Normality

Before we begin, let’s go ahead and activate our packages and load our data.

library(tidyverse)
chicks <- as_tibble(ChickWeight)

The quickest method of testing the normality of a variable is with the Shapiro-Wilk normality test. This will return two values, a W score and a p-value. FOr the purposes of this course we may safely ignore the W score and focus on the p-value. When p >= 0.05 we may assume that the data are normally distributed. If p < 0.05 then the data are not normally distrubted. Let’s look at all of the chicks data without filtering it:

shapiro.test(chicks$weight)

R> 
R>  Shapiro-Wilk normality test
R> 
R> data:  chicks$weight
R> W = 0.90866, p-value < 2.2e-16

Are these data normally distributed? How do we know? Now let’s filter the data based on the different diets for only the weights taken on the last day (21):

chicks %>% 
  filter(Time == 21) %>% 
  group_by(Diet) %>% 
  summarise(norm_wt = as.numeric(shapiro.test(weight)[2]))

R> # A tibble: 4 × 2
R>   Diet  norm_wt
R>   <fct>   <dbl>
R> 1 1       0.591
R> 2 2       0.949
R> 3 3       0.895
R> 4 4       0.186

How about now?

Homoscedasticity

Here we need no fancy test. We must simply calculate the variance of the variables we want to use and see that they are not more than 3-4 times greater than one another.

chicks %>% 
  filter(Time == 21) %>% 
  group_by(Diet) %>% 
  summarise(var_wt = var(weight))

R> # A tibble: 4 × 2
R>   Diet  var_wt
R>   <fct>  <dbl>
R> 1 1      3446.
R> 2 2      6106.
R> 3 3      5130.
R> 4 4      1879.

Well, do these data pass the two main assumptions?

Epic fail. Now what?

After we have tested our data for the two key assumptions we are faced with a few choices. The basic guidelines below apply to paired tests, one- and two-sample tests, as well as one- and two-sided hypotheses (i.e. t-tests and their ilk):

When we compare two or more groups we usually do an ANOVA, and the same situation is true. For ANOVAs our options include (but are not limited to):

See this discussion if you would like to know about some more advanced options when faced with heteroscedastic data.

Our tests for these two assumptions fail often with real data. Therefore, we must often identify the way in which our data are distributed (refert to Chapter 5) so we may better decide how to transform them in an attempt to coerce them into a format that will pass the assumptions.

Log transformation

This consists of taking the log of each observation. You can use either base-10 logs (log10(x)) or base-e logs, also known as natural logs (log(x)). It makes no difference for a statistical test whether you use base-10 logs or natural logs, because they differ by a constant factor; the base- 10 log of a number is just 2.303…× the natural log of the number. You should specify which log you’re using when you write up the results, as it will affect things like the slope and intercept in a regression. I prefer base-10 logs, because it’s possible to look at them and see the magnitude of the original number: \(log(1) = 0\), \(log(10) = 1\), \(log(100) = 2\), etc.

The back transformation is to raise 10 or e to the power of the number; if the mean of your base-10 log-transformed data is 1.43, the back transformed mean is \(10^{1.43} = 26.9\) (in R, 10^1.43). If the mean of your base-e log-transformed data is 3.65, the back transformed mean is \(e^{3.65} = 38.5\) (in R, exp(3.65)). If you have zeros or negative numbers, you can’t take the log; you should add a constant to each number to make them positive and non-zero (i.e. log10(x) + 1). If you have count data, and some of the counts are zero, the convention is to add 0.5 to each number.

Many variables in biology have log-normal distributions, meaning that after log-transformation, the values are normally distributed. This is because if you take a bunch of independent factors and multiply them together, the resulting product is log-normal. For example, let’s say you’ve planted a bunch of maple seeds, then 10 years later you see how tall the trees are. The height of an individual tree would be affected by the nitrogen in the soil, the amount of water, amount of sunlight, amount of insect damage, etc. Having more nitrogen might make a tree 10% larger than one with less nitrogen; the right amount of water might make it 30% larger than one with too much or too little water; more sunlight might make it 20% larger; less insect damage might make it 15% larger, etc. Thus the final size of a tree would be a function of nitrogen×water × sunlight × insects, and mathematically, this kind of function turns out to be log-normal.

Slightly skewed data

– sqrt(x) for positively skewed data

– sqrt(max(x+1) - x) or x^2 for negatively skewed data

Moderately skewed data

– log10(x) for positively skewed data,

– log10(max(x + 1) - x) or x^3 for negatively skewed data

Severely skewed data

– 1/x for positively skewed data

– 1/(max(x + 1) - x) or higher powers than cubes for negatively skewed data

Deviations from linearity and heteroscedasticity

– log(x) when the dependent variable starts to increase more and more rapidly with increasing independent variable values

– x^2 when the dependent variable values decrease more and more rapidly with increasing independent variable values

– Regression models do not necessarily require data transformations to deal with heteroscedasticity. Generalised Linear Models (GLM) can be used with a variety of variance and error structures in the residuals via so-called link functions. Please consult the glm() function for details.

– The linearity requirement specifically applies to linear regressions. However, regressions do not have to be linear. Some degree of curvature can be accommodated by additive (polynomial) models, which are like linear regressions, but with additional terms (you already have the knowledge you need to fit such models). More complex departures from linearity can be modelled by non-linear models (e.g. exponential, logistic, Michaelis-Menten, Gompertz, von Bertalanffy and their ilk) or Generalised Additive Models (GAM) — these more complex relation- ships will not be covered in this module. The gam() function in the mgcv package fits GAMs. After fitting these parametric or semi-parametric models to accommodate non-linear regressions, the residual error structure still does to meet the normality requirements, and these can be tested as before with simple linear regressions.

Knowing how to successfully implement transformations can be as much art as science and requires a great deal of experience to get right. Due to the multitude of options I cannot offer comprehensive ex- amples to deal with all eventualities — so I will not provide any examples at all! I suggest reading widely on the internet or textbooks, and practising by yourselves on your own datasets.