• Identify possible transformations of the response when your errors have unequal variance or are skewed

• Understand how to use common transformations and make inference from the resulting model

• Understand that there are alternatives to transformation that we will use later

We have made a number of assumptions about our models, which include

• the distribution of the errors (IID)

• linear relationship(s) between the response and predictor(s)

What can we do when these assumptions are not met?

It’s possible to transform both sides of our models to

• achieve constant variance \((y)\)

• correct for skewness \((y)\)

• linearize the relationship \((y, x)\)

The most common form is where \(y' = y^\lambda\)

and \(\lambda > 1\) (powers)

or \(0 < \lambda < 1\) (roots)

For example

\(\lambda = 2 \Rightarrow y' = y^2\)

\(\lambda = \frac{1}{2} \Rightarrow y' = \sqrt{y}\)

One can also use inverses where \(y' = y^{-\lambda}\)

and \(\lambda > 1\) (powers)

or \(0 < \lambda < 1\) (roots)

For example

\(\lambda = 2 \Rightarrow y' = \frac{1}{y^2}\)

\(\lambda = \frac{1}{2} \Rightarrow y' = \frac{1}{\sqrt{y}}\)

The Box-Cox transformation is a popular method for stabilizing the variance of errors

It is defined as

\[ y' = \frac{y^{\lambda} - 1}{\lambda} \]

for all \(y > 0\)

More specifically, because

\[ \lim\limits_{\lambda \to 0} \frac{y^{\lambda} - 1}{\lambda} = \log(y) \]

we instead use

\[ y' = \left\{ \begin{matrix} \frac{y^{\lambda} - 1}{\lambda} ~ \text{if} ~ \lambda \neq 0 \\ \log(y) ~ \text{if} ~ \lambda = 0 \end{matrix} \right. \]

How does one choose \(\lambda\)?

By using profile likelihoods (which we will see in a later lecture)

(We’ll use the boxcox() function in the MASS package)

Let’s return to the plant data from the Galapagos Archipelago where we modeled diversity as a function of island area

## get data
data(gala, package = "faraway")
## fit regression model
mm <- lm(Species ~ Area, gala)
## estimate lambda
MASS::boxcox(mm)

Here is the result of calling boxcox(mm)

After transformation, how do we interpret \(\lambda = 0.17\)?

Box-Cox transformations work well, but sometimes we can do better with an approximation to \(\lambda\)

General considerations

• The Box–Cox method gets upset by outliers

For example, if \(\hat{\lambda}\) = 5 there is little rationale for such an extreme transformation

General considerations

• The Box–Cox method gets upset by outliers

• If some \(y_i\) < 0, we can add a constant to all the \(y\)

This works if the constant is small, but it’s a “hack”

General considerations

• The Box–Cox method gets upset by outliers

• If some \(y_i\) < 0, we can add a constant to all the \(y\)

• If the range in \(y\) is small, then the Box–Cox transformation will not have much effect

Recall that linear models work well for local non-linear functions

Consider the fecundity of a fish versus its length

Here’s the fit from a linear regression

And here are the residuals from the fitted model

This \(\hat{\lambda}\) is really close to 0.5 (ie, a square root transform)

Here’s the fit from a linear regression to \(\sqrt{y}\)

And here are the residuals from the fitted model

Using predict() will give fits on the transformed scale

## expected sqrt(fecundity) for length = 5 dm
predict(ms, data.frame(ll = 5), interval = "confidence")

##        fit      lwr      upr
## 1 65.25383 64.48932 66.01835

We need to incude the back-transformation on predict()

\[ \begin{aligned} \sqrt{\hat{y}}_i &= x_i \hat{\boldsymbol{\beta}} \\ &\Downarrow \\ \hat{y}_i &= (x_i \hat{\boldsymbol{\beta}})^2 \\ \end{aligned} \]

Here’s the fit and prediction interval on the natural scale

Think back to an early lecture where we transformed a nonlinear polynomial into a linear model

\[ y_i = \beta_0 + \beta_1 x_{1,i} + \beta_2 x_{2,i}^2 + \epsilon_i \\ \Downarrow \\ y_i = \beta_0 + \beta_1 x_{1,i} + \beta_2 z_{2,i} + \epsilon_i \\ z_{2,i} = x_{2,i}^2 \]

Polynomials are an easy way to model nonlinearities in data, such as

• Seasonal effects on primary productivity

• Temperature effects on growth of poikilotherms

Many ecological observations only take positive values \((y > 0)\)

• length or mass or fecundity

• species counts/density

• latency periods for infectious diseases

The distributions of these data also tend to be “long-tailed”

Distribution of plant diversity data in the gala dataset

These long-tailed data often follow a log-normal distribution

A log-transformation is a really common way to deal with ecological data that are constrained to be positive

\[ y_i = \exp(\beta_0 + \beta_1 x_{i} + \epsilon_i) \\ \Downarrow \\ \log(y_i) = \beta_0 + \beta_1 x_{i} + \epsilon_i \]

Consider allometric scaling laws in ecology of the form

\[ y_i = \alpha x_i^\beta \epsilon_i \]

For example, body mass as a function of length

Log-log transformations are an easy way to linearize power models

\[ m_i = \alpha l_i^\beta \epsilon_i \\ \Downarrow \\ \log(m_i) = \log(\alpha) + \beta ~ \log(l_i) + \log(\epsilon_i) \\ \Downarrow \\ y_i = \alpha' + \beta x_i + \epsilon'_i \]

The response and predictor are linear on the log-log scale

• Box-Cox is good to help ID a power/root, but the transformed variable can be hard to interpret

• \(\sqrt{y}\) is good for equalizing variance

• \(\log(y)\) is good for skewed data

• \(\log(y + a)\) with \(a\) small relative to the data is good for skewed data with some 0’s

• We will see later that there are model alternatives to transformations (GLMs)