1. Sample Size Determination In the Context of Hypothesis Testing

2. Recall, in context of Estimation, Sample Size is based upon:
the width of the Confidence interval:
The confidence level (1 – a)  confidence coefficient, z1-a/2
The population standard error, s/n w ( )
x – z1-a/2(s/n) x
x + z1-a/2(s/n)

3. Sample size in Context of Hypothesis Testing:
Need to consider POWER as well as confidence level
Example: Suppose we have a hypothesis on one mean:
Ho: mo = vs Ha: mo 100
s = 10
a = .05
If the true mean is in fact ma = 105,
what size sample is required so that the power of the test is (1-b) = .80 ?

4. For our hypothesis test, we will reject Ho for x greater than C1
or less than C2
a/2 = .025
a/2 = .025
mo=100
C2 = mo-Z1-a/2(s/n)
C1 = mo+Z1-a/2(s/n)

5. Suppose, in fact, that ma = 105. We will reject Ho
Let’s look at these decision points (C1 and C2) relative to a specific alternative.
Suppose, in fact, that ma = 105.
We will reject Ho
if x is greater than C1
or x is less than C2
Distribution based on Ha C1
ma=105 C2

6. We want b = .20
for power=.80
We want b = .20
 zb = -.842 ma C2 C1

7. Note for sample size determination: a, b are set by the investigator
Both a specific null (mo) and a specific alternative (ma) must be specified
we assume that the same variance s2 holds for both the null and alternative distributions
a/2
a/2 b ma m0
C1 = mo+z1-a/2(s/n)
C1 = ma-z1- (s/n)

8. We now have:
C1 in terms of
both the Ho and Ha
distributions:
Setting these equal:
Then solve for n.

9. Sample Size is then:
Note:
Always use positive values for z1-a/2 and (we defined CI using  positive z)

10. In our example:
s = 10
a = .05  z1-a/2 = 1.96
b = .20  = .842
mo = 100
ma = 105
Or a sample size of n=32 is needed.

11. If we change the desired power to 1-b = .90:
Or a sample size of n=42 is needed.

12. In the context of hypothesis testing, sample size is a function of:
s2, the population variance
a = .05  z1-a/2 , Type I error rate
b = .20  , Type II error rate
Distance between mo , hypothesized mean and ma , a specific alternative

13. Using Minitab to estimate Sample Size:
Stat  Power and Sample Size  1-Sample Z
Difference between mo and ma
Desired power (separate by spaces if entering several)
2-sided test s

14. Power and Sample Size
1-Sample Z Test
Testing mean = null (versus not = null)
Calculating power for mean = null + difference
Alpha = Sigma = 10
Sample Target Actual
Difference Size Power Power

15. Sample size and power for comparing means of 2 independent groups.
In the example comparing LOS for elective vs. emergency patients, we observed a difference between sample means of 3.3 days – but found that this was NOT statistically significantly different from zero.
However 3.3 days is a large, expensive difference in length of stay. Our data had relatively large observed variance, and small n.
What was the power of our study to detect a difference of 3.3 days?
What sample size would be needed per group to find a difference of 3 days or more significantly different from zero?

16. Set a and 1- or 2- sided test, using options menu.
In Minitab: Stat  Power and Sample Size  2-sample t
To evaluate power:
Enter sample sizes
Enter observed difference in means
Enter standard deviation
Set a and 1- or 2- sided test, using options menu. s

17. In Minitab: Stat  Power and Sample Size  2-sample t
2-Sample t Test
Testing mean 1 = mean 2 (versus not =)
Calculating power for mean 1 = mean
Alpha = Sigma = 10
Sample
Size Power
Note: Minitab assumes equal n’s for the 2 groups, and only gives space for one value of s
Clearly, our power to detect a difference as large as 3.3 days was only about 12% -- not very good!

18. s In Minitab: Stat  Power and Sample Size  2-sample t
To estimate sample size:
Enter desired power
Enter desired significant difference in means
Enter standard deviation s

19. Power and Sample Size
2-Sample t Test
Testing mean 1 = mean 2 (versus not =)
Calculating power for mean 1 = mean 2 + 3
Alpha = Sigma = 10
Sample Target Actual Size Power Power
Note: sample sizes are per group. If this seems excessive – is your estimate of the standard deviation reasonable?
I used the larger of the 2 observed SD here. You might want to compute a pooled SD, and try that.