6. Assumptions and Transformations

Checking Model Requirements Before and After Inference

NoteIn This Chapter

• Why assumptions matter
• Normality, homoscedasticity, and independence
• Graphical versus formal assumption checks
• When data transformation is useful
• When transformation is not the right solution

1 Introduction

Most standard inferential methods rely on assumptions. These assumptions do not exist to make statistics difficult. They exist because every model is a simplified representation of the data-generating process.

In this chapter, we bring together two ideas that should be considered jointly:

1. assumptions, which tell us what the method expects, and
2. transformations, which are one possible response when the assumptions are not well met.

2 Key Concepts

The most important concepts in this chapter are these.

• Assumptions describe the conditions under which a model or test is intended to work well.
• Graphical diagnostics are usually the first and most informative way to assess assumptions.
• Formal tests can support diagnostics, but they do not replace judgement.
• Transformations can improve symmetry, variance structure, or linearity.
• Model choice is often a better response than transformation when the data structure is fundamentally different.

3 Core Assumptions

Across t-tests, ANOVA, and linear models, the most common assumptions are:

• approximate normality,
• homoscedasticity or constant variance,
• independence of observations,
• the correct functional form of the relationship.

Independence is usually a design issue. Normality and variance structure are more often checked after fitting a model.

4 How To Check Assumptions

Assumptions should be checked using both:

• graphs such as histograms, Q-Q plots, and residual plots, and
• formal tests such as Shapiro-Wilk, Levene’s test, or Breusch-Pagan.

Graphical assessment is often more informative than a single p-value, especially in large datasets where trivial deviations can appear statistically significant.

5 Transformations

Common transformations include:

• log transformation,
• square-root transformation,
• reciprocal transformation.

Transformations may help:

• stabilise variance,
• reduce skewness,
• make relationships more nearly linear.

But transformations are not a universal fix. If the data are counts, proportions, dependent observations, or clearly governed by a different error structure, a different model may be more appropriate than transforming the response.

6 Practical Principle

Use transformation when it improves the match between the model and the biology. Do not transform reflexively just to force the data into a preferred method.

7 Links

This chapter replaces the previous split treatment in:

• 06-assumptions.qmd
• 14-transformations.qmd