Unchecked and unreported statistical assumptions are one of the most common reasons a quantitative results chapter is questioned in a viva or sent back for revision. The pattern is well documented: audits of theses and published papers repeatedly find that assumption checks are either not performed or not reported, and examiners know exactly where to look. Yet the fix is not to run every possible test and report a wall of p-values. The mark of a strong analysis is knowing which assumptions each test rests on, checking them properly, and, just as important, understanding when a violation actually matters and when it does not. A candidate who can say "normality was violated, but with a sample of this size the test is robust, and here is the source that establishes that" is in a far stronger position than one who abandoned a good analysis at the first significant Shapiro-Wilk result. This guide sets out each assumption, how to check it, how to report it, and when it truly threatens your conclusions. When you want the checks run and documented to examiner standard, our dissertation data analysis service does exactly that.
Quick Answer:
The assumptions examiners ask about are normality, homogeneity of variance, linearity and homoscedasticity, multicollinearity, independence of residuals, and sphericity. Each has a standard check and a standard remedy. The sophisticated points are that significance tests of assumptions (Shapiro-Wilk, Levene's, Mauchly's) over-detect trivial violations in large samples and miss real ones in small samples; that t-tests, ANOVA, and regression are robust to non-normality in large samples through the central limit theorem; and that most cited thresholds (VIF above 10, Durbin-Watson near 2, skewness and kurtosis near 0) are conventions, not laws. Report every assumption you checked, lead with graphical methods over single p-values, and be ready to justify why a violation does or does not compromise your findings.
Why This Is Where Chapters Get Sent Back
Assumption testing sits at the exact point where a technically capable analysis can still fail review, because it is invisible in the final numbers. A t-test result looks the same whether or not you checked that its assumptions held, which is precisely why examiners probe it: it is where corners get cut. The empirical record shows how routinely this happens. Reviews of the health and biomedical literature find that only a minority of papers using regression report checking any of its assumptions, and studies of these find similar gaps, with a large share of candidates applying parametric tests without confirming the conditions those tests require. Reporting your assumption checks, even the ones that passed, is therefore one of the highest-value things you can do to preempt criticism, and it is a distinction our overview of the ten most common dissertation mistakes returns to.
The Assumptions, One by One
Each parametric test rests on a specific set of conditions. The six below are the ones examiners ask about most, with the standard check and the standard remedy for each.
Table 1: The Assumptions Examiners Ask About, With Checks and Remedies
Assumption | Standard Check | Standard Remedy |
|---|---|---|
Normality | Q-Q plot, histogram, skewness/kurtosis, Shapiro-Wilk | Transformation, non-parametric test, or rely on robustness in large samples |
Homogeneity of variance | Levene's test (or Brown-Forsythe) | Welch's t-test or Welch's ANOVA |
Linearity & homoscedasticity | Scatterplots; residual-vs-fitted plot | Transformation, robust standard errors, weighted least squares |
Multicollinearity | Variance inflation factor (VIF), tolerance | Combine or remove redundant predictors (with caution) |
Independence of residuals | Durbin-Watson statistic (~2 = no autocorrelation) | Mixed/multilevel model or time-series model |
Sphericity (repeated measures) | Mauchly's test (unreliable; consider skipping) | Greenhouse-Geisser or Huynh-Feldt correction; mixed model |
Normality
Many parametric tests assume that a variable, or more precisely the residuals of a model, is approximately normally distributed. You check this with a combination of methods rather than any single one. Use a Q-Q plot and a histogram to see the shape, skewness, and kurtosis as summary indices, both near zero for a normal distribution. Add the Shapiro-Wilk test as the most powerful formal test (Shapiro & Wilk, 1965). The critical thing to understand is that the Shapiro-Wilk p-value is sample-size dependent. In large samples, it flags trivial, immaterial departures as significant; in small samples, it lacks the power to detect real ones. As Ghasemi and Zahediasl put it in their widely cited guide for non-statisticians, visual methods and skewness and kurtosis should be used alongside the formal test rather than the test being treated as decisive (Ghasemi & Zahediasl, 2012). Do not, therefore, abandon a well-chosen analysis simply because Shapiro-Wilk returned p less than .05 in a large sample.
