Exploring Links for the Gaussian Distribution

knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )

Overview

Compared to the linear model, one advantage of the generalized linear model is its ability to model different relationships between the response variable and the predictors. One challenge is knowing which link to use. In this vignette, we will explore how different relationships affect correlation and the visual appearance of scatter plots.

Inverse Link

Mathematical Background

For the identity link, the underlying model is $$Y = \beta_2X_2 + \beta_1X_1 + \beta_0$$ Note there is no need to rearrange for Y because the link is the identity function.

Using the inverse link function, the underlying model is $$ 1/Y = \beta_2X_2 + \beta_1X_1 + \beta_0$$.

Rearranging for Y, we get $$ Y = 1 / (\beta_2X_2 + \beta_1X_1 + \beta_0) $$ We see the relationship between Y and X is different between the two models. This is the beauty of the glm framework. It can handle many different relationships between Y and X.

Exploring Data

First, lets generate some data.

library(GlmSimulatoR) library(ggplot2) library(stats) set.seed(1) simdata <- simulate_gaussian(N = 1000, weights = c(1, 3), link = "inverse", unrelated = 1, dispersion = .005)

Next, lets do some basic data exploration. We see the response is gaussian.

ggplot(simdata, aes(x=Y)) + geom_histogram(bins = 30)

The connection between Y and X1 is not obvious. There is only a slight downward trend. One might see it as unrelated.

ggplot(simdata, aes(x=X1, y=Y)) + geom_point()

There is a connection between Y and X2. No surprise as the true weight is three.

ggplot(simdata, aes(x=X2, y=Y)) + geom_point()

The scatter plot between the unrelated variable and Y looks like random noise. It is interesting to note the scatter plot for X1 looks more similar to this one than X2's scatter plot despite being included in the model.

ggplot(simdata, aes(x=Unrelated1, y=Y)) + geom_point()

We see the correlation is very strong between Y and X2. This is no surprise considering the above graph. The correlation between Y and X1 is somewhat larger in absolute value than the unrelated variable. However, I would not see this as particularly good news in predicting Y if I did not know the correct model.

cor(x=simdata$X1, y = simdata$Y) cor(x=simdata$X2, y = simdata$Y) cor(x=simdata$Unrelated1, y = simdata$Y)

Building Models

Pretending we don't know the correct answer, lets see if we can find the correct model. We will try three models. One with just X2, one with X1 and X2, and one with everything. Will the correct model stand out?

glmInverseX2 <- glm(Y ~ X2, data = simdata, family = gaussian(link = "inverse")) glmInverseX1X2<- glm(Y ~ X1 + X2, data = simdata, family = gaussian(link = "inverse")) glmInverseX1X2U1<- glm(Y ~ X1 + X2 + Unrelated1, data = simdata, family = gaussian(link = "inverse")) summary(glmInverseX2) summary(glmInverseX1X2) summary(glmInverseX1X2U1)

Inspecting the 3 summaries, we see a few things. First, the correct model has the lowest AIC! The slope estimates are very close to the true values. Overall, the mathematics behind the glm framework worked very well.

Log Link

Mathematical Background

Above we saw using the identity link assumes an additive relationship between Y and X. $$Y = \beta_2X_2 + \beta_1X_1 + \beta_0$$

For the log link, the underlying model is $$\ln(Y) = \beta_2X_2 + \beta_1X_1 + \beta_0$$

Rearranging for Y, we get $$ Y = \exp (\beta_2X_2 + \beta_1X_1 + \beta_0) $$ Splitting up the exponent, we get $$ Y = \exp (\beta_2X_2) \exp (\beta_1X_1) \exp (\beta_0) $$ Thus the relationship between Y and X is multiplicative, not additive for the log link.

Exploring Data

First, lets generate some data.

library(GlmSimulatoR) library(ggplot2) library(stats) set.seed(1) simdata <- simulate_gaussian(N = 1000, weights = c(.3, .8), link = "log", unrelated = 1, dispersion = 1)

We see the response is somewhat gaussian.

ggplot(simdata, aes(x=Y)) + geom_histogram(bins = 30)

The connection between Y and X1 is not obvious...

ggplot(simdata, aes(x=X1, y=Y)) + geom_point()

There is a connection between Y and X2. No surprise as the true weight is .8 on the log scale.

ggplot(simdata, aes(x=X2, y=Y)) + geom_point()

The scatter plot between the unrelated variable and Y looks like random noise.

ggplot(simdata, aes(x=Unrelated1, y=Y)) + geom_point()

Again X2's correlation is large. X1 is in the gray area. The unrelated variable's correlation is near zero.

cor(x=simdata$X1, y = simdata$Y) cor(x=simdata$X2, y = simdata$Y) cor(x=simdata$Unrelated1, y = simdata$Y)

Building Models

Pretending we don't know the correct answer, lets see if we can find the correct model. For a change we will compare the the link functions for the gaussian family.

glmIdentity <- glm(Y ~ X1 + X2, data = simdata, family = gaussian(link = "identity")) glmInverse<- glm(Y ~ X1 + X2, data = simdata, family = gaussian(link = "inverse")) glmLog<- glm(Y ~ X1 + X2, data = simdata, family = gaussian(link = "log")) summary(glmIdentity) summary(glmInverse) summary(glmLog)

Again, the correct model has the lowest AIC and the estimated weights are very close to the true values.

Summary

We have explored different links for the gaussian distribution, but the gaussian distribution is not a special case. Everything that was done here could be done for any distribution in the glm framework. Once you understand one distribution, you are very far along in understanding the other distributions. The glm framework can handle categorical response variables (binomial), integer response variables (poisson), and right skewed response variables (gamma and inverse gaussion).