Linear models, as their name implies, relates an outcome to a set of predictors of interest using linear assumptions. Regression models, a subset of linear models, are the most important statistical analysis tool in a data scientist’s toolkit. This course covers regression analysis, least squares and inference using regression models. Special cases of the regression model, ANOVA and ANCOVA will be covered as well. Analysis of residuals and variability will be investigated. The course will cover modern thinking on model selection and novel uses of regression models including scatterplot smoothing.
Multivariable Regression Examples part II
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Regression Models
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Week 3: Multivariable Regression, Residuals, & Diagnostics
This week, we'll build on last week's introduction to multivariable regression with some examples and then cover residuals, diagnostics, variance inflation, and model comparison.
與講師見面
Brian Caffo, PhDProfessor, Biostatistics
Bloomberg School of Public Health
Roger D. Peng, PhDAssociate Professor, Biostatistics
Bloomberg School of Public Health
Jeff Leek, PhDAssociate Professor, Biostatistics
Bloomberg School of Public Health
So you might be surprised to find out how flexible linear regression models are.
For example, you can fit factor variables as regressors and
come up with things like analysis of variance,
if you've heard of that before, as a special case of linear models.
Let's go through an example where we have one covariant, X equal to zero or
one, and let's see what happens when we put that into a linear regression model.
So here I have my model, Y, my outcome, is beta-naught,
an intercept, plus X times beta one plus an error term.
Where here now my X only takes the value zero for, let's say,
people in a control group and one for, say, people who received a treatment.
Then, for the people who will receive the treatment, the group of people where
their X value is one, their mean is beta-naught plus beta one.
For the people who are in the control group,
those people where their covariant X is zero, their mean is beta-naught.
If you were to fit this, as you would expect, the estimated mean for
the treated group is just the mean of the people who are treated.
So that beta one hat plus beta not hat works out to just be the mean for
the treated group.
Similarly, beta not hat, by itself, works out to be the mean for the control group.
Beta one, now, is interpreted as the increase, or decrease if it's negative, in
the mean response for those that had the X value of one, for those that were treated.
So that's just a nice way to be able to fit,
factor a two-level factor variable as a linear regression variable.
And it gives you, not only the fitted values tell you about the means for
both of the groups, but
it gives you an inference for comparing the two groups automatically.
That T test, by the way, the T test for
beta one, is exactly identical to a two-group T test where you
assume a common variance if you happen to have taken the inference class.
We can extend this to more than two levels.
For example, imagine if you had a three-level variable.
For example, you have some outcome but you wanna compare it to U.S.
political party affiliation.
In this case, let's say you were only considering those that were Democrats,
Republicans, or registered Independents.
Well, you can do that by having a variable X1, that's one for
Republicans and zero for otherwise, a variable X2 that's one for
Democrats and zero for otherwise, and then, I'll tell you here in a minute
why we omit the X3 that would be one for Independents and zero otherwise.
That one, it would happen to be redundant.
Okay, so if a person is a Republican,
then their mean is gonna be beta-naught, plus this first X term is gonna be one.
So plus beta one and the second X term is gonna be zero and
so their main will be beta not plus beta one.
If the person is a Democrat, then it's gonna beta-naught, then X1 will be zero,
so that term will drop out, then X2 will be one and so it'll be plus beta two.
So for the Democrat,
their mean from the regression model would be beta-naught plus beta two.
And then if they're an independent, both these S terms will be zero and
it'll just be beta-naught.
And that's why we can't include a third time, right?
Because if we know that you're Republican, in the way that we've set up the variable,
if we know that you're not Republican and not a Democrat,
then you must be an Independent in our data set the way we've set things up.
And so it would be redundant to have a third variable in there,
it wouldn't have any new information.
Here we have three means, Republican, Democrat, and Independent, and
three parameters, Beta-naught, Beta one, and Beta two.
If we were to add an extra parameter, it would kind of break the model.
And I'll show you in R what happens when you do that in a minute.
So if we look at our means here, if we compare beta-naught and
the mean for the Independents versus the mean for the Republicans,
so we subtract those two, we get beta one.
So beta one compares Republicans to Independents.
And beta two, similarly, compares Democrats to Independents.
Then, of course, beta one minus beta two compares Democrats to Republicans.
