User:Al83tito/Stats

MGTECON 603: Econometrics (Part II)

Hypothesis Testing

Introduction

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Truth/Decision	H0	H1
H0	×	Type I error
H1	Type II error	×

Critical Value and Test Statistic

Type I/II Errors; Power Function

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)

– P
_θ(type 2 error)=1−β(θ),∀θ∈Θ
₁.

(i.e. Given H
₁ is true the probability of accepting H
₀)

Extended content

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Test Statistics

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Hypothesis Testing

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