28.180 a curve something like that? 00:03:28.180 --> 00:03:29.740 What is appropriate? 00:03:29.740 --> 00:03:31.480 What are appropriate kinds of functions 00:03:31.480 --> 00:03:34.180 to use for fitting of these things? 00:03:34.180 --> 00:03:37.660 Now, hopefully you've got some intuition. 00:03:37.660 --> 00:03:40.250 And I'm not going to try to destroy any of that intuition 00:03:40.250 --> 00:03:40.750 today. 00:03:40.750 --> 00:03:42.820 I'm just going to try to extend it. 00:03:42.820 --> 00:03:44.580 One question for you-- 00:03:44.580 --> 00:03:46.420 hopefully you have an intuitive feel-- 00:03:46.420 --> 00:03:50.530 would you be justified in fitting this blue curve, 00:03:50.530 --> 00:03:53.020 in most cases, to this data? 00:03:56.390 --> 00:03:58.060 Well, I've now made it a red curve. 00:03:58.060 --> 00:03:58.780 Yeah? 00:03:58.780 --> 00:04:00.236 AUDIENCE: Well, [INAUDIBLE]. 00:04:04.288 --> 00:04:05.080 DUANE BONING: Yeah. 00:04:05.080 --> 00:04:07.900 AUDIENCE: So [INAUDIBLE] a physical model, or something, 00:04:07.900 --> 00:04:10.030 [INAUDIBLE] 00:04:10.030 --> 00:04:13.000 DUANE BONING: Good-- because usually the answer 00:04:13.000 --> 00:04:16.420 is it depends, and you already got the depends in there. 00:04:16.420 --> 00:04:19.552 If all you had was the pure data, certainly, 00:04:19.552 --> 00:04:21.260 you've only got really two sample points, 00:04:21.260 --> 00:04:23.320 so all you can fit is a line. 00:04:23.320 --> 00:04:25.770 But your intuition is very good there. 00:04:25.770 --> 00:04:28.210 If you have a lot of physical knowledge 00:04:28.210 --> 00:04:32.260 that, in fact, the relationship would have to be parabolic, 00:04:32.260 --> 00:04:34.390 two data points might give you enough information 00:04:34.390 --> 00:04:39.610 to be able to justifiably fit some more complicated function. 00:04:39.610 --> 00:04:41.170 We're going to be mostly focusing 00:04:41.170 --> 00:04:44.290 on empirical data and just sort of the limits of what you can 00:04:44.290 --> 00:04:46.270 infer from the data although. 00:04:46.270 --> 00:04:50.410 We're not going to be, at least today, 00:04:50.410 --> 00:04:54.670 trying to mix in both the physical functional 00:04:54.670 --> 00:04:59.260 dependencies and empirical. 00:04:59.260 --> 00:05:01.480 So one key question then is, what 00:05:01.480 --> 00:05:04.922 functional form might be justified by the data? 00:05:04.922 --> 00:05:06.880 In this case, if I've only got two data points, 00:05:06.880 --> 00:05:11.650 maybe it's only a very simple linear model. 00:05:11.650 --> 00:05:14.110 If I had more data, well, I might 00:05:14.110 --> 00:05:19.850 have enough to fit parabolic or other, more complicated data. 00:05:19.850 --> 00:05:22.770 But in addition now, when we start to go to two inputs-- 00:05:22.770 --> 00:05:25.280 this is this is, in fact, the same data, 00:05:25.280 --> 00:05:28.190 where we have not only hold time, 00:05:28.190 --> 00:05:30.560 as we did on the previous screen, 00:05:30.560 --> 00:05:33.530 but we may also be playing with the velocity 00:05:33.530 --> 00:05:36.770 and then doing some number of replicates-- something like 14 00:05:36.770 --> 00:05:40.880 or 15 runs at the low condition for velocity 00:05:40.880 --> 00:05:42.590 and low condition for hold time-- 00:05:42.590 --> 00:05:45.450 and then measure the data. 00:05:45.450 --> 00:05:47.660 And what we have here is conditions 00:05:47.660 --> 00:05:51.650 where we've changed both of those conditions, 00:05:51.650 --> 00:05:56.690 and it may well be the case that these two inputs interact-- 00:05:56.690 --> 00:06:01.250 that the effective velocity gets amplified 00:06:01.250 --> 00:06:05.720 when I've got a longer hold time than a shorter hold time. 00:06:05.720 --> 00:06:08.550 So it's not necessarily just an additive model. 00:06:08.550 --> 00:06:12.860 So we also need to mix in this issue of interaction, when 00:06:12.860 --> 00:06:14.930 we start thinking about model form-- 00:06:14.930 --> 00:06:19.670 not just sort of the functional dependence terms of powers 00:06:19.670 --> 00:06:24.320 or for the shape of the dependence on a single input, 00:06:24.320 --> 00:06:28.880 but also when we have multiple inputs, if there 00:06:28.880 --> 00:06:31.670 are interactions between them. 00:06:34.262 --> 00:06:35.970 So we can also turn this question around. 00:06:35.970 --> 00:06:40.020 I can say, depending on how much data I have, what kind of model 00:06:40.020 --> 00:06:41.160 might I fit? 00:06:41.160 --> 00:06:42.630 Can turn it around and say, I would 00:06:42.630 --> 00:06:46.670 like to be able to fit a model up to a certain kind of form-- 00:06:46.670 --> 00:06:49.270 you can turn it around and ask the question, 00:06:49.270 --> 00:06:50.760 what data points do I need? 00:06:50.760 --> 00:06:53.730 How much data do I need, and what's 00:06:53.730 --> 00:06:56.430 the location of those data points 00:06:56.430 --> 00:06:59.250 in order to be able to fit different functional forms? 00:06:59.250 --> 00:07:03.180 And that becomes the design of the experiment 00:07:03.180 --> 00:07:09.430 in order to be able to meet your modeling or estimation needs. 00:07:09.430 --> 00:07:11.220 So what we'll be doing is dealing 00:07:11.220 --> 00:07:15.630 with some models of the process, where 00:07:15.630 --> 00:07:19.110 we have multiple inputs or multiple treatments-- 00:07:19.110 --> 00:07:22.690 not just different conditions or levels at these, but in fact, 00:07:22.690 --> 00:07:26.070 we may have multiple different kinds of treatments, 00:07:26.070 --> 00:07:28.720 such as the velocity and the hold time. 00:07:28.720 --> 00:07:31.080 And so in general, we might talk about processes 00:07:31.080 --> 00:07:34.920 that have k input. 00:07:34.920 --> 00:07:38.310 So in our previous case, we had two inputs. 00:07:38.310 --> 00:07:40.470 And then we can look at different combinations 00:07:40.470 --> 00:07:44.310 of the data, depending on how many levels 00:07:44.310 --> 00:07:46.750 we pick for each of those inputs. 00:07:46.750 --> 00:07:49.110 So a level is simply where you're 00:07:49.110 --> 00:07:55.050 sampling along that axis of that particular factor or input. 00:07:55.050 --> 00:07:58.740 And we'll be talking especially about full factorial models, 00:07:58.740 --> 00:08:01.140 where we're looking at every combination 00:08:01.140 --> 00:08:07.140 of the discrete selection of those levels for those factors 00:08:07.140 --> 00:08:08.770 that we're interested in. 00:08:08.770 --> 00:08:12.690 So if I've got a two level selection or choice 00:08:12.690 --> 00:08:15.460 for each of our factor levels, then, in general, 00:08:15.460 --> 00:08:18.450 we can have 2 to the k combinations 00:08:18.450 --> 00:08:20.700 of different discrete sample points, 00:08:20.700 --> 00:08:24.670 different discrete treatment sets. 00:08:24.670 --> 00:08:26.070 You can extend this, of course. 00:08:26.070 --> 00:08:29.790 You might be looking at tree level experiments for each 00:08:29.790 --> 00:08:33.960 of our factor, and then I would have 3 to the k combinations. 00:08:33.960 --> 00:08:36.059 And you'll detect a trend here that's not 00:08:36.059 --> 00:08:38.880 especially encouraging. 00:08:38.880 --> 00:08:43.799 What happens as k gets big? 00:08:43.799 --> 00:08:45.600 We have exponential growth in the number 00:08:45.600 --> 00:08:48.010 of experimental points that we need. 00:08:48.010 --> 00:08:51.150 Similarly, as the number of levels go up, 00:08:51.150 --> 00:08:53.140 you can fit more powerful models, 00:08:53.140 --> 00:08:56.790 but again, you need a lot more data fairly rapidly. 00:08:56.790 --> 00:08:58.890 Today we'll be just pretty much focusing 00:08:58.890 --> 00:09:03.540 on relatively simple small numbers of k 00:09:03.540 --> 00:09:05.020 and small numbers of level. 00:09:05.020 --> 00:09:07.890 In fact, traditional 2 to the k design 00:09:07.890 --> 00:09:11.070 is what we'll be talking the most about. 00:09:11.070 --> 00:09:12.940 When you get back from spring break, 00:09:12.940 --> 00:09:16.320 we'll also come back to clever approaches 00:09:16.320 --> 00:09:20.400 out of design of experiments for subsampling the space-- 00:09:20.400 --> 00:09:24.780 basically, not having to explore all of the combinations 00:09:24.780 --> 00:09:26.640 because, in fact, all of the combinations 00:09:26.640 --> 00:09:29.580 may give you more data and fit a more powerful model than you're 00:09:29.580 --> 00:09:30.690 even really interested in. 00:09:34.520 --> 00:09:38.270 If I were doing, say, a 3 to the k, 00:09:38.270 --> 00:09:40.670 I might be able to actually estimate 00:09:40.670 --> 00:09:43.760 all kinds of very subtle three-factor interactions 00:09:43.760 --> 00:09:47.360 or four-factor interactions, but I might actually not 00:09:47.360 --> 00:09:48.530 believe they exist. 00:09:48.530 --> 00:09:50.380 I may had prior knowledge they don't exist. 00:09:50.380 --> 00:09:58.280 And so we'll look at reduced models next time. 00:09:58.280 --> 00:10:01.100 There is kind of just a general question. 00:10:01.100 --> 00:10:04.700 Why would you even want to model or use more than one input? 00:10:04.700 --> 00:10:07.190 Why wouldn't you want to simplify your process, 00:10:07.190 --> 00:10:09.170 make it as simple as possible? 00:10:09.170 --> 00:10:11.600 Why might more than one input be especially 00:10:11.600 --> 00:10:14.180 important for manufacturing process control? 00:10:16.750 --> 00:10:19.745 These are fairly obvious, so don't overthink too much. 00:10:19.745 --> 00:10:21.370 Why would you need more than one input? 00:10:25.538 --> 00:10:28.080 AUDIENCE: Because you have more than one input in the world-- 00:10:28.080 --> 00:10:30.130 DUANE BONING: You have more than one input in the real world. 00:10:30.130 --> 00:10:32.380 Why do you have more than one input in the real world? 00:10:34.880 --> 00:10:36.507 AUDIENCE: [INAUDIBLE] complex. 00:10:36.507 --> 00:10:38.340 You can manipulate them in a number of ways, 00:10:38.340 --> 00:10:44.550 so [INAUDIBLE] process might have temperature and pressure, 00:10:44.550 --> 00:10:47.730 different chemical baths you could use, [INAUDIBLE] 00:10:47.730 --> 00:10:48.588 processing. 00:10:48.588 --> 00:10:49.380 DUANE BONING: Good. 00:10:49.380 --> 00:10:52.360 AUDIENCE: So they all create slightly different outputs, 00:10:52.360 --> 00:10:56.420 but [INAUDIBLE] very similar outputs at the same time. 00:10:56.420 --> 00:10:57.170 DUANE BONING: Yes. 00:10:57.170 --> 00:11:00.020 OK, so I think you've touched on both of them there, 00:11:00.020 --> 00:11:01.790 as a freeform spoke. 00:11:01.790 --> 00:11:06.650 One is you often have more than one output. 00:11:06.650 --> 00:11:08.820 So it would actually be odd if you 00:11:08.820 --> 00:11:11.720 were trying to achieve just one parameter-- say, 00:11:11.720 --> 00:11:15.440 a length parameter-- with a dozen different inputs 00:11:15.440 --> 00:11:16.110 all at once. 00:11:16.110 --> 00:11:19.370 In fact, that would be a really challenging design problem. 