Is your data normal enough for the test?

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What exactly has to be normal?

For a t-test or ANOVA, the values within each group should be roughly normal. For regression, the residuals should be. The outcome column with all groups pooled together does not have to be normal.

“Roughly normal” matters most for small samples. The t-test compares means, and the mean of many patients follows a nearly normal distribution even when individual values do not. This is the central limit theorem, explained below. So the question is not only “are these data normal?” but “are they far enough from normal, for this sample size, to change the p-value and CI?”

Checking by eye

How do you check the shape by looking?

A histogram shows the shape, a box plot shows the spread and outliers, and a Q-Q plot compares your values with those a normal sample of the same size would have.

Interactive

Skew the population and watch the Q-Q plot and the Shapiro-Wilk p-value at 20 and 500 patients

The same population is sampled twice. Only the number of patients changes.

Normality tests

What do the Shapiro-Wilk and Kolmogorov-Smirnov tests actually test?

Both test the null hypothesis that the data come from a normal distribution. A small p-value says the data are unlikely to be exactly normal. A large p-value does not prove that they are normal.

The Shapiro-Wilk test is based on the correlation between the ordered data and the corresponding normal scores, the same comparison a Q-Q plot shows. W close to 1 means the points lie close to a straight line. The Kolmogorov-Smirnov test measures the largest vertical gap between the cumulative proportion of your data and the cumulative normal curve. When the mean and SD are estimated from the same data, as they almost always are, the plain Kolmogorov-Smirnov p-value is not correct and the Lilliefors correction is needed.

This is why normality tests are least useful exactly where you would want them. For small samples they have little power and most samples pass. For large samples even a small deviation from normality gives a significant result, although it will not affect a parametric test (Ghasemi and Zahediasl, 2012). Look at the plots, and use the test as supporting information.

Running a normality test and then choosing between a t-test and a Mann-Whitney test on the same data changes the error rates of the second test. Rochon, Gondan and Kieser (2012) found that this two-stage procedure seriously altered the conditional type I error rates, although the overall error rate and power stayed within acceptable limits in their examples. Planning the analysis from the type of outcome and earlier data avoids the problem.

The central limit theorem

Why does a t-test still work with skewed data in large samples?

Because it compares means, and means of many patients are close to normally distributed even when individual values are skewed.

Lumley and colleagues (2002) put it directly: the t-test and linear regression are valid even in very small samples if the outcome is normal, and in large samples they are valid for any distribution. Some guides give 30 or 40 patients as the size above which non-normality should not cause major problems. The number really needed depends on the shape: in the figure the mean of 30 still has some skew.

What the central limit theorem does not do

• It does not make individual values normal. A reference range or a prediction for one patient still needs the real distribution.
• It does not help much in small samples, which is where most thesis groups are.
• One extreme value still pulls the mean and inflates the SD.
• It does not make the mean the right summary. With very skewed data, a median or a ratio may answer the clinical question better.

Transformations

What does taking the log do to skewed data?

It turns multiplication into addition. Values that double in steps become equally spaced, so a long right tail is pulled in towards the rest.

Choosing a transformation

The Box-Cox method (Box and Cox, 1964) chooses a power from the data, and includes the log, square root and reciprocal as special cases. Whatever you choose, ICH E9 asks for transformations of key variables to be specified in the protocol with a rationale.

Back-transformation: what the numbers mean on the original scale

• The mean of log values, back-transformed (e^{mean of ln x}), is the geometric mean, not the arithmetic mean.
• A difference between two means of log values, back-transformed, is the ratio of the geometric means. Its CI back-transforms to a CI for that ratio.
• The SD of log values cannot be back-transformed into an SD in mg/L. Report the geometric mean with its CI, or the median and IQR.

Interactive

Choose a transformation and watch the residuals, the t-test and the effect estimate change

The same 20 CRP values from the worked example below, analysed on four scales.

Outliers

What should you do with an outlier?

First find out why it is extreme. Correct data-entry and measurement errors. Keep genuine values, and show how much they change the result.

1. Look for extreme values on box plots, Q-Q plots and residual plots.
2. Check each one against the case record. Correct errors, such as a haemoglobin of 112 entered instead of 11.2.
3. Do not delete a genuine value because it is inconvenient. It may be the patient who matters most.
4. Run the analysis with the actual values and again without them (or with a method less affected by them), and report both.

In a normal distribution only 0.27% of values lie more than 3 SD from the mean, so a standardised residual far beyond ±3 deserves a check. In a box plot, points beyond 1.5 × IQR from the quartiles are drawn separately.

Regression

What has to be checked after fitting a regression?

The residuals: each observed value minus the value the model predicts. If the model is right, residuals scatter evenly around zero with no pattern.

When the spread is not constant, the least-squares coefficients are still unbiased, but their ordinary standard errors, CIs and p-values are not correct. Heteroscedasticity-consistent (“sandwich”) standard errors correct them without changing the coefficients.

Interactive

Add spread in older patients or a curve, and watch the residual plot

The same 60 made-up patients each time. The fitted line changes little, but the residual plot shows what is wrong.

