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.

The critical step is being able to spot a gamma distribution when you see one. Theatrical skewness is \(\frac{2}{\sqrt(shape)}\). If shape is small, the gamma distribution is right skewed. If shape increases, the gamma becomes more symmetrical

```
library(GlmSimulatoR)
library(ggplot2)
library(dplyr)
library(stats)
set.seed(1)
#Very right skewed. Skewness 2
<- rgamma(1000, shape = 1, scale = 1)
Gamma <- tibble(gamma = Gamma)
temp ggplot(temp, aes(x=gamma)) +
geom_histogram(bins = 30)
```

```
#Very right skewed and spread out more. Skewness 2
<- rgamma(1000, shape = 1, scale = 5)
Gamma <- tibble(gamma = Gamma)
temp ggplot(temp, aes(x=gamma)) +
geom_histogram(bins = 30)
```

```
#Hump moves slightly towards the middle. Skewness 1.414214
<- rgamma(1000, shape = 2, scale = 1)
Gamma <- tibble(gamma = Gamma)
temp ggplot(temp, aes(x=gamma)) +
geom_histogram(bins = 30)
```

```
#Hump moves slightly more towards the middle. Skewness 1.154701
<- rgamma(1000, shape = 3, scale = 1)
Gamma <- tibble(gamma = Gamma)
temp ggplot(temp, aes(x=gamma)) +
geom_histogram(bins = 30)
```

```
#Hump moves slightly more towards the middle. Skewness 0.8944272
<- rgamma(1000, shape = 5, scale = 1)
Gamma <- tibble(gamma = Gamma)
temp ggplot(temp, aes(x=gamma)) +
geom_histogram(bins = 30)
```

```
#Nearly gaussian. Very slightly right skewed. Skewness .2
<- rgamma(1000, shape = 100, scale = 1)
Gamma <- tibble(gamma = Gamma)
temp ggplot(temp, aes(x=gamma)) +
geom_histogram(bins = 30)
```

To show the generalized linear model can handle skewness, lets make some data and train a model and calculate mean squared error.

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

```
<- glm(Y ~ X1 + X2 + X3, data = simdata, family = Gamma("inverse"))
glm
#Mean Squared Error
mean((simdata$Y - predict(glm, newdata = simdata, type = "response"))^2)
#> [1] 0.004147222
```