00:11:19.370 --> 00:11:22.010 Often, they do interact, but very often, 00:11:22.010 --> 00:11:24.320 what you're actually after are fairly 00:11:24.320 --> 00:11:26.660 good solid one-to-one correspondences where, 00:11:26.660 --> 00:11:31.770 if I play with this knob, I have good control over this output. 00:11:31.770 --> 00:11:36.290 So you often strive for almost univariate, very simple 00:11:36.290 --> 00:11:38.390 direct control. 00:11:38.390 --> 00:11:42.470 You can't always achieve that, and so often you've 00:11:42.470 --> 00:11:44.780 got a multiple input and multiple output 00:11:44.780 --> 00:11:47.120 problem with interactions. 00:11:47.120 --> 00:11:49.370 But even if you only had one output, 00:11:49.370 --> 00:11:53.060 you might actually want two inputs, because one of them 00:11:53.060 --> 00:11:57.230 may actually give you control over the mean of the parameter 00:11:57.230 --> 00:12:02.750 and another input might let you deal with, or compensate 00:12:02.750 --> 00:12:08.490 for, or generate, or change the variance of the process. 00:12:08.490 --> 00:12:10.340 And so it's actually very nice to be 00:12:10.340 --> 00:12:12.410 able to deal with the process robustness, 00:12:12.410 --> 00:12:15.800 to be able to freely recenter the process without changing 00:12:15.800 --> 00:12:19.250 your variance, for example. 00:12:19.250 --> 00:12:24.363 So just highlighting that the practical world-- you often do 00:12:24.363 --> 00:12:26.780 have more than one input, and there are good reasons where 00:12:26.780 --> 00:12:30.140 you can actually use those. 00:12:30.140 --> 00:12:33.140 So we're going to mostly be dealing with empirical models 00:12:33.140 --> 00:12:36.950 here, and here's an example of a fairly simple 00:12:36.950 --> 00:12:38.480 general linear model. 00:12:42.080 --> 00:12:42.580 Question? 00:12:46.200 --> 00:12:48.260 Is there a question from Singapore? 00:12:48.260 --> 00:12:52.430 Somebody maybe pushed their button? 00:12:52.430 --> 00:12:55.490 [INAUDIBLE] like the microphone was-- 00:12:55.490 --> 00:12:57.020 we got a close-up view of you. 00:12:57.020 --> 00:13:00.440 We know you're awake, though-- 00:13:00.440 --> 00:13:02.750 good. 00:13:02.750 --> 00:13:06.328 OK, so here's just a quick overview of the kind of models 00:13:06.328 --> 00:13:07.370 that we'll be looking at. 00:13:07.370 --> 00:13:10.190 We've got some output. 00:13:10.190 --> 00:13:13.010 Here it's expressed in a univariate case. 00:13:13.010 --> 00:13:16.790 We'll extend this to multiple outputs in a minute. 00:13:16.790 --> 00:13:19.520 But I might have a single output, 00:13:19.520 --> 00:13:23.600 and the true value here may be eta. 00:13:23.600 --> 00:13:28.560 Again, I'm using a little bit the Greek to indicate truth, 00:13:28.560 --> 00:13:29.780 if you will. 00:13:29.780 --> 00:13:34.070 And it's got some overall mean. 00:13:34.070 --> 00:13:36.380 We might model that with a single coefficient, 00:13:36.380 --> 00:13:37.550 some beta 0. 00:13:37.550 --> 00:13:43.460 And then it's got single factor direct dependencies 00:13:43.460 --> 00:13:47.390 on some x sub i input. 00:13:47.390 --> 00:13:54.680 So x sub i here is indicating which factor it is. 00:13:54.680 --> 00:13:59.300 So I might have x1 and x2 would be a two-factor-- 00:13:59.300 --> 00:14:01.190 two different kinds of input, sort of 00:14:01.190 --> 00:14:03.875 like our injection molding problem. 00:14:03.875 --> 00:14:05.900 I might have a direct linear dependence 00:14:05.900 --> 00:14:09.270 and coefficient on each of those. 00:14:09.270 --> 00:14:13.190 And then we'll also be looking at interaction terms, ways 00:14:13.190 --> 00:14:17.570 of looking at a synergistic effect between the two 00:14:17.570 --> 00:14:18.740 of these terms-- 00:14:18.740 --> 00:14:20.450 still first-order. 00:14:20.450 --> 00:14:27.290 These are only direct x1, x2 kinds of interactions, 00:14:27.290 --> 00:14:30.240 but we'll see that actually gives you quite a bit of power 00:14:30.240 --> 00:14:35.000 for dealing with some interesting behavior. 00:14:35.000 --> 00:14:39.240 Then we'll also talk about are these higher order terms. 00:14:39.240 --> 00:14:43.070 So in truth, the model may have a more complicated functional 00:14:43.070 --> 00:14:44.330 dependence. 00:14:44.330 --> 00:14:47.240 What we're hoping is, in many cases, 00:14:47.240 --> 00:14:51.920 that the linear version of the model 00:14:51.920 --> 00:14:55.730 captures most of the behavior, and that these higher 00:14:55.730 --> 00:15:00.020 order model terms are smaller effects. 00:15:00.020 --> 00:15:03.950 And so often, we'll actually be ignoring these higher order 00:15:03.950 --> 00:15:10.430 terms, absorbing whatever errors there are as a result of that, 00:15:10.430 --> 00:15:13.640 and referring to those as a model form or model structure 00:15:13.640 --> 00:15:18.080 error, and distinguish that from residual error 00:15:18.080 --> 00:15:22.160 or other kinds of experimental error, 00:15:22.160 --> 00:15:23.630 or just replication error. 00:15:26.440 --> 00:15:28.930 So let's see-- couple other definitions just that we've got 00:15:28.930 --> 00:15:29.725 in here-- 00:15:29.725 --> 00:15:34.780 in here, you'll notice the k subscript indicates 00:15:34.780 --> 00:15:38.050 the number of factors, just like we were describing before-- 00:15:38.050 --> 00:15:41.770 two-factor, three-factor kinds of experiments. 00:15:41.770 --> 00:15:48.340 And then we often are forming sums over some particular-- 00:15:48.340 --> 00:15:52.040 either factor or input levels within those factors. 00:15:52.040 --> 00:15:56.300 So I'll try to be a little bit careful on what we've got then. 00:15:56.300 --> 00:16:02.080 So with that simple two-order or two input model, 00:16:02.080 --> 00:16:06.230 where k is equal to 2 in this case, 00:16:06.230 --> 00:16:08.500 we've got something that we might 00:16:08.500 --> 00:16:11.440 try to use with our injection molding case. 00:16:11.440 --> 00:16:15.400 That was a two input model-- 00:16:15.400 --> 00:16:19.030 injection velocity and hold time. 00:16:19.030 --> 00:16:22.940 We've got four coefficients. 00:16:22.940 --> 00:16:25.040 Quick question for you-- how many 00:16:25.040 --> 00:16:27.200 data points do you think you need 00:16:27.200 --> 00:16:31.280 at least a minimum of in order to be able to fit those four 00:16:31.280 --> 00:16:32.397 coefficients? 00:16:38.740 --> 00:16:42.960 If I did just one experiment and got one output, 00:16:42.960 --> 00:16:46.890 would that be enough data to estimate four coefficients? 00:16:46.890 --> 00:16:47.390 No. 00:16:51.130 --> 00:16:52.720 Four? 00:16:52.720 --> 00:16:54.398 Or was that just a wave of your hand? 00:16:54.398 --> 00:16:54.940 AUDIENCE: No. 00:16:54.940 --> 00:16:56.950 DUANE BONING: It was close. 00:16:56.950 --> 00:16:57.588 AUDIENCE: Yeah. 00:16:57.588 --> 00:16:59.380 DUANE BONING: So this is a simple extension 00:16:59.380 --> 00:17:00.700 of that linear case. 00:17:00.700 --> 00:17:05.290 If you had only one input and you wanted to fit a line, 00:17:05.290 --> 00:17:08.770 you would need at least two data points. 00:17:08.770 --> 00:17:10.038 Yeah? 00:17:10.038 --> 00:17:12.310 AUDIENCE: [INAUDIBLE] 00:17:12.310 --> 00:17:15.030 DUANE BONING: Oh, OK. 00:17:15.030 --> 00:17:17.740 AUDIENCE: Because that's a zero-- 00:17:17.740 --> 00:17:20.140 DUANE BONING: Yes. 00:17:20.140 --> 00:17:27.310 AUDIENCE: [INAUDIBLE] relative to your [INAUDIBLE]?? 00:17:27.310 --> 00:17:28.060 DUANE BONING: Yes. 00:17:28.060 --> 00:17:28.900 That's a good point. 00:17:32.230 --> 00:17:37.060 So if these are normalized in some way, 00:17:37.060 --> 00:17:41.290 such that there is a 0 in between, such 00:17:41.290 --> 00:17:46.030 that also the average of these are 0, 00:17:46.030 --> 00:17:47.330 then this would be the mean. 00:17:47.330 --> 00:17:50.185 So this would be more general-- 00:17:50.185 --> 00:17:51.490 some offset term. 00:17:51.490 --> 00:17:56.890 It may not necessarily be the mean-- good point, good point. 00:17:59.740 --> 00:18:03.430 OK, so part of the question is I think there's-- it's fairly 00:18:03.430 --> 00:18:05.950 clear intuition that you're probably going to need at least 00:18:05.950 --> 00:18:09.290 four data points in order to be able to fit a four-coefficient 00:18:09.290 --> 00:18:09.790 model. 00:18:09.790 --> 00:18:13.000 We need at least that number, but there 00:18:13.000 --> 00:18:16.960 are some additional subtleties. 00:18:16.960 --> 00:18:19.170 In this case, we've got at least two factors, 00:18:19.170 --> 00:18:21.520 so we know we've got two factors at work here. 00:18:21.520 --> 00:18:24.400 And I've got to at least sample across both 00:18:24.400 --> 00:18:28.600 of those input factors of both of those treatment things. 00:18:28.600 --> 00:18:30.610 But then there's a question, especially 00:18:30.610 --> 00:18:32.620 with this crazy interaction term-- 00:18:32.620 --> 00:18:37.000 how many levels do you really need in each case? 00:18:37.000 --> 00:18:41.740 Notice also here that, if I only have four data points, 00:18:41.740 --> 00:18:45.115 there's no way for me to really estimate higher order terms. 00:18:47.720 --> 00:18:50.010 And we'll also chat a little bit, 00:18:50.010 --> 00:18:53.520 because if I have exactly four data points, 00:18:53.520 --> 00:19:00.220 it's also not easy for me to estimator terms, 00:19:00.220 --> 00:19:03.370 because if I have four data points, I can exactly fit-- 00:19:03.370 --> 00:19:08.020 I get a direct solution for the four coefficients. 00:19:08.020 --> 00:19:10.660 And I can perfectly fit those-- 00:19:10.660 --> 00:19:13.570 I may be wrong in terms of actually interpolating 00:19:13.570 --> 00:19:16.480 between those, but I have a direct solution 00:19:16.480 --> 00:19:18.350 for those four coefficients. 00:19:18.350 --> 00:19:19.780 So part of the thing we'll get at 00:19:19.780 --> 00:19:27.160 is, how might you select levels and data samples 00:19:27.160 --> 00:19:30.700 so that you have more data to be able to do 00:19:30.700 --> 00:19:34.570 additional inferences on your model-- 00:19:34.570 --> 00:19:38.530 things like, what is the actual confidence level I 00:19:38.530 --> 00:19:41.080 have that an effect is real? 00:19:41.080 --> 00:19:45.760 What's my estimation not just on a point estimate for a model 00:19:45.760 --> 00:19:49.030 coefficient-- a beta 0 here or a beta 1-- 00:19:49.030 --> 00:19:51.850 but also, what is an integral estimate 00:19:51.850 --> 00:19:53.930 for those kinds of coefficients? 00:19:53.930 --> 00:19:56.410 So we'll get to many of those things. 00:19:56.410 --> 00:19:59.050 OK, here's back to our injection molding case-- 00:19:59.050 --> 00:20:01.330 dimension versus old time. 00:20:01.330 --> 00:20:06.010 I might try to fit that with a very simple one input linear 00:20:06.010 --> 00:20:13.210 model, some offset beta 0 and some beta 1 linear dependence. 00:20:13.210 --> 00:20:15.190 And we've already said it looks like I'm going 00:20:15.190 --> 00:20:16.630 to need at least two levels. 