Decide

A check has failed: what do you do now?

Answer two questions. Each answer links to the section that explains it.

Comparison

Log transformation or a rank test: which answers your question?

Both deal with right-skewed data, but they estimate different things.

The formula

Skewness, back-transformation and a worked example

Teaching example with made-up data: CRP (mg/L) on day 3 in two groups of 10 patients.

g₁ = [Σ(x − x̄)³ ÷ n] ÷ [Σ(x − x̄)² ÷ n]^1.5

g₁

moment coefficient of skewness: 0 for a symmetric shape, positive for a long right tail

G₁

the small-sample adjusted version some software reports: G₁ = g₁ × √[n(n − 1)] ÷ (n − 2)

geometric mean = e^{mean of ln x} · ratio of geometric means = e^{(mean ln x₁ − mean ln x₂)}

Group A: 12, 18, 25, 31, 40, 52, 66, 90, 140, 230. Group B: 6, 9, 11, 15, 19, 24, 30, 38, 55, 95.

1. Original scale. Means 70.4 and 30.2 mg/L. Medians 46.0 and 21.5. SDs 68.1 and 27.2. The mean is well above the median and the SD is close to the mean: right skew. g₁ = 1.44 and 1.46 (G₁ = 1.71 and 1.73). Shapiro-Wilk p = 0.019 and 0.026.
2. Natural log. Means of ln CRP 3.874 and 3.082, SDs 0.921 and 0.851. g₁ = 0.20 and 0.19. Shapiro-Wilk p = 0.99 and 0.997.
3. t-test on the log values. Difference 0.792, t = 2.00, df 18, p = 0.061. 95% CI for the difference −0.041 to 1.625.
4. Back-transform. Geometric means e^3.874 = 48.1 and e^3.082 = 21.8 mg/L. Ratio e^0.792 = 2.21, 95% CI e^−0.041 = 0.96 to e^1.625 = 5.08.
5. For comparison. t-test on the original values: difference 40.2 mg/L (95% CI −8.5 to 88.9), p = 0.100. Mann-Whitney U = 74, exact p = 0.075.

ratio of geometric means = e^0.792 = 2.21 → 95% CI 0.96 to 5.08

A mean ± 2 SD “normal range” on the original scale for group A would run from 70.4 − 2 × 68.1 = −65.8 mg/L, an impossible value. That is the practical sign that the mean and SD do not describe these data.

In a real paper

Checking normality and log-transforming in a randomised trial

How to run it

How to check assumptions in SPSS, R and jamovi

Every step below was checked against the official documentation. Steps that could not be checked are left out.

SPSS Statistics

1. Analyze > Descriptive Statistics > Explore. Select the outcome as the dependent variable and the group as the factor variable.
2. In the Display group, select Both or Plots, then click Plots and select Normality plots with tests. The output gives normal probability and detrended normal probability plots, the Kolmogorov-Smirnov statistic with a Lilliefors significance level, and the Shapiro-Wilk statistic (for 3 to 5,000 cases with no weights or integer weights).
3. For regression, in the Linear Regression dialog box click Plots. Plot the standardised residuals (*ZRESID) on the vertical axis against the standardised predicted values (*ZPRED) on the horizontal axis, and request the histogram and normal probability plot of standardised residuals.

R

# Shape of each group: Q-Q plot and Shapiro-Wilk test
a <- crp[group == "A"]
qqnorm(a); qqline(a)            # qqline passes through the 1st and 3rd quartiles
shapiro.test(a)                 # needs 3 to 5000 non-missing values

# t-test on log values, then back-transform
fit <- t.test(log(crp) ~ group, var.equal = TRUE)   # default var.equal = FALSE gives Welch's test
exp(fit$conf.int)               # 95% CI for the ratio of geometric means

# Regression diagnostics
m <- lm(sbp ~ age)
plot(m)                         # residuals vs fitted, Q-Q, scale-location, residuals vs leverage
cooks.distance(m)
lmtest::bptest(m)               # Breusch-Pagan, Koenker's studentized version by default
lmtest::coeftest(m, vcov. = sandwich::vcovHC(m))   # vcovHC default type is "HC3"
MASS::boxcox(m)                 # profile likelihood for the Box-Cox power

jamovi

The options below are documented for jmv, the R package of jamovi's own analyses, under the same analysis names.

• Descriptives: Shapiro-Wilk p-value, skewness, kurtosis, histograms, box plots and Q-Q plots.
• Independent Samples T-Test: Shapiro-Wilk tests of normality, Q-Q plots of residuals, Levene's test for homogeneity of variances, and Welch's and Mann-Whitney U tests.
• Linear Regression: Shapiro-Wilk test on the residuals, Q-Q plot of residuals, residual plots, the Durbin-Watson test for autocorrelation and summary statistics for Cook's distance.

How to report it

What to write in the methods and results

Say how you checked the assumptions, what you did about them, and give results on the scale you analysed plus the back-transformed values.

Methods:

“The distribution of CRP in each group was examined with histograms and normal Q-Q plots. CRP was right skewed and was analysed on the natural log scale, as planned in the protocol. Results were back-transformed and are presented as geometric means and their ratio with 95% confidence intervals.”

