Instructor: Hello students, and welcome
to Biostat ER Chapter Eight, Example Eight.
In this example, we will learn how to calculate sample size
for Two-Independent Samples Dichotomous Outcome
to perform Hypothesis Testing.
This example is from our textbook
and it is Problem Four.
So first, I'm going to read the problem.
"A clinical trial was conducted comparing
a new compound designed to improve wound healing
in trauma patients to a placebo.
After treatment for five days,
58% of the patients taking the new compound
had a substantial reduction in the size of their wound
as compared to 44% in the placebo group.
The trial failed to show significance.
So how many subjects would be required
to detect the difference in proportion observed
in the trial with 80% power?
A two-sided test is planned at alpha equal to 0.05."
So first here, I have inserted the formula for you,
and the next thing we really need to do here
is determine the Effect Size,
which is the denominator for this formula.
Now for the Effect Size,
the numerator here is the absolute difference between
the two proportions,
which is provided to us, 58% and 44%.
But the denominator is the square root
of the pooled proportion multiplied by one minus
that pooled proportion,
and that is not provided to us.
However, below, I have shown you the step-by-step process
on how to determine the pooled proportion.
Here, the formula is,
p equal to x1 plus x2 divided by n1 plus n2.
So the x1 and x2 is the success,
our number of successes,
and n1 and n2 is the sample size.
But here's the issue.
We are actually here trying to determine the sample size.
So what we need to do
is solve this a little bit algebraically.
So we also know that p1 is equal to x1 divided by n1
and p2 is equal to x2 divided by n2.
So when we do the cross-multiplication here,
we find that x1 is actually equal to p1,
multiplied by n1 and x2 is equal to p2 multiplied by n2.
And given...
The problem here,
the n2 and the n1 is equal,
and we can just call them n.
So now, if we replace the n1 and n2 just by the n,
which I have done here,
then we can simplify everything
by inserting these new values.
And once we do that,
the ns all cancel out,
and all we are left with then is p1 plus p2 divided by two.
And we definitely have the value of p1 and p2,
because that is given to us,
which is 0.58 and 0.44.
And I have highlighted those values in yellow
in the problem.
So once we add those and divide it by two,
we get 0.51,
and now we will insert those values into the formula
to determine the Effect Size.
The numerator here is the absolute difference,
and that's exactly what I've inserted here.
And in the denominator,
we have the pooled proportion,
which is 0.51,
and then one minus the pooled proportion,
which is 0.51.
And once we algebraically solve this,
we get 0.28.
Now we need to move forward in terms of inserting
the other values into the formula,
so that we can determine the sample size.
So the value for Z1 minus alpha divided by two
is the Z of the 97.5th percentile,
and that is 1.96.
And the value for Z1 minus beta
is the Z of the 80th percentile,
because one minus beta is the power of the study,
and that is 0.84.
So once we insert those values
and perform the algebra,
we get 200.
So the sample size of n1 equal to 200,
and the sample size of n2 is also equal to 200,
will ensure that the test of hypothesis will have 80% power
to detect a difference in proportions
on the order of 0.58 versus 0.44 between groups.
(computer mouse clicking)