00:20:16.630 --> 00:20:21.100 I'm going to label those with an x underscore minus and an x sub 00:20:21.100 --> 00:20:21.640 plus-- 00:20:21.640 --> 00:20:25.680 so a high level and a low level. 00:20:25.680 --> 00:20:28.390 We'll actually talk about different labeling strategies 00:20:28.390 --> 00:20:31.780 for these things that make some of the calculations 00:20:31.780 --> 00:20:34.040 a little bit easier. 00:20:34.040 --> 00:20:36.580 So if I, again, do just one trial 00:20:36.580 --> 00:20:38.870 at each level, what I've got-- 00:20:38.870 --> 00:20:41.440 there's a little bit more notation for us 00:20:41.440 --> 00:20:43.750 in very simple vector notation-- 00:20:43.750 --> 00:20:48.010 what I've got is one factor, two levels, 00:20:48.010 --> 00:20:51.380 if I'm just doing a one-input model. 00:20:51.380 --> 00:20:55.450 So what I will have is two measurements, an eta 1 00:20:55.450 --> 00:20:56.530 and an eta 2. 00:20:56.530 --> 00:20:59.260 Those are my two actual sample points. 00:20:59.260 --> 00:21:04.340 And what I've done is run them at two different conditions. 00:21:04.340 --> 00:21:07.240 So in fact, these indicate the two different conditions 00:21:07.240 --> 00:21:08.710 that I ran at. 00:21:08.710 --> 00:21:12.340 And what I'm trying to do is fit the beta 0 and the beta 1 00:21:12.340 --> 00:21:13.810 for this case. 00:21:13.810 --> 00:21:16.510 Now, because I've got exactly the right amount of data, 00:21:16.510 --> 00:21:19.890 I'm going to have to estimate that the noise is 0, 00:21:19.890 --> 00:21:22.280 as we'll see in just a second. 00:21:22.280 --> 00:21:27.010 Now, I can express the two sets of samples-- 00:21:27.010 --> 00:21:30.340 so I've got, basically, an eta 1 is 00:21:30.340 --> 00:21:36.670 equal to some beta 0 plus beta 1x at the low condition. 00:21:36.670 --> 00:21:39.640 And I've, similarly, got eta 2. 00:21:39.640 --> 00:21:44.680 I'm estimating as beta 0 plus beta 1 at the high condition. 00:21:44.680 --> 00:21:47.140 That's just a system of two equations 00:21:47.140 --> 00:21:52.120 I can write it in matrix form, as shown down here. 00:21:52.120 --> 00:21:56.810 So I've got this condition times my two coefficient. 00:21:56.810 --> 00:22:00.910 So I've got my beta 0 and my beta 1. 00:22:00.910 --> 00:22:04.630 That's my first equation, giving me my eta 1-- 00:22:04.630 --> 00:22:07.570 and similarly, for my eta 2. 00:22:07.570 --> 00:22:12.100 Now, since we've got exactly the same number of model 00:22:12.100 --> 00:22:18.880 coefficients and model runs, my x is square, 00:22:18.880 --> 00:22:25.600 and I've got a direct solution to this equation-- 00:22:25.600 --> 00:22:29.410 where, again, eta is 0, because I don't have any replicates, 00:22:29.410 --> 00:22:31.930 I don't have any noise, and so I can simply 00:22:31.930 --> 00:22:39.050 directly invert that and solve directly for my coefficient. 00:22:39.050 --> 00:22:41.350 So my purpose here was both to remind us 00:22:41.350 --> 00:22:45.310 that there is some vector notation that just follows what 00:22:45.310 --> 00:22:49.750 you would think of naturally, and in the special case 00:22:49.750 --> 00:22:53.530 where you've only got exactly just barely enough data, 00:22:53.530 --> 00:22:56.720 there's a direct solution. 00:22:56.720 --> 00:22:58.840 So that's pretty straightforward. 00:22:58.840 --> 00:23:02.770 But what we'd often like to do is 00:23:02.770 --> 00:23:07.270 remember that there's noise in our process-- 00:23:07.270 --> 00:23:11.110 that, if I were to, in fact, run at the same condition 00:23:11.110 --> 00:23:14.980 multiple times, I've probably got some variance in there. 00:23:14.980 --> 00:23:17.200 It may be measurement error. 00:23:17.200 --> 00:23:20.420 It may be underlying process variation. 00:23:20.420 --> 00:23:24.130 So what do we do when we've got replicates? 00:23:24.130 --> 00:23:26.710 First off, we probably want to design 00:23:26.710 --> 00:23:30.950 the process or the experimental design for the process 00:23:30.950 --> 00:23:33.130 so that we do sample at multiple-- 00:23:33.130 --> 00:23:34.990 at the same condition multiple times 00:23:34.990 --> 00:23:37.525 so we can estimate these variances. 00:23:37.525 --> 00:23:39.400 I'm going to need that information for things 00:23:39.400 --> 00:23:42.230 like confidence intervals on the coefficients. 00:23:42.230 --> 00:23:45.970 So how do we deal with data where 00:23:45.970 --> 00:23:52.210 there is some spread, even at the same factor 00:23:52.210 --> 00:23:56.230 level for hold time? 00:23:56.230 --> 00:24:04.390 Well, now we're into an over-specified set 00:24:04.390 --> 00:24:05.380 of linear equations. 00:24:05.380 --> 00:24:09.250 I've got multiple sets of data for the same input. 00:24:09.250 --> 00:24:10.720 I've got multiple y's. 00:24:10.720 --> 00:24:12.880 I got multiple outputs. 00:24:12.880 --> 00:24:16.150 And what we'll generally be doing 00:24:16.150 --> 00:24:20.710 is trying to do our best fit in the same intuitive way you've 00:24:20.710 --> 00:24:25.090 probably been doing regression modeling or fitting of lines 00:24:25.090 --> 00:24:27.400 to data for a long time. 00:24:27.400 --> 00:24:30.280 We have a general approach of trying 00:24:30.280 --> 00:24:34.840 to get the best fit in a conceptual approach 00:24:34.840 --> 00:24:37.660 of minimizing some error metric. 00:24:37.660 --> 00:24:42.100 The most standard one is a sum of squared deviations 00:24:42.100 --> 00:24:45.140 of your data from your prediction. 00:24:45.140 --> 00:24:49.030 So we often talk about least squares or minimum 00:24:49.030 --> 00:24:51.460 squared error data sets. 00:24:51.460 --> 00:24:54.310 So in those kinds of conditions, what we're trying to do 00:24:54.310 --> 00:24:57.640 is find the data not and the one that gives us 00:24:57.640 --> 00:25:02.560 the best line that, in aggregate across all of our data, 00:25:02.560 --> 00:25:08.710 minimizes the sum of all squared error terms, where each error 00:25:08.710 --> 00:25:09.700 term-- 00:25:09.700 --> 00:25:13.880 each e sub i is-- 00:25:13.880 --> 00:25:17.200 so this would be my e sub i term. 00:25:17.200 --> 00:25:24.700 When I ran run sub i for input at, 00:25:24.700 --> 00:25:28.940 let's say, level 1 for my single factor here. 00:25:28.940 --> 00:25:32.620 This is my prediction. 00:25:32.620 --> 00:25:34.100 This is my data. 00:25:34.100 --> 00:25:38.680 So I've got a deviation between my best prediction 00:25:38.680 --> 00:25:41.590 at that input level and my actual measurement. 00:25:41.590 --> 00:25:44.080 I'm squaring that, and then what we're looking at 00:25:44.080 --> 00:25:47.830 is the sum of all of those across all of my data. 00:25:47.830 --> 00:25:51.940 And you can formulate it as sort of that optimization problem 00:25:51.940 --> 00:25:56.830 and do a search for the best beta 0, beta 1, conceptually. 00:25:56.830 --> 00:25:58.930 Turns out, in the simple linear cases, 00:25:58.930 --> 00:26:03.370 the power of linear models is that one can still 00:26:03.370 --> 00:26:07.240 solve the matrix formulation of the problem using 00:26:07.240 --> 00:26:10.960 a pseudo-inverse in the over-specified case, where 00:26:10.960 --> 00:26:15.250 I've got multiple samples at the same condition. 00:26:15.250 --> 00:26:17.560 And we need to do that, because now, 00:26:17.560 --> 00:26:19.750 if I've got multiple rows where I've 00:26:19.750 --> 00:26:23.650 sampled the same conditions multiple times, 00:26:23.650 --> 00:26:26.290 my x matrix is no longer square, so I 00:26:26.290 --> 00:26:27.490 can't do a direct solution. 00:26:31.000 --> 00:26:36.370 There's an aside here, which is simply 00:26:36.370 --> 00:26:41.740 the derivation of this relationship, this beta 00:26:41.740 --> 00:26:49.090 solution, based on the minimum squared error condition. 00:26:49.090 --> 00:26:51.310 Yes, question? 00:26:51.310 --> 00:26:52.690 AUDIENCE: Yes. 00:26:52.690 --> 00:26:54.970 For the previous equation, isn't it 00:26:54.970 --> 00:27:01.810 beta 0 plus beta 1 time x based on beta 0 minus beta 1 time x? 00:27:01.810 --> 00:27:02.620 DUANE BONING: Good. 00:27:02.620 --> 00:27:04.960 Yeah. 00:27:04.960 --> 00:27:06.880 I guess I could change the sign in the middle, 00:27:06.880 --> 00:27:08.500 but I probably shouldn't do that. 00:27:08.500 --> 00:27:12.510 Yeah, that should probably be a plus there. 00:27:12.510 --> 00:27:14.280 Thank you. 00:27:14.280 --> 00:27:20.490 So we could, again, formulate our problem just 00:27:20.490 --> 00:27:21.990 to give you a quick sense of where 00:27:21.990 --> 00:27:28.770 that funky description of this-- using this pseudo-inverse idea 00:27:28.770 --> 00:27:29.670 comes from. 00:27:29.670 --> 00:27:31.860 Again, if I have my linear model, 00:27:31.860 --> 00:27:37.920 I can express based on my measured data. 00:27:37.920 --> 00:27:40.050 These are my different conditions 00:27:40.050 --> 00:27:42.640 and these are my model terms. 00:27:42.640 --> 00:27:48.700 I've got a way to express the error for all of my runs, 00:27:48.700 --> 00:27:51.810 and now the squared error can be written in matrix form 00:27:51.810 --> 00:27:57.630 as the transpose of that error vector times the error vector. 00:27:57.630 --> 00:28:00.270 And now I simply substitute in, and you 00:28:00.270 --> 00:28:02.740 get something like that. 00:28:02.740 --> 00:28:05.250 And now you can do the derivative. 00:28:05.250 --> 00:28:07.380 What we're looking for is the point 00:28:07.380 --> 00:28:11.430 at which this quadratic error is minimum. 00:28:11.430 --> 00:28:15.870 So I can do the derivative there, and at that point, 00:28:15.870 --> 00:28:21.420 I'm looking for the bottom of that squared error 00:28:21.420 --> 00:28:24.430 curve, that quadratic error-- 00:28:24.430 --> 00:28:28.030 which, if I differentiate this, I get this. 00:28:28.030 --> 00:28:30.210 And now I can simply solve, and then, 00:28:30.210 --> 00:28:35.280 when you go ahead and plug that in, that drives for us-- 00:28:35.280 --> 00:28:38.820 this funky thing, where you take your data condition 00:28:38.820 --> 00:28:41.700 x transpose x, then you do the inverse 00:28:41.700 --> 00:28:50.060 of that times the x transpose with your measured outputs 00:28:50.060 --> 00:28:52.910 to solve for beta. 00:28:52.910 --> 00:28:56.600 So there is a matrix algebra formulation here 00:28:56.600 --> 00:29:01.130 that's basically giving us the best minimum error fit 00:29:01.130 --> 00:29:02.460 for all of our data. 00:29:02.460 --> 00:29:06.200 So that's a very nice, powerful general linear regression 00:29:06.200 --> 00:29:08.770 result. 00:29:08.770 --> 00:29:10.630 Here's another picture of the same thing. 00:29:10.630 --> 00:29:13.200 This is the same idea-- not the derivation, but what I meant 00:29:13.200 --> 00:29:17.430 by when I've got replicates. 00:29:17.430 --> 00:29:19.530 Again, this is my measured vector. 00:29:22.240 --> 00:29:24.010 These are my conditions. 