1 00:00:00,120 --> 00:00:05,700 Earlier, we briefly mentioned the chi square test statistic and now we want to look more at chi squared 2 00:00:05,700 --> 00:00:06,370 tests. 3 00:00:06,390 --> 00:00:10,080 There are many different kinds of chi square tests we can perform. 4 00:00:10,080 --> 00:00:14,160 We'll look at three of them and then in particular we'll focus on one. 5 00:00:14,160 --> 00:00:20,370 So one kind of chi squared test that we can do is a chi squared test for homogeneity, where we take 6 00:00:20,370 --> 00:00:25,350 a sample from two groups and essentially compare their probability distributions. 7 00:00:25,350 --> 00:00:31,800 So an example of a question we could answer here is whether or not gender has an effect on pet preference. 8 00:00:31,800 --> 00:00:39,570 So if we were to ask men and women whether they prefer cats, dogs or some other kind of pet and then 9 00:00:39,570 --> 00:00:45,210 create a table of our data, it might look something like this, We can see that what we're doing here 10 00:00:45,210 --> 00:00:46,890 is sampling from two groups. 11 00:00:46,890 --> 00:00:52,440 One group is men, one group is women, and we are comparing their probability distributions. 12 00:00:52,440 --> 00:00:58,110 So we have a probability distribution for men across cats, dogs and other pets, and then the same 13 00:00:58,110 --> 00:01:00,240 probability distribution for women. 14 00:01:00,240 --> 00:01:06,720 Now, if gender has no effect at all on pet preference, then we would expect those probability distributions 15 00:01:06,720 --> 00:01:07,890 to be similar. 16 00:01:07,890 --> 00:01:15,810 In other words, if men prefer cats at roughly a 25% rate, dogs at 50% and other pets at 25%, then 17 00:01:15,810 --> 00:01:22,500 we would expect roughly the same distribution for women, 25% for cats, 50% for dogs and 25% for other 18 00:01:22,500 --> 00:01:23,100 pets. 19 00:01:23,130 --> 00:01:29,370 Now, of course, we know that these two distributions are very unlikely to be identical, but if gender 20 00:01:29,370 --> 00:01:34,140 doesn't affect pet preference, then we could expect them to be roughly similar. 21 00:01:34,140 --> 00:01:39,450 But if gender does have an effect on pet preference, then we might see these probability distributions 22 00:01:39,450 --> 00:01:41,370 being extremely different. 23 00:01:41,370 --> 00:01:47,670 And the chi squared test statistic lets us look at how different the distributions are and then determine 24 00:01:47,670 --> 00:01:53,220 whether the difference is big enough to be called statistically significant, such that we could reject 25 00:01:53,220 --> 00:01:58,410 the null hypothesis that gender doesn't affect pet preference and therefore lend support to the alternative 26 00:01:58,410 --> 00:02:01,800 hypothesis that pet preference is affected by gender. 27 00:02:01,800 --> 00:02:04,770 So that's one kind of test We can run. 28 00:02:04,800 --> 00:02:11,220 Another kind of test we can run with chi squared is a chi squared test for association or independence. 29 00:02:11,220 --> 00:02:18,540 In this kind of test, we sample from one group and we compare characteristics for that same group. 30 00:02:18,540 --> 00:02:22,770 For instance, we could compare eye color and handedness. 31 00:02:22,770 --> 00:02:29,220 So whether someone is left handed or right handed for the same set of individuals, we might make a 32 00:02:29,220 --> 00:02:30,990 data table that looks like this. 33 00:02:30,990 --> 00:02:38,130 So we pull a sample of people all from the same population and we record whether they are left handed 34 00:02:38,130 --> 00:02:40,890 or right handed in contrast with their eye color. 35 00:02:40,890 --> 00:02:47,640 So out of our one group, we can see that we have 72 people who are left handed with brown eyes, 36 36 00:02:47,640 --> 00:02:52,650 people who are left handed with blue eyes, 130 people who are right handed with green eyes, etc.. 37 00:02:52,650 --> 00:02:58,500 So in this kind of test, our null hypothesis would be that eye color and handedness are independent, 38 00:02:58,500 --> 00:03:01,080 they're not associated, they have no association. 39 00:03:01,080 --> 00:03:06,510 And therefore the alternative hypothesis would state that eye color and handedness are associated, 40 00:03:06,510 --> 00:03:12,570 they are dependent and we would again, similar to the homogeneity test, look at the distributions 41 00:03:12,570 --> 00:03:15,270 of the left handed people and the right handed people. 