1 00:00:00,000 --> 00:00:04,120 Welcome back to practical time series analysis. 2 00:00:04,120 --> 00:00:06,280 In these introductory lectures, 3 00:00:06,280 --> 00:00:08,625 we're reviewing basic statistics. 4 00:00:08,625 --> 00:00:10,548 In this lecture in particular, 5 00:00:10,548 --> 00:00:13,740 we'll look at some inferential statistics. 6 00:00:13,740 --> 00:00:16,050 Now if your statistical background is strong, 7 00:00:16,050 --> 00:00:18,585 and if you're very comfortable in the R environment, 8 00:00:18,585 --> 00:00:21,630 you can move through these lectures very quickly. 9 00:00:21,630 --> 00:00:23,820 They're really meant for people who either haven't done 10 00:00:23,820 --> 00:00:27,645 statistics in any meaningful way in quite some time, 11 00:00:27,645 --> 00:00:34,745 or who are just new to R. Our objectives. 12 00:00:34,745 --> 00:00:37,610 We do say this is basic inferential statistics. 13 00:00:37,610 --> 00:00:40,310 Our objectives are to review some basics. 14 00:00:40,310 --> 00:00:43,280 Learn how to develop graphical intuition in new data set, 15 00:00:43,280 --> 00:00:46,155 based upon the commands available in R. 16 00:00:46,155 --> 00:00:50,930 And learn how to perform a simple hypothesis test. 17 00:00:50,930 --> 00:00:56,205 The data set we'll be using is a famous traditional data set. 18 00:00:56,205 --> 00:00:59,320 It's the Gossett Data on sleep. 19 00:00:59,320 --> 00:01:05,720 So, he's reporting on results that other researchers actually had already published, 20 00:01:05,720 --> 00:01:08,290 and using it to develop his techniques. 21 00:01:08,290 --> 00:01:11,618 But the basic data set looks at two soporific drugs, 22 00:01:11,618 --> 00:01:16,550 these are drugs meant to induce extra sleep in a patient. 23 00:01:16,550 --> 00:01:20,300 And there are 10 people in play in this data set. 24 00:01:20,300 --> 00:01:23,600 There are two drugs and we're looking at the increase over 25 00:01:23,600 --> 00:01:28,930 control for each of these 10 individuals with each drug. 26 00:01:28,930 --> 00:01:35,255 The formative data frame is 20 observations and we have three variables here. 27 00:01:35,255 --> 00:01:38,615 So, 20 observations, these are 10 people, 28 00:01:38,615 --> 00:01:42,930 and so we're looking at the effect of each of two drugs here. 29 00:01:42,930 --> 00:01:45,565 So we've got extra group and ID. 30 00:01:45,565 --> 00:01:48,485 Extra is going to be the extra amount of sleep. 31 00:01:48,485 --> 00:01:51,515 Group will tell you which drug is in play, 32 00:01:51,515 --> 00:01:54,380 and ID tells you which patient. 33 00:01:54,380 --> 00:01:59,055 Of course, always plot your data. 34 00:01:59,055 --> 00:02:02,340 The plot command is a rather powerful one, 35 00:02:02,340 --> 00:02:05,495 and it will make decisions based upon the kind of data 36 00:02:05,495 --> 00:02:09,300 you're presenting it as to what kind of plot it's going to return. 37 00:02:09,300 --> 00:02:13,065 We're going to plot the extra sleep on group. 38 00:02:13,065 --> 00:02:18,105 So, think of that as drug and main of course is going to put a title on a graph. 39 00:02:18,105 --> 00:02:22,105 We'll look at extra sleep in Gossett Data by group. 40 00:02:22,105 --> 00:02:24,515 Now, after that we'll do a couple other things. 41 00:02:24,515 --> 00:02:28,660 I want to have the data available to me very easily in a variable. 42 00:02:28,660 --> 00:02:30,635 So, I'm gonna say extra dot one, 43 00:02:30,635 --> 00:02:34,670 is the extra sleep for those in group one. 44 00:02:34,670 --> 00:02:40,500 So we're testing group identically equal to one or group identically equal to two, 45 00:02:40,500 --> 00:02:45,920 as we assign our numbers to each of these two vectors. 46 00:02:45,920 --> 00:02:48,762 When we look at the graph, 47 00:02:48,762 --> 00:02:50,420 we've got extra sleep. 48 00:02:50,420 --> 00:02:53,385 it looks like the second group, 49 00:02:53,385 --> 00:02:57,805 the second drug has a pretty clear advantage over the first. 