00:29:24.010 --> 00:29:28.780 Again, I'm either running at the low condition 00:29:28.780 --> 00:29:31.240 or at the high condition for each of my-- 00:29:31.240 --> 00:29:33.760 for my single input here. 00:29:33.760 --> 00:29:38.470 I've got two coefficients, so my x times my beta, 00:29:38.470 --> 00:29:44.440 again, gives me my beta 0 plus my either x high or x low times 00:29:44.440 --> 00:29:45.430 beta 1. 00:29:45.430 --> 00:29:48.160 And for each case, I get some measured output. 00:29:48.160 --> 00:29:52.540 And the point is now, if I have replicates-- 00:29:52.540 --> 00:29:56.800 there's an x low, here's another x low, and if I continued on, 00:29:56.800 --> 00:30:00.400 there may be multiple places where my conditions were 00:30:00.400 --> 00:30:02.030 exactly the same-- 00:30:02.030 --> 00:30:04.570 oops-- my coefficients are exactly the same, 00:30:04.570 --> 00:30:07.060 but I get different measured outputs. 00:30:07.060 --> 00:30:08.170 So that's how my-- 00:30:08.170 --> 00:30:12.690 what used to be my square matrix becomes non-square. 00:30:12.690 --> 00:30:19.090 I'm filling in additional data rows as I add more replicates. 00:30:19.090 --> 00:30:23.110 And so in that case, I still got the simple linear model, 00:30:23.110 --> 00:30:25.570 but now I have to use not a direct inverse, 00:30:25.570 --> 00:30:28.300 but I got to use this sort of pseudo-inverse 00:30:28.300 --> 00:30:31.153 in order to fit my beta coefficients. 00:30:34.540 --> 00:30:39.550 OK, so that's the very powerful, general linear regression 00:30:39.550 --> 00:30:43.420 kind of approach, and it's based on minimizing a squared error 00:30:43.420 --> 00:30:45.100 term. 00:30:45.100 --> 00:30:48.070 But there's a lot more intuitive way 00:30:48.070 --> 00:30:50.920 of looking at this and some other observations that 00:30:50.920 --> 00:30:56.590 make estimation easier, and also connect up, 00:30:56.590 --> 00:30:59.440 I think, very nicely with our intuition. 00:30:59.440 --> 00:31:03.410 One important point is, when I've only got two levels-- 00:31:03.410 --> 00:31:05.290 so our one input, two levels-- 00:31:05.290 --> 00:31:07.930 we know a line has to fit that data. 00:31:07.930 --> 00:31:12.610 And if I've got replicates, the line 00:31:12.610 --> 00:31:15.700 actually has to go through-- 00:31:15.700 --> 00:31:18.880 provably, and we'll see a very quick little derivation-- 00:31:18.880 --> 00:31:21.670 the line has to go through the mean at each 00:31:21.670 --> 00:31:23.740 of the data points, my replicate points, 00:31:23.740 --> 00:31:25.030 at each of my two levels. 00:31:28.340 --> 00:31:30.860 So in order to fit the data, one is 00:31:30.860 --> 00:31:32.420 you could put it all on a big matrix 00:31:32.420 --> 00:31:35.210 and send it off to Matlab, or Excel, 00:31:35.210 --> 00:31:38.540 or whatever to do this big matrix pseudo-inverse, 00:31:38.540 --> 00:31:44.120 or you can simply calculate, look at your data 00:31:44.120 --> 00:31:47.690 under your low condition, your data under your high condition, 00:31:47.690 --> 00:31:51.050 find the mean, and your line has to go 00:31:51.050 --> 00:31:55.650 through the mean of those two conditions. 00:31:55.650 --> 00:31:58.640 So one of the two might be a little bit easier on a quiz. 00:32:02.710 --> 00:32:04.740 And this is simply working out and showing you, 00:32:04.740 --> 00:32:10.860 if, in this simple two-input condition, where I only 00:32:10.860 --> 00:32:16.380 had two data points, then you could see how this works. 00:32:16.380 --> 00:32:19.320 But with the mean, what's very nice 00:32:19.320 --> 00:32:21.990 is now this is the mean of my output, 00:32:21.990 --> 00:32:25.920 and what that gives you is a direct expression using 00:32:25.920 --> 00:32:30.930 that fact to be able to calculate your offset term 00:32:30.930 --> 00:32:32.130 and your beta 1 term. 00:32:37.960 --> 00:32:38.580 Let me see. 00:32:38.580 --> 00:32:39.663 Am I going to do this yet? 00:32:39.663 --> 00:32:40.390 No. 00:32:40.390 --> 00:32:44.650 So here's a simple application of that for our real data, 00:32:44.650 --> 00:32:46.600 this injection molding case. 00:32:46.600 --> 00:32:50.590 Here now I'm doing something kind of sneaky. 00:32:50.590 --> 00:32:53.500 This is I'm treating all of that injection molding 00:32:53.500 --> 00:32:56.170 experimental design as if I'd only 00:32:56.170 --> 00:33:00.010 done an experimental design in hold time. 00:33:00.010 --> 00:33:04.240 So what's really hidden inside of this spread of data 00:33:04.240 --> 00:33:08.230 is it is all replicates at the low value x minus 00:33:08.230 --> 00:33:09.890 for hold time. 00:33:09.890 --> 00:33:13.720 Similarly, these are replicates at x plus, 00:33:13.720 --> 00:33:16.300 but they're not really good pure replicates, 00:33:16.300 --> 00:33:19.360 because other things were changing. 00:33:19.360 --> 00:33:21.890 First off, time might often be changing, 00:33:21.890 --> 00:33:24.250 but what was going on in here is I was also 00:33:24.250 --> 00:33:28.570 playing with the injection velocity at the same time. 00:33:28.570 --> 00:33:32.320 So in this case, this is saying I'm 00:33:32.320 --> 00:33:34.570 going to treat these other things that 00:33:34.570 --> 00:33:37.570 might be changing, like injection velocity, 00:33:37.570 --> 00:33:39.340 as just noise factors. 00:33:39.340 --> 00:33:45.880 They're what contributes to the spread at the condition 00:33:45.880 --> 00:33:47.710 that I'm actually interested in. 00:33:47.710 --> 00:33:50.680 And so you will often do that. 00:33:50.680 --> 00:33:53.650 I'm just trying to fit a very simple 00:33:53.650 --> 00:33:58.090 first-order linear model, just relating my one factor here 00:33:58.090 --> 00:34:01.910 of hold time to my output. 00:34:01.910 --> 00:34:04.900 And it's going to have some variance and error 00:34:04.900 --> 00:34:09.969 in my estimate because of all these other higher order 00:34:09.969 --> 00:34:12.969 terms that I'm not including, other factors that I'm not 00:34:12.969 --> 00:34:16.810 including, pure replication error, measurement error. 00:34:16.810 --> 00:34:19.030 But what I am trying to do is understand 00:34:19.030 --> 00:34:24.429 as closely as possible the effective that single factor. 00:34:24.429 --> 00:34:27.250 So very often, I really do want to know just the hold time 00:34:27.250 --> 00:34:31.060 effect fact, even when it's contaminated with or spread out 00:34:31.060 --> 00:34:33.009 with these other effects. 00:34:36.708 --> 00:34:38.500 This is just showing, for that actual data, 00:34:38.500 --> 00:34:41.469 I can fit my two means for the spread, 00:34:41.469 --> 00:34:46.810 and then fit a very simple linear model. 00:34:46.810 --> 00:34:50.560 Now, in this case, I'm treating hold time 00:34:50.560 --> 00:34:53.500 as kind of a continuous factor, so you might think, 00:34:53.500 --> 00:34:56.480 if these are hold time in minutes, 00:34:56.480 --> 00:35:00.160 now you can interpolate. 00:35:00.160 --> 00:35:03.820 And I could ask, what I think the hold 00:35:03.820 --> 00:35:06.490 time-- or the effective on my dimension 00:35:06.490 --> 00:35:09.850 might be at 1.5 minutes? 00:35:09.850 --> 00:35:11.980 So I'm interpolating within my data. 00:35:11.980 --> 00:35:15.130 I could also extrapolate and ask questions. 00:35:15.130 --> 00:35:19.035 Well, what do I think it'll be out here at 2.5 minutes? 00:35:19.035 --> 00:35:20.410 That might be a little bit risky, 00:35:20.410 --> 00:35:23.590 and we'll talk about that a little bit 00:35:23.590 --> 00:35:27.700 in a later lecture, when we talk about the confidence intervals. 00:35:27.700 --> 00:35:29.740 But at another point I want to make 00:35:29.740 --> 00:35:40.300 is this same picture would apply even if these 00:35:40.300 --> 00:35:43.760 weren't continuous parameters. 00:35:43.760 --> 00:35:46.670 And I find it easier to think about the problem in terms 00:35:46.670 --> 00:35:49.040 of continuous parameters, because you're often 00:35:49.040 --> 00:35:52.400 changing a pressure, or a velocity, or a hold time, 00:35:52.400 --> 00:35:54.950 and looking at a continuous variation in some output 00:35:54.950 --> 00:35:55.760 parameter. 00:35:55.760 --> 00:36:01.310 But a very important point that I do want to make here is, 00:36:01.310 --> 00:36:09.680 if I were to use exactly the same chart and relabel the axes 00:36:09.680 --> 00:36:13.730 and say something like, this is the dimension-- 00:36:13.730 --> 00:36:15.740 this is still dimension-- 00:36:15.740 --> 00:36:19.730 and I'm doing injection molding, but here I'm 00:36:19.730 --> 00:36:30.030 using it from a tool from different manufacturers. 00:36:30.030 --> 00:36:32.670 So here, I run the tool on-- 00:36:32.670 --> 00:36:36.330 I don't know-- Fritze Inc's injection molding tool, 00:36:36.330 --> 00:36:44.670 and over here it's on Xenon Corp's tool. 00:36:44.670 --> 00:36:48.960 I could have essentially the same model, the same formalism 00:36:48.960 --> 00:36:53.100 if these were discrete levels, discrete choices 00:36:53.100 --> 00:36:54.270 in some factor. 00:36:54.270 --> 00:36:56.670 Here my factor is, which choice of tool 00:36:56.670 --> 00:36:58.560 would I want to be making? 00:36:58.560 --> 00:37:00.330 I want to know, is there an effect? 00:37:00.330 --> 00:37:04.830 I maybe even want to build a model with model coefficients 00:37:04.830 --> 00:37:11.280 for the effect, depending on what factor level I pick. 00:37:11.280 --> 00:37:12.840 Now, if these were discrete choices, 00:37:12.840 --> 00:37:17.380 it no longer makes sense to interpolate or extrapolate, 00:37:17.380 --> 00:37:20.730 but I do want to just highlight that the same formalism-- 00:37:20.730 --> 00:37:23.340 especially for factorial designs-- 00:37:23.340 --> 00:37:26.670 applies to both continuous parameters 00:37:26.670 --> 00:37:28.388 and discrete parameters. 00:37:28.388 --> 00:37:30.180 I'm going to come back to that a little bit 00:37:30.180 --> 00:37:33.390 when we talk about what might-- 00:37:33.390 --> 00:37:35.670 slightly different ways of looking at the model 00:37:35.670 --> 00:37:37.260 coefficients be in those cases. 00:37:39.850 --> 00:37:42.030 But I just want to remind you or highlight 00:37:42.030 --> 00:37:46.530 that you can use these for both continuous and discrete kinds 00:37:46.530 --> 00:37:49.390 of designs and factors. 00:37:49.390 --> 00:37:54.870 OK, so continuing with our plan here-- 00:37:54.870 --> 00:37:57.270 we've talked about multiple inputs and effects-- 00:37:57.270 --> 00:38:03.030 what I want to do next is say, if I do have multiple inputs, 00:38:03.030 --> 00:38:06.030 how does our ANOVA formalism apply 00:38:06.030 --> 00:38:10.830 to being able to decide whether both factors matter, 00:38:10.830 --> 00:38:14.520 or not, or three factors, which factors actually matter. 00:38:14.520 --> 00:38:16.800 And based on my observations, do I 00:38:16.800 --> 00:38:20.080 think that that factor actually has an effect or not? 00:38:20.080 --> 00:38:22.950 So I want to derive first real quickly, 00:38:22.950 --> 00:38:25.200 or show for us the different ANOVA 00:38:25.200 --> 00:38:27.220 formalisms in these cases. 00:38:27.220 --> 00:38:29.160 And then we'll come back a little bit more 00:38:29.160 --> 00:38:34.720 to the estimation of these model coefficients. 