- A statistical test's output is only as trustworthy as the assumptions behind it — the software will never warn you when an assumption is violated; it will simply produce a number.
- Independence of observations must be verified before any other assumption — repeated measures, matched pairs, and clustered data (patients within hospitals) all violate it and require different methods entirely.
- Normality applies to the continuous outcome (or regression residuals), not to every variable in your dataset, and matters most in smaller samples.
- Different tests carry different assumption profiles — Cox regression needs the proportional hazards assumption checked; chi-square needs expected cell counts checked; neither of these applies to a t-test.
- Checking assumptions is part of running the analysis, not an optional validation step performed only if a reviewer asks.
Why Checking Assumptions Matters Before You Trust a P-Value
Every parametric statistical test is derived mathematically under a specific set of conditions — a particular distribution shape, a particular independence structure, a particular relationship between variables. The formulas that produce your p-value, your confidence interval, and your effect estimate are only valid proofs under those conditions. When the conditions don't hold, the formula still runs to completion and still returns a number, but that number is no longer the quantity the test was designed to estimate.
This is different from most errors in a data pipeline, which tend to announce themselves — a missing file throws an error, a wrong data type throws a warning. A violated statistical assumption does neither. SPSS, R, and every other statistical package will compute a t-test on badly non-independent data and hand back a p-value formatted identically to a p-value computed on perfectly valid data. The distinction between a trustworthy and an untrustworthy result exists only in whether the researcher checked the assumptions first — nothing about the output itself reveals which one you're looking at.
This is also why assumption-checking sits logically after study design and variable classification, but strictly before interpreting any test result — it is one of the last checks standing between your analysis and a conclusion you can actually defend to a thesis committee or peer reviewer.
Independence of Observations
Independence means that each observation in your dataset provides genuinely separate information — knowing one patient's value should tell you nothing about another patient's value. Most standard tests (independent t-test, one-way ANOVA, chi-square, standard linear and logistic regression) assume this by default, and it is arguably the single most consequential assumption to check first, because violating it doesn't just bias one number — it invalidates the entire mathematical basis for the standard error calculation.
Independence is most commonly broken by repeated measurements on the same patient (baseline, 4 weeks, 8 weeks), by matched or paired designs, and by clustered data, where patients treated by the same surgeon, admitted to the same ward, or enrolled at the same trial site tend to be more similar to each other than to patients elsewhere — a structure common enough in multi-center trials that it has its own name, the design effect.
Measuring pain scores in 30 patients before and after a nerve block and analyzing all 60 values with an independent t-test, as if they came from 60 separate patients, ignores that each pair of measurements comes from the same person — a paired t-test is required instead.
Normality
The normality assumption applies to the distribution of a continuous outcome variable (or, in regression, to the model's residuals) and underlies parametric tests including the t-test, ANOVA, Pearson correlation, and linear regression. It is formally checked with the Shapiro-Wilk test in small-to-moderate samples, and assessed visually with a histogram or Q-Q plot, as covered in full in our normality testing guide.
Many medical measurements are naturally skewed rather than normal — length of hospital stay, C-reactive protein, and most biomarker concentrations tend to have a long right tail, with a small number of very high values pulling the mean upward. Reporting mean ± SD and running a t-test on data like this can be misleading; median and interquartile range, alongside a non-parametric test, is usually the more honest summary.
Comparing ICU length of stay between two treatment groups: a handful of very long stays make the distribution right-skewed, so the median (IQR) and a Mann-Whitney U test better represent the typical patient than the mean and a t-test would.
Homogeneity of Variance (Homoscedasticity)
Homogeneity of variance means the spread (variance) of the outcome is similar across the groups being compared, or, in regression, similar across the range of predicted values. It applies to the independent t-test, one-way ANOVA, and the residuals of a linear regression model, and is checked formally with Levene's test or visually with a residual-versus-fitted plot.
Unequal variances distort the standard error used to calculate the p-value, and the distortion is worse when group sample sizes are also unequal — a common scenario in medical research when a rare-exposure or rare-outcome group is naturally much smaller than the comparison group.
Comparing a biomarker's variability between healthy controls (a tight, consistent range) and a disease group (a much wider range, reflecting disease heterogeneity) — Levene's test flags unequal variances, and Welch's t-test, which does not assume equal variances, is the more defensible choice.