Homogeneity of Variance
When you compare groups, tests like the independent-samples t-test and ANOVA assume the groups have roughly equal variances. You check this with Levene's test (Levene, 1960), or its more robust Brown-Forsythe variant, which uses deviations from the median. When variances are unequal, the standard remedy is not to abandon the comparison but to use a version that does not assume equal variances: Welch's t-test or Welch's ANOVA (Welch, 1947). A substantial methodological literature argues that Welch's test can reasonably be used by default, because it loses little power when variances are in fact equal (Delacre, Lakens, & Leys, 2017). It is worth being precise here, because the same authors later narrowed that recommendation. In a 2022 correction, they cautioned that their conclusion does not generalize to situations where sample sizes differ and are drawn from highly skewed distributions. Subsequent work showed that with unequal, small group sizes, Welch's test can inflate the false-positive rate. The defensible position is to use Welch where variances are unequal, but to justify the choice for your specific design rather than applying it mechanically.
Linearity and Homoscedasticity
Regression assumes that the relationship between predictors and outcome is linear, and that the residuals have constant variance across the range of fitted values, a property called homoscedasticity. You check linearity with scatterplots of the outcome against each predictor, and homoscedasticity with a plot of residuals against fitted values, looking for a formless cloud rather than a funnel or curve. When linearity fails, you can transform a variable or specify a non-linear model; when homoscedasticity fails, you can use heteroscedasticity-consistent (robust) standard errors, a weighted model, or a transformation. A common and revealing error, documented in the health literature, is checking the raw outcome variable for normality instead of checking the residuals, which is what regression actually assumes.
Multicollinearity
In multiple regression, multicollinearity, where predictors are highly correlated with one another, inflates the standard errors of the coefficients and makes them unstable. You check it with the variance inflation factor and its reciprocal, tolerance. The commonly cited flags are a VIF above 10, or sometimes above 5, with tolerance below 0.1. These are rules of thumb, not hard laws, and it is worth citing the source that says so. O'Brien (2007) cautions that mechanically dropping variables in reaction to a VIF above 10 can create problems more serious than the collinearity itself. This is especially true when the correlated terms are genuine parts of the model, such as polynomial or interaction terms. Report the VIF, interpret it in context, and be cautious about deleting predictors on the strength of a threshold alone.
Independence of Residuals
Most tests assume observations, and therefore residuals, are independent of one another. In regression, first-order autocorrelation is checked with the Durbin-Watson statistic (Durbin & Watson, 1950, 1951), which ranges from 0 to 4, with a value near 2 indicating no autocorrelation, values toward 0 indicating positive autocorrelation, and values toward 4 indicating negative autocorrelation. When independence is violated, often because data are clustered, repeated, or collected over time, the remedy is a model that accounts for the dependency: a mixed or multilevel model for nested data, or a time-series model for serially correlated data. Independence is an assumption you often protect by design, through your sampling, as much as by a post-hoc test.
Sphericity
Repeated-measures ANOVA adds a further assumption, sphericity, which concerns the equality of the variances of the differences between all pairs of within-subject conditions. It is checked with Mauchly's test (Mauchly, 1940). Here, the professional caveat is strong enough that many methodologists recommend not relying on the test at all: Mauchly's test has low power in small samples, where violations matter most, and over-detects in large ones. For this reason, the widely used afex package in R applies the Greenhouse-Geisser correction by default, on the reasoning that a small loss of power is preferable to overlooking a violation and producing anti-conservative, false-positive results (CRAN afex documentation). When sphericity is violated, you correct the degrees of freedom. Use the Greenhouse-Geisser correction (Greenhouse & Geisser, 1959), typically when the epsilon estimate is below 0.75, or the Huynh-Feldt correction (Huynh & Feldt, 1976) when it is higher. Alternatively, sidestep the assumption entirely with a mixed-effects model.
