Type I, Type II, and Power
Introduced| English | Meaning |
|---|---|
| Type I error/taɪp aɪ ˈerə/ | Rejecting a null hypothesis when it is true. |
| Type II error/taɪp ˈtuː ˈerə/ | Failing to reject a null hypothesis when it is false. |
| power/ˈpaʊə/ | The probability that a statistical test rejects the null hypothesis when a specified alternative is true. |
An audit can wrongly flag a reliable seller
- Suppose $H_0$ says the on-time proportion is 0.90 and $H_a$ says it is lower. Rejecting when the null is actually true is a Type I error 第一类错误.
- This is a false alarm about the population rate. A test decision can be wrong even when its calculations were carried out correctly.
An audit can miss a real shortfall
- A Type II error 第二类错误 occurs when the null is false but the study fails to reject it. For example, a true rate of 0.85 might go undetected.
- Non-rejection reports insufficient evidence; it is not a declaration that no shortfall exists. The missed-detection probability depends on the actual alternative rate.
A Type I error is...
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Type I = false alarm = reject a true null.
Treat the level as a conditional error target
- Under the null, a correctly calibrated test controls its rejection rate at the chosen $\alpha$. Normal-approximation procedures have approximately that rate.
- The rate is over repeated studies when the null is true. It is not the probability that the null is true or the probability of error after a particular rejection.
A Type II error is...
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Type II = missed detection = fail to reject a false null.
H0 says the delivery proportion is 0.90. Match the outcome to its meaning.
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Classify by the true state and the decision, not by whether the sample percentage looks large.
Power 功效 is linked to a specified effect
- Power is the probability of rejecting under a specified false-null value: $1-\beta$. Different true rates can give different power.
- For a fixed design and effect, larger samples generally improve power. A larger effect is usually easier to detect.
If β = 0.30 at p = 0.85, power is 0.70 at p = 0.85. This is not a power claim for every possible alternative.
For a test calibrated at significance level α, α is the nominal Type I error level.
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α is the chosen false-rejection level under H0. Approximate and discrete procedures need not have actual error probability exactly α.
The power of a test equals 1 minus ___ (the Greek letter for the Type II error rate).
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Power = 1 − β.
At a specified true proportion, the Type II error probability is 0.30. What is the power?
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Power = 1 - β = 1 - 0.30 = 0.70 for that particular alternative.
Changing the threshold changes the trade-off
- With the sample size and true effect fixed, lowering $\alpha$ makes rejection harder and typically lowers power.
- Increasing sample size can reduce missed detections while retaining the same nominal Type I level. It does not automatically lower that chosen level.
At fixed nominal α, a larger sample does not automatically reduce the chosen Type I error level.
Which increase the power of a test?
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Bigger n, bigger effect, or larger α all raise power.
Increasing n automatically lowers the nominal Type I error level when α stays at 0.05.
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The chosen nominal level remains 0.05. Larger n usually improves power for a specified effect under the same valid model; approximate actual error rates may vary.
Identify errors using truth and decision
- Check two things: whether the null is true and whether the test rejects it. Their combination identifies an error or a correct decision.
- Without knowing the true population rate, a study cannot label its own rejection a realised Type I error solely from the observed p-value.
Check two things: whether the null is true and whether the test rejects it. Their combination identifies an error or a correct decision.
Write from a power of 0.8
- The probability of missing a real shortfall is 0.2.
- That miss is a Type II error. The lesson power context uses 0.7.
What is the power?
Power is the chance of correctly rejecting a false null.
What is the power if beta is 0.2?
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1 minus 0.2 is 0.8. Other options — 0.2 (which is beta); 0.7 (the other study).