42 00:03:15,270 --> 00:03:21,210 If handedness and eye color are not associated, then the distributions for left handed and right handed 43 00:03:21,210 --> 00:03:22,320 should be similar. 44 00:03:22,320 --> 00:03:27,300 But if handedness and eye color are associated, if they're dependent on one another, then maybe we'll 45 00:03:27,300 --> 00:03:31,320 see different distributions for left handed and right handed people. 46 00:03:31,320 --> 00:03:37,230 And if that difference is significant enough, then we may be able to reject the null and lend support 47 00:03:37,230 --> 00:03:43,290 to our alternative hypothesis that handedness and eye color are associated variables. 48 00:03:43,290 --> 00:03:47,070 So you can see how these two kinds of tests are similar. 49 00:03:47,070 --> 00:03:51,570 The third kind of test will look at is a chi squared goodness of fit test. 50 00:03:51,570 --> 00:03:56,910 And here we're comparing actual versus expected values in a contingency table. 51 00:03:56,910 --> 00:04:03,060 For example, we might try to answer the question, are sales number of sales of a product affected 52 00:04:03,060 --> 00:04:04,230 by month? 53 00:04:04,230 --> 00:04:09,870 And maybe we would use this table here where we have the months of the year, January, February, March, 54 00:04:09,870 --> 00:04:14,370 etc. and we have the number of sales that we make in each month. 55 00:04:14,370 --> 00:04:21,510 So we call this a contingency table because the number of sales are contingent on the month of the year. 56 00:04:21,540 --> 00:04:27,870 To answer this kind of a question, we would think about the expected number of sales in each month 57 00:04:27,870 --> 00:04:30,480 based on the total number of sales for the year. 58 00:04:30,480 --> 00:04:37,080 In other words, if sales aren't really impacted by month, then we would expect a steady level or a 59 00:04:37,080 --> 00:04:39,660 consistent level of sales across the year. 60 00:04:39,660 --> 00:04:44,100 Month wouldn't have an effect on sales, and sales should remain fairly steady. 61 00:04:44,100 --> 00:04:50,340 And so in that sense, just like with these two tests, we would compare the distribution of actual 62 00:04:50,340 --> 00:04:53,190 sales to the distribution of expected sales. 63 00:04:53,190 --> 00:04:59,130 And if those two distributions are significantly different enough, then we might be able to conclude 64 00:04:59,130 --> 00:04:59,850 that sales. 65 00:04:59,910 --> 00:05:01,630 Are affected by month. 66 00:05:01,650 --> 00:05:06,230 So that gives you a little bit of an idea of what we're doing with chi squared tests. 67 00:05:06,240 --> 00:05:11,970 Let's dig into this goodness of fit test in a little more detail so that we can see an example and work 68 00:05:11,970 --> 00:05:14,310 through a full chi squared test. 69 00:05:14,670 --> 00:05:20,670 So we're starting here with an expanded version of the table that we had for the goodness of FIT test. 70 00:05:20,700 --> 00:05:23,850 We're trying to figure out if sales are affected by month. 71 00:05:23,850 --> 00:05:25,920 So we had the months of the year. 72 00:05:26,040 --> 00:05:29,880 These were the observed number of sales in each month. 73 00:05:29,880 --> 00:05:35,970 So in January we sold 60 products, in February we sold 80 products, etc. And when we total up all 74 00:05:35,970 --> 00:05:39,570 those sales, we have 1020 sales for the year. 75 00:05:39,600 --> 00:05:47,700 Now, if we take 1020 and we divide it by 12 months in the year, we get 85. 76 00:05:47,700 --> 00:05:55,170 And so our expected value for each month of the year is 85, because if a month has no effect on number 77 00:05:55,170 --> 00:06:02,430 of sales, then sales should remain steady and our expected value would be a steady, consistent 85 78 00:06:02,430 --> 00:06:07,470 sales per month for a total at the end of the year of 1020 sales. 79 00:06:07,470 --> 00:06:12,060 What we want to do now is use a chi squared goodness of fit test. 80 00:06:12,060 --> 00:06:21,060 We're going to pick a confidence level of 95% or an alpha value of 0.05 and output value of 5%. 81 00:06:21,060 --> 00:06:26,640 And we're going to use this chi squared test to make a determination about whether sales are affected 82 00:06:26,640 --> 00:06:27,870 by month of the year. 83 00:06:27,870 --> 00:06:35,460 So with all of these chi squared tests are null, hypothesis is that month has no effect or that the 84 00:06:35,460 --> 00:06:37,410 distributions aren't different. 