50 00:02:57,805 --> 00:03:02,880 I don't see a huge difference in heterogeneity here, there is some, 51 00:03:02,880 --> 00:03:07,770 but I think what's most pronounced in this graph is that, 52 00:03:07,770 --> 00:03:13,035 now this bar here in a box plot of course is telling you the median not the mean, 53 00:03:13,035 --> 00:03:17,675 but the median certainly seems to be higher in the second group. 54 00:03:17,675 --> 00:03:19,555 Now that's a visual impression, 55 00:03:19,555 --> 00:03:24,835 and what we'll do now is try to follow it up with a standard statistical test. 56 00:03:24,835 --> 00:03:27,495 As we test our hypothesis, 57 00:03:27,495 --> 00:03:30,550 we'll use the command t. test, 58 00:03:30,550 --> 00:03:34,740 we'll put in extra one and extra two as our data. 59 00:03:34,740 --> 00:03:36,375 You can do tests, 60 00:03:36,375 --> 00:03:40,650 you'll recall from elementary stats with independent samples, 61 00:03:40,650 --> 00:03:42,435 and there are different ways to go within 62 00:03:42,435 --> 00:03:45,680 independent samples depending upon your variability. 63 00:03:45,680 --> 00:03:48,900 Right now instead, we're going to treat these data as 64 00:03:48,900 --> 00:03:52,890 paired because remember there are only 10 people in the study, 65 00:03:52,890 --> 00:03:54,965 and there are two different drugs. 66 00:03:54,965 --> 00:03:59,550 We're going to do a two sided test rather than a one sided test, 67 00:03:59,550 --> 00:04:02,055 because coming in I had no theory, 68 00:04:02,055 --> 00:04:07,010 no intuition that one drug would be better than the other. 69 00:04:07,010 --> 00:04:09,120 Our results look like this. 70 00:04:09,120 --> 00:04:14,630 We'll obtain a t value of negative four or so. 71 00:04:14,630 --> 00:04:18,055 That's a fairly hefty t value. 72 00:04:18,055 --> 00:04:21,015 Now, if that were a z value from a normal distribution, 73 00:04:21,015 --> 00:04:23,350 it would be quite quite large. 74 00:04:23,350 --> 00:04:25,620 How large it is with a T distribution, 75 00:04:25,620 --> 00:04:27,775 depends upon your sample size. 76 00:04:27,775 --> 00:04:30,810 Our degrees of freedom in a pair of T-test like this with 77 00:04:30,810 --> 00:04:35,040 ten individuals remember is nine and minus one, 78 00:04:35,040 --> 00:04:39,650 and the p value we obtain is less than the standard nickel. 79 00:04:39,650 --> 00:04:42,015 Less than point zero five. 80 00:04:42,015 --> 00:04:44,205 It's even less than point zero one. 81 00:04:44,205 --> 00:04:48,673 And I think many people would say that these data are highly significant. 82 00:04:48,673 --> 00:04:51,120 R agrees and is going to go with 83 00:04:51,120 --> 00:04:59,035 the Alternative hypothesis that the there is a difference between the two drugs. 84 00:04:59,035 --> 00:05:03,085 It's also good to report a confidence interval. 85 00:05:03,085 --> 00:05:08,020 And another approach to a test like this would be to calculate a confidence interval and 86 00:05:08,020 --> 00:05:13,440 see if it includes zero as a plausible value. It does not. 87 00:05:13,440 --> 00:05:16,930 So, the 95 percent confidence interval 88 00:05:16,930 --> 00:05:24,140 here is between around negative two and a half to a negative point seven. 89 00:05:26,470 --> 00:05:30,480 Now, if it's been a little while since you've done 90 00:05:30,480 --> 00:05:33,600 a confidence interval or a hypothesis test, 91 00:05:33,600 --> 00:05:36,605 let's go back and remember what this is all about. 92 00:05:36,605 --> 00:05:42,570 In a standard hypothesis test we have a null hypothesis and an alternative hypothesis, 93 00:05:42,570 --> 00:05:45,570 traditionally labeled H sub zero, and H sub one. 94 00:05:45,570 --> 00:05:50,025 The null hypothesis will be no difference, 95 00:05:50,025 --> 00:05:54,290 just that the mean response is going to be the same for both drugs. 96 00:05:54,290 --> 00:06:01,245 The alternative since we're doing two tail tests will be that it's not the same. 97 00:06:01,245 --> 00:06:05,080 Alpha is what people, 98 00:06:05,080 --> 00:06:08,104 or researchers often set up before they conduct the tests, 99 00:06:08,104 --> 00:06:10,720 probability of a type one error. 