Checking Statistical Assumptions in Practice
What to do when assumptions fail
Every statistical test assumes a set of conditions holds in the data. These assumptions — normality, homogeneity of variance, independence, and linearity — when unmet can lead to invalid conclusions. Researchers must check these conditions systematically before analysis using both visual diagnostics and formal tests, and must respond to violations with data transformation, robust methods, or non-parametric alternatives rather than ignoring the problem.
What Is a Statistical Assumption and Why Does It Matter
A statistical test pre-supposes that certain mathematical conditions hold in the data. For example, the independent-samples t-test assumes that both groups are drawn from normally distributed populations and that the population variances are equal. When these conditions are violated, the test statistic and p-value become unreliable: the Type I error rate may deviate from the nominal level, and confidence intervals may be incorrectly sized. Assumption checking is not a formality — it is a foundational step that safeguards the validity of the entire analysis. Researchers are expected to know in advance which assumptions each test carries and to actively verify them for every dataset.
How to Check Core Assumptions
Four core assumptions and their practical checks are as follows. Normality: A Q-Q plot shows whether data points fall near the diagonal reference line; a histogram provides a visual shape assessment; for small samples (n below 50) the Shapiro-Wilk test offers a formal test. Homogeneity of variance: Levene's test compares variances across two or more groups; a funnel-shaped pattern in a residuals-versus-fitted plot signals heteroscedasticity. Independence: This is primarily a design issue — repeated measures, clustered, or time-series structures violate independence for standard tests; the Durbin-Watson statistic detects serial autocorrelation in regression residuals. Linearity: A scatterplot and a residual-versus-fitted plot that show a curved pattern suggest the linearity assumption is questionable.
Common Misuses and Misconceptions
The most common error is treating a non-significant Shapiro-Wilk result as proof of normality. In small samples the test has low power and may miss substantial departures; in large samples it may flag trivially small deviations as significant. The correct approach is to combine formal tests with visual inspection. A second common mistake is continuing the analysis despite detected violations without reporting them, which seriously undermines confidence in the findings. A third misconception is believing that large sample sizes automatically render assumptions irrelevant by appealing to the central limit theorem. The central limit theorem concerns only the sampling distribution of the mean; violations such as heteroscedasticity or autocorrelation continue to distort inference even with large samples.
What to Do When Assumptions Fail and How to Report It
When an assumption violation is detected, three main strategies are available. First, data transformation: a logarithmic or square-root transformation of right-skewed distributions can improve both normality and variance homogeneity, but the changed scale of interpretation must be stated explicitly. Second, select a non-parametric or robust alternative: methods such as Mann-Whitney U, Kruskal-Wallis, or heteroscedasticity-consistent standard errors (HC3) are more resistant to violations. Third, model the violation directly — for example, mixed-effects models can handle correlated observations. In the research report, clearly state which assumption was tested, the method used, the result obtained, and the remedial action taken.