Class 09: Multivariate Analysis with Regression

Learning Objectives

  • Understand the role of nuisance variables in hypothesis testing and why we need to control for them.
  • Apply the linear regression function in the tmodels package to test for relationships between explanatory and response variables while controlling for nuisance variables.
  • Identify and use the output of the function tmod_linear_regression for various types of explanatory variables.
  • Interpret and summarize the values provided by a confidence interval.

Linear regression

A linear regression model in a very common type of statistical estimator. In the simpliest case, it is built by assuming that we can approximately model a numeric variable y as a linear function of another numeric variably x:

The noise term just captures the fact that the model is not perfect and we make errors in trying to write the y variable as a linear function from the x variable.

Using the cars dataset again, let’s see a visual description of how city fuel efficency is effected by the size of a car’s engine (we will cover how to actually make these plots after the second exam) and how we can model this relationship using a line:

plot of chunk unnamed-chunk-2

The solid black line shows the regression line and the red dashed vertical lines show the noise values.

We can run an hypothesis test that uses the following null and alternative hypotheses using the tmodels package.

The way to do this should not come as a surprise to you at this point:

tmod_linear_regression(city ~ engine_size, data = cars)
## Linear regression; T-Test
## 	H0: Change in conditional mean is zero
## 	HA: Change in conditional mean is non-zero
## 	Test statistic: t = -6.6659
## 	P-value: 1.035e-07
## 	Parameter: change in city for unit change in engine_size
## 	            controlling for -- 
## 	Point estimate: -1.2579
## 	Confidence interval: [-1.64100, -0.87483]

The point estimate here is the point estimate for the term b (the slope) in the regression model.

Why is linear regression useful from an hypothesis testing standpoint? More specifically, we have two continuous variables, what is the difference between this and the correlation tests? Let’s compare the output of using Pearson’s product-moment correlation test:

tmod_pearson_correlation_test(city ~ engine_size, data = cars)
## Pearson's product-moment correlation test
## 	H0: True correlation is zero
## 	HA: True correlation is non-zero
## 	Test statistic: t(35) = -6.6659
## 	P-value: 1.035e-07
## 	Parameter: (Pearson) correlation coefficient
## 	Point estimate: -0.74792
## 	Confidence interval: [-0.86284, -0.55949]

The point estimate is completely different (it is trying to find the correlation not a slope), but look at the test statistic and p-value. They are exactly the same as the ones given by linear regression!

This is no fluke. The linear regression hypothesis test and Pearson’s product-moment correlation test are exactly equivalent. They provide different point estimates, but test the exact same hypotheses in exactly the same way. From a statistical inference perspective, there is not different between the two. So why then, do we need to concern ourselves with linear regression?

Controlling for Nuisance Variables

All of the hypothesis tests that we have so far studied have exactly two variables: an explanatory variable (IV) and the response variable (DV). Our tests have null hypotheses that indicate, in various ways, that there is no relationship between the two variables. We want to see if there is evidence to support having an actual effect between the two variables.

Linear regression provides a nice way to extend hypothesis testing to account for other variables that we believe may effect the response variable and that we want to factor out or control for in the analysis. For example, returning very briefly to the pea plants and colored light experiment. We assumed that the plants were all identical; what if instead some where snow peas (saccharatum) and others were snap peas (macrocarpon). One way to deal with this is to include the variety of the pea plant as a third variable into the model. The variety variable as used here is what we call a nuisance variable.

Specifically, we can write down a multivariate linear regression with a response variable y, independent variable x, and nusiance variable z as:

Here, we are accounting for the variable z and isolating the effect of x in the slope term b. To run this in R, we require only a slight modification of the code we have used for the other hypothesis tests. Simply add (with the plus sign) the nusiance variable(s) as additional factors to the model:

tmod_linear_regression(city ~ engine_size + highway, data = cars)
## Linear regression; T-Test
## 	H0: Change in conditional mean is zero
## 	HA: Change in conditional mean is non-zero
## 	Test statistic: t = -2.22
## 	P-value: 0.03319
## 	Parameter: change in city for unit change in engine_size
## 	            controlling for -- highway
## 	Point estimate: -0.3299
## 	Confidence interval: [-0.631890, -0.027908]

The hypothesis tests here are:

The point estimate here is an estimate of the parameter b. The output even tells us explictily what we are controlling for (here, the highway fuel efficency). The p-value says that, even controlling for the highway fuel efficency, there is a significant relationship between engine size and city fuel efficency.

