ECON 61001: Hypothesis Testing: Power
Alastair R. Hall
The University of Manchester
Alastair R. Hall ECON 61001: Hypothesis Testing: Power 1 / 11
Hypothesis testing
A statistical hypothesis is a conjecture about the distribution of one or more random variables.
The classical theory of hypothesis testing provides a framework for deciding whether a particular hypothesis is correct.
Within this framework, there are only two possible decisions: the hypothesis is true or it is not. A decision procedure for such a problem is called a test.
Alastair R. Hall ECON 61001: Hypothesis Testing: Power 2 / 11
Basic framework
Assume: our hypothesis involves θ, the parameter vector indexing distribution of V, and Θ denote the parameter space with Θ ⊂ Rp.
Divide Θ into two mutually exclusive and exhaustive parts:
Θ0 = {θ : such that the hypothesis is true},
Θ1 = {θ : such that the hypothesis is false}.
Using this partition, we can state the object as being to test the
null hypothesis,
against the alternative hypothesis,
H0 : θ∈Θ0 H1 : θ ∈ Θ1.
Alastair R. Hall ECON 61001: Hypothesis Testing: Power 3 / 11
Basic framework
Base inference on some test statistic; denoted by ST . Decision rule:
ST ∈ R0 ⇒ H0 is accepted or rather not rejected ST ∈ R1 ⇒ H0 is rejected in favour of H1
In the companion podcast discussed how R0 and R1 are chosen to control the probability of a Type I error.
Now consider the properties of the test under H1.
Alastair R. Hall ECON 61001: Hypothesis Testing: Power 4 / 11
Power of a test
Let Pθ ( · ) denote the probability of the event in parentheses if the parameter vector takes the value θ.
Define β(θ) = Pθ(R0) = 1 − Pθ(R1) that is, β(θ) describes the probability of a type II error for values of θ that satisfy H1.
The power function of the test is:
π( · ) = 1 − β( · ).
→ for θ∗ ∈ Θ1, π(θ∗) is:
the probability of correctly rejecting H0 when θ = θ∗.
the power of the test against the alternative θ = θ∗.
Alastair R. Hall ECON 61001: Hypothesis Testing: Power 5 / 11
Example 2.8 in Lecture Notes
Suppose that:
vt ∼ IN(θ,σ2), t = 1,2,…T; assume σ2 known. wish to test H0 : θ = 0 versus H1 : θ ̸= 0.
Decisionrule: rejectH0 : θ=0infavourofH1 : θ̸=0atthe 5% significance level if |τT | > 1.96.
So power of the test is given by
π(θ) = P(|τT| > 1.96|θ,θ ∈ Θ1).
Alastair R. Hall ECON 61001: Hypothesis Testing: Power 6 / 11
Example 2.8 in Lecture Notes
To evaluate π(θ) we need the distribution of τT if θ ̸= 0. We have
v ̄T v ̄T−θ θ
τT = σ2/T = σ2/T + σ2/T ∼ N(μ,1),
where μ = θ/(σ2/T).
Power is a function of μ.
Next slide shows power for θ = σ/√T, i.e. μ = 1.
Alastair R. Hall ECON 61001: Hypothesis Testing: Power 7 / 11
Example 2.8 in Lecture Notes: power of test when θ = σ/√T
Alastair R. Hall ECON 61001: Hypothesis Testing: Power 8 / 11
Example 2.8 in Lecture Notes: power curve
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3
π (θ )
θ
Alastair R. Hall
ECON 61001: Hypothesis Testing: Power
9 / 11
Unbiased tests
From plot of power function for this two-sided test we see that: power depends on |θ| in this case
Pθ(R1)>αforallθ∈Θ1 →testissaidtobeunbiased.
Alastair R. Hall ECON 61001: Hypothesis Testing: Power 10 / 11
Further reading
Further discussion of the topics in this podcast can be found in Section 2.8.1 of the Lecture Notes.
Alastair R. Hall ECON 61001: Hypothesis Testing: Power 11 / 11