Type I Error and the Significance Level

Why the Significance Level Matters

A hypothesis test turns sample data into a decision: reject the null hypothesis or do not reject it. But a decision rule can sometimes reject \(H_0\) even when \(H_0\) is true. This possibility matters because a test is not judged only by what it decides in one sample; its rule can also be evaluated across repeated samples under specified conditions.

The significance level, written \(\alpha\), describes how often a testing procedure is allowed to make one particular error: rejecting a true null hypothesis. It is chosen as part of the procedure, before the data are used to decide whether to reject. The interpretation depends on holding that procedure fixed, including its sampling plan, statistical model, and rejection rule.

The phrase long-run probability is essential. Imagine repeating the same study many times under the same null condition and applying the same test rule each time. The fraction of those repetitions that reject a true \(H_0\) estimates the Type I error probability. The significance level controls that repeated-use behavior; it does not tell us the probability that a particular rejection is wrong after observing the data.

Decisions and Errors

Whether an error has occurred depends on two things: the test's decision and whether the null hypothesis is in fact true. A test cannot determine the truth directly; the outcomes in the table describe the possibilities that would be known only in principle.

This tutorial focuses on the first error. “Do not reject \(H_0\)” is a description of the decision, not proof that the null is true. Likewise, a rejection is not automatically a Type I error: it is a Type I error only in the situation where the null hypothesis really is true.

Significance Level as a Probability of the Rejection Rule

A test's rejection rule identifies which possible data outcomes lead to rejection. Under \(H_0\), the statistical model assigns probabilities to those outcomes. The Type I error probability is the probability that the data fall in the rejection region when the null condition holds.

For example, if a test rejects for large values of a statistic \(T\), then its Type I error probability at a specified null value is \(P(T\text{ is in the rejection region}\mid H_0\text{ is true})\). If \(H_0\) permits several parameter values, that probability may differ across them. A level-\(\alpha\) test keeps the probability at or below \(\alpha\) throughout the null parameter space. In some tests the maximum equals \(\alpha\); in others it is smaller.

One-Sided and Two-Sided Rejection Regions

The alternative hypothesis determines where the rejection region lies. A one-sided test puts the rejection region in one tail; a two-sided test places it in both tails. The significance level is the total probability assigned to the rejection region when the null model holds. For a symmetric two-sided test, that probability is split between the two tails.

Why a Test's Actual Error Rate Can Be Below Alpha

The outcomes of some tests are discrete: the statistic can take only separated values. In that case, it may be impossible to choose a rejection region whose null probability is exactly \(\alpha\). A standard level-\(\alpha\) procedure can instead use a rejection region whose probability is less than \(\alpha\). The difference between the stated level and the actual probability is not a contradiction; the level is an upper bound.

A Short Justification: Why the Tail Probability Is the Error Rate

The link between a rejection threshold and Type I error follows directly from the definition of the error and the probability model. Let \(R\) denote the event that the fixed rule rejects \(H_0\). When the null is true, a Type I error occurs exactly when \(R\) occurs. Therefore, at a particular null parameter value \(\theta\),

For a level-\(\alpha\) procedure, the design requirement is \(P(R\mid\theta)\le\alpha\) for every \(\theta\) allowed by the null. If the null is simple, it specifies one parameter value, so there is one probability to evaluate. If the null is composite, there can be several such probabilities; controlling their maximum ensures the bound holds throughout \(H_0\). This explains both why the rejection rule must be fixed and why the null model is needed: changing the rule or the model changes the probability of \(R\).

What Alpha Does—and Does Not—Say

The significance level describes a property of a testing procedure under a true null hypothesis. It is not a measure of the size or importance of an observed effect, and it is not a guarantee that a particular conclusion is correct. A 5% level means the procedure is designed so that its long-run Type I error rate is no greater than 5% under the null conditions.

A p-value answers a different question: assuming \(H_0\) and the model, it measures how unusual results at least as incompatible with \(H_0\) as the observed result would be. Comparing a p-value with a chosen \(\alpha\) determines whether the observed result falls in the test's rejection region when the test is built using that rule. Neither quantity is the probability that \(H_0\) is true. In particular, an \(\alpha\) of 0.05 does not mean that a rejection has a 5% chance of being a Type I error.

Common Mistakes

• Wrong: “Alpha is the probability that the null hypothesis is true.” Correction: Alpha is a probability of a rejection decision under the condition that the null is true; it does not assign a probability to the truth of the null.
• Wrong: “A 5% significance level means that 5% of this study's rejection is likely to be wrong.” Correction: The 5% describes long-run behavior over repetitions under the null, not the probability that a particular rejection is erroneous.
• Wrong: “A test at level 0.05 must have an actual Type I error probability of exactly 0.05.” Correction: The probability must be no greater than 0.05; discreteness or conservative procedures can make it smaller.
• Wrong: “A rejection is always a Type I error.” Correction: It is a Type I error only when \(H_0\) is actually true.
• Wrong: “A two-sided 5% test has 5% in each tail.” Correction: For a symmetric test, each tail contains 2.5%, so the total rejection probability under the null is 5%.