So, a few of our examples.
This short hint, because I always get asked this question.
How many humans do I need when I know nothing?
Sometimes the basic element has to do more with stability
of the estimating equation than with really solving the problem.
So, you may run one of these calculations
that tells you, you only need three people per arm.
You will need more than three people per arm.
Fifteen people per arm or rats per arm, et cetera, tend to be the minimum number.
This is a good rule of thumb for early studies.
Why is this?
Well, when you have 15 -- 12 to 15 values, you actually get a somewhat stable
variance equation. I'm not saying it's right.
It could be that you have weird sampling, and the variance is off because of bias,
but the equation itself for variance stabilizes between --
sometimes as low as 10, but usually 12 to 15 numbers in it.
If you're using a Bayesian analysis technique,
sometimes you need at least 70 people per arm before the analysis
is, in fact, stable.
A lot of our studies,
a lot of our analyses and regressions float to something called the Z test.
However, if you have a smaller sample size below
20 or 30, even sometimes below 100, you need to check t-distribution.
If you're doing something like logistic regression, principal components
analysis, et cetera, you usually need a minimum of 10 participants per variable.
These are individual participants per variable.
And in some cases, you may need 100 per variable in order to get stable estimates.
This is for like -- the math will spew out of the computer,
it doesn't mean the functions are stable.
So, there's this balance of wanting to find significant results.
What do I need to if I think these are the differences I'm aiming for?
And then also, how many people do I need for the math to be sound?
So, now, we're going to do
-- it's not truly a live statistical consult, because I took screenshots.
But we're going to look at sample size and power calculations.
And actually, this isn't from your -- this year's hypothesis testing
lecture now, because Paul gave it for you all last week instead.
But I'm going to talk about an example of cholesterol and hypertensive men.
Let's say you have data on 25 hypertensive men,
mean is 220 for their total cholesterol,
and you have a sample standard deviation of 38.6.
If I go out and I look at national population data, 20 to 74-year-old male population,
the mean serum cholesterol is 211 with a standard deviation of 46.
So, their standard deviation of population is actually much higher
than in my little sample.
Not common, but that happens.
The question is, my hypertensive men, is their cholesterol different
from the average population?
Now, it could be, because sometimes as we're treating hypertension, we're
treating a lot of other things including hypercholesterolemia.
So, we're going to calculate the power with the numbers that are given.
So, what's the power to see a 9-point difference in mean cholesterol
with 25 people?
There's the question, is this a single sample or a two-sample test?
If I'm comparing the kind of population level information, this is really a one
sample test.
So, Russ Lenth has set up a whole bunch of different software what's called Piface.
There are lots of different software programs
you can use for this, but this is free and it has a cute cow picture.
So, how can I resist?
He also has some really great FAQs
or frequently asked questions and other elements on his website.
He also has a view that I deeply agree with that, "Please don't ask me
for post hoc power."
So, it's
a lovely article on why you shouldn't do post hoc power.
Anyway, you've got this drop down for what type of analysis you want to do.
So, let's say you had a two-sample test.
Let's say I actually had 25 people that were normal that I pulled out
medical records on, and 25 people
that were hypertensive that I pulled medical records on.
I can set up all this information and figure out what my power is.
So, I can either set my sample size and calculate the power,
or I can set my power and calculate my sample size.
Well, let's do it for the one sample test.
If I put in my sigma, and I put in the true difference of means,
if I went 90 percent power, I need 170 people.
I basically have no power with my little sample of
25 to test for any types of difference in cholesterol.
But you know, I kind of chose the 9 as a difference,
because that's what my difference was.
But what is a clinically meaningful difference in total cholesterol?
What should I be testing as that difference?
These are all questions we would ask each other in advance of the study
and not afterwards when you come to me and say, "I have 25 medical records.
This is the data I have." So, what's nice about trying to design
the study is I can move all these values around.
Okay, you can't get 170 medical records.
What can you do? What other assumptions can we make?
So, you can kind of see
if it's not tenable at all to do this study, or what your trade-offs are.
So, a different study,
going back to the two sample which I meant to put that slide down here.
Let's say I have a standard deviation of 2 in each arm.
