One-Sided and Two-Sided Tests

Overview

A two-sided test considers departures from the null in both directions.

A one-sided test considers evidence in one prespecified direction only.

For a mean difference $\Delta$:

• two-sided alternative: $\Delta\ne0$;
• lower one-sided alternative: $\Delta<0$;
• upper one-sided alternative: $\Delta>0$.

Sidedness affects:

• the hypothesis;
• critical value;
• p-value;
• confidence interval;
• power; and
• scientific claim.

It must be chosen before the results are known. Choosing a one-sided test after observing the favourable direction invalidates the stated type I error rate [1].

Why this matters

A one-sided test can turn a result from $p=0.08$ two-sided into $p=0.04$ one-sided when the effect is in the prespecified direction.

This does not make the evidence stronger. It changes the question and moves all type I error into one tail.

The opposite direction may still be clinically important. An antihypertensive intervention could lower or raise blood pressure. A treatment could benefit or harm patients. Therefore, two-sided testing is usually the appropriate default in medical superiority research [1], [2].

Two-sided test

The hypotheses are:

$$H_0:\Delta=0$$ $$H_1:\Delta\ne0$$

With $\alpha=0.05$, approximately 0.025 is allocated to each tail of a symmetric standard-normal reference distribution.

The critical values are approximately −1.96 and 1.96.

A two-sided test can reject for a sufficiently large effect in either direction.

One-sided test

For a beneficial reduction:

$$H_0:\Delta\ge0$$ $$H_1:\Delta<0$$

Operationally, the boundary null value 0 is commonly used to calculate the test statistic.

With lower one-sided $\alpha=0.05$, all rejection probability is placed in the lower tail. The standard-normal critical value is approximately −1.645.

A large effect in the opposite direction does not support the stated one-sided alternative.

Worked example

When may a one-sided test be defensible?

A one-sided test may be considered when all of the following are satisfied:

• only one direction can support the intended claim;
• the opposite direction would not lead to a different favourable claim;
• the direction is specified before data examination;
• the design and protocol justify the decision;
• harms in the opposite direction remain monitored and reported; and
• relevant regulatory or disciplinary standards permit it.

Examples can include:

• some non-inferiority hypotheses;
• quality-control rules where only excess contamination triggers action;
• a mechanistic parameter constrained by design; or
• a prespecified directional scientific question with no symmetric claim.

Even in these settings, a two-sided confidence interval may be preferred for transparent effect reporting.

When is a two-sided test preferable?

Use a two-sided test when:

• either benefit or harm would be scientifically important;
• the direction is uncertain;
• the opposite direction would change clinical conclusions;
• the study is a conventional superiority trial;
• readers expect two-sided inference; or
• no strong prospective justification exists for one-sided testing.

A belief that the intervention cannot be harmful is not sufficient. Unexpected directions are common and clinically relevant.

Relationship between p-values

For a continuous symmetric test statistic and an observed effect in the prespecified direction:

$$p_{\text{two-sided}}\approx2p_{\text{one-sided}}$$

This relationship does not apply universally when:

• tests are discrete;
• distributions are asymmetric;
• exact methods use special definitions;
• the effect is opposite to the prespecified direction; or
• composite hypotheses differ.

If the observed effect is opposite to a lower one-sided alternative, the corresponding one-sided p-value is large, not half the two-sided value.

Effect direction and sidedness

simulated data

Table 1. Interpreting a lower one-sided test

Observed result	Two-sided interpretation	Lower one-sided interpretation
Large negative statistic	Evidence in either-direction test	Evidence for prespecified reduction
Statistic near zero	Little evidence against null	Little evidence for reduction
Large positive statistic	Evidence of increase or harm	Does not reject for reduction

Note: The opposite-direction result must still be reported clinically even when the one-sided primary test cannot reject there.

Confidence intervals and sidedness

A two-sided test at $\alpha=0.05$ corresponds to a two-sided 95% CI under compatible methods.

A one-sided test at $\alpha=0.025$ is commonly paired with a one-sided 97.5% confidence bound or a two-sided 95% CI.

A one-sided test at $\alpha=0.05$ corresponds to a one-sided 95% bound or, under compatible symmetric methods, a two-sided 90% CI. This convention can be unfamiliar in clinical reporting.

The chosen confidence level and sidedness must be explicit.

Non-inferiority

In a non-inferiority trial, the null hypothesis usually states that the new treatment is worse than the comparator by at least a prespecified margin.

For an effect where larger values favour the new treatment and margin $-\Delta_{NI}$:

$$H_0:\theta\le-\Delta_{NI}$$ $$H_1:\theta>-\Delta_{NI}$$

The one-sided test is built into the scientific objective. It is not selected merely to reduce sample size.

Non-inferiority requires:

• a justified margin;
• assay sensitivity;
• suitable comparator evidence;
• careful adherence and analysis populations; and
• reporting of the full confidence interval [3]. General clinical-trial principles likewise require the test direction and analysis strategy to be specified prospectively [4].

Sample size implications

A one-sided test requires a smaller critical value than a two-sided test at the same nominal $\alpha$. It therefore needs fewer observations for the same directional alternative and power.

This efficiency is obtained by giving up the ability to reject in the opposite direction under that procedure.

Sample-size convenience is not a scientific justification for one-sided testing.

Tests with composite null hypotheses MMed extension

A one-sided null such as $H_0:\Delta\ge0$ contains many parameter values.

For regular monotone tests, the boundary value $\Delta=0$ gives the largest rejection probability under the null and is used to calibrate the test.

This is why the calculation often appears to test equality even though the scientific null includes an inequality.

Suggested reporting

Common mistakes

Knowledge check

What is the main difference between one-sided and two-sided tests?

Answer and reasoning

A one-sided test places the rejection region in one prespecified direction. A two-sided test allows rejection for sufficiently extreme effects in either direction.

Why can z = −1.75 give values of 0.080 two-sided and 0.040 one-sided?

Answer and reasoning

The two-sided test counts extreme values in both tails, while the lower one-sided test counts only the prespecified lower tail. The data are unchanged; the hypothesis and error allocation differ.

Can researchers switch to a one-sided test after observing a favourable result?

Answer and reasoning

No. The resulting nominal p-value would omit the data-dependent choice and would not retain the stated type I error rate.

Why is two-sided testing usually preferred in superiority trials?

Answer and reasoning

Benefit and harm can both be clinically important, and unexpected effects can occur. A two-sided procedure represents both directions.

Why is non-inferiority a legitimate one-sided problem?

Answer and reasoning

Its claim is whether the new treatment is not worse than the comparator by more than a prespecified margin. The directional margin and hypothesis are built into the design before data collection.

Key take-home messages

References

1. Bland JM, Altman DG. One and two sided tests of significance. BMJ. 1994;309(6949):248. https://doi.org/10.1136/bmj.309.6949.248
2. Greenland S, Senn SJ, Rothman KJ, Carlin JB, Poole C, Goodman SN, et al. Statistical tests, P values, confidence intervals, and power: a guide to misinterpretations. Eur J Epidemiol. 2016;31(4):337–350. https://doi.org/10.1007/s10654-016-0149-3
3. International Council for Harmonisation of Technical Requirements for Pharmaceuticals for Human Use. Choice of control group and related issues in clinical trials E10. Geneva: ICH; 2000. https://database.ich.org/sites/default/files/E10_Guideline.pdf
4. International Conference on Harmonisation of Technical Requirements for Registration of Pharmaceuticals for Human Use. Statistical principles for clinical trials E9. Geneva: ICH; 1998. https://database.ich.org/sites/default/files/E9_Guideline.pdf