Lecture transcript
Recognizing whether you are asking
an inferential question or a prediction question
is really important. Because the type of question that you are asking can
greatly influence the modeling strategy
that you pursue in any data analysis. So just to recap,
with inferential questions the goal is to estimate the association between an outcome and a key predictor.
While potentially trying to adjust for all these confounding factors. There is usually a very small number, or even a single key predictor that we're interested in and it's relationship to the outcome. And then there may be all these other variables. And the key goals for the modeling is to estimate this association and to make sure that you appropriately adjust for any other kinda factors. You often do sensitivity analyses to see how the association might change in the presence of other factors.
With the prediction question, the goal is to develop a model that best predicts the outcome using whatever information you have available to you.
use all variable no predictors (variables) favoured , not care mechanism.
So typically you don't put any weight on one predictor over another. And so there's no notion of let's say a key predictor and confounder. All the predictors might be equally important before you look at the data. And any good prediction algorithm will tell you which predictors or which variables are more or less important for predicting the outcome. But beforehand, we don't necessarily divide them into different groups based on importance. Finally, we usually don't care about the mechanism or the specific details of the relationships between the various predictors or variables. We just wanna find a kind of a model form that predicts the outcome with a high accuracy and low error.
So I wanted to give a brief example of how these
different types of questions can lead to different conclusions, even on the same dataset.
I'll start first with an inferential question. Suppose I want to know
how is air pollution in New York City related to mortality in New York City.
Okay? So the data I'm gonna use comes from this national morbidity and mortality in air pollution study, and here's a picture of daily mortality in New York from 2001 to 2005.
And you can see that it has highly seasonal components, the mortality tends to be higher in the winter and lower in the summer, and it's very specific to kind of to pattern across every year.
Here is what particulate matter data looks like.
And so for the same time period you can see this has also a seasonal structure to it. In particular, particulate matter is high in the summer. It tends to be low in the winter in New York City. That's a very, that pattern is kind of repeated across the years.
So the first step I'm gonna do is just to try the easy solution. The easiest solution is just a scatter plot of mortality and PM10 on the x-axis, and here you can see what the relationship looks like.
There's quite a bit of noise because we wouldn't necessarily expect particulate matter to explain all the vulnerability in mortality. And so, we may need to resort to some modeling, some formal modeling to see if there's any sort of association here.
So the first thing we're going to do is to take that scatter plot and just fit a simple linear aggression to the data in that scatter plot. This is basically regressing mortality as the outcome and PM10 as our key predictor, without any other factors, just as kind of a baseline model. Now here are the results that we get from a regression model of that nature.
And you can see that the coefficient for PM10 here is 0.00004, etc. And so basically is close to zero. You can tell by the size of the standard error, which is much bigger than the estimate, that there's a lot of variability around this estimate and so it's effectively zero. The association between the two variables is zero. Okay. So that's kind of what our basic first cut analysis tells us.
Now one thing we know about pollution and mortality, just from the pictures that we just saw of the data is that they're highly seasonal. Season plays a big role in explaining variability in both mortality and in air pollution okay? Remember mortality was high in the winter and low in the summer, and pollution was high in the summer and low in the winter. So it seems like season is clearly related to both mortality and pollution. So it might be a reasonable thing to include as a potential confounding factor. So we can fit a secondary model to the data, which includes pm10 as our key predictor, and then maybe we'll include the season of the year as a potential confounding factor, so the season will just be, you know, there'll be four seasons, and we'll have a categorical value with a category for each season. So, here is the results of fitting that model to the data.
And you can see that I highlighted the coefficient for pm10 here is actually quite a bit larger now, is 0.00149, and you can see that the standard error is quite a bit smaller relative to the estimate. So, this suggests that the coefficient for pm10 is quite a bit bigger than zero, maybe statistically significant.
So the addition of this confounding factor dramatically changed the association we estimate between pm10 and mortality.
And so, part of this is because season is very strongly related to both, but it's kind of positive, it's correlated in one way with mortality, but it's correlated in a different way with pollution. That's part of why we saw it when we didn't include season we saw no relationship. But when we do include it, we see this kinda quite a bit stronger relationship.
Now there are other potential factors that we might wanna consider in terms of things that might both be related to mortality and to air pollution. So another one is the weather. Right? So weather is associated with mortality and it's also highly associated with various air pollutants and so we can characterize the weather with something like temperature or dew point temperature to think to just capture a piece of kind of what weather is and so we can include that into our model.
So here's the results of including temperature which is the tmpd variable and dew-point temperature which is the dptp variable.