When a Violation Actually Matters
This is the section that separates a defensible analysis from an anxious one, and it is where citing the right sources earns you real credibility. Not every assumption violation threatens your conclusions, and the single most important reason is the robustness of common tests to non-normality in large samples.
The central limit theorem means that as the sample size grows, the sampling distribution of the mean approaches normality regardless of the shape of the underlying data. The practical consequence, established authoritatively by Lumley and colleagues, is that the belief that the t-test and linear regression are valid only for normally distributed outcomes is simply incorrect; in large samples, these methods are valid for almost any distribution (Lumley, Diehr, Emerson, & Chen, 2002). More recent work reaches the same conclusion, noting that Gaussian models are remarkably robust to non-normality and that switching to a more complex model to fix a normality violation can introduce worse problems than it solves (Knief & Forstmeier, 2021). The common working threshold, that samples above roughly 30 per group are large enough for this robustness to hold, is itself a convention rather than a precise cutoff, and it weakens when distributions are severely skewed and samples are small.
The corollary is that in small samples, robustness arguments are weaker, and violations matter more, so non-parametric alternatives or bootstrapping become more attractive. The judgment, then, is not "was the assumption violated?" but "given my sample size and the severity of the violation, does it threaten my inference?" Being able to answer that question, with a source, is what a strong defense sounds like. Our dissertation defense preparation guide helps candidates rehearse exactly this kind of reasoning.
Table 2: Does the Violation Actually Matter?
Situation | How Much It Matters | Why |
|---|---|---|
Non-normality, large sample | Usually minor | Central limit theorem; t/ANOVA/regression are robust (Lumley et al., 2002) |
Non-normality, small sample | Matters more | Robustness is weaker; consider non-parametric or bootstrap |
Significant Shapiro-Wilk, large sample | Often ignorable | The test over-detects trivial deviations (Ghasemi & Zahediasl, 2012) |
Unequal variances | Matters; fixable | Use Welch, justified for your design |
Violated independence | Matters a lot | Biases standard errors; needs a model that accounts for the structure |
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General Remedies When an Assumption Fails
When a violation is severe enough to matter, you have a small set of standard, defensible responses, and naming the right one is part of a rigorous chapter. Transformations of the outcome, such as the log, square-root, and inverse transformations formalized in the Box-Cox family (Box & Cox, 1964), can simultaneously improve normality and stabilize variance. Non-parametric alternatives, such as the Mann-Whitney U test in place of the t-test or the Kruskal-Wallis test in place of one-way ANOVA, make fewer distributional assumptions, though they do not automatically offer more protection than the parametric test in large samples. Robust methods, including heteroscedasticity-consistent standard errors and robust estimators, handle specific violations without discarding the analysis. And bootstrapping produces distribution-free confidence intervals by resampling your own data. The right choice depends on which assumption failed and how badly, which is a judgment our broader dissertation support is built to make with you rather than for you.
How to Report Assumption Checks
Reporting is where the credit is earned, because an assumption you checked but did not report is invisible to your examiner. The APA Journal Article Reporting Standards for quantitative research, JARS-Quant, are explicit on this. You should report any problems with statistical assumptions or data distributions that could affect the validity of your findings. You should also describe your planned data diagnostics, including the handling of outliers and any transformations used (American Psychological Association, JARS-Quant). In practice, this means a short, structured account in your methods or results: state which assumptions each test requires, how you checked them, what you found, and what you did about any violation. Lead with the graphical evidence, the Q-Q plots and residual plots, rather than a bare significance test, and where you relied on robustness rather than remedying a violation, say so and cite the basis for it. Reporting the checks that passed as well as those that failed signals thoroughness, and it turns a question about normality or variance in the viva into one you have already answered on the page. This reporting connects directly to the rest of your results chapter, which our dissertation chapter support is built to make coherent from methods through findings.
