Every statistical test comes with assumptions, and a significant result built on a violated assumption is not trustworthy. The good news is that the common assumptions are few, the checks are quick, and each one has a standard fallback when it fails. This guide walks through them and points you to the reporting guide for each test.
Why assumptions matter
A test's p value is only valid if the data meet the conditions the test was derived under. Ignore a violated assumption and you can get a false positive, a false negative, or a biased estimate. Checking assumptions first, and switching methods when needed, is the difference between a defensible analysis and one a reviewer sends back.
The assumptions, one by one
Normality
Many parametric tests (t-tests, ANOVA, regression) assume the outcome, or the model residuals, are approximately normal.
- How to check: a histogram and a Q-Q plot are the first look; the Shapiro-Wilk test is the common formal test. For regression, check the normality of the residuals, not the raw outcome.
- If it fails: use a non-parametric alternative (for example the Mann-Whitney U test instead of an independent t-test, or the Kruskal-Wallis test instead of a one-way ANOVA), or transform the variable. In large samples, mild non-normality is usually fine.
Homogeneity of variance
Tests that compare groups (independent t-test, ANOVA) assume the groups have similar variances.
- How to check: Levene's test. A significant Levene's test means the variances differ.
- If it fails: use a version that does not assume equal variances, such as Welch's t-test or Welch's ANOVA, and report the adjusted degrees of freedom.
Independence
Almost every test assumes observations are independent: one participant's value does not depend on another's.
- How to check: this is a matter of study design, not a test. Repeated measures, clustered sampling, or siblings in the same dataset break independence.
- If it fails: use a method built for it, such as a paired or repeated-measures test, or a mixed (multilevel) model.
Linearity
Correlation and regression assume the relationship between variables is linear.
- How to check: a scatterplot of the two variables, or a plot of residuals against fitted values for regression.
- If it fails: use Spearman correlation for a monotonic but non-linear pattern, or add polynomial or transformed terms to the model.
Homoscedasticity (constant variance of residuals)
Regression assumes the residuals have constant variance across the range of predictions.
- How to check: a residuals-versus-fitted plot; a funnel shape signals a problem.
- If it fails: transform the outcome, or use robust standard errors.
No severe multicollinearity
Multiple regression and logistic regression assume predictors are not too highly correlated with each other.
- How to check: the variance inflation factor (VIF). A common threshold for concern is a VIF above 5 or 10.
- If it fails: drop or combine redundant predictors.
Sphericity
Repeated-measures ANOVA assumes sphericity, that the variances of the differences between conditions are equal.
- How to check: Mauchly's test.
- If it fails: apply a Greenhouse-Geisser or Huynh-Feldt correction to the degrees of freedom.
How to report assumption checks in APA 7
You do not need a paragraph per assumption. State the checks briefly in the analysis section, and only elaborate when something failed and changed your method. For example:
Levene's test indicated unequal variances, F(2, 87) = 4.62, p = .012, so Welch's ANOVA is reported.
The quick reference
| Assumption | How to check | If it fails |
|---|---|---|
| Normality | Q-Q plot, Shapiro-Wilk | Non-parametric test, or transform |
| Homogeneity of variance | Levene's test | Welch's t-test or ANOVA |
| Independence | Study design | Paired / repeated-measures / mixed model |
| Linearity | Scatterplot, residual plot | Spearman, or transform / add terms |
| Homoscedasticity | Residuals vs fitted plot | Transform, or robust SEs |
| Multicollinearity | VIF | Drop or combine predictors |
| Sphericity | Mauchly's test | Greenhouse-Geisser correction |
Let KyroStat check the assumptions for you
KyroStat checks the relevant assumptions before it reports a result, runs the correct fallback when one is violated (for example switching to Welch or a non-parametric test), and tells you what it did and why. If you are still choosing a test, start with the decision guide. Upload your spreadsheet, and the analysis, with its assumption checks, is done in seconds.
Frequently asked questions
Do I have to test normality of the raw data or the residuals? For group comparisons, check normality within each group; for regression, check the residuals. The residuals are what the model assumes to be normal.
My Shapiro-Wilk test is significant. Is my analysis ruined? Not necessarily. In large samples Shapiro-Wilk flags trivial departures. Look at the Q-Q plot too, and consider whether the deviation is severe before switching methods.
What if several assumptions fail at once? Move to a method with fewer assumptions, such as a non-parametric test or a robust/bootstrap approach, rather than patching each violation separately.
Where do I report the assumption checks? Briefly in the analysis or results section, usually one sentence, expanding only when a violation changed the test you ran.