User:Al83tito/Stats
MGTECON 603: Econometrics (Part II)
Hypothesis Testing
[edit]Introduction
[edit]
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| Truth/Decision | H0 | H1 |
|---|---|---|
| H0 | × | Type I error |
| H1 | Type II error | × |
Critical Value and Test Statistic
[edit]Type I/II Errors; Power Function
[edit]Type I error: the rejection of a true null hypothesis
Type II error: retaining a null hypothesis when it’s false.
Remark: fix the H0 and H1 first, then discuss what the type I/II errors are. i.e. the errors are relative.
- Whether a small significant level is desired: If the consequences of a type I error are serious or expensive, then a very small significance level is appropriate; e.g. Two drugs are being compared for effectiveness in treating the same condition. Drug 1 is very affordable, but Drug 2 is extremely expensive. The null hypothesis is ”both drugs are equally effective,” and the alternate is ”Drug 2 is more effective than Drug 1.” In this situation, a Type I error would be deciding that Drug 2 is more effective, when in fact it is no better than Drug 1, but would cost the patient much more money. That would be undesirable from the patient’s perspective, so a small significance level is warranted (reference: “common statistical errors”, Martha Smith (UT Austin)).
- The trade-off between two types of errors:
- The general approach (in the textbook but not taught here) is in “statistical decision theory” (S.8.3.5).
- Neyman approach: Choose a critical value s.t. P
θ(type I errors) is “small”, ∀θ ∈ Θ
0. - typically “at most 5%”: truncating data - e.g. 6% is thrown away.
- “P-hacking”: Once take a rule, we need to prolong the rule regardless the results - for example, we cannot push the p-value slowly by changing experiment parameters. (more on “prolonging the rule”: medical trials?)
Power Function: The power function of a test with a critical region C is the function β :
- Θ → [0, 1] given by β(θ) = P
θ((X
1, ..., X
n)′ ∈ C) = P
θ(reject H
0), θ ∈ Θ. - – P
θ(type 1 error) = β(θ),∀θ ∈ Θ
0.- (i.e. Given H
0 is true the probability of rejecting it)
- (i.e. Given H
- – P
θ(type 2 error)=1−β(θ),∀θ∈Θ
1.- (i.e. Given H
1 is true the probability of accepting H
0)
- (i.e. Given H
- Θ → [0, 1] given by β(θ) = P
Extended content
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Test Statistics
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Hypothesis Testing
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