100 00:06:10,720 --> 00:06:15,075 The probability that we're going to reject a true null hypothesis is 101 00:06:15,075 --> 00:06:17,110 fairly standard to say that alpha equal to 102 00:06:17,110 --> 00:06:21,010 the two values point zero five or point zero one. 103 00:06:21,010 --> 00:06:23,530 The t value that we calculated here, 104 00:06:23,530 --> 00:06:27,280 is we're going to look at the average of 105 00:06:27,280 --> 00:06:30,055 the differences which is the same 106 00:06:30,055 --> 00:06:33,930 if you follow the language as the difference of the averages. 107 00:06:33,930 --> 00:06:36,130 So, we're looking at the bar here. 108 00:06:36,130 --> 00:06:38,465 Essentially, we're just taking the average, 109 00:06:38,465 --> 00:06:41,110 the mean value with the first group, 110 00:06:41,110 --> 00:06:43,630 and subtracting off the average on the second. 111 00:06:43,630 --> 00:06:45,845 It's a very intuitive thing to do. 112 00:06:45,845 --> 00:06:50,280 We'll compare that to our null hypothesis value of zero. 113 00:06:50,280 --> 00:06:53,590 Downstairs we're going to look at variability. 114 00:06:53,590 --> 00:06:58,540 So, we're looking at the variability of the averages here not of individuals. 115 00:06:58,540 --> 00:07:02,170 We're going to take s sub d. So, 116 00:07:02,170 --> 00:07:07,840 this is the sample standard deviation of the differences. 117 00:07:07,840 --> 00:07:09,910 Now, be careful, if you take 118 00:07:09,910 --> 00:07:13,330 the differences for these 10 individuals between the two drugs, 119 00:07:13,330 --> 00:07:17,575 take their response on the first drug and subtract off the response in the second, 120 00:07:17,575 --> 00:07:19,315 and do that for all 10, 121 00:07:19,315 --> 00:07:21,955 and then take the standard deviation of that, 122 00:07:21,955 --> 00:07:25,255 that's the standard deviation of the differences. 123 00:07:25,255 --> 00:07:28,090 That's not going to be the same number generally speaking, 124 00:07:28,090 --> 00:07:31,105 as if you take the standard deviation of the first data set, 125 00:07:31,105 --> 00:07:34,015 and subtract off the standard deviation of the second. 126 00:07:34,015 --> 00:07:36,445 We just have to be a little bit careful here. 127 00:07:36,445 --> 00:07:41,345 But the standard test is to look at the standard error down here, 128 00:07:41,345 --> 00:07:44,165 standard deviation divided by the square of N, 129 00:07:44,165 --> 00:07:47,245 to give us a measure of variability. 130 00:07:47,245 --> 00:07:49,000 And that's how we calculate, 131 00:07:49,000 --> 00:07:50,640 you can follow through the numbers, 132 00:07:50,640 --> 00:07:53,875 negative for t value. 133 00:07:53,875 --> 00:07:56,485 So as we just said, 134 00:07:56,485 --> 00:07:59,100 d bar is the average of the differences, 135 00:07:59,100 --> 00:08:02,285 or the difference of the averages however you like, 136 00:08:02,285 --> 00:08:07,645 and Sd is the standard deviation of the differences from the sample. 137 00:08:07,645 --> 00:08:10,080 Now R also got a p-value. 138 00:08:10,080 --> 00:08:11,470 So what's the p-value? 139 00:08:11,470 --> 00:08:17,225 It's the likelihood of seeing data this extreme under the null hypothesis. 140 00:08:17,225 --> 00:08:20,675 And, what we'll do here is look at twice, 141 00:08:20,675 --> 00:08:23,725 it was a two tailed test, twice. 142 00:08:23,725 --> 00:08:26,395 Now the t distribution is the one in play for us, 143 00:08:26,395 --> 00:08:28,270 that's that letter t right there. 144 00:08:28,270 --> 00:08:31,130 And p is just short for probability. 145 00:08:31,130 --> 00:08:34,990 So, what we're trying to do is get some tail areas here. 146 00:08:34,990 --> 00:08:38,300 So, we're going to take twice the tail area, 147 00:08:38,300 --> 00:08:41,030 I'll look down at the left tail, 148 00:08:41,030 --> 00:08:43,300 I'll pop my negative four in there. 149 00:08:43,300 --> 00:08:47,360 Nine degrees of freedom and calculate a p-value. 150 00:08:47,360 --> 00:08:49,725 If your p value is small, 151 00:08:49,725 --> 00:08:54,893 you'll reject your null hypothesis and our p value is really quite small. 152 00:08:54,893 --> 00:08:59,516 So we rejected the null hypothesis. 