1
00:00:08,174 --> 00:00:10,910
So, a few of our examples.

2
00:00:10,910 --> 00:00:15,448
This short hint,
because I always get asked this question.

3
00:00:15,448 --> 00:00:19,986
How many humans do I need
when I know nothing?

4
00:00:19,986 --> 00:00:24,524
Sometimes the basic element has to do more
with stability

5
00:00:24,524 --> 00:00:29,062
of the estimating equation
than with really solving the problem.

6
00:00:29,062 --> 00:00:32,699
So, you may run one of these calculations

7
00:00:32,699 --> 00:00:37,237
that tells you,
you only need three people per arm.

8
00:00:37,704 --> 00:00:41,141
You will need more than three people per arm.

9
00:00:41,141 --> 00:00:47,313
Fifteen people per arm or rats per arm,
et cetera, tend to be the minimum number.

10
00:00:47,313 --> 00:00:51,151
This is a good rule of thumb
for early studies.

11
00:00:51,151 --> 00:00:52,285
Why is this?

12
00:00:52,285 --> 00:00:58,458
Well, when you have 15 -- 12 to 15 values,
you actually get a somewhat stable

13
00:00:58,458 --> 00:01:01,127
variance equation. I'm not saying it's right.

14
00:01:01,127 --> 00:01:07,267
It could be that you have weird sampling,
and the variance is off because of bias,

15
00:01:07,267 --> 00:01:10,737
but the equation itself
for variance stabilizes between --

16
00:01:10,737 --> 00:01:15,742
sometimes as low as 10,
but usually 12 to 15 numbers in it.

17
00:01:17,043 --> 00:01:19,712
If you're using a Bayesian analysis
technique,

18
00:01:19,712 --> 00:01:24,284
sometimes you need at least 70 people per arm
before the analysis

19
00:01:24,284 --> 00:01:27,353
is, in fact, stable.

20
00:01:28,621 --> 00:01:30,890
A lot of our studies,

21
00:01:30,890 --> 00:01:37,263
a lot of our analyses and regressions float
to something called the Z test.

22
00:01:37,263 --> 00:01:41,367
However,
if you have a smaller sample size below

23
00:01:41,367 --> 00:01:46,806
20 or 30, even sometimes below 100,
you need to check t-distribution.

24
00:01:46,806 --> 00:01:50,910
If you're doing something like
logistic regression, principal components

25
00:01:50,910 --> 00:01:56,850
analysis, et cetera, you usually need
a minimum of 10 participants per variable.

26
00:01:56,850 --> 00:01:59,552
These are individual participants
per variable.

27
00:02:00,687 --> 00:02:06,626
And in some cases, you may need 100 per
variable in order to get stable estimates.

28
00:02:06,626 --> 00:02:11,464
This is for like --
the math will spew out of the computer,

29
00:02:11,464 --> 00:02:14,067
it doesn't mean the functions are stable.

30
00:02:14,067 --> 00:02:17,770
So, there's this balance
of wanting to find significant results.

31
00:02:17,770 --> 00:02:23,343
What do I need to if I think these are
the differences I'm aiming for?

32
00:02:23,343 --> 00:02:29,115
And then also, how many people do I need for
the math to be sound?

33
00:02:29,115 --> 00:02:31,551
So, now, we're going to do

34
00:02:31,551 --> 00:02:36,389
-- it's not truly a live statistical consult,
because I took screenshots.

35
00:02:36,389 --> 00:02:40,627
But we're going to look at sample size
and power calculations.

36
00:02:40,627 --> 00:02:44,864
And actually, this isn't from your --
this year's hypothesis testing

37
00:02:44,864 --> 00:02:49,502
lecture now, because Paul gave it for you
all last week instead.

38
00:02:49,502 --> 00:02:54,507
But I'm going to talk about an example
of cholesterol and hypertensive men.

39
00:02:54,507 --> 00:02:57,977
Let's say
you have data on 25 hypertensive men,

40
00:02:58,745 --> 00:03:02,148
mean is 220 for their total cholesterol,

41
00:03:02,148 --> 00:03:06,519
and you have a sample
standard deviation of 38.6.

