1 00:00:00,150 --> 00:00:04,410 Remember earlier in the course that we looked at scatter plots when we talked through different types 2 00:00:04,410 --> 00:00:05,920 of charts and graphs. 3 00:00:05,939 --> 00:00:10,970 Now we want to revisit scatter plots and specifically this idea of regression. 4 00:00:10,980 --> 00:00:16,980 So earlier when we talked about scatter plots, remember that we talked about this least squares line 5 00:00:16,980 --> 00:00:18,550 or the line of best fit. 6 00:00:18,570 --> 00:00:24,420 We also called this the regression line, and we learned to calculate the equation of the regression 7 00:00:24,420 --> 00:00:29,400 line or determine the equation of the regression line by finding M, the slope which was found with 8 00:00:29,400 --> 00:00:37,650 this formula and B the y intercept which we found using this formula and we indicated the equation of 9 00:00:37,650 --> 00:00:43,800 the regression line with this y hat symbol to make the specific point that this is sort of a line of 10 00:00:43,800 --> 00:00:48,410 estimation and it doesn't actually run through all of the data points in the scatterplot. 11 00:00:48,420 --> 00:00:54,900 Instead, it charts a course through the scatterplot that minimizes the error or that minimizes the 12 00:00:54,900 --> 00:00:55,590 residuals. 13 00:00:55,590 --> 00:00:57,620 And so we indicate it with y hat. 14 00:00:57,630 --> 00:01:03,780 So technically regression is just the process of estimating the value of the dependent variable for 15 00:01:03,780 --> 00:01:06,450 some given value of the independent variable. 16 00:01:06,450 --> 00:01:12,540 And in order to estimate the value of the dependent variable, we're going to go through the process 17 00:01:12,540 --> 00:01:13,650 of curve fitting. 18 00:01:13,650 --> 00:01:20,490 Now this equation is the regression equation for a line, but the curve that we use to approximate the 19 00:01:20,490 --> 00:01:27,720 data does not necessarily have to be a line, which is why we'll often refer to the regression equation 20 00:01:27,720 --> 00:01:29,610 as just the approximating curve. 21 00:01:29,610 --> 00:01:34,980 Or instead of saying that we're always finding the linear regression equation, we'll instead just say 22 00:01:34,980 --> 00:01:36,960 that we're going through the curve fitting process. 23 00:01:36,990 --> 00:01:42,540 That being said, as we're getting into this idea of regression, we really want to talk through the 24 00:01:42,540 --> 00:01:46,700 four different ways that we identify the regression curve. 25 00:01:46,710 --> 00:01:53,130 In other words, there are four different ways that we normally describe the trend, and those four 26 00:01:53,130 --> 00:01:59,520 ways are the form, the direction, the strength, and whether or not we have outliers in our data. 27 00:01:59,520 --> 00:02:03,900 So let's talk first about the form of the trend. 28 00:02:03,900 --> 00:02:05,460 Through the scatterplot. 29 00:02:05,490 --> 00:02:11,009 We can think about three different examples here, and I've made these examples pretty extreme so that 30 00:02:11,009 --> 00:02:13,020 the trends are obvious. 31 00:02:13,020 --> 00:02:19,380 But in this first example here, we can clearly see that the trend is linear. 32 00:02:19,410 --> 00:02:26,610 We have linear correlation in the data because the shape of the curve that best fits this data is a 33 00:02:26,610 --> 00:02:27,090 line. 34 00:02:27,090 --> 00:02:31,620 And so in this case, if we were going to find the trend through the data, we would find the trend 35 00:02:31,620 --> 00:02:35,940 line and we would find this linear regression equation. 36 00:02:35,940 --> 00:02:37,440 But that's not always the case. 37 00:02:37,440 --> 00:02:43,080 Sometimes the trend through the data follows a parabolic shape or an exponential shape like this one. 38 00:02:43,080 --> 00:02:52,800 We might describe this trend or this correlation as parabolic or exponential correlation because of 39 00:02:52,800 --> 00:02:54,930 the shape of this approximating curve. 40 00:02:54,930 --> 00:03:01,380 And we can see clearly here that this exponential parabolic shape does a better job of fitting the data 41 00:03:01,380 --> 00:03:03,540 than just using a line. 42 00:03:03,570 --> 00:03:06,750 These are certainly not the only two shapes. 43 00:03:06,750 --> 00:03:13,620 For instance, the data points might follow a trend similar to a sign curve, something like this. 44 00:03:13,620 --> 00:03:18,600 And so maybe our approximating curve looks like this and we would call the trend sinusoidal because 45 00:03:18,600 --> 00:03:20,280 it follows a sine curve. 46 00:03:20,280 --> 00:03:26,970 And maybe that curve does a better job than specifically a line or this parabolic shape at approximating 47 00:03:26,970 --> 00:03:28,320 the trend in the data. 48 00:03:28,320 --> 00:03:32,040 So we're looking for the form of the trend. 49 00:03:32,040 --> 00:03:40,470 And for a scatterplot like this one, we might say here that we see no correlation because the data 50 00:03:40,470 --> 00:03:47,880 is so scattered in such a randomized way that it's difficult for us to see any kind of trend whatsoever 51 00:03:47,880 --> 00:03:50,100 in the data, at least visually. 52 00:03:50,100 --> 00:03:55,830 It almost doesn't make sense to fit this with a linear trend line or a parabolic trend line or some 53 00:03:55,830 --> 00:03:57,810 other approximating curve. 54 00:03:57,810 --> 00:03:59,130 And that can be the case. 55 00:03:59,130 --> 00:04:04,710 We can just have a data set where there's really no correlation at all so we can talk through different 56 00:04:04,710 --> 00:04:06,540 forms that approximate the trend. 