So what happens is by omitting the regression variable for the Independents,
then the intercept became the value for the Independents, and
all of the other coefficients have become interpreted relative to Independents.
The beta one, in fact, the one in front of the Republican covariant is now
interpreted as the change between Republicans and Independents.
The beta two, the one in front of the Democrat covariant
is now interpreted as the change between Democrats and Independents.
And this was all a consequence of having omitted the one regressor for
Independents.
If we had included the regressor for Independents and excluded the one for
Republicans, then the intercept would be for Republicans, and
the coefficient in front of the Democratic one would be Democrats versus Republicans.
The coefficient in front of the Independent one,
would be Independent versus Republican,
and we'll go through some more examples just to illustrate how this works.
And R kinda does this on purpose, or R kinda does this automatically for you,
if you include it as a factor value.
It picks one of the levels to be the reference level.
And so, let's go through some examples, hopefully, that'll shore this up,
but the main point I'd like to get across is, whenever you're dealing with factor
variables in linear models, what you set at your reference level has a big effect.
These coefficient are interpreted quite differently,
depending on how you sent them up and what you set up as your reference level.
Okay, so let's go through an example in R, where we look at a factor variable and
see how R is treating it.
So, I want to make sure I require the data sets package.
We've already loaded that in in this lecture but
let's just do it again just to remind ourselves.
And then I have this data InsectSprays, and then I'm requiring the stats package.
I don't know if that's technically necessary for what I'm doing, but
if you do help InsectSprays, InsectSprays,
here, it gives you the help file for this data set, and
the outcome is a count is a numeric insect count.
So, presumably, number left after applying the spray,
and the spray factor is the type of spray, okay?
And then it gives some examples of working with this data, but
we don't need that cuz we're gonna build our own examples.
So let's first plot some of the data.
So I want to do a ggplot and I've already loaded ggplot2,
but just to remind you, in case you're restarting your R session from earlier,
you want to make sure that you acquire ggplot2.
There, it's loaded.
And then I have my ggplot and then my data is InsectSprays.
And then for my aesthetic, my Y is the count, the number of insects.
My X is the spray.
They don't give you too much information about the sprays, but
there's a couple of different sprays that they use.
And then I want to fill the objects I'm creating with the factor variable spray.
So there I've created my ggplot.
And then I wanna do a violin plot.
A violin plot is kind of like a histogram but sort of tilted on its side.
And then they repeat it on both sides so it looks a little like a violin.
Well, it looks like a violin if you're data cooperates.
Otherwise, it looks like a blob.
Okay, there's our violin plot.
And then I wanna set my labels.
And then if you wanna actually see the plot, you gotta bring it up.
Okay, so, here's my violin plot.
So you see there's sprays eight labeled spray A, B, C, D, E, and F?
Okay. And you can see the insect counts, so
I presume they applied the spray to numerous batches of insects and
they, >> It's unfortunate they're not telling me
whether or not the count is the count of the number of alive or the number dead.
So we don't know if this is a better spray, a better, let's say
it's a mosquito spray or something like that, cuz no one likes mosquitoes.
We don't know if this is a better spray or a worse spray.
But let's talk about how we can test the difference between
different factor levels in this case using linear models.
And then, at the end I'll talk about some shortcomings of the approach that I'm
proposing here.
But here's a violin plot.
And let me just do head Insect sprays
to just show you the data, what it looks like, to see we have a bunch of counts.
And then, the spray label's a very simple data set.
And so, let's look at what happens when we include insect spray as a linear model and
y as an outcome.
So, let's fit our model.
And now, what we're fitting is,
our outcome is the count, the number of insects.
Our predictor is the spray, which spray was used as a factor variable.
It's already a factor variable.
And then, I give it the data-set.
And then, here, I just want the summary of the output from lm.
Again, normally you wanna assign your lm to a variable so
you can keep it for later.
And then, I just wanna, for, to keep the printing a little bit self-contained,
I'm grabbing the coefficient table.
And so, there you see, the Intercept, spray B, spray C, spray D,
spray E, and spray F.
And spray A is conspicuously missing.
And the idea is that everything here is now in comparison with spray A.
So, this 0.833 is the change in
the mean between spray B and spray A.
In this case, 14.5, the intercept is the mean for spray A.
And if you look over here at our plot, that seems about right.
Look at our violin plot.