42
00:03:06,519 --> 00:03:14,260
If I go out and I look at national population
data, 20 to 74-year-old male population,

43
00:03:14,260 --> 00:03:20,066
the mean serum cholesterol is 211
with a standard deviation of 46.

44
00:03:20,066 --> 00:03:24,904
So, their standard deviation of population
is actually much higher

45
00:03:24,904 --> 00:03:27,340
than in my little sample.

46
00:03:27,340 --> 00:03:29,742
Not common, but that happens.

47
00:03:29,742 --> 00:03:34,614
The question is, my hypertensive men,
is their cholesterol different

48
00:03:34,614 --> 00:03:36,549
from the average population?

49
00:03:36,549 --> 00:03:41,421
Now, it could be, because sometimes as we're
treating hypertension, we're

50
00:03:41,421 --> 00:03:44,991
treating a lot of other things
including hypercholesterolemia.

51
00:03:44,991 --> 00:03:50,663
So, we're going to calculate the power
with the numbers that are given.

52
00:03:50,663 --> 00:03:55,735
So, what's the power to see
a 9-point difference in mean cholesterol

53
00:03:55,735 --> 00:03:57,003
with 25 people?

54
00:03:57,003 --> 00:04:02,041
There's the question, is this a single sample
or a two-sample test?

55
00:04:02,041 --> 00:04:07,914
If I'm comparing the kind of population
level information, this is really a one

56
00:04:07,914 --> 00:04:08,881
sample test.

57
00:04:08,881 --> 00:04:15,488
So, Russ Lenth has set up a whole bunch
of different software what's called Piface.

58
00:04:16,022 --> 00:04:18,157
There are lots of different software programs

59
00:04:18,157 --> 00:04:23,062
you can use for this, but this is free
and it has a cute cow picture.

60
00:04:23,062 --> 00:04:24,597
So, how can I resist?

61
00:04:24,597 --> 00:04:26,733
He also has some really great FAQs

62
00:04:26,733 --> 00:04:29,802
or frequently asked questions
and other elements on his website.

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00:04:29,802 --> 00:04:34,407
He also has a view that I deeply agree
with that, "Please don't ask me

64
00:04:34,407 --> 00:04:35,642
for post hoc power."

65
00:04:39,646 --> 00:04:40,446
So, it's

66
00:04:40,446 --> 00:04:44,784
a lovely article on why you shouldn't do post
hoc power.

67
00:04:44,784 --> 00:04:50,923
Anyway, you've got this drop down for what
type of analysis you want to do.

68
00:04:50,923 --> 00:04:54,294
So, let's say you had a two-sample test.

69
00:04:54,294 --> 00:04:59,832
Let's say I actually had 25 people
that were normal that I pulled out

70
00:04:59,832 --> 00:05:02,201
medical records on, and 25 people

71
00:05:02,201 --> 00:05:05,772
that were hypertensive
that I pulled medical records on.

72
00:05:07,473 --> 00:05:13,646
I can set up all this information
and figure out what my power is.

73
00:05:13,646 --> 00:05:18,918
So, I can either set my sample size
and calculate the power,

74
00:05:18,918 --> 00:05:23,923
or I can set my power and calculate
my sample size.

75
00:05:23,923 --> 00:05:28,161
Well, let's do it for the one sample test.

76
00:05:28,161 --> 00:05:34,767
If I put in my sigma,
and I put in the true difference of means,

77
00:05:34,767 --> 00:05:39,172
if I went 90 percent power,
I need 170 people.

78
00:05:39,172 --> 00:05:43,576
I basically have no power
with my little sample of

79
00:05:43,576 --> 00:05:47,980
25 to test
for any types of difference in cholesterol.

80
00:05:47,980 --> 00:05:53,252
But you know,
I kind of chose the 9 as a difference,

81
00:05:53,252 --> 00:05:55,922
because that's what my difference was.

82
00:05:56,956 --> 00:06:00,927
But what is a clinically
meaningful difference in total cholesterol?

83
00:06:00,927 --> 00:06:04,130
What should I be testing as that difference?