Linearity
Linearity is the assumption that the relationship between a continuous predictor and the outcome follows a straight line — for Pearson correlation and linear regression, this means the outcome itself; for logistic regression, it means the log-odds of the outcome; for Cox regression, it means the log-hazard. It is checked with a scatterplot of predictor against outcome, or with a residual-versus-predictor plot in a fitted model.
Many physiological relationships are not linear across their full range. Age and a given biomarker, for example, may rise, plateau, and fall again across a lifespan, producing a curved or even U-shaped true relationship. Forcing a straight line through a genuinely curved relationship can produce a near-zero, non-significant slope even when a real, strong, non-linear association exists — a false negative created entirely by an unchecked assumption, not by an absence of true effect.
Modeling the relationship between maternal age and risk of chromosomal abnormality as a straight line would badly misrepresent a relationship that is actually roughly flat through the 20s and 30s before rising sharply after 35 — a quadratic term or a spline captures this shape far better than a single linear slope.
Outliers and Influential Points
An outlier is a data point with an extreme value relative to the rest of the dataset. An influential point is a data point whose removal would meaningfully change the model's results — not every outlier is influential, and not every influential point looks extreme at first glance; formal detection in regression uses measures like Cook's distance rather than eyeballing the raw values alone.
The instinct to simply delete outliers is usually the wrong first move. An extreme value can be a genuine, clinically important observation (a real, unusually severe case), a data entry error (a decimal point in the wrong place), or evidence the point belongs to a different population entirely (a lab value from a sample that was actually hemolyzed). The correct first step is always to investigate the cause, not to remove the point and move on.
One patient in a chronic kidney disease cohort has a creatinine level far above the rest of the group due to a documented episode of acute kidney injury — this is a genuine, clinically meaningful value, not an error, and the appropriate response is a sensitivity analysis reporting results with and without that patient, rather than silent deletion.
Multicollinearity
Multicollinearity occurs in multiple regression (linear or logistic) when two or more predictor variables are highly correlated with each other. The overall model can still fit reasonably well, but the individual coefficient estimates become unstable — standard errors inflate, confidence intervals widen, and which of the correlated predictors appears "significant" can flip unpredictably with small changes to the data or model specification.
Multicollinearity is detected using the Variance Inflation Factor (VIF) for each predictor; a VIF near 1 indicates no problematic correlation, while a VIF above 5, and especially above 10, signals a predictor whose effect estimate can no longer be trusted in isolation. A simple correlation matrix among candidate predictors before modeling is a useful, faster first screen.
Including both BMI and waist circumference as separate predictors of cardiovascular risk in the same regression model — the two measures are highly correlated, and their individual coefficients become unreliable; choosing one, or combining them into a single composite measure, produces a more interpretable model.
Adequate Sample Size and Expected Cell Counts
Beyond the assumptions specific to individual tests, every statistical method also assumes the sample is large enough for its underlying approximation to hold. An underpowered study risks a Type II error regardless of how carefully every other assumption was checked — a topic covered fully in our sample size and power guide.
The chi-square test has its own specific version of this assumption: no more than 20% of cells in the contingency table should have an expected count below 5, and no cell should have an expected count below 1. When this is violated — common with rare outcomes or small subgroups — Fisher's Exact test should be used instead, since it calculates exact probabilities rather than relying on chi-square's large-sample approximation. Multivariable logistic regression carries a related rule of thumb, commonly cited as at least 10 outcome events per predictor variable included in the model.
A 2×2 table comparing a rare surgical complication (only 3 events in one arm) between two procedures has expected cell counts below 5 — chi-square is not appropriate here; Fisher's Exact test is the correct choice.
The Proportional Hazards Assumption
The proportional hazards assumption is specific to Cox regression: it requires that the hazard ratio comparing any two groups (or any two levels of a covariate) stays constant across the entire follow-up period. Visually, this means the survival curves for compared groups should never cross, and log-minus-log survival plots for each group should run roughly parallel throughout follow-up.
The formal check uses a test based on Schoenfeld residuals, which examines whether each covariate's residuals show a significant relationship with time — a significant result flags a violation. When the assumption is violated, a single hazard ratio no longer meaningfully summarizes the relationship, since the true risk difference between groups is actually changing across follow-up; the fix is a stratified Cox model, a covariate-by-time interaction term, or reporting time-varying coefficients instead of a single pooled hazard ratio.