00:38:34.720 --> 00:38:35.890 Oh, what happened? 00:38:35.890 --> 00:38:37.750 There we go. 00:38:37.750 --> 00:38:38.980 OK. 00:38:38.980 --> 00:38:41.380 So back to our injection molding problem, 00:38:41.380 --> 00:38:45.310 now with two factors, where we have the velocity 00:38:45.310 --> 00:38:47.960 and we have the hold time-- 00:38:47.960 --> 00:38:50.860 and this is where I said I was being a little bit sneaky 00:38:50.860 --> 00:38:53.360 when I was showing you the one factor data. 00:38:53.360 --> 00:38:56.980 In fact, the setup here was to do four different combinations 00:38:56.980 --> 00:38:58.960 of these two factors-- 00:38:58.960 --> 00:39:01.690 two-factor, two-level experimental design-- 00:39:01.690 --> 00:39:04.240 with replicates at each factor-- 00:39:04.240 --> 00:39:06.970 something like 14 or 15 replicates. 00:39:06.970 --> 00:39:09.580 And here nothing was intentionally changing, 00:39:09.580 --> 00:39:12.880 so these are more like pure replicates 00:39:12.880 --> 00:39:16.450 within that set of experimental conditions. 00:39:16.450 --> 00:39:19.490 And what we can do is look at, in that case, where I've 00:39:19.490 --> 00:39:22.210 got pure replicates, what is the average 00:39:22.210 --> 00:39:24.620 at that test condition-- 00:39:24.620 --> 00:39:27.700 so test condition 1, where it was low and low 00:39:27.700 --> 00:39:32.710 for the two factors, and so on for test condition 2, 3, and 4. 00:39:32.710 --> 00:39:34.480 And now we can ask the question-- 00:39:34.480 --> 00:39:38.440 not just in the univariate case, but in the multivariate case, 00:39:38.440 --> 00:39:41.350 what kind of model works? 00:39:41.350 --> 00:39:43.870 And this is actually a description 00:39:43.870 --> 00:39:46.330 of the model that might be more appropriate if they 00:39:46.330 --> 00:39:51.720 were discrete choices and you weren't 00:39:51.720 --> 00:39:54.150 trying to do interpolation. 00:39:54.150 --> 00:39:56.160 But it's also implicitly the model 00:39:56.160 --> 00:39:58.890 that we're using for ANOVA analysis, 00:39:58.890 --> 00:40:01.710 because the ANOVA doesn't care whether it's 00:40:01.710 --> 00:40:05.460 a continuous parameter or a discrete parameter 00:40:05.460 --> 00:40:07.510 that we're using. 00:40:07.510 --> 00:40:10.890 So the basic idea for a two-- 00:40:10.890 --> 00:40:13.230 this is a two-factor kind of model. 00:40:13.230 --> 00:40:18.940 This is very close to what we saw with the ANOVA on Tuesday. 00:40:18.940 --> 00:40:20.760 One of the factor might be-- 00:40:20.760 --> 00:40:23.880 or treatment effect that we might be looking for 00:40:23.880 --> 00:40:26.790 might be the velocity effect. 00:40:26.790 --> 00:40:28.530 The other might be the hold time effect, 00:40:28.530 --> 00:40:31.740 and then we may be looking for interactions between the two. 00:40:31.740 --> 00:40:34.068 And the picture you should have in your mind 00:40:34.068 --> 00:40:35.610 is that what we're going to try to do 00:40:35.610 --> 00:40:37.830 is model this with an overall mean-- 00:40:37.830 --> 00:40:42.360 some mu-- and then we're asking the question-- 00:40:42.360 --> 00:40:44.910 think back to the ANOVA question-- 00:40:44.910 --> 00:40:48.330 is there a deviation from the grand mean, 00:40:48.330 --> 00:40:54.840 from the overall mean, if I have velocity at the high condition? 00:40:54.840 --> 00:40:57.240 Is there a statistically significant shift 00:40:57.240 --> 00:41:00.580 up for a high setting? 00:41:00.580 --> 00:41:05.070 And similarly, is there a statistically significant 00:41:05.070 --> 00:41:06.780 low effect? 00:41:06.780 --> 00:41:10.020 So is there an up or down shift, depending on 00:41:10.020 --> 00:41:14.160 whether I pick the high or low condition for velocity? 00:41:14.160 --> 00:41:17.550 And similarly, I would ask the question, is there an effect-- 00:41:17.550 --> 00:41:23.400 a beta 1 effect or a beta 2 effect-- 00:41:23.400 --> 00:41:27.735 depending on whether I selected the low-- 00:41:31.910 --> 00:41:35.420 I guess this is the high-- the high condition 00:41:35.420 --> 00:41:41.730 or the low condition for that actual variable? 00:41:41.730 --> 00:41:45.860 So these are discrete offsets, depending 00:41:45.860 --> 00:41:50.600 on whether I was at the low or the high condition. 00:41:50.600 --> 00:41:53.450 So i indicates, in our earlier notation, 00:41:53.450 --> 00:41:54.950 either the plus or minus. 00:41:54.950 --> 00:41:57.020 Here, we're switching rotation a little bit 00:41:57.020 --> 00:41:59.690 to just be a numerical index-- 00:41:59.690 --> 00:42:02.690 either 1 or 2 in this case, but if I 00:42:02.690 --> 00:42:07.370 had multiple levels of that factor, I could extend that. 00:42:07.370 --> 00:42:12.440 I can have little deltas with a coefficient that 00:42:12.440 --> 00:42:15.680 would go for each delta, depending on the discrete value 00:42:15.680 --> 00:42:18.250 that I picked for that level. 00:42:18.250 --> 00:42:18.750 Yeah? 00:42:18.750 --> 00:42:20.542 AUDIENCE: What if it's a shift of variance, 00:42:20.542 --> 00:42:23.442 and not a shift in the mean altering one of them? 00:42:23.442 --> 00:42:24.900 DUANE BONING: It's a good question. 00:42:24.900 --> 00:42:27.200 The question was, what if it's a shift in variance? 00:42:27.200 --> 00:42:31.580 Here we are looking for just-- 00:42:31.580 --> 00:42:37.650 we're just modeling the mean value of the output with this. 00:42:37.650 --> 00:42:41.870 You could similarly start looking at building up 00:42:41.870 --> 00:42:43.190 models of variance. 00:42:47.030 --> 00:42:49.280 You would probably not be thinking of it 00:42:49.280 --> 00:42:52.980 as additive variance terms in those cases, 00:42:52.980 --> 00:42:55.520 so you might also end up doing some other kinds 00:42:55.520 --> 00:42:57.860 of transformations on your data to get 00:42:57.860 --> 00:43:02.480 to additive models of variance. 00:43:02.480 --> 00:43:06.020 But basically, if I had enough data, 00:43:06.020 --> 00:43:10.670 you could start to try to model variance. 00:43:10.670 --> 00:43:13.280 If you think back to the ANOVA formulation, 00:43:13.280 --> 00:43:17.360 the simple ANOVA is all assuming your variance 00:43:17.360 --> 00:43:21.120 is exactly the same across all of your conditions. 00:43:21.120 --> 00:43:30.650 There are wonderful books on more powerful modeling 00:43:30.650 --> 00:43:33.440 and how ANOVA extends to those. 00:43:33.440 --> 00:43:34.730 In fact, one of my favorites-- 00:43:34.730 --> 00:43:37.280 I lent it to Hayden-- 00:43:37.280 --> 00:43:40.730 is a book by George Milliken, the title of which 00:43:40.730 --> 00:43:44.240 is Analysis of Messy Data. 00:43:44.240 --> 00:43:46.790 And it deals with all kinds of wonderful conditions-- 00:43:46.790 --> 00:43:49.640 not only modeling of variance, but also 00:43:49.640 --> 00:43:53.060 when you've got very complicated design situations 00:43:53.060 --> 00:43:57.500 with partially replicated, data unbalanced design. 00:43:57.500 --> 00:43:59.720 You run the whole experimental design and you're 00:43:59.720 --> 00:44:02.750 missing a data point, because the experiment crashed-- 00:44:02.750 --> 00:44:05.570 what do you do now? 00:44:05.570 --> 00:44:07.620 Looks at every possible thing-- 00:44:07.620 --> 00:44:12.740 so here, we're really looking just at mean modeling. 00:44:12.740 --> 00:44:14.420 Just to finish this up, I should also 00:44:14.420 --> 00:44:17.510 point out that there would be discrete offsets, depending 00:44:17.510 --> 00:44:24.480 on every combination of our interaction factors as well. 00:44:24.480 --> 00:44:30.690 And so I might have discrete model terms for those as well. 00:44:30.690 --> 00:44:34.220 Now, an important point just to make here is that, 00:44:34.220 --> 00:44:38.390 in this kind of a model-- and it'll maybe become clearer 00:44:38.390 --> 00:44:39.770 a little bit later-- 00:44:39.770 --> 00:44:42.260 each of these model coefficients-- 00:44:42.260 --> 00:44:48.770 this tau 1 and a tau 2 effect, for example, for the velocity-- 00:44:48.770 --> 00:44:52.010 they're not really independent. 00:44:52.010 --> 00:44:54.560 And this gets back to, I think, your question about, 00:44:54.560 --> 00:44:56.540 in this model, this-- 00:44:56.540 --> 00:44:59.030 if I were modeling that as a beta 0 term, 00:44:59.030 --> 00:45:01.010 this really is the mean, and that 00:45:01.010 --> 00:45:03.050 imposes an additional linear constraint 00:45:03.050 --> 00:45:04.700 on my other coefficients. 00:45:04.700 --> 00:45:07.400 And in particular, that tau 1 and tau 2 00:45:07.400 --> 00:45:09.380 would have to be equal and opposite here. 00:45:13.370 --> 00:45:17.570 This is a great model for being able to conceptualize and plug 00:45:17.570 --> 00:45:20.750 in, depending on what factor levels you have. 00:45:20.750 --> 00:45:23.540 You just have to be careful in fitting the model, 00:45:23.540 --> 00:45:27.500 and recognize that, in this kind of a formulation, 00:45:27.500 --> 00:45:33.650 there's some implied dependencies within this-- 00:45:33.650 --> 00:45:36.620 in particular, that each of the factor levels-- 00:45:36.620 --> 00:45:39.830 factor effects have to be 0 mean overall, 00:45:39.830 --> 00:45:41.690 and the interaction terms, similarly, 00:45:41.690 --> 00:45:43.640 have to be, in aggregate, 0 mean. 00:45:46.160 --> 00:45:49.740 That kind of makes sense, because-- 00:45:49.740 --> 00:45:55.110 think of the case where I've got two inputs, two outputs-- 00:45:55.110 --> 00:45:59.010 or one output, got my two factors-- 00:45:59.010 --> 00:46:02.220 velocity and hold time-- and I'm just doing two levels for each. 00:46:02.220 --> 00:46:04.710 I've got four different conditions. 00:46:04.710 --> 00:46:06.270 I may have replication, but I've only 00:46:06.270 --> 00:46:07.900 got four different conditions. 00:46:07.900 --> 00:46:15.300 How could I possibly fit a mean, two taus for factor 1, 00:46:15.300 --> 00:46:18.240 two coefficients-- taus for-- 00:46:18.240 --> 00:46:23.010 or betas here for factor 2, and four interaction terms? 00:46:23.010 --> 00:46:25.020 That's four, five, six, seven, eight-- 00:46:25.020 --> 00:46:28.545 nine model coefficients, four data points. 00:46:31.780 --> 00:46:34.860 So clearly, there's got to be some additional constraints 00:46:34.860 --> 00:46:40.220 on the model coefficients hidden under that. 00:46:40.220 --> 00:46:46.040 OK, so that's one way of looking at the experimental design 00:46:46.040 --> 00:46:47.610 and the data. 00:46:47.610 --> 00:46:52.640 Here's another way, simply pointing out-- 00:46:52.640 --> 00:46:56.690 what we'll often be doing is building tables 00:46:56.690 --> 00:47:00.110 to show the high and low condition 00:47:00.110 --> 00:47:02.840 for our different factors and the levels. 