Interpreting Coefficients

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En provenance du cours de Johns Hopkins University

Regression Models

2121 notes

Johns Hopkins University

Regression Models

2121 notes

Cours 7 sur 10 dans Specialization Data Science

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.

À partir de la leçon

Week 2: Linear Regression & Multivariable Regression

This week, we will work through the remainder of linear regression and then turn to the first part of multivariable regression.

Statistical Linear Regression Models2:38

Interpreting Coefficients3:47

Linear Regression for Prediction10:34

Rencontrer les enseignants

Brian Caffo, PhD
Professor, Biostatistics
Bloomberg School of Public Health
Roger D. Peng, PhD
Associate Professor, Biostatistics
Bloomberg School of Public Health
Jeff Leek, PhD
Associate Professor, Biostatistics
Bloomberg School of Public Health

Now that we have a formal statistical framework, we can interpret our regression

coefficients with respect to that framework.

Take for example, the intercept.

It's the expected value Y given that the regressor is 0.

And here I'm simply plugging into the formula.

Note that the regressor being equal to zero is often

not of interest in the study.

For example, if the regression variable is blood pressure, probably

you're not interested in the response for among people with blood pressure of zero.

However, there is an easy fix for this.

Consider just shifting our regression variable by a constant a.

And here in this equation, I've just subtracted off an a here.

Which means I have to add an a beta 1 over here to have done nothing,

to have simply added 0.

Well this is exactly rewriting the aggression equation

as beta not tilde plus beta 1 times x minus a.

A new regression line with a new intercept, the same slope, and

my regressor now shifted by a constant a.

So this goes to show you that if you shift your regressor by a constant,,

it doesn't change the slope, however it does change the intercept.

Now the intercept is inter, interpreted not when x is zero, but

when x is this value that you shifted it by a.

A very common constant to due is the average.

Set a equal to the average of the X variable.

Then your intercept will be interpreted as the expected

value of the response when the regressor is at the average value of the Xs.

Let's see if we can figure out a model-based interpretation of our

regression coefficient.

Now, it's a slope so it must have an interpretation that is of the form,

the change in Y per unit change in X.

So let's see if we can work with that.

Take the expected value of Y,

given that X takes the value x plus 1, a one unit increase in X.

And subtract off from the expected value of Y, given that X takes the value x.

Well, that works out to be beta naught plus beta 1 times x plus1,

minus beta naught plus beta 1 times x.

Do the arithmetic and that works out to be beta 1.

So, beta 1 is the expected change in the response per unit

change in the regression variable.

And that, so it has a nice interpretation, it's not that different from the regular

interpretation of a line, just now with the expected value,

the conditional expected value of the response.

Now, remember, if we shift the X variable, it doesn't change the slope,

we discussed that on the previous slide.

What if we scaled the regression variable?

And it usually only we're usually scaling by number, say a, that's positive.

So, what we can do is simply multiply and divide by a, and we can rewrite our

progression equation as beta naught plus beta 1 tilde times X times a plus epsilon.

So, we've multiplied X by a and

our regression slope has now been divided by a.

So in other words, multiple, multiplication of a regression variable by

a factor a results in dividing the coefficient by a factor of a.

Let me give you a simple example, so to hopefully solidify these points.

Imagine if X's height measured in meters and

Y's weight measured in kilograms, a pretty simple regression problem.

Then beta 1 is interpreted in kilograms per meter.

Converting X to centimeters implies multiplying X by

100 centimeters per meter.

To get beta 1 in the right units, now we simply divide

it by 100 centimeters per meter to get it, to, to, to get it back on the right scale.

And here I just wrote a, a little equation to remind us.

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