The appropriateness of a statistical test, which depends on underlying distributional assumptions, is generally not a problem if the population distribution is known in advance. If the assumption of normality is known to be wrong, a nonparametric test may be used that does not require normally distributed data. Difficulties arise if the population distribution is unknown—which, unfortunately, is the most common scenario in medical research. Many statistical textbooks and articles state that assumptions should be checked before conducting statistical tests, and that tests should be chosen depending on whether the assumptions are met (e.g., [22, 28, 47, 48]). Various options for testing assumptions are easily available and sometimes even automatically generated within the standard output of statistical software (e.g., see SAS or SPSS for the assumption of variance homogeneity for the *t* test; for a discussion see [42–45]). Similarly, methodological guidelines for clinical trials generally recommend checking for conditions underlying statistical methods. According to ICH E3, for example, when presenting the results of a statistical analysis, researchers should demonstrate that the data satisfied the crucial underlying assumptions of the statistical test used [49]. Although it is well-known that decision-making after inspection of sample data can lead to altered Type I and Type II error probabilities and sometimes to spurious rejection of the null hypothesis, researchers are often confused or unaware of the potential shortcomings of such two-stage procedures.

### Conditional Type I error rates

We demonstrated the dramatic effects of preliminary testing for normality on the conditional Type I error rate of the main test (see Tables 1 and 2, and Figures 1 and 2). Most of these consequences were qualitatively similar for Strategy I (separate preliminary test for each sample) and Strategy II (preliminary test based on residuals), but quantitatively more pronounced for Strategy II than for Strategy I. On the one hand, the results replicated those found for the one-sample *t* test [41]. On the other hand, our study revealed interesting new findings: Preliminary testing not only affects the Type I error of the *t* test on samples from non-normal distributions but also the performance of Mann-Whitney’s *U* test for equally sized samples from uniform and normal distributions. Since we focused on a two-stage procedure assuming homogenous variances, it can be expected that an additional test for homogeneity of variances should lead to a further distortion of the conditional Type I error rates (e.g., [39, 42–45]).

Detailed discussion on potential reasons for the detrimental effects of preliminary tests is provided elsewhere [30, 41, 50]; therefore, only a global argument is given here: Exponentially distributed variables follow an exponential distribution, and uniformly distributed variables follow a uniform distribution. This trivial statement holds, regardless of whether a preliminary test for normality is applied to the data or not. A sample or a pair of samples is not normally distributed just because the result of the Shapiro-Wilk test suggests it. From a formal perspective, a sample is a set of fixed ‘realizations’; it is not a random variable which could be said to follow some distribution. The preliminary test cannot alter this basic fact; it can only select samples which *appear* to be drawn from a normal distribution. If, however, the underlying population is exponential, the preliminary test selects samples that are not representative of the underlying population. Of course, the Type I error rates of hypotheses tests are strongly altered if they are based on unrepresentative samples. Similarly, if the underlying distribution is normal, the pretest will filter out samples that do not appear normal with probability *α* _{pre}. These latter samples are again not representative for the underlying population, so that the Type I error of the subsequent nonparametric test will be equally affected.

In general, the problem is that the distribution of the test statistic of the test of interest depends on the outcome of the pretest. More precisely, errors occurring at the preliminary stage change the distribution of the test statistic at the second stage [38]. As can be seen in Tables 1 and 2, the distortion of the Type I error observed for Strategy I and II is based on at least two different mechanisms. The first mechanism is related to the power of the Shapiro-Wilk test: For the exponential distribution, Strategy I considerably affects the *t* test, but Strategy II does so even more. As both tables show, distortion of the Type I error, if present, is most pronounced in large samples. In line with this result, Strategy II alters the conditional Type I error to a greater extent than Strategy I, probably because in Strategy II, the pretest is applied to the collapsed set of residuals, that is, the pretest is based on a sample twice the size of that used in Strategy I.