00:47:02.840 --> 00:47:06.170 We might label those, and then often, we'll 00:47:06.170 --> 00:47:12.260 illustrate our design space or experimental space 00:47:12.260 --> 00:47:18.360 with a design matrix or a design plot. 00:47:18.360 --> 00:47:20.450 So here I'm showing simply-- 00:47:20.450 --> 00:47:23.990 the four different conditions for velocity and hold 00:47:23.990 --> 00:47:26.250 time either, low or high. 00:47:26.250 --> 00:47:29.840 and I'm also doing something a little bit subtle, 00:47:29.840 --> 00:47:33.260 which is labeling them with not just a plus or a minus, 00:47:33.260 --> 00:47:37.325 but in fact, giving them a normalized value of plus 1 00:47:37.325 --> 00:47:38.120 or minus 1. 00:47:41.150 --> 00:47:43.350 There's some subtle reasons for doing that. 00:47:43.350 --> 00:47:50.720 One is, by definition, when I now fit for a two-level 00:47:50.720 --> 00:47:52.550 experiment-- 00:47:52.550 --> 00:47:54.560 when I fit a model coefficient to that, 00:47:54.560 --> 00:47:57.410 it will always be 0 mean, and so now 00:47:57.410 --> 00:48:02.270 that beta 0 term really does turn into the mean estimate. 00:48:02.270 --> 00:48:06.120 But-- might see that a little bit later. 00:48:06.120 --> 00:48:08.330 But the point is here now this is basically 00:48:08.330 --> 00:48:11.780 expressing all four of my conditions, my test conditions 00:48:11.780 --> 00:48:15.350 from the previous design. 00:48:15.350 --> 00:48:18.500 And I can also plot that, each one of these test conditions, 00:48:18.500 --> 00:48:21.930 on the x1 axis. 00:48:21.930 --> 00:48:24.830 We said this is velocity, basically. 00:48:24.830 --> 00:48:30.490 And this is my x2 axis, which is hold time. 00:48:30.490 --> 00:48:33.760 So it's either at the plus 1 or the minus 1 level 00:48:33.760 --> 00:48:35.950 for that factor. 00:48:35.950 --> 00:48:39.340 And so I've got an experimental design, 00:48:39.340 --> 00:48:42.280 where I've got four sample points 00:48:42.280 --> 00:48:46.330 at the plus minus every combination 00:48:46.330 --> 00:48:48.790 of those particular factors. 00:48:48.790 --> 00:48:53.590 And so we'll often talk about this as a corner 00:48:53.590 --> 00:49:04.400 point DOE or a full factorial DOE, 00:49:04.400 --> 00:49:08.270 where I've got every combination of those discrete factor 00:49:08.270 --> 00:49:11.990 levels being mapped out. 00:49:11.990 --> 00:49:14.070 And then I go and I run my tests. 00:49:14.070 --> 00:49:17.330 And here I've just written down the results 00:49:17.330 --> 00:49:21.200 of the average of the output over all replicates 00:49:21.200 --> 00:49:22.770 for that point. 00:49:22.770 --> 00:49:27.500 So this was over the 14 or 15 replicates. 00:49:27.500 --> 00:49:30.320 Notice here, this is just mapping out the design space, 00:49:30.320 --> 00:49:31.970 my x1 versus x2. 00:49:31.970 --> 00:49:37.730 It's not really showing you on a third axis what the output is. 00:49:37.730 --> 00:49:40.610 We could also do that and have a 3D plot, 00:49:40.610 --> 00:49:43.680 and we'll see that in a second. 00:49:43.680 --> 00:49:47.660 But the point for doing this formulation, this full effects 00:49:47.660 --> 00:49:52.130 model with these effects offsets and just 00:49:52.130 --> 00:49:55.460 reminding us what the model exploration here 00:49:55.460 --> 00:49:59.270 is, is that what we would also like to do, 00:49:59.270 --> 00:50:01.200 even when I've got more than one factor-- 00:50:01.200 --> 00:50:03.410 I've got these x1 and x2 factors-- 00:50:03.410 --> 00:50:07.880 is ask the ANOVA question, which would be-- 00:50:07.880 --> 00:50:11.300 from an ANOVA point of view, I would like to ask, 00:50:11.300 --> 00:50:15.830 does velocity have an effect? 00:50:15.830 --> 00:50:19.490 If I change from the low to the high, 00:50:19.490 --> 00:50:23.720 do I get an appreciable and significant change 00:50:23.720 --> 00:50:25.380 in the output? 00:50:25.380 --> 00:50:27.590 So from an ANOVA point of view, if I'm just 00:50:27.590 --> 00:50:31.530 looking at velocity, I would like to ask a question-- 00:50:31.530 --> 00:50:34.670 just looking together across all of the data 00:50:34.670 --> 00:50:37.040 that I've got at the low condition for x1 00:50:37.040 --> 00:50:39.980 and all of the data from the condition at x2, 00:50:39.980 --> 00:50:42.660 I might ask that question and say, 00:50:42.660 --> 00:50:47.000 is there a shift from that perspective or not? 00:50:47.000 --> 00:50:50.220 So really, we're back to a hypothesis test-- 00:50:50.220 --> 00:50:52.530 oh, what happened there? 00:50:52.530 --> 00:50:57.200 This should be a dot, dot, dot. 00:50:57.200 --> 00:51:00.680 [INAUDIBLE]---- which is basically asking the question, 00:51:00.680 --> 00:51:03.140 in the null hypothesis, there is no effect-- 00:51:03.140 --> 00:51:11.780 all of my taus are equal and 0, and all my interactions are 0. 00:51:11.780 --> 00:51:14.450 And in the alternative hypothesis, 00:51:14.450 --> 00:51:17.070 something has an effect. 00:51:17.070 --> 00:51:19.460 So that's the key question. 00:51:19.460 --> 00:51:22.130 Now, what you can do is start formulating, exactly 00:51:22.130 --> 00:51:25.880 like we did with ANOVA, sums of squared deviations 00:51:25.880 --> 00:51:29.630 across, for example, all of the A levels 00:51:29.630 --> 00:51:33.170 averaged over the B factors and any replicates. 00:51:33.170 --> 00:51:36.130 Remember, this is replicates. 00:51:36.130 --> 00:51:37.670 This is the B factor. 00:51:37.670 --> 00:51:43.390 So for example, this sum right here would be-- 00:51:43.390 --> 00:51:45.460 let's see-- forgot which one-- 00:51:48.160 --> 00:51:50.740 I'm looking at the responses for the first level, 00:51:50.740 --> 00:51:53.080 so call it the A level. 00:51:53.080 --> 00:51:58.990 So that would be basically the big red thing 00:51:58.990 --> 00:52:00.700 that I had drawn here. 00:52:00.700 --> 00:52:06.370 I basically call this first one factor A, 00:52:06.370 --> 00:52:08.650 and what I'm simply doing is averaging over 00:52:08.650 --> 00:52:11.950 all of the n replicates at each of these points 00:52:11.950 --> 00:52:15.190 and over the B levels at that point. 00:52:18.110 --> 00:52:23.860 So I can form a sum of squared set of deviations from the mean 00:52:23.860 --> 00:52:26.680 as I change factor A. I can similarly 00:52:26.680 --> 00:52:29.180 do that as I change factor B. 00:52:29.180 --> 00:52:31.240 Then I can also look at these interaction 00:52:31.240 --> 00:52:34.840 sum of squared terms, and once I have these, 00:52:34.840 --> 00:52:37.690 I can now start to break down the total sum 00:52:37.690 --> 00:52:45.310 of squared deviations in my data into components due to factor 00:52:45.310 --> 00:52:50.530 A, sum of squares from-- 00:52:50.530 --> 00:52:56.830 deviations from the grand mean for factor B, and terms 00:52:56.830 --> 00:52:59.260 here from interactions. 00:53:03.470 --> 00:53:08.910 And then this is residuals, sum of squared error. 00:53:08.910 --> 00:53:11.420 So basically, it's just like we did with the single factor 00:53:11.420 --> 00:53:15.410 ANOVA, but now I can do that, well, looking and seeing, 00:53:15.410 --> 00:53:16.760 did factor A have an effect? 00:53:16.760 --> 00:53:19.160 What's the sum of squared deviations 00:53:19.160 --> 00:53:21.230 that I observed in my data from the mean 00:53:21.230 --> 00:53:23.480 as I change that factor-- 00:53:23.480 --> 00:53:26.000 and similarly, for factor B. And I 00:53:26.000 --> 00:53:28.490 could extend this on to more factors, 00:53:28.490 --> 00:53:30.590 if I wanted-- if I had three factors or four. 00:53:33.920 --> 00:53:36.230 This is also just reminding us or laying out 00:53:36.230 --> 00:53:39.500 that there are different degrees of freedom in each case, which 00:53:39.500 --> 00:53:41.960 we're going to need when we want to estimate 00:53:41.960 --> 00:53:44.540 the mean square, because what we'll do 00:53:44.540 --> 00:53:47.480 is we'll take the sum of squared deviations, 00:53:47.480 --> 00:53:50.880 divide it by its degrees of freedom-- 00:53:50.880 --> 00:53:53.840 which is the number of factor levels I had minus 1 00:53:53.840 --> 00:53:55.640 in each of those cases-- 00:53:55.640 --> 00:54:00.230 and put these together to get a mean square estimate 00:54:00.230 --> 00:54:03.680 for that factor, which is essentially 00:54:03.680 --> 00:54:08.450 an estimate of the variance due to changing factor A, 00:54:08.450 --> 00:54:11.240 just like in the ANOVA formulation. 00:54:11.240 --> 00:54:14.660 And you can plot that into an ANOVA table 00:54:14.660 --> 00:54:17.030 for multiple factors, where, again, we 00:54:17.030 --> 00:54:19.400 have the sum of squared deviations 00:54:19.400 --> 00:54:21.050 due to each of those factors. 00:54:21.050 --> 00:54:24.770 This is just plugging in those formulas that we saw earlier. 00:54:24.770 --> 00:54:29.330 And the interaction-- if I've got pure replicates, that's 00:54:29.330 --> 00:54:34.730 a great case where I can separate out exactly just 00:54:34.730 --> 00:54:36.950 the replication error, perhaps coming 00:54:36.950 --> 00:54:41.390 from measurement or pure process replication error. 00:54:41.390 --> 00:54:44.540 And then, with the right appropriate degree of freedom, 00:54:44.540 --> 00:54:47.810 I can now form the mean square estimates. 00:54:47.810 --> 00:54:51.230 Remember, these are variance estimates. 00:54:51.230 --> 00:54:57.030 And then I can form the right F factor to decide, 00:54:57.030 --> 00:55:00.830 does this mean square, with respect to my 00:55:00.830 --> 00:55:04.580 within group mean square-- 00:55:04.580 --> 00:55:08.870 is that large enough effect to be 00:55:08.870 --> 00:55:12.780 statistically significant using an F-test? 00:55:12.780 --> 00:55:16.250 So it's exactly the same formulation for the ANOVA, 00:55:16.250 --> 00:55:18.740 but now where we can break it out 00:55:18.740 --> 00:55:23.720 across multiple factors or multiple interactions. 00:55:23.720 --> 00:55:24.740 OK. 00:55:24.740 --> 00:55:28.830 You actually have to try it and use this to get it, 00:55:28.830 --> 00:55:32.150 but the key point here is, when we 00:55:32.150 --> 00:55:35.300 are building these multiple factor models, 00:55:35.300 --> 00:55:39.620 it's still an ANOVA formalism to decide whether or not 00:55:39.620 --> 00:55:43.010 that factor is statistically significant or not. 00:55:46.180 --> 00:55:50.560 So let's look at this in the case of a linear model. 00:55:50.560 --> 00:55:53.830 And I'm going to give you a little bit more terminology 00:55:53.830 --> 00:55:58.060 here and start to get to this idea of contrasts, which 00:55:58.060 --> 00:56:02.620 are easy ways of estimating some of these models. 00:56:02.620 --> 00:56:08.140 So this is still our injection molding data. 00:56:08.140 --> 00:56:12.460 Factor x1 here was velocity. 00:56:12.460 --> 00:56:16.930 This was hold time. 00:56:16.930 --> 00:56:19.060 And I've got every combination. 00:56:19.060 --> 00:56:22.840 What's new on this page that I want to draw your attention to 00:56:22.840 --> 00:56:26.470 is some additional terminology that we often 00:56:26.470 --> 00:56:30.040 use that's shorthand for indicating 00:56:30.040 --> 00:56:36.370 the different combinations of those factor levels, that 00:56:36.370 --> 00:56:40.360 tell us exactly what point, what condition, 00:56:40.360 --> 00:56:44.260 combination we're running for each of our tests. 