The Sophisticated Stance in One Sentence
If there is a single principle to carry into your analysis and your defense, it is the one the afex developers state directly: an appropriate exploratory data analysis is often better than blindly applying statistical assumption tests. Check your assumptions with graphics and judgment, not just p-values. Report what you checked and what you found. Remedy the violations that matter, and where a violation does not threaten your inference, say why, with a source. That posture, informed rather than mechanical, is what examiners are actually looking for when they ask about your assumptions.
Frequently Asked Questions
What statistical assumptions do examiners ask about most?
The six most common are normality, homogeneity of variance, linearity and homoscedasticity, multicollinearity, independence of residuals, and sphericity in repeated-measures designs. Each parametric test rests on a specific subset of these. Examiners ask about them because assumption checks are invisible in the final results and are therefore where analytical corners are most often cut, so demonstrating that you checked and reported them is one of the clearest signals of a careful analysis.
Do I have to abandon my t-test if the Shapiro-Wilk test is significant?
Not necessarily. The Shapiro-Wilk p-value is sample-size dependent: in large samples, it flags trivial departures from normality that do not affect the result, and t-tests, ANOVA, and regression are robust to non-normality in large samples through the central limit theorem. Check normality with Q-Q plots and skewness and kurtosis, as well as the formal test, and judge whether the violation is severe enough to matter, given your sample size, rather than reacting to the p-value alone.
How do I check the normality assumption?
Use several methods together rather than one. Inspect a Q-Q plot and a histogram for the shape of the distribution, examine skewness and kurtosis (both near zero for a normal distribution), and run the Shapiro-Wilk test as the most powerful formal test. Because the formal test is sample-size dependent, the graphical methods and summary indices should carry weight in your judgment, especially in large samples where the test over-detects and in small samples where it lacks power.
What is multicollinearity, and what does VIF mean?
Multicollinearity occurs when predictors in a regression are highly correlated with each other, which inflates and destabilizes their coefficients. It is measured with the variance inflation factor. A VIF above 10, or sometimes above 5, is the common flag, but these are rules of thumb, not laws. O'Brien (2007) cautions that reacting mechanically to a high VIF, by dropping variables, can cause worse problems than the collinearity, especially when the correlated terms are legitimate parts of the model. Report the VIF and interpret it in context.
What do I do when the sphericity assumption is violated?
When Mauchly's test indicates a sphericity violation in repeated-measures ANOVA, correct the degrees of freedom rather than abandoning the analysis: use the Greenhouse-Geisser correction, typically when the epsilon estimate is below 0.75, or the Huynh-Feldt correction when it is higher. Because Mauchly's test is unreliable in both small and large samples, many methodologists recommend applying the Greenhouse-Geisser correction by default, or using a mixed-effects model that does not require sphericity at all.
How should I report assumption checks in my dissertation?
Report every assumption you checked, including those that passed, in a short, structured account in your methods or results. State which assumptions each test requires, how you checked them, what you found, and what you did about any violation. The APA JARS-Quant standards require reporting any problems with assumptions that could affect validity, along with your data diagnostics and any transformations. Lead with graphical evidence over single p-values, and where you relied on a test's robustness, say so and cite the basis.
When does an assumption violation actually matter?
It depends on the assumption, the severity of the violation, and your sample size. For normality, large samples make t-tests, ANOVA, and regression robust, so violations matter less, as Lumley and colleagues established. In small samples or with severe violations, the same violation matters more, and non-parametric or bootstrap methods become preferable. The right question is not whether an assumption was violated but whether, given your data, the violation threatens your inference, and you should be able to answer that with a source.
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Checking With Judgment, Not Reflex
The assumption checks that satisfy an examiner are not the ones that mechanically run every test and report every p-value. They are the ones that identify which assumptions each analysis requires, check them with graphics and judgment as well as formal tests, report what was found, remedy the violations that matter, and explain, with a source, why the ones left unremedied do not threaten the conclusions. Do that, and the assumptions section stops being the soft spot in your results chapter and becomes evidence that you understand your own analysis well enough to defend it.
If you would rather a PhD statistician run every check, apply the right remedy, and document each decision so you can defend it, send us your data and research questions. You will have an itemized quote within 2 to 4 business hours, with no obligation.