84
00:06:04,130 --> 00:06:09,669
These are all questions we would ask
each other in advance of the study

85
00:06:09,669 --> 00:06:15,641
and not afterwards when you come to me
and say, "I have 25 medical records.

86
00:06:15,641 --> 00:06:21,214
This is the data I have." So,
what's nice about trying to design

87
00:06:21,214 --> 00:06:25,184
the study is
I can move all these values around.

88
00:06:25,618 --> 00:06:28,121
Okay, you can't get 170 medical records.

89
00:06:28,121 --> 00:06:31,758
What can you do?
What other assumptions can we make?

90
00:06:31,758 --> 00:06:33,926
So, you can kind of see

91
00:06:33,926 --> 00:06:39,499
if it's not tenable at all to do this study,
or what your trade-offs are.

92
00:06:39,499 --> 00:06:41,134
So, a different study,

93
00:06:41,134 --> 00:06:46,572
going back to the two sample
which I meant to put that slide down here.

94
00:06:46,572 --> 00:06:50,877
Let's say
I have a standard deviation of 2 in each arm.

95
00:06:50,877 --> 00:06:52,345
And you may not

96
00:06:52,345 --> 00:06:56,949
-- sometimes you have different
standard deviations for each study arm, I'm

97
00:06:56,949 --> 00:07:02,789
going to run a two-sided alpha 0.05 test,
And we'll look at a true difference

98
00:07:02,789 --> 00:07:07,627
of means of 1, 90 percent power,
solve for my sample size.

99
00:07:07,627 --> 00:07:11,697
So, sigma is to 2. 2, I'd say, have equal.

100
00:07:11,697 --> 00:07:14,801
I'm going to have two-tailed test
on equivalence.

101
00:07:14,801 --> 00:07:18,304
I defined my alpha. Difference of means is 1.

102
00:07:18,304 --> 00:07:22,975
And if I went 90 percent power,
and I just hit solve

103
00:07:22,975 --> 00:07:28,781
for a sample size, it's going to tell me
I need 85 in each arm.

104
00:07:28,781 --> 00:07:32,351
But you can also make these
nice little graphs.

105
00:07:32,351 --> 00:07:37,924
So, you can say along the y-axis,
this tall vertical axis, "That's my power."

106
00:07:37,924 --> 00:07:44,664
And then I can see for the sample size
in a single study arm on my x-axis

107
00:07:44,664 --> 00:07:50,603
how much power -- oops,
I have depending on where I am in the curve.

108
00:07:50,603 --> 00:07:55,741
So, if I only recruit 50 people,
what's my power going to be?

109
00:07:56,742 --> 00:07:57,510
These are

110
00:07:57,510 --> 00:08:02,014
nice curves to put in your IRB applications
or your grant applications.

111
00:08:02,014 --> 00:08:06,552
So, especially if you're asking for money,
then you can say, "Listen,

112
00:08:06,552 --> 00:08:08,821
if you give me less money,

113
00:08:08,821 --> 00:08:14,093
and I have fewer people in my study,
this is the tradeoff that's happening.

114
00:08:14,093 --> 00:08:18,598
Are you sure you want me to make that
tradeoff?" So,

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00:08:18,598 --> 00:08:21,601
let's talk about dose escalation
a little bit.

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00:08:22,935 --> 00:08:26,939
Those -- again, remember,
we have basic sample size formulas.

117
00:08:26,939 --> 00:08:30,943
Most of us live in Phase I and Phase II.

118
00:08:30,943 --> 00:08:35,748
We're not going to use
those basic sample size formulas as often.

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00:08:35,748 --> 00:08:42,121
One of the simple Phase I types of studies
that we do are called dose escalation

120
00:08:42,121 --> 00:08:42,522
studies.

121
00:08:42,522 --> 00:08:46,526
We're looking many times
at least first in human studies

122
00:08:46,526 --> 00:08:51,731
with a new compound for something
called a dose limiting toxicity or DLT.

123
00:08:52,465 --> 00:08:56,569
Now, it might be that this drug
or this compound has been used

124
00:08:56,569 --> 00:09:00,673
in humans before, but now I'm trying it
in a different patient group.