14.5 seems about right for spray A.
And spray B, it seems reasonable that it would be off by,
it would be changed just by a little bit from spray A.
Now, spray C looks like it has a much lower count, okay?
And look, it's coefficient is minus 12.
Okay? And that looks like about right.
So this one's at 14.5.
And somewhere around two seems about right for this one, spray C.
And so, that's exactly what this coefficient is saying.
This negative 12 here is the different between spray C minus spray A.
Now, if we wanted to compare, for example, spray B and spray C,
we would have to subtract this, 0.833, and this negative12.
Now, we wouldn't have a standard error for that comparison immediately.
However, that would give us the estimate.
If I were to take the average count for the sprays,
for those with spray A, I would get 14.5.
If I were to take the average count for
spray B, I would get 14.5 plus 0.833.
So, I'd like now to show you how I can hard code the same model and
not rely on r to actually pick the reference level.
So, remember what I did last time is I did count was my outcome and
my factor variable spray was my predictor.
And what r does is it picks the spray level that's the lowest alphanumerically.
So, in this case, spray level A, to set as the reference level.
So let me show how you can hard code that myself manually.
So, here count is my outcome.
And then, I'm gonna create a variable using the I function which in lm
actually performs the operation inside the regression, inside the model statement.
So, here I just wanna look at the instances where the spray is equal to B.
And then, I multiply that times 1 to change it from Boolean to numeric.
And then, here's a variable that's 1 when spray is C, and 0 otherwise.
And here's a variable that's 1 when the spray is D, and 0 otherwise.
And here's one for E, and here's one for F.
So, I've included all of them except A.
So, I've forced A to be my reference level.
And I'm going to run this model.
And it should give me the same result as with r did.
It's just now I've shown you exactly how r is creating the regression variables.
So, let me just remind ourselves what r gives us when we run,
and let it handle the factor variable by itself.
And then, let me do the same thing where I've created my own factor variables.
And then, you can see 14.5, 14.5, 0.833,
0.833, you can see that it's identical.
So this is what r is doing behind the scenes.
And let's keep exploring this because this is kind of an important point.
If you mess this up with factor variables, you get very incorrect conclusions.
Now, let me show you what happens if I do include spray A here.
So, I've done the same model I did before where I include all the variables,
but now I'm also including an extra variable for spray A.
And let me just do the lm part.
And notice it gives an NA in front of the spray A coefficient.
And the reason for that is because it's redundant.
We have six means, right?
For six sprays.
And we have seven parameters in intercept.
And then, now I've tried to put in six regression parameters.
I have six means to fit seven parameters.
It can't do that, so it's gonna drop one of them.
Now, what if I do want my coefficients,
instead of being interpreted as levels referenced to a control level,
what if I want my coefficients to be the mean for each of the groups?
Well, you can do that, but you have to remove the intercept.
So, watch what happen when I say count is my outcome, and spray is my predictor, but
I remove the intercept.
Then notice what happens
is that now I get a different set of coefficients, one for each spray level.
So, it includes A, B, C, D, E and F.
It hasn't dropped any levels.
And it can do that now because it has six parameters, and six means to work with.
And these are exactly equal to the means for each spray in the data.
So, if I were to just go ahead and calculate the means for each spray, right?
It works out to be the same numbers.
14.5, 15.3, 2.08 in both, and so on.
Now, I want to emphasize this model is
no different than my model that included an intercept.
Why don't I go back to my model with my intercept just to illustrate this.
So, now it's just that the coefficients have a different interpretation.
Now, the intercept from the model, when I fit count as spray but
included the intercept, my intercept now is interpreted, 14.5,
as the mean for spray A.
And you can see that it's exactly the empirical mean for
spray A when I calculate the mean.
It works out that way.
And then, spray B, we talked about earlier,
was the comparison between the reference level spray A and spray B.
Okay? So, if I add these together,
14.5 and 0.833, I should get the mean for spray B.
Okay, and that's what you see, 14.5 plus 0.833, that gets me 15.33, and so on.
So, If I add 14.5 and minus 12, I'm gonna get 2.08.
If I add 14.5 and negative 9, I get 4.9, and so on.
So, this model, where I've included an intercept,
has all the same information as the model where I omitted the intercept,
the only difference is how the coefficients are interpreted.
