So, we can go a little bit through this Phase III example.
And here, this is actually a survival example.
So, our primary objective here was to determine
if patients with metastatic melanoma, who underwent this procedure
had a different overall survival compared with patients receiving standard of care.
So, remember, we have to then define standard of care.
But they were trying to do all this and the idea was
that we're going to have a two-arm randomized Phase III single institution trial.
We don't see many Phase III trials done at one site, but this one was going to be.
So, they came to us and said, "How many patients do
we need to enroll?" We want to have a 1-to-1 ratio between the arms.
So, we're going to equally randomize to the two study arms.
We want 80 percent power
to detect a difference between eight-month median survival and 16-month median survival.
Again, then you need to look and say, "Okay, what's currently on the market?
What's going on?
Do these values still make sense?" They added two-tailed 0.05 test they wanted to run.
And they wanted to do 24 months of follow-up after the last patient had been enrolled.
And they were planning on doing 36 months of accrual.
These lovely things are called nomograms.
So, I have plenty of books with lots of formulas.
And I have to say, actually, of all sample size calculations,
probably I like survival analysis ones the most
because you get to move a lot of different things.
But some of the best statisticians I work with keep photocopying this picture.
There's this paper that basically has different figures
and the number of patients per treatment group.
So, you set -- they have different pictures for alpha and beta combinations.
The dashed lines are for two-sided tests and the solid lines are for one-sided tests.
So, what do you do?
And I said, "Okay.
First, my follow-up period is two years." So, you draw a line that is perpendicular
to this down to intersect the line for your accrual period, which was 36 months.
So, you draw that line out.
And then where line one and line two intersect, you draw
straight up parallel to this y-axis with the number of patients.
You draw straight up. You say, "All right.
R is the ratio between the two median survival values." They said that they wanted
to basically double median survival from eight to 16 months, they had a two-sided
test.
So, we'd go from the dash line for R equals 2.0.
We draw a line parallel to this x-axis, boom, 37 patients.
So, they have all these pictures.
And no joke, you will see the statisticians like drawing with their lines.
You can do it online now.
They have a little app.
Schoenfeld and Richter made these pictures back
before we had computers that easily calculated those for us.
Now, most statisticians will honestly run a bunch of simulations and simulate
your sample size for you taking into account a lot of different issues
about your study design. But it's a nice check.
I think for folks, it's a low-tech check that you can use.
You can use formulas, you can use simulations.
Almost everybody can draw straight lines, as long as you have a ruler or something
to draw them against.
Oh, I'm sorry.
It wasn't 37. It's 38 per arm that we needed.
So, the decision was to try to recruit 42 people per arm for a total of 84 studies,
because they assumed, you know,
that they were going to have censoring, people were going to drop out, et cetera.
They said that they anticipated approximately 30 patients
a year entering the trial, we said, "Yay, that works." Because sometimes they come back
and you're like, "Oh, you had an accrual period of 36 months.
Well, maybe that's not going to work anymore, because of the number that you think
you can get per year.
We're going to have to jigger and do this estimation back and forth." So,
30 patients a year.
You think in three years
I'll get 90 patients, I should hopefully clear my 84 that I need.
So, again, nomograms.
I showed this picture because you're not always -- sometimes you're up here.
You're not always down here. So, let's think of non-survival simple sample size.
Let's start off with one arm or one sample study.
Then we're going to move into a two-arm sample or a two-arm study.
Cheat trick.
If you have three plus arms, calculate the per arm sample size for two-arm study.
Use that per arm.
It doesn't always work, but it's typically a reasonable estimate
of how many people you need per study arm.
So, let's go back to the single sample test.
I've got a new sleep aid.
I want to look at baseline sleep time after taking the medication for a week.
I'm going to run a two-sided test, alpha equals 0.05, power of 90 percent.
The difference is, on average, 1.
So, I think on average, people are going to have four hours of sleep at baseline.
And I'm going to move them to five hours of sleep after my intervention for one week.
Standard deviation, 2 hours.
These are not real numbers.
They are numbers to make this thing move.
So, now, I've got my equations.
I've got this one sample test.
You'll notice that there's no 4 in front of this.
That's because the 2 and the 2 before were because we had a two-sample test.
So, 2 times 2 equals 4, which is why that 4 is there usually.
But I have kind of that critical value for my Type I error.
It's a two-sided test that was divided by two.
Critical value for the power, variance, and the difference of interest.
So, for an alpha equals 0.05 two-sided test, my Z value is 1.960.
For the 80 percent power, my Z value is 1.282.
Add those together, and square it.
My standard deviation was 2, so I'm going to square that.
So, I have 4 for my variance.