To illustrate the second mechanism, asymmetry, we consider the interesting special case of Strategy I applied to samples from uniform distribution. In Strategy I, Mann-Whitney’s *U* test was chosen if the pretest for normality failed in at least one sample. Large violations of the nominal significance level of Mann-Whitney’s *U* test were observed for small samples and numerically low significance levels for the pretest (23.3% for *α* _{pre} = .005 and *n* = 10). At *α* _{pre} = .005 and *n* = 10, the Shapiro-Wilk test has low power, so that only samples with extreme properties will be identified. In general, however, samples from the uniform distribution do not have extreme properties, such that, in most cases, only one member of the sample pair will be sufficiently extreme to be detected by the Shapiro-Wilk test. Consequently, pairs of samples are selected by the preliminary test for which one member is extreme and the other member is representative; the main significance test will then indicate that the samples differ indeed. For these pairs of samples, the Shapiro-Wilk test and the Mann–Whitney *U* test essentially yield the same result because they test similar hypotheses. In contrast, in Strategy II, the pretest selected pairs of samples for which the set of residuals (i.e., the two samples shifted over each other) appeared non-normal. This result mostly corresponds to the standard situation in nonparametric statistics, so that the conditional Type I error rate of Mann-Whitney’s *U* test applied to samples from uniform distribution was unaffected by the asymmetry mechanism.

### Type I error and power of the entire two-stage procedure

On the one hand, our study showed that conditional Type I error rates may heavily deviate from the nominal significance level (Tables 1 and 2). On the other hand, direct assessment of the unconditional Type I error rate (Table 3) and power (Table 4) of the two-stage procedure suggests that the two-stage procedure as a whole has acceptable statistical properties. What might be the reason for this discrepancy? To assess the consequences of preliminary tests for the entire two-stage procedure, the power of the pretest needs to be taken into account,

\begin{array}{c}\hfill \text{P}\left(\text{TypeIerror}\right)=\text{P}\left(\text{TypeIerror}\cap \text{Pretest}n.s.\right)+\text{P}\left(\text{TypeIerror}\cap \text{Pretest}sig.\right)\hfill \\ \hfill =\text{P}\left(\text{TypeIerror}|\text{Pretest}n.s.\right)\times \text{P}\left(\text{Pretest}n.s.\right)\hfill \\ \hfill +\text{P}\left(\text{TypeIerror}|\text{Pretest}sig.\right)\times \text{P}\left(\text{Pretest}sig.\right)\text{,}\hfill \end{array}

(2)

with P(Type I error | Pretest *n.s.* ) denoting the conditional Type I error rate of the *t* test (Tables 1 and 2 left), P(Type I error | Pretest *sig.* ) denoting the conditional Type I error rate of the *U* test (Tables 1 and 2 right), and P(Pretest *sig.* ) and P(Pretest *n.s.* ) denoting the power and 1 – power of the pretest for normality. In Strategy I, P(Pretest *sig.* ) corresponds to the probability to reject normality for at least one of the two samples, whereas in Strategy II, it is the probability to reject the assumption of normality of the residuals from both samples.

For the *t* test, unacceptable rates of false decisions due to selection effects of the preliminary Shapiro-Wilk test occur for large samples and numerically high significance levels *α* _{pre} (e.g., left column in Table 2). In these settings, however, the Shapiro-Wilk test detects deviations from normality with nearly 100% power, so that the Student’s *t* test is practically never used. Instead, the nonparametric test is used that seems to protect the Type I error for those samples. This pattern of results holds for both Strategy I and Strategy II. Conversely, it was demonstrated above that Mann-Whitney’s *U* test is biased for normally distributed data if the sample size is low and the preliminary significance level is strict (e.g., *α* _{pre} = .005, right columns of Tables 1 or 2). For samples from normal distribution, however, deviation from normality is only rarely detected at *α* _{pre} = .005, so that the consequences for the overall Type I error of the entire two-stage procedure are again very limited.

A similar argument holds for statistical power: For a given alternative, the overall power of the two-stage procedure corresponds, by construction, to the weighted sum of the conditional power of the *t* test and *U* test. When populations deviate only slightly from normality, the pretest for normality has low power, and the power of the two-stage procedure will tend towards the unconditional power of Student’s *t* test; this fact only does not hold in those rare cases in which the preliminary test indicates non-normality, so that the slightly less powerful Mann–Whitney *U* test is applied. When the populations deviate considerably from normality, the power of the Shapiro-Wilk test is high for both strategies, and the overall power of the two-stage procedure will tend towards the unconditional power of Mann-Whitney’s *U* test.

Finally, it should be emphasized that the conditional Type I error rates shown in Tables 1 and 2 correspond to the rather unlikely scenario in which researchers would continue sampling until the assumptions are met. In contrast, the unconditional Type I error and power of the two-stage procedure are most relevant because in practice, researchers do not continue sampling until they obtain normality. Researchers who do not know in advance whether the underlying population distribution is normal, usually base their decision on the samples obtained. If by chance a sample from a non-normal distribution happens to look normal, the researcher could falsely assume that the normality assumption holds. However, this chance is rather low because of the high power of the Shapiro-Wilk test, particularly for larger sample sizes.