Power Analysis for a 2 x 2 Contingency Table
I received this email in September of 2009:
I'm an orthopaedic surgeon and I'm looking for someone that can help me with a
power analysis I'm doing for a study.
We want to know if there is any increased risk of infection in giving steroid injections
before a total hip arthroplasty.
It is generally accepted that the risk of infection after a hip arthroplasty is 1%. A
recent paper has demonstrated an infection rate of 10% (4 of 40) in patients that had a hip
replacement after they had a steroid injection. Another paper cites an infection rate of
1.34% in the injected group (3 of 224) versus 0.45% in the non-injected group (1 of 224)
We want to match two groups of patients (one with and one without steroid
injection) and look at the difference in infection rate. Who can I calculate how many
patients we have to include to get 80% power (p < 0.05)?
Here is my response:
As always, the power is critically dependent on how large the effect is. I shall
first assume that the papers highlighted in yellow, above, provides a good estimate of
the actual effect of the steroid injection. I shall also assume that the overall risk of
infection is 1%. I shall also assume that half of patients are injected and half are not.
Under the null hypothesis, the cell proportions are:
Injected?
Infected?
No
Yes
No
.495
.495
Yes
.005
.005
Marginal
.500
.500
Under the alternative hypothesis, the cell proportions are:
Injected?
Infected?
No
Yes
No
.4978
.493
Yes
.0022
.007
These were entered into G*Power, as shown below.
Notice that the effect size statistic has a value of about 0.049. In the behavioral
sciences the conventional definition of a small but not trivial value of this statistic is 0.1,
but this may not apply for the proposed research.
G*Power next computes the required number of cases to have an 80% chance of
detecting an effect of this size, employing the traditional .05 criterion of statistical
significance.
As you can see, 3,282 cases would be needed. I suspect the surgeon to be
disappointed with this news.
What if the actual effect is larger than assumed above? Suppose the rate of
infection is 10% in injected patients and 1% in non-injected patients. The contingency
table under the alternative hypothesis is now
Injected?
Infected?
No
Yes
No
.495
.45
Yes
.005
.05
Now the effect size statistic is about .64. By convention, in the behavioral
sciences, .5 is the benchmark for a large effect.
χ² tests - Goodness-of-fit tests: Contingency tables
Analysis: A priori: Compute required sample size
Input:
Effect size w
= 0.6396021
α err prob
= 0.05
Power (1-β err prob)
= 0.80
Df
= 1
Output: Noncentrality parameter λ = 8.1818169
Critical χ²
= 3.8414588
Total sample size
= 20
Actual power
= 0.8160533
As you can see, now only 20 cases are necessary to have an 80% chance of
detecting this large effect.
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