153 00:08:59,516 --> 00:09:02,630 In general, if you have a hypothesis test, 154 00:09:02,630 --> 00:09:05,170 different books have different details here, 155 00:09:05,170 --> 00:09:06,950 but it's all they all rhyme. 156 00:09:06,950 --> 00:09:09,065 They're all basically telling the same thing. 157 00:09:09,065 --> 00:09:12,105 You're going to state clearly what your variables are 158 00:09:12,105 --> 00:09:15,425 so that everybody knows including you what you're talking about. 159 00:09:15,425 --> 00:09:19,340 State your null and a whole alternative hypotheses, 160 00:09:19,340 --> 00:09:25,330 and then divide, decide upon rather a level of significance. 161 00:09:25,330 --> 00:09:30,395 Once you've got that basic framework down those organizing principles, 162 00:09:30,395 --> 00:09:31,630 go ahead and look at your data, 163 00:09:31,630 --> 00:09:33,440 compute a test statistic, 164 00:09:33,440 --> 00:09:39,155 and you'll run across very often z's and z's chi square as an f's. 165 00:09:39,155 --> 00:09:43,055 These are kind of the big four in an elementary statistics course. 166 00:09:43,055 --> 00:09:47,100 You'll find the p value corresponding to your test statistic, 167 00:09:47,100 --> 00:09:49,880 and then you'll form a conclusion, 168 00:09:49,880 --> 00:09:55,655 you'll reject or not reject typically. 169 00:09:55,655 --> 00:10:00,075 Confidence intervals are, 170 00:10:00,075 --> 00:10:04,620 there is a difference between the word confidence and probability. 171 00:10:04,620 --> 00:10:10,035 Many people get very sticky on this and say that once an event has occurred, 172 00:10:10,035 --> 00:10:12,390 you really can't talk about probability anymore. 173 00:10:12,390 --> 00:10:14,940 Instead you must talk about confidence. 174 00:10:14,940 --> 00:10:17,790 The basic idea is we're trying to give 175 00:10:17,790 --> 00:10:22,215 a good indication of where we believe the actual meaning would be, 176 00:10:22,215 --> 00:10:25,080 here it's going to be a mean difference. 177 00:10:25,080 --> 00:10:30,825 The common form that you'll see for many confidence intervals is estimate. 178 00:10:30,825 --> 00:10:34,065 That was our D Bar here, plus and minus, 179 00:10:34,065 --> 00:10:38,725 some sort of table value multiplied by an estimated standard error. 180 00:10:38,725 --> 00:10:41,640 It's not hard to really demonstrate where this comes from. 181 00:10:41,640 --> 00:10:43,103 In our particular case, 182 00:10:43,103 --> 00:10:44,265 we'll look at d bar, 183 00:10:44,265 --> 00:10:45,975 plus and minus the T value, 184 00:10:45,975 --> 00:10:50,200 times our standard error. 185 00:10:50,200 --> 00:10:54,825 We already saw that R will print this out for you. 186 00:10:54,825 --> 00:10:58,098 If you like to follow along and do a hand calculation yourself, 187 00:10:58,098 --> 00:10:59,900 we've got the numbers right here. 188 00:10:59,900 --> 00:11:04,150 It's just a direct substitution. 189 00:11:05,580 --> 00:11:10,700 Standard error is just a shorthand notation 190 00:11:10,700 --> 00:11:14,710 for the standard deviation of a sampling distribution. 191 00:11:14,710 --> 00:11:17,150 And since we're dealing with means not individuals, 192 00:11:17,150 --> 00:11:20,410 standard error is the right term to use. 193 00:11:20,410 --> 00:11:25,635 Also, statistics are things that you compute from data. 194 00:11:25,635 --> 00:11:30,624 Parameters are usually the things that you're trying to draw inferences on, 195 00:11:30,624 --> 00:11:33,015 and it's a numerical descriptor about 196 00:11:33,015 --> 00:11:39,110 either a theoretical distribution or an actual population. 197 00:11:39,110 --> 00:11:42,795 And we can discuss type one and type two errors. 198 00:11:42,795 --> 00:11:47,030 A lot of this is done in some more detail in the reading. 199 00:11:47,030 --> 00:11:51,590 In this lecture you've learned how to use R to develop 200 00:11:51,590 --> 00:11:56,401 graphical intuition into a data set as you're trying to answer a question, 201 00:11:56,401 --> 00:12:00,040 and we've learned how to perform a statistical hypothesis, 202 00:12:00,040 --> 00:12:06,000 we've reviewed the concept from basic statistics and shown how to do it in R.