57 00:04:06,540 --> 00:04:10,050 We can also talk about the direction of the trend. 58 00:04:10,050 --> 00:04:19,560 So to take two obvious examples here in this first example, the trend here is positive because as we 59 00:04:19,560 --> 00:04:21,870 move to the right, the graph moves up. 60 00:04:21,870 --> 00:04:25,830 Whereas with this second example here the trend is. 61 00:04:26,520 --> 00:04:32,550 Negative, because as we move to the right, the graph moves down and that holds not just for linear 62 00:04:32,550 --> 00:04:35,660 relationships, but also for relationships of other shapes. 63 00:04:35,670 --> 00:04:42,900 So instead of this parabolic curve here, if we had something that looked like this, that approximating 64 00:04:42,900 --> 00:04:48,180 curve could be parabolic or exponential, but the trend would be negative, we would describe the direction 65 00:04:48,180 --> 00:04:49,350 as negative. 66 00:04:49,560 --> 00:04:54,600 So we talk about direction, we talk about strength of the relationship. 67 00:04:54,600 --> 00:05:01,410 So in these two examples here, this first one, we might call this a strong relationship, whereas 68 00:05:01,410 --> 00:05:07,800 the relationship among this data here we might call either moderate or maybe even. 69 00:05:08,560 --> 00:05:09,340 Weak. 70 00:05:09,340 --> 00:05:15,190 And really the judgment we're making here is based on how tightly clustered the data points are around 71 00:05:15,190 --> 00:05:16,730 the approximating curve. 72 00:05:16,750 --> 00:05:21,110 In these cases, the trend line in this first scatterplot. 73 00:05:21,130 --> 00:05:25,360 All of the data points are very, very close to this trend line. 74 00:05:25,360 --> 00:05:28,560 And so the relationship in the data is strong. 75 00:05:28,570 --> 00:05:33,520 But in this example, the data points are spread further away from the trend line. 76 00:05:33,550 --> 00:05:36,130 They're not all packed in close to the line. 77 00:05:36,130 --> 00:05:39,190 And instead we see many data points that are further from the line. 78 00:05:39,190 --> 00:05:45,340 And so we would maybe say that this relationship is a moderate relationship or maybe even a weak relationship, 79 00:05:45,340 --> 00:05:51,310 but we would certainly say that it is not as strong of a relationship as the relationship we see in 80 00:05:51,310 --> 00:05:53,180 this first scatterplot. 81 00:05:53,200 --> 00:05:56,310 And then the last thing we always want to describe is outliers. 82 00:05:56,320 --> 00:06:02,530 So, for instance, in a graph like this one, once we have the approximating curve, if it's linear, 83 00:06:02,530 --> 00:06:07,750 once we have the line of best fit or the linear regression line here, we want to be able to identify 84 00:06:07,750 --> 00:06:11,800 what appear to be outliers in the data in this particular scatterplot. 85 00:06:12,280 --> 00:06:15,420 This point right here appears to be an outlier. 86 00:06:15,430 --> 00:06:22,180 It is by far the furthest point from the regression line, and so it's the biggest outlier in the data 87 00:06:22,180 --> 00:06:22,630 set. 88 00:06:22,660 --> 00:06:28,660 Obviously, that being said, the more outliers there are in the data set, the weaker the relationship 89 00:06:28,660 --> 00:06:29,170 is. 90 00:06:29,170 --> 00:06:33,820 The fewer outliers we have then clearly the stronger the relationship. 91 00:06:33,850 --> 00:06:41,380 So taking account of all four of these characteristics or descriptors, whenever we have the scatterplot 92 00:06:41,410 --> 00:06:47,500 of a data set, we want to be able to describe the trend in the data using these four characteristics. 93 00:06:47,500 --> 00:06:55,060 So for example, if we're looking at this scatterplot right here, we might say that the data displays 94 00:06:55,060 --> 00:07:03,550 a moderate, positive linear trend where the most significant outlier is this point here, which appears 95 00:07:03,550 --> 00:07:12,430 to be at maybe the point, let's say 618 roughly, that would be a fairly comprehensive way to describe 96 00:07:12,430 --> 00:07:19,420 the trend in this data as a moderate, positive linear relationship with an outlier at this point. 97 00:07:19,420 --> 00:07:29,440 Whereas this scatterplot here we might describe as a strong negative linear relationship with no noticeable 98 00:07:29,440 --> 00:07:30,220 outliers. 99 00:07:30,220 --> 00:07:37,060 So the takeaway here is just that the purpose of regression is to be able to estimate the value of the 100 00:07:37,060 --> 00:07:42,010 dependent variable y for any value of the independent variable X. 101 00:07:42,010 --> 00:07:46,350 And the way that we do that is by finding a trend through the data. 102 00:07:46,360 --> 00:07:51,280 Earlier when we talked about scatter plots, we learned how to use these equations to find the equation 103 00:07:51,280 --> 00:07:56,890 of the least squares line or the line of best fit, that regression line that's given by this equation. 104 00:07:56,890 --> 00:08:03,220 So we know how to find this simple equation of the regression line, and now we know how to describe 105 00:08:03,220 --> 00:08:09,010 that regression line based on its form, direction, strength and any outliers in the data set. 106 00:08:09,010 --> 00:08:14,560 With that foundation out of the way, we'll take the rest of this section to talk about some more advanced 107 00:08:14,560 --> 00:08:16,630 regression calculations.