Comparing surgical versus medical management for a condition: surgery may carry a higher early risk from procedural complications but a lower risk later from durable treatment benefit — the hazard curves cross, proportional hazards is violated, and a single reported "HR = 0.8" would misrepresent a relationship that actually reverses direction over time.
Which Assumptions Apply to Which Test
Not every assumption applies to every method — this table maps the assumptions covered above to the tests they actually govern, as a quick reference before you run your analysis.
| Assumption | Applies To |
|---|---|
| Independence of observations | Nearly all standard tests — t-test, ANOVA, chi-square, standard regression |
| Normality | t-test, ANOVA, Pearson correlation, linear regression (residuals) |
| Homogeneity of variance | Independent t-test, one-way ANOVA, linear regression |
| Linearity | Pearson correlation, linear regression, logistic regression (logit scale), Cox regression (log-hazard scale) |
| Outliers / influential points | Correlation, regression, and any mean-based comparison |
| Multicollinearity | Multiple linear regression, multiple logistic regression |
| Adequate sample size / expected cell counts | Chi-square test, logistic regression |
| Proportional hazards | Cox regression |
Checking only normality for every analysis, regardless of which test is actually being run, and assuming a "normal-looking" histogram means the analysis is assumption-safe.
Identify the specific test first, then check only the assumptions that test actually depends on — a chi-square table needs expected cell counts checked, not normality; a Cox model needs proportional hazards checked, not homogeneity of variance.
Common Mistakes When Checking Assumptions
Mistake 1: Skipping Normality Because "n Is Large Enough"
Treating the Central Limit Theorem as a blanket excuse to skip checking normality entirely, even when a subgroup within the larger sample is small or the data are severely skewed.
Mistake 2: Ignoring Clustering or Repeated Measures
Analyzing repeated or clustered observations as if each row in the dataset were an independent patient, artificially shrinking standard errors and inflating false-positive risk.
Mistake 3: Deleting Outliers Without Investigation
Removing an extreme value simply because it looks unusual or because it changes the p-value, without first checking whether it's a data error, a genuine case, or a different population.
Mistake 4: Reporting a Single Hazard Ratio Without Checking Proportional Hazards
Running a Cox model and reporting the hazard ratio without ever testing whether it's actually constant across follow-up time.
Mistake 5: Running Chi-Square on Small Expected Cell Counts
Applying a standard chi-square test to a contingency table with several cells expected to contain fewer than 5 observations, silently violating the test's large-sample approximation.
Mistake 6: Interpreting Individual Coefficients Under Severe Multicollinearity
Drawing conclusions about which of two highly correlated predictors is "the real driver" of an outcome, when multicollinearity has made both coefficients individually unstable.
Practical Workflow for Checking Assumptions Before Analysis
Final Checklist Before Running Your Analysis
Confirm observations are independent
Identify repeated measures, matching, or clustering before choosing a test.
Test normality of the continuous outcome (or model residuals)
Shapiro-Wilk plus a visual check (histogram or Q-Q plot).
Check homogeneity of variance for group comparisons
Levene's test; use Welch's correction if variances differ.
Verify linearity for correlation and regression
Scatterplot or residual-versus-predictor plot before fitting the final model.
Screen for outliers and influential points
Investigate the cause before deciding whether to keep, transform, or flag a point.
Check multicollinearity in any multivariable model
VIF for every predictor before interpreting individual coefficients.
Confirm adequate sample size and expected cell counts
The 20% rule for chi-square; events-per-predictor for logistic regression.
Test proportional hazards if running Cox regression
Schoenfeld residuals or log-minus-log plots before reporting a hazard ratio.
Further Reading
For deeper methodological background on statistical assumptions and reporting standards in medical research, these external resources are widely regarded as authoritative:
- The EQUATOR Network ↗ maintains reporting guidelines (STROBE, CONSORT, PRISMA) that specify how statistical methods and their assumptions should be documented.equator-network.org
- The Cochrane Handbook for Systematic Reviews of Interventions ↗ covers statistical assumptions in the context of evidence synthesis and meta-analysis.training.cochrane.org
Frequently Asked Questions
Related Articles
Once your assumptions are confirmed, these guides take you through the next steps:
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