And you may not
-- sometimes you have different standard deviations for each study arm, I'm
going to run a two-sided alpha 0.05 test, And we'll look at a true difference
of means of 1, 90 percent power, solve for my sample size.
So, sigma is to 2. 2, I'd say, have equal.
I'm going to have two-tailed test on equivalence.
I defined my alpha. Difference of means is 1.
And if I went 90 percent power, and I just hit solve
for a sample size, it's going to tell me I need 85 in each arm.
But you can also make these nice little graphs.
So, you can say along the y-axis, this tall vertical axis, "That's my power."
And then I can see for the sample size in a single study arm on my x-axis
how much power -- oops, I have depending on where I am in the curve.
So, if I only recruit 50 people, what's my power going to be?
These are
nice curves to put in your IRB applications or your grant applications.
So, especially if you're asking for money, then you can say, "Listen,
if you give me less money,
and I have fewer people in my study, this is the tradeoff that's happening.
Are you sure you want me to make that tradeoff?" So,
let's talk about dose escalation a little bit.
Those -- again, remember, we have basic sample size formulas.
Most of us live in Phase I and Phase II.
We're not going to use those basic sample size formulas as often.
One of the simple Phase I types of studies that we do are called dose escalation
studies.
We're looking many times at least first in human studies
with a new compound for something called a dose limiting toxicity or DLT.
Now, it might be that this drug or this compound has been used
in humans before, but now I'm trying it in a different patient group.
A lot of these study designs came out of oncology.
So, they came from the cancer world.
We've been modifying them over the years for kind of, you know, less severe illnesses.
And there are a lot of different ways to do Phase I and Phase II studies.
But I'm going to go through what's called sometimes the 3x3, or there are
a lot of other names, Fibonacci sequence, although it's not a real Fibonacci sequence.
But these are kind of the basic models used for several decades for Phase
I studies, and probably large portion of studies still today use them.
But these have been modified
in the last couple of decades just a little more sophisticated studies.
But I want you to understand kind of like the basement where we all started.
So, in Phase I, you decide on a handful of dose levels.
You decide kind of what toxicities are so bad,
you would stop, and you wouldn't give people that dose anymore.
So, it might be that people die.
It might be that,
you know, they have uncontrolled vomiting, whatever you want to call it.
Usually, you use something like the common toxicity criteria.
They're AE grading, you'd say, like anything that's a Grade 3 or Grade 4
AE or something like that.
You know, if you have a disease that's not that bad,
you aren't going to be less willing to tolerate adverse events.
And then we have some interventions that never cost toxicities.
So, you've got to figure out something else that you want to measure.
Typically, what happens in these Phase III, 3x3 designs, is at least
three patients will be treated on each dose level, in each cohort.
That's where this 3 comes from.
This isn't a power and sample size calculation issue.
This is when it comes down to it, how many willing -- how many people are
you willing to have
on a totally unknown substance and basically kill at any given point in time?
That sounds very ugly, and I've said it before, but I think it's
really important to understand the gravity of some of the work that we're doing.
And sometimes when there's
no choice, you put more people on and you are taking a higher risk.
They need to understand those risks.
And we'll talk about that more in the next segment of lectures.
But important element here is sometimes even though we want to try three people
in a given dose, we do it one by one, just in case, you know, two people die,
and they weren't supposed to, there's no reason to try it on the third person.
So, kind of the old way that we would do this is we would enroll three patients.
If zero out of three patients have these dose limiting toxicities, we'd escalate
to the new dose.
If you saw a dose limiting toxicity in one of the three patients,
we expanded the cohort to six.
So, we added three more people to it, because maybe it was a fluke.
It may not be related to this drug that we're giving them.
And we would escalate to a new dose if zero of the next three new patients
didn't develop that DLT.
In other words, only one out of three at a given --
or one out of six at a given dose developed a dose limiting toxicity.
The idea is that you are trying to work your way to what's called a maximum
tolerated dose or MTD.
This is the dose level immediately below the level at which two
or more patients in a cohort of three or six experienced that dose limiting toxicity.
We're trying to go for a safe dose.
Typically, the maximum dosage is pre-specified in the protocol.