The difference of interest is 1 squared is 1.
If I calculate this, it says I need 42.04 people
in my sleep study, which means I need at least 43.
Let's say now I want a two-hour difference in sleep.
So, I want to measure and then go from 4 to 6, not 4 to 5.
My sample size shifts down to 11.
How did I get 11?
Well, remember what I said, it's easier to see big differences.
It's easy to see if people have on average two hours more of sleep.
It's easier to see that than one hour.
It might be harder in reality to make that change happen, however.
But I took this value in the denominator.
Remember, it's a value squared on top of that.
So, you can -- if you're more math algebra-oriented, remember the formula.
If you're not, just remember, if you increase the difference of interest,
you decrease your sample size for the same power.
But the sample size of 11 leads us back to one of those little tricks in the front.
I found this using the Z critical values.
But it's a small number.
So, I need to do this iteration with the T's.
If you are not a math-y person, do not stress.
Your computer will typically do it for you.
If you are trying to plug numbers in, you need to stress and remember this.
So, use -- you first calculate it using your regular formula.
Then you take that sample size,
and you're going to have N minus 1 degrees of freedom, down here.
For a simple one sample test, you're going to find
the critical value of T for 10 degrees of freedom in this case.
Recalculate the sample size using the T's,
instead of the Z's, you're going to find a sample size of 13.
You repeat this whole thing.
You iterate it down until you get a single sample size that sticks.
Sometimes you have to iterate this a lot.
Again, only for those
that bother to do this essentially by hand are going to program their own.
Do you care about this?
Although, you should check, you know, this auto formula.
Is it using the T or is it using the Z?
If you use this T's, you're good.
If you use the Z for the critical values, you may want to use a different formula.
So, let's say my power -- before, I wanted a power of 90 percent.
Now, I want 80 percent.
Oops, I'm sorry, I had the wrong Z there before.
My sample size goes from 11 to eight.
Again, I need rethinking the T distribution on Z distribution.
But this value in the numerator just got smaller because my power got smaller.
If your power you're entrusted in is lowered, your sample size as necessary as lowered.
Standard deviation -- oh, and I should say here, you do not need
an actual sample size of eight.
You need something larger because your math for all your regressions are going to fail.
But this is kind of your starting point.
It's the eight.
Standard deviation goes from 2 to 3.
My sample size has now moved from eight to 13 -- I'm sorry, eight to 18. Why?
Because I squared that 3. It's in my numerator.
Increased variance means you need an increased sample size.
Let's say you remembered that Dr.
Johnson told you to do a randomized two-arm design, don't do these single arm studies.
You can if you need to, but you're like, "I'm going to do better." If you take
the original design -- remember, we needed 43 people in the one sample design.
We have -- we stuck with our original numbers.
Two-sample study needs twice of that in each group in order to account for
the variance, because now you have variance in two different study arms.
You have even more variance.
So, because of that, I need
85 people in each study arm or 170 people total.
And then you go "Well, didn't Dr.
Johnson say sometimes single arm studies were okay?" So, again,
you're going back and forth on this.
Here are your 2s.
Those 2s are because you have a two-sample study.
Again, five-arm study, calculate that sample size per study arm, multiply it by 5.
It doesn't consider all the multiple comparisons.
It depends on the analysis techniques you're using.
But this is your rough estimate to get you started.
So, think about it again.
You change the difference of interest from one hour to two hours,
that 170-person study goes to 44 people total.
Sounds pretty sweet.
Changing my power from 90 percent to 80 percent,
the sample size goes down to some more.
My standard deviation maybe needed to be a little bit higher,
my sample size has gone back up.
So, in conclusion, a lot of things drive your sample size.
You can either remember the directions, or you can remember the formula
and that will help you figure out the directions.
But just remember, basically everything about your study is driving that sample size.
So, are there other designs? Sure, there are tons of designs.
If you do a matched pair design, essentially this looks like the one sample formula.
If you're doing means, you're going to do the formula for a paired t-test.
It's going to look at the mean difference from the paired data
and the variance of the differences.
And proportions, proportions
are actually based on discordant pairs for your sample size calculation.
The textbook has various examples with paired designs, two and one sample
means, the various proportions.
And we also talk a little bit about how you take pilot data and design your next study.
I mentioned I don't like Cohen's effect sizes, but it is very popular
to use them in sample size calculations when you don't have any other information.
And Cohen himself who has a very commonly used textbooks in the psychological
literature says, "Don't use them unless you don't have a choice."
A large effect
size is typically 0.8, medium is 0.5, small is 0.2.
Medium -- again, you'll use the same sample size regardless of what you're measuring.
That's the reason Cohen says, "Please don't do this."