Hi, and welcome to our second to the last lecture. This lecture is on Poisson GLNs, and I should give some credit to Jeff Leek who I got much of this content from, from an earlier version of this class. So modeling count data arises quite frequently in applications. For example, the number of calls to a call center, the number of flu cases. And in each of these cases, the counts are unbounded in the sense of well, there might be some theoretical bounded account, the total number of people in the world or whatever. However, we don't really know what that is or that number is really large, relative to the count that we're looking at. So in addition to count, data can come in the form of rates or proportions, such as the percentage of people passing a test, or in terms of rates, think about the number of cases or something like that, that occur over a unit of time. My favorite example is from a nuclear pump failure experiment where we're looking at the number of instances that nuclear pumps failure, per, failed per unit time. So that would be a rate. A very common rate that occurs in bio statistics and public health. Where I work in is, the so called incidence rate, which is the number of newly developed cases per person time at risk. Okay so all of these are instances of counts and rates and proportions are also you can think of as counts. Both because the numerator is a count and whatever you're dividing by either the percent time at risk or the total time or the total sample or the number of trials or something like that. That's a second number that we're going to show you how to deal with as well when looking at the numerator, the count part, okay? And all of these can be handled with Poisson GLMs. So the Poisson Distribution is a useful model for counts and rates. And again a rate is a count per some monitoring time. So, often incidence rates and web traffic and all these other things are modeled by Poisson distributions. A very common use of the Poisson distribution is approximating binomial probabilities where the success probability is very small and the end is very large, so you can think of that as an instance of the sort of approximated and unbounded count, even though the actual count is bounded. And then an application that I like quite a bit is so called Contingency Table data. So if you have an instance where you just counted the number of occurrences of a different collection of variables. So if took a random sample of people and I counted the number of people that had blonde hair, brown hair and black hair and I cross tabulated that with the number of people who had blue eyes, brown eyes and hazel eyes. Okay that table of counts is called a contingency table and Poisson models are very useful for modeling contingency table data. They give a very elegant framework for doing that. I give the Poisson mass function here and so the rate of counts per unit time is lambda, whereas t is the total time. If x is a plus on with this mean, then its expected value is t times lambda. So the expected value is plus sign Is that's is the t times lambda. So our natural estimate of the rate would be the count over the total time okay? So x over t and it's nice to know in this case that the expected value of x over t the expected value of our rate estimate is exactly lambda. The rate that we would like the estimate. So, that's the useful property associated with the Poisson. The variance is equal to the mean, so the variance is e lambda. So that's the assumption of our model that we can check and we have some potential solutions of it's doesn't hold. And another interesting fact is the Poisson tends to a normal as the mean gets large. So you can think of this in several ways. All that has to happen is for t lambda to get large. This could occur if t is fixed and lambda gets large, if lambda is fixed and t gets large, or both of them get large. And in a lot of different applications the way in which the mean gets large could vary but as long as it gets large in some sense then the Poisson is going to approximate a normal distribution. And here I show you this via simulation. I simulate three different collections of Poisson random variables as the mean of the Poisson distribution gets larger and larger and you can see by the right most panel that it's nearly identical to a normal distribution at that point. And then, we can actually show that we don't, if this isn't the appropriate class the fact to show the mathematics that the meaning of variants are equal theoretically so, a way could do that by simulation and I do that here where I right, are not, I'm sorry this is an access simulation. We're actually try to show it using the density and summing up the density in the right way. So if you're interested try that experiment and it will prove to you that the meaning of variance or equal, try it for bunch of different scenarios. Or you could just believe me or you could take for example mathematical biostatistics boot camp one or two are my other course or classes. Where we cover how to do the actual mathematics for this. So as an example, let's look at Jeff Leek, his web traffic. So this is his website, www.biostat, or I'm sorry, biostat.jhsph.edu/~jleek. And the place I mean in this case is the interpretive a number of web hits per day. So our unit our time in this case is T equal to one. Now for the one to interpret the length that we estimate as web hits per hour we would have to put the T equal 24. So I hope you