Two Tailed Test

A two tailed test is a statistical hypothesis test in which the alternative hypothesis specifies that the parameter of interest differs from the null value in either direction, without specifying whether the difference is positive or negative. In contrast to a one tailed test, which tests for a difference in only one direction, a two tailed test distributes the significance level equally across both tails of the sampling distribution, rejecting the null hypothesis if the test statistic falls in either the upper or lower extreme. For a significance level of 0.05, each tail contains a rejection region of 0.025. Two tailed tests are the default in most scientific research because they do not assume in advance which direction the effect will take.

The choice between a one tailed and two tailed test has direct consequences for the statistical power and the p value of a study. A two tailed test is more conservative because it requires a larger test statistic to achieve significance at the same alpha level. However, it protects against the risk of missing an effect in the unexpected direction. If a researcher uses a one tailed test assuming that a treatment will be beneficial and the treatment is actually harmful, the harmful effect will not be detected. Two tailed testing at 0.05, or a one sided test at the equivalent 0.025, is the standard in clinical trials, regulatory submissions, and most academic research because it maintains objectivity and avoids the appearance of testing only for desired results. Using a one tailed test without strong a priori justification is widely viewed as a questionable research practice that inflates the apparent significance of results.

In a two tailed test, the null hypothesis states that there is no difference between groups (or that a parameter equals a specified value), and the alternative hypothesis states that there is a difference in either direction. The researcher calculates the test statistic (such as a t statistic or z statistic) from the data and compares it to the critical values at both tails of the sampling distribution. For a symmetric sampling distribution such as the normal or t distribution, the p value is twice the probability of obtaining a test statistic as extreme as the observed value in one tail, reflecting the possibility of extreme values in either direction. For example, with a standard normal distribution and an observed z statistic of 2.10, the one tailed p value would be 0.018, but the two tailed p value is 0.036 (2 times 0.018). A result is statistically significant at the 0.05 level if the two tailed p value is less than 0.05, corresponding to a test statistic more extreme than plus or minus 1.96 standard deviations from the mean in a z test.

Use a two tailed test as the default for virtually all hypothesis tests unless there is a strong, prespecified scientific rationale for predicting the direction of the effect and no interest in detecting an effect in the opposite direction. Superiority trials usually use two tailed tests because it is important to detect both beneficial and harmful effects. Regulators do not strictly require them: ICH E9, the statistical guideline adopted by the FDA and EMA, accepts one sided tests in confirmatory trials but prefers that their significance level be half the conventional two sided level (0.025 rather than 0.05), so the evidence needed in the direction of benefit is the same as for a two sided test at 0.05. Beyond such cases, a one tailed test may be appropriate only where an effect in one direction is theoretically impossible or scientifically meaningless, or where the question itself is one sided, as in a noninferiority trial, and such a decision must be specified and justified in the study protocol before data collection begins.

A common mistake is switching from a two tailed to a one tailed test after seeing the data to achieve a smaller p value, which is a form of p hacking that inflates the Type I error rate. Another error is using a one tailed test without a priori justification simply because it makes results significant that would not be significant under two tailed testing. Researchers sometimes report two tailed p values but compare them to a significance threshold of 0.10 (i.e., treating alpha as 0.05 for each tail), which doubles the overall Type I error rate to 0.10 without stating so: in whichever direction the data fall, it gives the same verdict as a one tailed test at 0.05. Confusing the p value from a two tailed test with that from a one tailed test when interpreting published results can lead to incorrect conclusions about statistical significance. Failing to specify whether the test was one or two tailed in the methods section leaves readers unable to properly evaluate the results.

A randomized trial tests whether a new anti nausea medication affects postoperative nausea scores compared to placebo. The researchers use a two tailed test because they want to detect both potential improvement and potential worsening. The treatment group (n = 50) has a mean nausea score of 3.2 (SD = 1.8) and the placebo group (n = 50) has a mean of 4.1 (SD = 2.0). The two tailed t test yields t(98) = 2.37, p = 0.020. Because 0.020 < 0.05, the null hypothesis is rejected, and the researchers conclude that the medication significantly reduces postoperative nausea. Had they used a one tailed test, the p value would have been 0.010, but the two tailed test was the appropriate and prespecified choice.