So, we're going to stop at the MTD or the maximum dosage.
This is the big picture.
We're going to go through it in steps.
So, first, I enroll three people.
One of two to three or three to four things can happen.
Maybe none of them have a dose limiting toxicity.
One of them or two or three of them have a dose
limiting toxicity.
Of course,
I drink water or my voice gets worse.
So, nobody has a DLT, I escalate to the new dose.
One out of three have a DLT, I enroll three more people at the same dose.
And if none of those new three people have it, I escalate.
If one or more in this new group
have the DLT, depending on my risk tolerance, I may stop.
But let's say I only have three people on that lower dose.
I don't really know that much.
So, what I might do is in fact drop down a dose and start over.
So, what about two or three people out of my first three have a dose limiting toxicity?
You may stop, or I might drop down a dose, fill
in three more people at that lower dose, make sure it's actually okay.
And I might start over.
I might say, "Well, you know, it was flu season.
I did have my flu shot." But it was flu season.
These people actually where they were already immunosuppressed,
and it seemed like they died of influenza, so we kind of want to restart this.
We're not so sure how much of this is really due to our therapy.
So, you have rules.
I don't want you to talk yourself out of your rules.
You need to think to have reasonable rules in the first place.
But sometimes something weird has happened.
And you went to kind of plan these different starts.
I will also tell you, sometimes this happens on your first dose.
You should always have a backup plan to drop down even from your lowest
-- supposedly lowest dose.
Here it is in a nice chart for those of you who prefer the charts.
This is actually taken out of something that Peter Thall at MD Anderson
did, which is interesting because Peter Thall really doesn't like this design.
But the basic idea is how many patients do you have with a DLT and the decision
you should make. A few thoughts.
So, if you have zero in three and zero in three, you really had to get here through
some de-escalation rule that had to have been applied at a higher dose.
Again, sometimes if I had one in three and then zero in three, I might escalate
unless it was a de-escalation rule that got me to this zero in three.
But you have to
decide what's your risk of a DLT that you're willing to take.
Is it, you know, 16 to 17 percent, or is it 33 percent?
So, what's the risk of having that severe adverse event?
It may or may not be that severe depending on how you set your toxicity levels.
What percentage of patients are you willing to have that?
When you're working with so few patients here, the confidence interval around
it is really huge.
This is the reason a lot of statisticians now don't use these.
They use these different adaptive designs to enroll more people and get more information.
But it is important to figure out like, if there is nothing available except death,
you might be willing to tolerate a lot of toxicity, or you may say no.
So, it's important to figure out what percent of toxicity
and what toxicities should be your dividing line.
So, those stars in that previous table, again, you're implicitly targeting
a dose with the probability of toxicity less than or equal to 17 percent.
But if you're going to look at the two out of six, then you're saying you're willing
to have an MTD with a probability of toxicity at about 33 percent.
Key point in these trials, patients do not go on to the next dose level
until all the patients at the previous dose level are beyond
the time frame where you plan to be looking for toxicity.
Do not tell me you need to keep enrolling them, "But
they're here." You have to make sure it's safe to put them on the next level.
Your patients have to be there.
You have to finish analyzing -- you have to finish following them.
You have to clean their data.
Analyze their data. And make a decision, do we escalate?
Yes or no?
If you do not have the data to decide if you can escalate,
you cannot ethically escalate.
Again, not a power or sample size calculation issue.
There are lots and lots and lots of new methods.
Almost all of them are better than the method
I just told you, but they are also all based on that method.
Sometimes in Phase I now I'm randomizing to multiple different arms.
I may have various control arms, might be placebo, might be active control. Why?
Because when I'm expecting a lot of toxicity,
again, I need to have this comparison to figure out, is this additional toxicity
because of my new therapy, or is this my general background rate of toxicity?
Especially when I'm dealing with diseases
that maybe I don't want to have a lot of toxicity,
I want to make sure that they're safer, I might put six to 15 plus people per arm.
Also, if I'm expecting a wide heterogeneity in the effects and the toxicities,
I may want to include more people in this Phase I layer of work.
But there are CRM, Bayesian methods, and lots of different methods you can use here.