In the model with the intercept now the intercept is interpreted as
the sprayA mean and all the coefficients are interpreted
as relative to sprayA differences from sprayA.
And then if I would have fit it without the intercept then I get the mean for
each spray.
And if I want differences then I have to subtract the coefficients.
And you usually want one of them to be a reference level because then you
can do test.
So now my p values are testing whether or not, for the t test whether or
not A is different from B, and A is different from C, and A is different from
D, and so on, whereas the p values from this test are just testing whether or
not those means are different from 0, which is a very different test.
Did sprayA kill any insects is what this is testing, where in this one,
the sprayB row is testing whether sprayA is different from sprayB.
So, what I'm trying to illustrate is that, how you play around with
factor variables in LM is very important in terms of how you interpret it.
It's not just a conceptual or theoretical thing to worry about it.
It's a very practical thing to worry about.
What your intercept means changes dramatically depending on what your
reference level is or whether or not you include an intercept.
Let me do one last thing.
Where now I show you how you can re-level and
in this case sprayA was my reference level you can very easily re-level it.
So say sprayC is your reference level.
So now here I just use the re level command so
now inspect spray, the reference level is sprayC but
now I've just created a new variable where that spray2.
And now I'm gonna do my linear model where my outcome is my count.
And spray2 is my predictor now instead of spray.
And this is the one that has C as the reference level.
And then R knows not to do the one that has the lowest alphanumeric letter,
but instead has the reference level that I set.
And there when I do it notice sprayA is present,
sprayC is gone cuz now it's the reference level.
My intercept is interpreted as the mean for sprayC, and
you can see 2.0833 that's exactly the mean for group C.
And then this coefficient 12.41 here
is interpreted as the comparison of sprayA to sprayC.
This 13.25 is the comparison of sprayB to sprayC, and
so if I want to test sprayA versus sprayC I've got to look at this P value.
If I wanted to test sprayB versus sprayC, I would look at this p value.
So let me just recap, since this is a very important point.
If we include a factor of that level, factor variable like spray in R,
then R automatically includes an intercept, and
treats the first level of the factor as the reference level.
So the intercept now is interpreted as the mean for that reference level,
in our example, the intercept is interpreted as the mean for sprayA.
Then each spray, other effects, so sprayB effect, sprayC effects,
sprayD effect, sprayE effect, and so on,
is the comparison of that spray level versus the reference level which is A.
If we want the means for those,
it has to be the intercept plus their specific coefficients.
So the intercept plus the sprayB coefficient will be the mean for sprayB.
All of the tests then, the test for the intercept will be a test for whether or
not the mean for sprayA is zero.
The test of all the other levels, the sprayB, sprayC, and
other coefficients, will be a test for the comparison versus sprayA.
If, on the other hand, we omit an intercept, then we're gonna get a mean for
each level, including sprayA.
And then all the test would be for whether or not the sprayA coefficient is different
from zero, the on for b would be whether or
not the sprayB coefficient was zero, and so on, which may or may not be relevant.
Usually you want the comparisons and that's why r's default
is to pick one of the levels and treat it as the reference level.
However, if you want a different one.
If you want B to be the reference level, you just need to use the re-level command.
Or if you wanna get involved a little bit more in linear models,
then you need to go into how you calculate standard errors in more general settings.
But that's a little bit more advanced.
For right now if you want to do the comparisons say between B and C then my
current suggestion is just to re-level so that now B is the reference level and
the coefficient for C will now be comparing spray B and spray C.
I wanna give some caveats about this data analysis that I presented,
it's not exactly a complete data analysis I think the modeling the data
as if they are normally distributed is perhaps problematic they're count data so
they're bounded from below by zero.
I think it would probably be a little more natural to model this data as if it was
Poisson or at least over dispersed Poisson or
something like that which we're going to cover In our GLM version of the class.
In addition the variance, it's clearly not constant.
So what I mean by the variance is I mean the variance around the mean.
And it's clearly not constant as our regression models would assume.
So this is a potential problem.
And so our means are probably correct.
Our estimates are probably correct.
But our inferences assuredly not.
So that creates an issue that needs to be handled at some level.
And later on in the class we'll talk about some things for handling this and
some of the rest you may have to take some further
statistical inferences classes to deal with some of the more advanced topics.
Like when the variances are unequal they call that heteroscedasticity.