85 00:06:37,410 --> 00:06:40,650 Basically that the two variables don't affect each other. 86 00:06:40,650 --> 00:06:44,130 In this case, that observed and expected values aren't different. 87 00:06:44,130 --> 00:06:47,280 So we would say here that sales. 88 00:06:48,240 --> 00:06:51,060 Aren't affected by month. 89 00:06:51,090 --> 00:06:57,570 That means that our null hypothesis then is that sales are affected by month or that the month of the 90 00:06:57,570 --> 00:06:59,900 year does have an effect on sales. 91 00:06:59,910 --> 00:07:06,090 So again, if sales aren't affected by month, then it would make sense that they would be evenly distributed 92 00:07:06,090 --> 00:07:11,550 over each month and we would see something fairly consistent like this expected value of 85. 93 00:07:11,580 --> 00:07:17,370 Now, keep in mind here that we can't just lend support to the alternative hypothesis that sales are 94 00:07:17,370 --> 00:07:23,610 affected by month just by having different observed values then the value that we expect. 95 00:07:23,610 --> 00:07:31,140 For instance, if our observed values were 84 for January, 86, for February 84, for March, 86, 96 00:07:31,140 --> 00:07:37,380 for April, 84, for May, 86 for June, etc. Yes, those values are technically different than the 97 00:07:37,380 --> 00:07:44,370 expected values of 85 for every single month, but the values are so similar, they're consistent enough 98 00:07:44,370 --> 00:07:50,820 that the difference will not be statistically significant and we will not have enough evidence to reject 99 00:07:50,820 --> 00:07:54,120 the null hypothesis and therefore lend support to the alternative. 100 00:07:54,150 --> 00:07:59,460 These observed values here not only have to be different, they have to be different enough to meet 101 00:07:59,460 --> 00:08:05,610 our level of statistical significance, and that's what will determine with our chi squared test statistic. 102 00:08:05,610 --> 00:08:08,970 So to calculate chi squared, that's our next step. 103 00:08:08,970 --> 00:08:14,640 For each of our data points, we find the difference between the observed and expected values. 104 00:08:14,640 --> 00:08:17,760 So we take observed value minus expected value. 105 00:08:17,760 --> 00:08:22,230 That should kind of remind us of the idea of the residual that we've been looking at as we've been working 106 00:08:22,230 --> 00:08:23,520 through regression here. 107 00:08:23,550 --> 00:08:27,780 So we take the observed minus expected and we get that difference. 108 00:08:27,780 --> 00:08:32,990 And so we can see here that the third row in our table is observed minus expected. 109 00:08:33,150 --> 00:08:36,450 Then we square that value to make all the values positive. 110 00:08:36,450 --> 00:08:39,480 That's the last row of our table, all those squared values. 111 00:08:39,480 --> 00:08:45,750 Once we have all those squared values, we add them all up and then divide by the expected value. 112 00:08:45,750 --> 00:08:52,050 We divide each one of them by the expected value, and then we add all of those results together in 113 00:08:52,050 --> 00:08:56,670 this particular chi squared test, because the expected values are all the same. 114 00:08:56,670 --> 00:09:03,390 That means we can add all these squared values in the fourth row and then just divide by 85 all at once. 115 00:09:03,390 --> 00:09:11,130 So the sum here of this fourth row is 6850, which means that the chi squared test statistic for this 116 00:09:11,130 --> 00:09:22,470 particular test will be 6850 divided by our expected value, 85, which is approximately 80 point. 117 00:09:23,150 --> 00:09:24,010 Five nine. 118 00:09:24,020 --> 00:09:31,790 So this is our chi squared test statistic, just like we would calculate a PT test statistic or a Z 119 00:09:31,790 --> 00:09:37,400 score, we calculate this chi squared test statistic and then, as you might have suspected, just like 120 00:09:37,400 --> 00:09:42,050 the T score or the Z score, we need to look this up in a chi squared table. 121 00:09:42,080 --> 00:09:48,500 Chi squared has its own distribution and so it has its own table and the chi squared table looks like 122 00:09:48,500 --> 00:09:49,040 this. 123 00:09:49,040 --> 00:09:56,900 It's very similar to the t table in the sense that we have a degrees of freedom value down this left 124 00:09:56,900 --> 00:09:57,560 hand side. 125 00:09:57,590 --> 00:10:03,650 Now when we think about degrees of freedom for a chi squared test, we always need to go back to the 126 00:10:03,650 --> 00:10:09,290 table that we're starting with and we want to look at the original data that we had, the observed data, 127 00:10:09,290 --> 00:10:10,730 including the total. 