Note: We select the independent variable of interest by putting it first (after the tilda) in the formula.

Linear regression with categorical variables

It is possible to also include categorical variables as terms in a linear regression. R will handle categorical nusiance variables without any change to the input or output. Here we control for the class of the car in the cars regression model:

tmod_linear_regression(city ~ engine_size + highway + class, data = cars)
## Linear regression; T-Test
## 	H0: Change in conditional mean is zero
## 	HA: Change in conditional mean is non-zero
## 	Test statistic: t = -1.0751
## 	P-value: 0.2906
## 	Parameter: change in city for unit change in engine_size
## 	            controlling for -- highway; class
## 	Point estimate: -0.37451
## 	Confidence interval: [-1.08500,  0.33596]

The model operates by allowing each class of car to have its own intercept term. Visually, it looks something like this:

plot of chunk unnamed-chunk-7

Finally, we can also have a categorical variable be the independent variable of the regression model. If there are only two categories, this will give an analog to the two-sample t-test. If there are multiple categories, it does a test similar to one-way ANOVA:

tmod_linear_regression(city ~ class + highway, data = cars)
## Linear regression; F-Test
## 	H0: Difference in conditional mean is zero
## 	HA: Difference in true means is non-zero
## 	Test statistic: F(1, 3) = 4.6126
## 	P-value: 0.008595

As with one-way ANOVA, we have a p-value but there is no consistent way to provide the analog of a point estimate.

Confidence Intervals

We have seen that a number of hypothesis tests return an object called a confidence interval. A p% confidence interval provides a range of guesses for a point estimate based on a dataset that will “capture the true parameter with probably p%”. All of the intervals in the tmodels package provide 95% confidence intervals.

We say that we are 95% percent confident that the true population parameter lies somewhere in the confidence interval.

Assumptions of linear regression models

The linear regression model assumes the two standard assumptions we have across all of our models: uniform sampling and independent response values. There are additionally a host of other assumptions that all well beyond the scope of this course. Just know that as long as you (approximately) have the two basic assumptions, you are usually safe using linear regression for testing the relationship between a continuous response and any explanatory variable.

A linear regression model is able to control for some nusiance variables, but can never control for all variables that might effect a response. Therefore, it can help more strongly suggest causal claims from observational data. However, it can never replace the strong causal proof provided by a randomized experimental design.

Note that there is no simple non-parametric version of linear regression. In order for the core ideas of linear regression to work, we need to make strong assupmtions about how the response data values y are generated. This makes non-parametric tests difficult to extend in a consistant way that mirrors the other univariate tests we have seen.

Regression table

The output of the function tmod_linear_regression only tells us about the parameter b (the slope of the independent variable of interest). Sometimes it is useful to see estimates and information for all of the parameters.

The function tmod_lin_reg_table provides this information:

tmod_lin_reg_table(city ~ highway + engine_size, data = cars)
##             Estimate CI Low CI High Pr(>|t|)
## (Intercept)    5.880  2.280  9.4900 2.19e-03
## highway        0.483  0.372  0.5940 2.40e-10
## engine_size   -0.330 -0.632 -0.0279 3.32e-02

When, there is a categorical variable, you’ll see a number of terms for each of the intercepts:

tmod_lin_reg_table(city ~ highway + class, data = cars)
##              Estimate  CI Low CI High Pr(>|t|)
## (Intercept)    -2.430 -6.6400   1.780 2.49e-01
## highway         0.719  0.5530   0.885 4.42e-10
## classcompact    1.000 -0.0815   2.080 6.88e-02
## classmidsize    1.310  0.1540   2.460 2.75e-02
## classsuv        2.790  1.1100   4.480 1.96e-03

One category (the first alphabetically) is selected as the baseline value and the other categories are given offsets for how much higher or lower their lines should be.