# Introduction

When the response variable is right skewed, many think regression becomes difficult. Skewed data is generally thought of as problematic. However the glm framework provides two options for dealing with right skewed response variables. For the gamma and inverse gaussian distributions, a right skewed response variable is actually helpful.

# Different Shapes Of A Gamma Distribution

The critical step is being able to spot a gamma distribution when you see one. Theatrical skewness is $$\frac{2}{scale}$$. Note the gamma distribution is always positive.

library(GlmSimulatoR)
library(ggplot2)
library(dplyr)
library(stats)

set.seed(1)
#Very right skewed. Skewness 2
Gamma <- rgamma(1000, shape = 1, scale = 1)
temp <- tibble(gamma = Gamma)
ggplot(temp, aes(x=gamma)) +
geom_histogram(bins = 30) #Very right skewed and spread out more. Skewness 2
Gamma <- rgamma(1000, shape = 1, scale = 5)
temp <- tibble(gamma = Gamma)
ggplot(temp, aes(x=gamma)) +
geom_histogram(bins = 30) #Hump moves slightly towards the middle. Skewness 1.414214
Gamma <- rgamma(1000, shape = 2, scale = 1)
temp <- tibble(gamma = Gamma)
ggplot(temp, aes(x=gamma)) +
geom_histogram(bins = 30) #Hump moves slightly more towards the middle. Skewness 1.154701
Gamma <- rgamma(1000, shape = 3, scale = 1)
temp <- tibble(gamma = Gamma)
ggplot(temp, aes(x=gamma)) +
geom_histogram(bins = 30) #Hump moves slightly more towards the middle. Skewness 0.8944272
Gamma <- rgamma(1000, shape = 5, scale = 1)
temp <- tibble(gamma = Gamma)
ggplot(temp, aes(x=gamma)) +
geom_histogram(bins = 30) #Nearly gaussian. Very slightly right skewed. Skewness .2
Gamma <- rgamma(1000, shape = 100, scale = 1)
temp <- tibble(gamma = Gamma)
ggplot(temp, aes(x=gamma)) +
geom_histogram(bins = 30) # Building A Models With Very Skewed Data

To show the generalized linear model can handle skewness, I will build a model that has a very low mean squared error.

#Make data
set.seed(1)
simdata <- simulate_gamma(N = 10000, link = "inverse",
weights = c(1, 2, 3), ancillary = .05)
#Confirm Y~gamma
ggplot(simdata, aes(x = Y)) +
geom_histogram(bins = 30) glm <- glm(Y ~  X1 + X2 + X3, data = simdata, family = Gamma("inverse"))

#Mean Squared Error
mean((simdata$Y - predict(glm, newdata = simdata, type = "response"))^2) #>  0.004147222 # Different Shapes Of A Inverse Gaussian Above we saw the gamma distribution take on many different shapes. The inverse gaussian distribution is not as flexible. It tends to maintain it’s skewness for a variety of parameters. library(statmod) set.seed(1) Invgauss <- rinvgauss(1000, mean = 1, shape = .2) temp <- tibble(Invgauss = Invgauss) ggplot(temp, aes(x=Invgauss)) + geom_histogram(bins = 30) Invgauss <- rinvgauss(1000, mean = 1, shape = 1) temp <- tibble(Invgauss = Invgauss) ggplot(temp, aes(x=Invgauss)) + geom_histogram(bins = 30) Invgauss <- rinvgauss(1000, mean = 1, shape = 3) temp <- tibble(Invgauss = Invgauss) ggplot(temp, aes(x=Invgauss)) + geom_histogram(bins = 30) Invgauss <- rinvgauss(1000, mean = 10, shape = .2) temp <- tibble(Invgauss = Invgauss) ggplot(temp, aes(x=Invgauss)) + geom_histogram(bins = 30) Invgauss <- rinvgauss(1000, mean = 10, shape = 10) temp <- tibble(Invgauss = Invgauss) ggplot(temp, aes(x=Invgauss)) + geom_histogram(bins = 30) Invgauss <- rinvgauss(1000, mean = 10, shape = 100) temp <- tibble(Invgauss = Invgauss) ggplot(temp, aes(x=Invgauss)) + geom_histogram(bins = 30) # Building A Model With Very Skewed Data Using Inverse Gaussian Distribution Similar to above, I will build a model that has a very low mean squared error. #Make data set.seed(1) simdata <- simulate_inverse_gaussian(N = 10000, link = "inverse", weights = c(1, 2, 3), ancillary = 10) #Confirm Y is right skewed ggplot(simdata, aes(x = Y)) + geom_histogram(bins = 30) glm <- glm(Y ~ X1 + X2 + X3, data = simdata, family = inverse.gaussian(link = "inverse")) #Mean Squared Error mean((simdata$Y - predict(glm, newdata = simdata, type = "response"))^2)
#>  0.005872676