128 00:10:10,730 --> 00:10:19,100 So in this case we would be looking at this part right here of our table, all the original observed 129 00:10:19,100 --> 00:10:22,400 data plus the total. 130 00:10:22,400 --> 00:10:27,860 Now the degrees of freedom is the number of values that we would have to include in the body of the 131 00:10:27,860 --> 00:10:34,490 table, not including the total, in order to be able to fill in any other missing values in the table. 132 00:10:34,490 --> 00:10:36,020 So here's what we mean. 133 00:10:36,020 --> 00:10:41,810 If we look at the body of this table, we have all these values in the body 60, 80, 65, all the way 134 00:10:41,810 --> 00:10:44,120 up to 65 here in December. 135 00:10:44,120 --> 00:10:45,710 And then we have the total. 136 00:10:45,740 --> 00:10:51,950 The question is, how many values can we remove from the body of the table before we would no longer 137 00:10:51,950 --> 00:10:54,500 know what to put in each cell? 138 00:10:54,500 --> 00:11:02,720 Well, if we remove the value here in December and we pretend now that we don't have the value for December, 139 00:11:02,720 --> 00:11:06,230 the only values we have are January through November. 140 00:11:06,230 --> 00:11:12,620 And the total the question is could we figure out what value to put back in for December? 141 00:11:12,620 --> 00:11:19,220 And of course, the answer is yes, because we could take the total 1020 and subtract from it all of 142 00:11:19,220 --> 00:11:22,160 these other values for January through November. 143 00:11:22,160 --> 00:11:27,290 And what we would be left with is 65, which has to be December's value. 144 00:11:27,290 --> 00:11:33,830 So we can remove the value for December and we can still know what all of the values in the table will 145 00:11:33,830 --> 00:11:34,270 be. 146 00:11:34,280 --> 00:11:40,490 Now the question becomes, can we remove another value from the body of the table and still know how 147 00:11:40,490 --> 00:11:42,080 to fill in the rest of the table? 148 00:11:42,080 --> 00:11:44,810 So let's say we remove November's value. 149 00:11:44,810 --> 00:11:48,830 Now the question is, can we accurately fill in November and December? 150 00:11:48,830 --> 00:11:53,000 And the answer is no, because we have the total 1020. 151 00:11:53,000 --> 00:12:00,530 But if we subtract out all these values January through October, we're left with a missing 125. 152 00:12:00,530 --> 00:12:05,030 But we don't know how to split 125 between November and December. 153 00:12:05,030 --> 00:12:11,000 We have no way of telling how much of that 125 goes in November and how much goes in December, which 154 00:12:11,000 --> 00:12:17,990 means that we can only remove one value from the table before we would no longer be able to finish filling 155 00:12:17,990 --> 00:12:18,410 it in. 156 00:12:18,410 --> 00:12:24,770 So if we can only remove one value, that means we have 11 values remaining in the table, which means 157 00:12:24,770 --> 00:12:28,190 in this case the degrees of freedom is 11. 158 00:12:28,190 --> 00:12:33,440 Degrees of freedom is the remaining number of values that we have in the table. 159 00:12:33,440 --> 00:12:39,170 And since that's a little confusing, let's look at one of the tables we had from earlier in the video. 160 00:12:39,200 --> 00:12:42,110 This handedness versus I color table. 161 00:12:42,110 --> 00:12:47,780 Really what this comes down to is that when we look at the body of the table, we're always able to 162 00:12:47,780 --> 00:12:50,720 remove the last row and or column of the table. 163 00:12:50,720 --> 00:12:55,580 So in this table we could remove all of these values. 164 00:12:56,230 --> 00:13:01,330 Because you can see that even if we didn't have any of these values here, we could find left handed 165 00:13:01,330 --> 00:13:08,770 Hazel by subtracting 72, 36 and 20 from this 140 total on the right hand side that would allow us to 166 00:13:08,770 --> 00:13:11,230 fill in this left handed Hazel here. 167 00:13:11,230 --> 00:13:13,720 So then we would have this value. 168 00:13:13,750 --> 00:13:23,460 And then once we have that value, we can fill in right handed and brown by subtracting 72 from 532. 