00:56:44.260 --> 00:56:46.150 So I already showed you this table 00:56:46.150 --> 00:56:49.490 which was saying, OK, for my first factor-- 00:56:49.490 --> 00:56:52.330 which we've also labeled as factor A-- 00:56:52.330 --> 00:56:59.490 factor x2 is our second factor. 00:56:59.490 --> 00:57:02.850 So note here, we've got three different kinds 00:57:02.850 --> 00:57:05.400 of-- ways of referring to the same factor. 00:57:05.400 --> 00:57:09.750 The first factor, factor a x1 value-- 00:57:09.750 --> 00:57:11.850 it's the velocity term. 00:57:11.850 --> 00:57:17.220 What I can also do is indicate in shorthand which 00:57:17.220 --> 00:57:19.950 condition is actually at work. 00:57:19.950 --> 00:57:23.200 And so it takes a little bit of getting used to, 00:57:23.200 --> 00:57:24.960 but what you'll see here is-- 00:57:24.960 --> 00:57:29.340 let's look at this condition first. 00:57:29.340 --> 00:57:32.190 What I've selected here for test number two 00:57:32.190 --> 00:57:34.770 is the high condition for factor A 00:57:34.770 --> 00:57:38.130 and the low condition for factor B. 00:57:38.130 --> 00:57:39.840 So what we're doing in shorthand is 00:57:39.840 --> 00:57:44.580 indicating with a lowercase version of that factor level 00:57:44.580 --> 00:57:51.550 all of the high selections for that factor. 00:57:51.550 --> 00:57:54.690 So the high here meant I picked the high condition 00:57:54.690 --> 00:57:58.110 for factor A, and I denote that with a little a. 00:57:58.110 --> 00:58:01.740 And the low factors we just leave off, or think 00:58:01.740 --> 00:58:04.960 of those as a 1 setting. 00:58:04.960 --> 00:58:07.600 So my high condition for factor B-- 00:58:07.600 --> 00:58:12.360 I'll indicate that test setup as-- with a little low b. 00:58:12.360 --> 00:58:14.790 And this interaction, where I vary and pick 00:58:14.790 --> 00:58:17.040 the high condition for both of those, 00:58:17.040 --> 00:58:20.490 gives me the AB combination. 00:58:20.490 --> 00:58:24.990 And the overall low condition we often refer to 00:58:24.990 --> 00:58:29.410 as the 1 test condition. 00:58:29.410 --> 00:58:33.130 Now, the reason for doing this will maybe not 00:58:33.130 --> 00:58:34.810 be completely clear, but-- 00:58:34.810 --> 00:58:36.010 give you some intuition. 00:58:36.010 --> 00:58:39.880 When we extend this to larger numbers of factors, where 00:58:39.880 --> 00:58:45.370 you want a very shorthand way of describing which combination-- 00:58:45.370 --> 00:58:48.250 which test condition, which combination of your factor 00:58:48.250 --> 00:58:52.720 levels you want to do, and as ways of forming 00:58:52.720 --> 00:58:56.920 sums on the outputs at those conditions 00:58:56.920 --> 00:59:00.520 in order to be able to do our estimates. 00:59:00.520 --> 00:59:02.950 So that's a little bit of additional notation. 00:59:02.950 --> 00:59:07.660 Now, the model that we might want to fit 00:59:07.660 --> 00:59:10.090 is, in fact, perhaps a regression model. 00:59:10.090 --> 00:59:16.900 It's a little bit different than the factor effects model, 00:59:16.900 --> 00:59:20.140 which becomes a very nice way to do it 00:59:20.140 --> 00:59:26.470 when we've normalized our inputs. 00:59:26.470 --> 00:59:30.790 This works best with normalized input levels-- 00:59:33.310 --> 00:59:39.210 in other words, my x sub i taking on values of plus 1 00:59:39.210 --> 00:59:40.200 and minus 1. 00:59:45.690 --> 00:59:50.160 And we'll see how those relate to the offset or effects model 00:59:50.160 --> 00:59:51.820 in a second. 00:59:51.820 --> 00:59:57.990 So this is now the same plot, my input x1 and my input x2, 00:59:57.990 --> 01:00:00.880 but now I'm also adding the output. 01:00:00.880 --> 01:00:05.560 So I'm getting to my 3D view of the results. 01:00:05.560 --> 01:00:07.560 This is the same graph as before-- 01:00:07.560 --> 01:00:10.830 my factor A and my factor B. 01:00:10.830 --> 01:00:13.170 But then the height here-- each of the values 01:00:13.170 --> 01:00:19.770 is the y bar at my n replicates averaged together 01:00:19.770 --> 01:00:22.350 to give me a sense of what my actual process 01:00:22.350 --> 01:00:29.040 output was at each of my four input conditions. 01:00:29.040 --> 01:00:33.270 So we can talk about here the mean values 01:00:33.270 --> 01:00:36.120 of each of those different treatment conditions, 01:00:36.120 --> 01:00:40.600 again, where each of those is an output across those. 01:00:40.600 --> 01:00:42.430 And now we can start to look and say, 01:00:42.430 --> 01:00:47.520 OK, are there trends as a function of factor A, 01:00:47.520 --> 01:00:49.770 are there trends as a function of factor B, 01:00:49.770 --> 01:00:52.020 and fit model coefficients that go 01:00:52.020 --> 01:00:56.190 or capture with those trends? 01:00:56.190 --> 01:01:02.030 So a couple more pieces of terminology-- we can be asking, 01:01:02.030 --> 01:01:05.270 is there an effect in the ANOVA sense-- 01:01:05.270 --> 01:01:07.030 and we don't want to estimate what 01:01:07.030 --> 01:01:11.140 the effective is from a model coefficient effect-- 01:01:11.140 --> 01:01:15.110 is there an affect or a main effect-- 01:01:15.110 --> 01:01:17.860 which is the direct linear term-- 01:01:17.860 --> 01:01:19.960 and are there these interaction effects, 01:01:19.960 --> 01:01:28.510 or these AB interaction or multiple factor interactions? 01:01:28.510 --> 01:01:31.960 And the perspective that we take is very close 01:01:31.960 --> 01:01:35.620 now to that sneaky trick that I showed-- 01:01:35.620 --> 01:01:38.560 described with the injection molding data. 01:01:38.560 --> 01:01:43.840 If I'm trying to estimate the main effective my first factor, 01:01:43.840 --> 01:01:47.440 I basically am treating my second factor 01:01:47.440 --> 01:01:48.970 as just noise conditions. 01:01:48.970 --> 01:01:52.240 And I form my aggregate perspective for, 01:01:52.240 --> 01:01:55.540 is there a velocity effect, averaging together 01:01:55.540 --> 01:01:59.860 all of the other data, just considering 01:01:59.860 --> 01:02:02.860 holding the velocity at its low condition or the velocity 01:02:02.860 --> 01:02:04.150 at its high condition. 01:02:04.150 --> 01:02:06.130 And then I basically do a linear fit 01:02:06.130 --> 01:02:09.020 through the mean of those two cases. 01:02:09.020 --> 01:02:12.250 So it's just like we were doing that the linear case, 01:02:12.250 --> 01:02:16.270 but ignoring or lumping together all of the other data 01:02:16.270 --> 01:02:20.030 to estimate the main effect in each case. 01:02:20.030 --> 01:02:23.170 So there's a way to look at this then in our data 01:02:23.170 --> 01:02:26.260 here for estimating the main effect. 01:02:26.260 --> 01:02:30.600 We would label the main effect for A 01:02:30.600 --> 01:02:33.340 as basically looking and saying, what I'd like to do 01:02:33.340 --> 01:02:37.570 is take both of my high settings for A-- 01:02:37.570 --> 01:02:40.600 this is the plus setting for A-- 01:02:40.600 --> 01:02:43.705 average the outputs together for those two cases-- 01:02:46.360 --> 01:02:51.770 similarly, take the low condition for A-- 01:02:51.770 --> 01:02:54.830 let me use a slightly different color here, make this blue-- 01:02:54.830 --> 01:02:59.600 look at the low condition, and average those together, 01:02:59.600 --> 01:03:02.870 and then look at in aggregate as I 01:03:02.870 --> 01:03:07.760 went from the low to the high for condition A. 01:03:07.760 --> 01:03:09.830 Is there an average jump up? 01:03:09.830 --> 01:03:14.660 Is there an effect of factor A in aggregate? 01:03:14.660 --> 01:03:19.460 And so what you can actually do is very simply form-- 01:03:19.460 --> 01:03:23.465 or model that overall delta coming from factor A 01:03:23.465 --> 01:03:27.680 as just the overall average between these two points 01:03:27.680 --> 01:03:30.380 and the overall average between these two points, 01:03:30.380 --> 01:03:37.850 and simply fit the line through the mean of those two cases. 01:03:37.850 --> 01:03:40.270 So essentially, I project all of my data 01:03:40.270 --> 01:03:46.240 just down onto my A axis and ask the univariate question, 01:03:46.240 --> 01:03:49.690 does my y output change as a function of A-- 01:03:49.690 --> 01:03:53.000 lumping together all of the other data? 01:03:53.000 --> 01:03:56.290 And so that's simply what we're doing with this equation. 01:03:56.290 --> 01:03:59.290 And now that terminology that I introduced in the earlier 01:03:59.290 --> 01:04:03.770 picture comes into play. 01:04:03.770 --> 01:04:08.110 You can talk about now the outputs at y sub 01:04:08.110 --> 01:04:13.480 A high as being those cases where 01:04:13.480 --> 01:04:19.030 I had the plus value for A and the low value for B. 01:04:19.030 --> 01:04:22.540 And the other one would be the plus value for A 01:04:22.540 --> 01:04:25.990 and the high condition for B. So these two points 01:04:25.990 --> 01:04:27.900 I'm simply referring to-- 01:04:27.900 --> 01:04:29.650 it's getting kind of messy here, isn't it? 01:04:32.660 --> 01:04:38.030 I can refer to this average right 01:04:38.030 --> 01:04:43.310 here as the A and AB outputs averaged. 01:04:43.310 --> 01:04:45.260 I've got two of those levels, and then n 01:04:45.260 --> 01:04:48.030 replicates at each of those points. 01:04:48.030 --> 01:04:51.020 So that's just a way of expressing that average, 01:04:51.020 --> 01:04:53.630 using our test condition terminology. 01:04:56.940 --> 01:04:57.960 OK. 01:04:57.960 --> 01:04:59.220 Everybody get mean effects? 01:05:02.670 --> 01:05:05.310 We can also do interaction effects. 01:05:05.310 --> 01:05:10.050 And they're a little bit more subtle to form the contrast, 01:05:10.050 --> 01:05:16.410 but it's essentially asking a different set of averages, 01:05:16.410 --> 01:05:21.690 and trying to say, as I change both A and B together, 01:05:21.690 --> 01:05:25.590 do I get something more than the A effect alone or the B effect 01:05:25.590 --> 01:05:26.400 alone? 01:05:26.400 --> 01:05:30.610 And what we actually do in this case is take the AB condition-- 01:05:30.610 --> 01:05:32.070 this one right here-- 01:05:32.070 --> 01:05:35.940 take the low-- so I'm taking the high-high and the low-low, 01:05:35.940 --> 01:05:38.770 forming its average. 01:05:38.770 --> 01:05:44.940 Then I take the other corner, the just A high and the just B 01:05:44.940 --> 01:05:46.050 high-- 01:05:46.050 --> 01:05:49.920 take its average, and then basically take the difference 01:05:49.920 --> 01:05:51.460 between those two. 01:05:51.460 --> 01:05:56.400 So I'm averaging the corners together, 01:05:56.400 --> 01:05:59.130 in contrast to the main effect, where I just averaged 01:05:59.130 --> 01:06:02.700 along the two factor levels. 01:06:02.700 --> 01:06:05.130 And that forms an estimate of the AB offset. 01:06:11.280 --> 01:06:16.237 Basically, this average of the A high and the A 01:06:16.237 --> 01:06:25.410 low we'll refer to as the contrast, the right combination 01:06:25.410 --> 01:06:27.690 of the output values that give me 01:06:27.690 --> 01:06:30.570 an estimate for the A effect. 