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.
Residuals and Diagnostics part III
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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.
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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
Okay, so let's go through some simulation experiments
to understand how some of these diagnostic measures work.
In this first case,
I'm looking at an instance where there's a big cloud of uncorrelated data and
I did this by just generating a bunch of pairs of independent random normals.
Then I added random standard then I added the point at ten,ten which
clearly does not fit the rest of the trend of data.
What we can see in this case is that there's a strong correlation
estimated by the data merely because of the existence of this point
otherwise the correlation would be estimated to be zero.
So you can see the code right here that I use to generate and plot the data.
So let's look at a couple of the diagnostic values and
the first thing, why we look at the DF datas, so here's the dfbetas,
the round statement here just rounds into the third decimal place when I put
the three there and notice this first point which is the ten, ten point is or
as the magnitude larger of a dfbeta then, the remaining point.
Let's look at the hat values.
The hat value for this point is much larger than the hat values for
the remaining points.
The hat values have to be between zero and one.
So and it's of course much larger than the other points.
So if we are looking at these, we would obviously single this point out.
Now let's look at another point, another instance where there's a clear
regression relationship, here I just generated the data along a line, and
then I generated another outlier that's similarly distant from the cloud of data.
However it adheres very nicely to the regression line.
So let's see how our diagnostic values look in this specific case.
Okay, so here's my dfbetas and if you look this first point,
which was that outlying point.
It's still large but
nowhere near as distinctively large as in the other case.
So it still does appear to have some influence in the fit, but
nothing like in the other case.
However if you look at the hat values right.
If you look at the hat values, this has a much larger hat value than and
the remaining all the other points.
A factor of ten than most of the other points.
Why?
Well, if we go back what we see is that this point
is outside of the range of the X values.
It's outside of the range of the X values.
But it does adhere to the direction relationship.
So it's going to have a large leverage value but
not a large DF beta or DFS or some of these other things.
Let's look at this example by Stefanski that to me shows why we do residual plots.
Basically, the reason we look at residual plots is because they zoom
on potential problems with our model.
In this case you can download the data from their website.
The pair Y scatter plots are given here they show kind of
nothing interesting other than a lot of over plotting.
Here I show my linear model fit and if you look every single P
values highly significant for every single coefficient here.
I set all of the data points but, for this to work for this model,
you have to try to intercept in the way that he generate with the data.
Okay, should we be done, is this fine, well here's the problem.
So the residual plot that you intend to do when you have multi variable examples
because you can't plop the residuals versus the only
acts as you can in linear regression is you need to pick a number
of the X with the most common ways to plot the residuals versus the fitted values but
residuals which are e vs y hat, okay.
So what happens in this particular instance when we plot
the residuals versus y hat.
Now you can see you get this very clever little picture that comes out.
So what's happened?
Without looking at the residuals we wouldn't have seen anything.
But looking at the residuals has zoomed in on this very clear aspect of
poor model fit and
is a very clear systematic pattern in our residuals that we've missed.
And remember in most cases in regression we want to model the systematic things and
everything that we can explain, we want to leave to,
we want to model this as if we're noise but the systematic things like
obviously this picture is systematic we want to actually be able to model.
So in this case without having to look at the residual plot,
we would have missed this pattern.
It is really just created by Stephansky and his co-authors to
described for us why it is that we do residual plots and
that is that they zoom in ,so finally on aspects of poor model fair.
So let's go back to the swiss data and just look at the different
examples of diagnostic plots that can spit out by default by art.
Remember at the beginning of the lecture we started out with this so
let's see if now we can interpret these things.
Well, for the residual plot versus the fitted values
that's the same plot as we just saw with the oriliow just a slide ago.
So what you're looking at in that plot is trying to find anything that's systematic.
For example, if you saw the data look something like that.
That would suggest heteroscedasticity
that the variance is either increasing or decreasing.
In a way that you wouldn't like and so on.
So we in this plot it doesn't look so bad.
There doesn't seem to be too many aspects of absence
of model departures there.
The Q-Q plot is specifically designed to test normality or
not to test to evaluate normality of the error terms, okay?
This scale location plot is plotting the standardized residuals, remember we
talked about standardized residuals they're the ordinary residuals but
standardize, so they have a more kind of comparable scale across subject,
across experiments and the scale to try and make them to be like a TY statistic.
So again, this is a lot like the residual plot, you're applying them against
the fitted values but, now you just mostly, you've change the scale.
So that's potentially useful for looking at these across different experiments.
The final is plot of the Residuals vs Leverage.
So here's the standardized Residuals on this scale and
then here's the Leverage on that scale and in this plot
again you're trying to look at any sort of systematic pattern any reason why points
with higher leverage are having higher or particularly small residual values.
If you had an instance like where you have plots like this and
you have one very high leverage point and you get something like that.
That point might have a very small residual but
unnecessarily while it has happened to have very large leverage or
you might have an instance like this where even though it's really impacted
the regression line, it still has high leverage and also has a high residual.
So at any rate these are many, these are just a couple of examples of the kind of
plots you might want to look at in this data set none of them seem
to look too inherently bad but when you go through these things ideally you
would have something where you could click on the individual points and it would It
would describe the aspects of the point when you hover over it with your mouse.
Some of the other software systems can do that.
R can do that now, we'll talk in the data products
class how you can actually create those kinds of plots.
So that's the end of the lecture and
I look forward to seeing you next time, but I hope now that you know a little bit
about diagnostic measures, and influence diagnostics, and leverage diagnostics that
you can incorporate them into your analysis in the future.
All right. See you next time.
