M. Drew LaMar

April 4, 2016

Introduction to Biostatistics, Spring 2016

- Ignore the violations of assumptions
~~Transform the data~~- Use a nonparametric method
- Use a permutation test (computer-intensive methods)

Definition:Adata transformation changes each measurement by the same mathematical formula.

Common transformations:

- Log transformation (
**data skewed right**) \[ Y^{\prime} = \ln[Y] \] - Arcsine transformation (
**data are proportions**) \[ p^{\prime} = \arcsin[\sqrt{p}] \] - Square-root transformation (
**data are counts**) \[ Y^{\prime} = \sqrt{Y + 1/2} \]

Other transformations:

- Square transformation (
**data skewed left**) \[ Y^{\prime} = Y^2 \] - Antilog transformation (
**data skewed left**) \[ Y^{\prime} = e^{Y} \] - Reciprocal transformation (
**data skewed right**) \[ Y^{\prime} = \frac{1}{Y} \] - Box-Cox transformation (
**skew**) \[ Y^{\prime}_{\lambda} = \frac{Y^{\lambda} - 1}{\lambda} \]

- Measurements are ratios or products
- Frequency distribution is skewed right
- Group having larger mean also has larger standard deviation
- Data span several orders of magnitude

- Measurements are ratios or products
- Frequency distribution is skewed right
~~Group having larger mean also has larger standard deviation~~- Data span several orders of magnitude

- Measurements are ratios or products
- Frequency distribution is skewed right
~~Group having larger mean also has larger standard deviation~~- Data span several orders of magnitude

**Hypothesis testing**

```
marine <- read.csv("http://whitlockschluter.zoology.ubc.ca/wp-content/data/chapter13/chap13e1MarineReserve.csv")
shapiro.test(log(marine$biomassRatio))
```

```
Shapiro-Wilk normality test
data: log(marine$biomassRatio)
W = 0.93795, p-value = 0.06551
```