understand that and if you want it to have it to be seconds you need to put a T equal 24 times or minutes it would have to be T equal 24 times 60 and so on. Let's look at the data I show here how you can download it and I convert the date from a standard character date time format to a Julian date. Julian date counts the number of days since January 1st 1970 I believe. So the Julian date is nice to think about because it's just a count. It's the number of days whereas the date is kind of a complicated format because it's characters. So when you do the head of the date here, you see the date which is in character format. You see the number of visits, and he is not doing so well. These early dates with 0 visits on all those dates. The number of visits that originate from simply statistics and the julian date. So here's a plot of the data set. The Julian date is on the x axis and the number of visits is on the y axis. Now, we've covered in the last lecture what linear regression, some of the shortfalls of linear regression is try to model count data or in that case, binary data. So let's not just re-hatch that same topic, there are some issues with modelling count data as if it was with a linear model directly. However, as we saw a couple of slides ago, as the mean of the counts gets larger and larger are concerned over this decreases quite a bit simply because it's going to trend to a normal distribution. So, if you have extremely large counts, this becomes a lot less objectionable. So, that's just for notation number of heads, NH is going to be our outcome JD, is the Julian day, that's going to be our predictor and this would be a linear regression model, we can plot it and see the fitted line that we would get. It has some issues. Clearly there's some curvature there, maybe we should have put an x squared term in. But that would be our first approach to this, and honestly it wouldn't be that bad. But the counts are kind of small, so it's not the best thing in the world. The interpretation isn't great for linear models, then we'll see some ways which in the next couple of slides, how we can tweak linear models to maybe get a slightly better interpretation. I think that of counts in web hits and things like that as things that you would want to think about on a relative scale and the linear model really treats it on a linear additive scale. So let's think about how we could get relative interpretations from our linear model. The first thing we might try is taking the log of the outcome, here I knew the natural log. And this would be our model, log of NH is the linear regression model. B0 + B1(JDi) + ei. Now let me speak a little bit about log and what it's accomplishing. The quantity e to the expected value of the log of a random variable is what I would call the population geometric mean. And the reason I would call it the population geometric mean is the empirical or just geometric mean is the product of a sample, product Yi, raised to the one over n power. So this the way to think about this, the product of yi to the one over n power. If we take a log of that, we get the arithmatic mean, the ordinary mean of the log data. So the geometric mean is just exponentiating the arithmatic mean of the log data. And we know that if we collect a lot of data, a lot more data in our sample, the arithmetic mean will converge to something. So the geometric mean is what this quantity, the product of the data, rays to the one over nth power, what it converges to. So, what, it turns out, when you take the log of the natural log of the outcome in a linear regression then, your exponentiated coefficients are interpretable with respect to geometric means. So, for example, E to the Beta of zero is the estimated geometric mean hits on day zero and I should reiterate the point from earlier on in the class. This intercept doesn't mean that much because January first 1970 is not a date that we care about in terms of number of web hits. So probably to make the intercept more interpretable, what we should have done is subtracted off the earliest date that we saw and started counting days from there. From all of the remaining days in our data set and then the intercept would be the e to the inner estimated intercept would be the geometric mean hits on the first day of this data set. Okay. So that's a small point but it doesn't change the fitted model. It doesn't change the slope or anything like that to shift around the intercept however nonetheless, if you want an interpretable intercept as we know from earlier on in the class, you have to do something like that. E to the beta1 on the other hand is the estimated relative increase or decrease in the geometric mean hits per day, okay? So the increase per day. So I should also mention there's a problem with logs. If you have zero counts you have to do something because you can't take the log of zero, so you need to add a constant. A very common constant that is plus one. So we do log of the outcome plus one. So if we do that, here I fit the linear model to the log of the outcome plus one versus the Julian date. We get the intercept which is kind of irrelevant in this case as we talked about before. And then we get 1.002. This is on the exponentiated scale. Okay so what that means is our model is estimating a 0.2% increase in web traffic per day. Okay? And that's a nice interpretation. If you added other covariates then that would be 0.02% increase per day holding the other covariates fixed.