And you can see now. And actually I've highlighted the coefficient for pm10, it's even bigger than it was before and the standard error is similar, so it's actually more statistically significant in some sense than in the previous models. Again, temperature and dew point are strong factors that are related to both mortality and pollution.
Finally, another type of factor that may be related to both mortality and PM, particulate matter, is other pollutants, right? So there are other pollutants that may affect health, and they may be correlated with particulate matter because they may share common sources. So, one common source in a city like New York is gonna be traffic. Traffic can produce particles, it can also produce other pollutants. So one of the pollutants that we'll look at is nitrogen dioxide. And so nitrogen dioxide tends to be correlated with particles. So if you are interested in association between particles and mortality, you might wonder well is what we're seeing actually the association between nitrogen dioxide and mortality? And pm10 is kind of getting mixed up in the two. And so we can include nitrogen dioxide in our model as a potential confounding factor, and see how the estimate for particulate matter changes when we do that.
So here are the results from that model, and you can see that compared to the previous model the coefficient for pm10 drops a little bit, it goes down a little bit, so the effect weakens a little bit when we include nitrogen dioxide in the model. Now it's still quite a strong effect, relative to the standard error that we estimate but it's not estimate, that is not as strong as it was before we entered, we included no2 in the model.
And so, no2 and pm10 might be sharing a lot of the same effect on mortality and it may be difficult to completely disentangle them.
However, even with no2 in the model we still see a reasonably strong association between pm10 and mortality in New York City. So when we put all these results together from the primary model which just had pm10 in mortality, and then the various secondary models we've shown, you can see that the primary model had a zero association effectively, and then the other three models had a relatively strong positive association with the outcome. And here I've plotted the 95% confidence intervals for each of these associations.
So this is the kind of analysis that we're interested in. We're looking at pm10 as our key predictor. And mortality is our outcome. And we want to see how that association changes under different scenarios. Under different sets of models, including different sets of confounding factors.
Now what we do with this information will depend on what the goal of the analysis is, who the stakeholders are, and what we might do with this information afterwards.
But we won't talk about that now. The point I want to make is the kind of the analysis that you do for an associational type of analysis is very much along these lines.
So another question that we could ask, is what best predicts mortality in New York City, using the data that we have available, okay?
So now I've changed the question to a prediction type of question, and we want to know what predicts the outcome best. So one of the things that we could do is fit a complex prediction algorithm, all right? We don't need to know, we don't need to worry about estimating associations, or adjusting for confounding factors. We're just gonna put all the data that we have available to us and see how well that predicts the outcome mortality.
So here I'm gonna use a random forest algorithm to make predictions of mortality in New York City.
And one of the aspects of random forest algorithm is that it allows for a summary statistic called variable importance. And this gives you a sense of how important a variable is in increasing the skill of the algorithm in predicting the outcome. So, I've plotted here, the variable importance plot that rank orders all the variables in terms of how important they are to improving the prediction skill of the algorithm.
And you can see at the very top here is temperature, which is kind of ranked as the most important variable. Followed by dew-point temperature, followed by the date, and then no2, ozone which is o3. Season. And then finally, pm10 and then dow, which is the day of the week.
So you can see that from this prediction model, particularly matter actually is second from the bottom in terms of improving the prediction skill of the algorithm.
So if you were to look at this analysis and then ask the original question, and say, how is pm10 related to mortality? You might think well, it's not related to mortality.
It's not important for predicting. But that's true, but it doesn't necessarily mean that pm10 is not associated with mortality, from an associational standpoint. It may have an important association with mortality. But that the association is inherently weak. And so it's not necessarily going to be good for be predicting the outcome, but it may, nevertheless, have an important association with the outcome.
And so, separating these kinds of questions out, and the goals of these questions is very important because
if you were to ask an inferential question, but then do an analysis that's really tuned for prediction you might be lead to come to the wrong conclusion
that, oh, pm10 is not important. And it doesn't have an important association with the outcome.
And particularly, if you think about something like pollution, it may have an inherently very small association with the outcome, but because everyone in New York City ostensibly breathes, and is exposed to this polluted air. The effect, the ultimate effect, of such an association, could be quite large given the large population of a city like New York.
So it's important to not kind of conflate the magnitude of the association with the ultimate effect on the population that you're interested in.
Using prediction algorithms for prediction questions, and associational analyses for associational questions, is really important so that you can draw the right conclusions from your data, and not mistake the results of one question for another.
So framing the question correctly is really important for developing an important modeling strategy, and for kinda drawing the right conclusions.