01:06:30.570 --> 01:06:33.780 And similarly, we'll talk about the contrast here 01:06:33.780 --> 01:06:35.700 for the B factor. 01:06:35.700 --> 01:06:37.980 Which combinations of output values 01:06:37.980 --> 01:06:42.330 do you use to estimate that effect and the contrast 01:06:42.330 --> 01:06:45.420 for the interaction terms? 01:06:45.420 --> 01:06:51.540 And what's cool, if I go back to that early slide where I said, 01:06:51.540 --> 01:06:53.910 if I'm just fitting a linear output, y 01:06:53.910 --> 01:06:58.770 as a function of one factor, and I got two data points, 01:06:58.770 --> 01:07:05.430 my line goes through the average of those two conditions. 01:07:05.430 --> 01:07:09.450 If I form the contrast now in a multiple output case, 01:07:09.450 --> 01:07:13.170 again, the best estimate of the linear line 01:07:13.170 --> 01:07:15.430 goes through the averages-- 01:07:15.430 --> 01:07:19.500 these are the averages at each of those conditions. 01:07:19.500 --> 01:07:22.710 And what I've simply got is, directly 01:07:22.710 --> 01:07:26.130 from the contrast, a direct estimate 01:07:26.130 --> 01:07:33.330 of the model coefficient as a function in the normalized axis 01:07:33.330 --> 01:07:34.500 space. 01:07:34.500 --> 01:07:37.740 If our x sub i's are in this normalized plus 1 01:07:37.740 --> 01:07:41.070 to minus 1 range, directly calculating 01:07:41.070 --> 01:07:43.080 the contrast of my outputs gives me 01:07:43.080 --> 01:07:47.250 directly what an estimate of the effects 01:07:47.250 --> 01:07:50.250 are-- the main effects as well as the interaction effects. 01:07:53.337 --> 01:07:54.920 So I didn't actually have to go and do 01:07:54.920 --> 01:07:59.160 the optimization problem, do the linear regression. 01:07:59.160 --> 01:08:01.760 I'm using the fact that the linear regression forces 01:08:01.760 --> 01:08:03.770 the data to go through the averages. 01:08:03.770 --> 01:08:06.740 And I can simply look at the data conditions that I've got, 01:08:06.740 --> 01:08:11.000 the averages of the output at each of my four combinations, 01:08:11.000 --> 01:08:13.790 and then look at it either from the A factor perspective 01:08:13.790 --> 01:08:17.840 and estimate the A affect, with a B factor perspective 01:08:17.840 --> 01:08:20.779 and estimate the B effect, and then also look and see 01:08:20.779 --> 01:08:23.750 if there's an interaction between the two 01:08:23.750 --> 01:08:28.529 because of cross-coupling between those two factors. 01:08:28.529 --> 01:08:32.510 So what you've got is the regression perspective 01:08:32.510 --> 01:08:36.080 and this contrast perspective are exactly the same. 01:08:38.819 --> 01:08:40.819 Now, I said this was a linear model, 01:08:40.819 --> 01:08:44.840 and what's cool is it actually is from either the A 01:08:44.840 --> 01:08:45.760 perspective-- 01:08:48.859 --> 01:08:50.210 I guess that's the B factor. 01:08:52.779 --> 01:08:56.020 That's the slope, if you will, versus my B 01:08:56.020 --> 01:08:59.680 factor, my x2 variable. 01:08:59.680 --> 01:09:04.450 I could similarly have the slope from my A factor. 01:09:04.450 --> 01:09:08.439 This was my A axes, or x1 axes. 01:09:08.439 --> 01:09:13.060 So I've got a simple linear dependence of output 01:09:13.060 --> 01:09:16.390 on the input in each of those two cases, 01:09:16.390 --> 01:09:23.350 and the interaction basically gives me linear dependencies 01:09:23.350 --> 01:09:28.029 that add additional offsets, but that are locally linear 01:09:28.029 --> 01:09:29.180 in each case. 01:09:29.180 --> 01:09:35.529 So what we get is this so-called 3D ruled surface that 01:09:35.529 --> 01:09:39.850 is linear every time I project it 01:09:39.850 --> 01:09:42.399 on either the A axis or the B axis. 01:09:42.399 --> 01:09:44.649 But it's got this kind of funky ability 01:09:44.649 --> 01:09:49.510 to shift because of the interaction, 01:09:49.510 --> 01:09:52.920 so it's kind of a subtle non-linearity at work. 01:09:52.920 --> 01:09:55.470 It's not a curvature non-linearity 01:09:55.470 --> 01:09:59.790 in the sense of quadratic curvature. 01:09:59.790 --> 01:10:02.580 It is a ruled surface projection down 01:10:02.580 --> 01:10:11.020 to linear dependencies on A and B for each of my combinations. 01:10:11.020 --> 01:10:15.690 So here's an example of the surface 01:10:15.690 --> 01:10:18.180 that results if I don't have the interaction. 01:10:18.180 --> 01:10:23.820 And a very simple model example is x1 and x2 01:10:23.820 --> 01:10:26.610 with some linear coefficient, and then 01:10:26.610 --> 01:10:30.090 what I've simply got for my ruled surface is a plane. 01:10:30.090 --> 01:10:36.420 It's a slanted plane as a function of x and x2 factors. 01:10:36.420 --> 01:10:40.020 Sometimes we also draw these interaction plots, 01:10:40.020 --> 01:10:43.320 and all of that this is just looking at one of the factors-- 01:10:43.320 --> 01:10:49.260 say, along x2-- and picking some value for the other factor-- 01:10:49.260 --> 01:10:54.930 say, x1 is-- let's see-- x1 is low-- 01:10:54.930 --> 01:10:58.320 that would be this line right there-- 01:10:58.320 --> 01:11:03.060 or x1 is high, which would be this line right there-- 01:11:03.060 --> 01:11:07.170 and then simply plotting y just as the univariate function 01:11:07.170 --> 01:11:11.760 of x2, where I picked or set some other value for my x1. 01:11:11.760 --> 01:11:15.030 And in this case without interaction, 01:11:15.030 --> 01:11:19.380 you'll notice those lines have to be parallel-- 01:11:19.380 --> 01:11:27.050 that what I've got as a simple linear additive model 01:11:27.050 --> 01:11:33.200 without interaction is I just get off sets 01:11:33.200 --> 01:11:35.600 because of whatever value of x when I pick 01:11:35.600 --> 01:11:39.510 or offsets because of whatever value of x2 I pick. 01:11:39.510 --> 01:11:45.620 So this is simply saying the x1 effect 01:11:45.620 --> 01:11:50.150 is shown right here as an additional shift in the output 01:11:50.150 --> 01:11:53.960 dependence on x2. 01:11:53.960 --> 01:11:57.860 So this is just another slice through the 3D picture, 01:11:57.860 --> 01:11:59.510 but what's important here is you would 01:11:59.510 --> 01:12:04.790 see this kind of interaction pattern on your output 01:12:04.790 --> 01:12:07.070 as a function of your input, if there's 01:12:07.070 --> 01:12:11.285 no cross term between x1 and x2. 01:12:14.200 --> 01:12:18.580 And in contrast, here's the response surface of output 01:12:18.580 --> 01:12:20.930 as a function of my two inputs. 01:12:20.930 --> 01:12:23.350 This is, again, a ruled surface. 01:12:23.350 --> 01:12:25.900 Each one of these is just a line, 01:12:25.900 --> 01:12:30.490 but now it's got this funky non-linearity because 01:12:30.490 --> 01:12:32.990 of this interaction term. 01:12:32.990 --> 01:12:37.090 And in this case, if I take a slice at different low and high 01:12:37.090 --> 01:12:39.970 of x1, you can see that there's something at work 01:12:39.970 --> 01:12:44.470 more than just an additive effect from x1, 01:12:44.470 --> 01:12:46.810 depending on my value of x2. 01:12:46.810 --> 01:12:51.040 There is synergy between whatever x1 value I pick 01:12:51.040 --> 01:12:53.500 and x2 value I picked that together-- 01:12:53.500 --> 01:12:57.040 I need both to explain the output 01:12:57.040 --> 01:12:59.650 and how they work together. 01:12:59.650 --> 01:13:02.140 You can also have negative interactions, 01:13:02.140 --> 01:13:08.710 where the cross term causes a negative synergistic effect 01:13:08.710 --> 01:13:11.930 between the two. 01:13:11.930 --> 01:13:15.680 OK, so what we'll pick up on next time 01:13:15.680 --> 01:13:21.380 is extending this a little bit to a more general way 01:13:21.380 --> 01:13:23.870 of talking about these contrasts. 01:13:23.870 --> 01:13:25.940 I think hopefully now you've got the picture 01:13:25.940 --> 01:13:30.560 from a univariate kind of modeling and regression picture 01:13:30.560 --> 01:13:33.890 up to a two-factor kind of picture 01:13:33.890 --> 01:13:35.460 with these interactions. 01:13:35.460 --> 01:13:38.330 Next time, what we'll do is extend that-- 01:13:38.330 --> 01:13:43.280 oops-- extend that or generalize that to when I might have three 01:13:43.280 --> 01:13:44.210 factors-- 01:13:44.210 --> 01:13:47.680 an A, B, and a C factor, for example. 01:13:47.680 --> 01:13:51.500 And how do you visualize your design space in those cases 01:13:51.500 --> 01:13:55.190 and how do you form the contrast to very quickly estimate model 01:13:55.190 --> 01:13:58.110 coefficients in those cases? 01:13:58.110 --> 01:13:59.660 And then we'll also get to-- 01:14:02.940 --> 01:14:04.980 oops-- why'd that happen? 01:14:04.980 --> 01:14:10.470 We'll also get next time to looking a little bit more at, 01:14:10.470 --> 01:14:13.710 how do I check the adequacy of these model terms, 01:14:13.710 --> 01:14:16.020 are they significant, as well as, 01:14:16.020 --> 01:14:21.780 what are good estimates for confidence intervals 01:14:21.780 --> 01:14:24.540 on these things-- and then come to these other subtle points, 01:14:24.540 --> 01:14:28.410 like what happens if I don't want 01:14:28.410 --> 01:14:32.580 to do 2 to the k, when k is 5? 01:14:32.580 --> 01:14:35.160 I don't necessarily need every combination. 01:14:35.160 --> 01:14:39.720 And we'll talk about fractional factorial designs. 01:14:39.720 --> 01:14:46.700 So one thing you should do in preparation for next time 01:14:46.700 --> 01:14:49.640 is start reading the experimental design chapter. 01:14:49.640 --> 01:14:52.460 And I can't remember which chapter it is in Montgomery. 01:14:52.460 --> 01:14:55.550 I'll post that on the-- 01:14:55.550 --> 01:14:57.560 as an announcement on the website. 01:14:57.560 --> 01:15:00.530 But there's a lot of lingo to get used to, 01:15:00.530 --> 01:15:03.870 but it all comes back to nice intuitive relationships, 01:15:03.870 --> 01:15:04.370 I think. 01:15:04.370 --> 01:15:06.620 But you do need to start at least scanning 01:15:06.620 --> 01:15:11.750 that chapter in Montgomery for the experimental design stuff. 01:15:11.750 --> 01:15:15.290 I think Montgomery has a more thorough description than we 01:15:15.290 --> 01:15:17.480 can find [INAUDIBLE]. 01:15:17.480 --> 01:15:19.280 Actually, [INAUDIBLE] might be a good place 01:15:19.280 --> 01:15:21.260 to get the quick read on it first, 01:15:21.260 --> 01:15:25.230 and then Montgomery in more detail. 01:15:25.230 --> 01:15:29.660 So with that, I hope you guys have a great spring break. 01:15:29.660 --> 01:15:34.580 And enjoy the MIT spring break, for our MIT students 01:15:34.580 --> 01:15:35.630 in Singapore. 01:15:35.630 --> 01:15:38.720 You can rub it in to all your classmates 01:15:38.720 --> 01:15:43.550 who are still meeting for other classes in Singapore. 01:15:43.550 --> 01:15:48.380 You get a break to work on your projects out in Singapore. 01:15:48.380 --> 01:15:51.490 So we'll see you in two weeks.