169 00:13:23,470 --> 00:13:24,580 So we'd have that value. 170 00:13:24,580 --> 00:13:30,640 We would find right handed and blue by subtracting 36 from 251 and so on. 171 00:13:30,640 --> 00:13:39,550 We'd take 150 -20 to get this cell and then 67 -12 to find this cell, and suddenly we'd have the whole 172 00:13:39,550 --> 00:13:40,420 table again. 173 00:13:40,420 --> 00:13:48,670 So if we're able to remove all of these values here, that means degrees of freedom for this table, 174 00:13:48,670 --> 00:13:54,430 for this scenario is three, because the number of remaining values in the body of our table is three. 175 00:13:54,430 --> 00:13:56,590 We have 72, 36 and 20. 176 00:13:56,590 --> 00:14:01,000 And so degrees of freedom here would be three in the same way that degrees of freedom up here would 177 00:14:01,000 --> 00:14:01,890 be 11. 178 00:14:01,900 --> 00:14:06,130 So back to our original example, our sales affected by month. 179 00:14:06,130 --> 00:14:11,650 We have 11 degrees of freedom and we're interested in this 5% significant level. 180 00:14:11,650 --> 00:14:19,240 So we come here to 11 degrees of freedom and we work our way across to an upper tail probability of 181 00:14:19,240 --> 00:14:21,190 0.05. 182 00:14:21,190 --> 00:14:26,980 Whatever our level of significance is, we find that in the upper tail probability hetero, and then 183 00:14:26,980 --> 00:14:33,490 we come here to their intersection and we see this value 19.68 from here. 184 00:14:33,490 --> 00:14:38,980 The only thing left to say is whether our chi squared test statistic that we calculated is less than 185 00:14:38,980 --> 00:14:42,700 or greater than the value that we find in our chi squared table. 186 00:14:42,760 --> 00:14:52,660 In this case, we can see obviously that 80.59 is greater than 19.68 and whenever we find a chi squared 187 00:14:52,660 --> 00:14:58,120 test statistic that is greater than this critical value that we find in the chi squared table, that 188 00:14:58,120 --> 00:15:04,780 means that the test statistic we found is significant enough to allow us to reject the null hypothesis. 189 00:15:04,780 --> 00:15:11,860 In our case, there's a massive difference between 80.59 and 19.68, which tells us that we have very, 190 00:15:11,860 --> 00:15:17,590 very strong evidence that sales are indeed affected by month of the year. 191 00:15:17,590 --> 00:15:25,090 So we can reject this null hypothesis that sales aren't affected by month and we give a lot of support 192 00:15:25,090 --> 00:15:26,980 because of the massive difference here. 193 00:15:26,980 --> 00:15:34,090 We give a lot of support to this alternative hypothesis, the hypothesis that sales are indeed affected 194 00:15:34,090 --> 00:15:40,210 by month of the year, that the distribution of sales doesn't just hold steady and constant unaffected 195 00:15:40,210 --> 00:15:46,450 as we go month to month, and that instead they fluctuate with some level of significance and the number 196 00:15:46,450 --> 00:15:49,870 of sales we make is dictated by the month of the year. 197 00:15:49,870 --> 00:15:51,700 It's affected by month of the year. 198 00:15:51,700 --> 00:15:57,280 So in a way we could say that month of the year and sales are dependent variables. 199 00:15:57,280 --> 00:16:01,420 They are associated, they are not independent of one another. 200 00:16:01,420 --> 00:16:08,560 So even though we looked at different kinds of Chi square tests, they all follow the same kind of general 201 00:16:08,560 --> 00:16:09,160 pattern. 202 00:16:09,160 --> 00:16:16,480 We're always comparing two different distributions and we sort of start with this idea that the distributions 203 00:16:16,480 --> 00:16:20,860 are the same and so we compute this expected value. 204 00:16:20,890 --> 00:16:27,040 Then we look at the difference between actual value or observed value and expected value, and we use 205 00:16:27,040 --> 00:16:30,220 this formula to calculate the chi squared test statistic. 206 00:16:30,220 --> 00:16:36,190 Once we have the chi squared test statistic and the number of degrees of freedom, we can look for our 207 00:16:36,190 --> 00:16:42,040 level of significance in the chi squared table and compare our test statistic to the value that we find 208 00:16:42,040 --> 00:16:47,620 in the table and use the relationship between the test statistic and the value from the table to make 209 00:16:47,620 --> 00:16:52,780 a conclusion about whether or not we can reject the null hypothesis.