Results:

“The geometric mean CRP on day 3 was 48.1 mg/L in group A and 21.8 mg/L in group B, a ratio of 2.21 (95% CI 0.96 to 5.08; t-test on log values, p = 0.061).”

The SAMPL guidelines (Lang and Altman) ask authors to identify procedures that modify raw data before analysis, such as transforming measurements to make distributions closer to normal, to confirm that the assumptions of the test or analysis were met (for linear regression, whether an analysis of residuals confirmed linearity), and to report how any outlying values were treated.

By specialty

Assumption problems by specialty

Common myths

Common mistaken beliefs

• ✗Myth: Shapiro-Wilk p > 0.05 proves the data are normal.

  ✓It means the test did not detect non-normality. In the simulation above, with 10 patients it missed strongly skewed data in 71% of samples.

• ✗Myth: A significant normality test in a large sample rules out a t-test.

  ✓With 500 patients the test flagged mild skew in 98% of samples. At that size the t-test on means is still valid (Lumley et al., 2002). Look at the plots.

• ✗Myth: The whole outcome column must be normal.

  ✓The values within each group, or the residuals of a regression, are what matter. Two normal groups with different means make a two-humped column.

• ✗Myth: Non-parametric tests have no assumptions.

  ✓They still need independent observations, and they answer a question about ranks, not means.

• ✗Myth: Outliers should be deleted.

  ✓Only proven errors should be corrected or removed. ICH E9 asks for an analysis with the actual values and one that reduces the outlier effect, with the differences discussed.

• ✗Myth: A log transformation always fixes skew.

  ✓It needs positive values and suits right skew. In the worked example the reciprocal left the residual skewness at 1.16, with the tail now coming from the smallest values.

• ✗Myth: The central limit theorem makes the data normal.

  ✓It concerns the distribution of the mean. Individual values, reference ranges and small samples are not covered by it.

Why this matters to you

Five consequences of getting assumption checks wrong

1. Protocol sent back

   ICH E9 expects transformations of key variables to be specified in the protocol with a rationale. "Normality will be checked" is not a plan.

2. A different test for every variable

   Choosing the test from the p-value of a normality test on pooled data gives an analysis that changes with sample size, not with the clinical question.

3. Questions you cannot answer at the viva

   "Why Mann-Whitney?" needs an answer about the shape of each group, the sample size and what the result estimates.

4. Misreading papers

   A geometric mean ratio of 0.76 means 24% lower. Reading it as a difference of 0.76 units is a common error.

5. Harm to patients

   A mean ± 2 SD range on skewed data can give an impossible negative lower limit, as in the worked example (−65.8 mg/L), and misclassify patients.

History

History of checking assumptions

1. 1964

   George Box and David Cox

   Published "An analysis of transformations", the family of power transformations that includes the log, square root and reciprocal as special cases.

2. 1965

   Samuel Shapiro and Martin Wilk

   Published "An analysis of variance test for normality (complete samples)" in Biometrika: the Shapiro-Wilk test.

3. 1967

   Hubert Lilliefors

   Published the Kolmogorov-Smirnov test for normality with the mean and variance unknown, that is, estimated from the sample.

4. 1968

   Martin Wilk and Ram Gnanadesikan

   Published "Probability plotting methods for the analysis of data" in Biometrika, still cited as a reference for Q-Q plots of residuals.

5. 1995

   Patrick Royston

   Published a remark on the algorithm for the W-test (AS R94). The documentation of R's shapiro.test(), which accepts 3 to 5000 values, refers to it for the accuracy of its approximate p-value.

6. 2002

   Thomas Lumley and colleagues

   Showed by simulation in extremely non-normal data that the t-test and linear regression are valid for any distribution in large samples.

7. 2012

   Justine Rochon, Matthias Gondan and Meinhard Kieser

   Showed that a preliminary Shapiro-Wilk test used to choose between the t-test and the Mann-Whitney test alters the conditional type I error rates of the main test.

Guidelines

Assumptions, transformations and outliers in guidelines

Reading a paper

Five checks on how a paper handled assumptions

• Was normality judged within each group, or on the residuals, and not on the pooled outcome column?
• Were plots used, or only a normality test whose result depends heavily on the sample size?
• If the data were transformed, were results back-transformed and described correctly (geometric means, ratios)?
• For regression, do the authors say how the residuals were checked for linearity, constant spread and outliers?
• Were outliers handled as stated in advance, with results given with and without them?

Why is it called that?

Word surgery

Exam answers

Model answer outlines

Past papers

TNMGRMU questions on checking assumptions

Questions from MD, MS, DM, MCh and Ph.D. papers that this chapter helps you answer. Tap a sitting to open the university's own question paper.

1. ×1Assumptions in ANOVAMD Community Medicine · Oct 2013
2. ×1Distinguish between Parametric and Non Parametric Tests. What are the assumptions of the Parametric Tests?Ph.D. RM · Jul 2013

Further reading

Related Stateazy articles