125
00:09:00,673 --> 00:09:03,843
A lot of these study designs
came out of oncology.

126
00:09:03,843 --> 00:09:06,045
So, they came from the cancer world.

127
00:09:06,045 --> 00:09:10,950
We've been modifying them over the years
for kind of, you know, less severe illnesses.

128
00:09:10,950 --> 00:09:16,188
And there are a lot of different ways
to do Phase I and Phase II studies.

129
00:09:16,188 --> 00:09:21,060
But I'm going to go through
what's called sometimes the 3x3, or there are

130
00:09:21,060 --> 00:09:25,932
a lot of other names, Fibonacci sequence,
although it's not a real Fibonacci sequence.

131
00:09:25,932 --> 00:09:30,770
But these are kind of the basic models
used for several decades for Phase

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00:09:30,770 --> 00:09:34,941
I studies, and probably large
portion of studies still today use them.

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00:09:34,941 --> 00:09:36,676
But these have been modified

134
00:09:36,676 --> 00:09:40,846
in the last couple of decades
just a little more sophisticated studies.

135
00:09:40,846 --> 00:09:46,218
But I want you to understand kind of like
the basement where we all started.

136
00:09:46,218 --> 00:09:50,556
So, in Phase
I, you decide on a handful of dose levels.

137
00:09:50,556 --> 00:09:53,693
You decide
kind of what toxicities are so bad,

138
00:09:53,693 --> 00:09:57,496
you would stop, and you wouldn't
give people that dose anymore.

139
00:09:57,496 --> 00:09:59,932
So, it might be that people die.

140
00:09:59,932 --> 00:10:01,300
It might be that,

141
00:10:01,300 --> 00:10:05,471
you know, they have uncontrolled vomiting,
whatever you want to call it.

142
00:10:06,672 --> 00:10:09,976
Usually, you use something like
the common toxicity criteria.

143
00:10:09,976 --> 00:10:15,114
They're AE grading, you'd say, like anything
that's a Grade 3 or Grade 4

144
00:10:15,114 --> 00:10:16,949
AE or something like that.

145
00:10:16,949 --> 00:10:20,853
You know, if you have a disease
that's not that bad,

146
00:10:21,087 --> 00:10:24,991
you aren't going to be less
willing to tolerate adverse events.

147
00:10:24,991 --> 00:10:28,661
And then we have some interventions
that never cost toxicities.

148
00:10:28,661 --> 00:10:33,599
So, you've got to figure out something else
that you want to measure.

149
00:10:33,599 --> 00:10:38,204
Typically, what happens in these Phase
III, 3x3 designs, is at least

150
00:10:38,204 --> 00:10:42,575
three patients will be treated on each dose
level, in each cohort.

151
00:10:42,842 --> 00:10:44,844
That's where this 3 comes from.

152
00:10:44,844 --> 00:10:47,813
This isn't a power and sample size
calculation issue.

153
00:10:47,813 --> 00:10:53,119
This is when it comes down to it,
how many willing -- how many people are

154
00:10:53,119 --> 00:10:54,453
you willing to have

155
00:10:54,453 --> 00:10:59,091
on a totally unknown substance and basically
kill at any given point in time?

156
00:10:59,091 --> 00:11:03,396
That sounds very ugly,
and I've said it before, but I think it's

157
00:11:03,396 --> 00:11:08,034
really important to understand the gravity
of some of the work that we're doing.

158
00:11:09,301 --> 00:11:10,603
And sometimes when there's

159
00:11:10,603 --> 00:11:15,207
no choice, you put more people on
and you are taking a higher risk.

160
00:11:15,207 --> 00:11:17,143
They need to understand those risks.

161
00:11:17,143 --> 00:11:21,080
And we'll talk about that
more in the next segment of lectures.

162
00:11:21,080 --> 00:11:25,685
But important element here is sometimes
even though we want to try three people

163
00:11:25,685 --> 00:11:31,590
in a given dose, we do it one by one,
just in case, you know, two people die,

164
00:11:31,590 --> 00:11:36,762
and they weren't supposed to, there's
no reason to try it on the third person.

165
00:11:36,762 --> 00:11:42,334
So, kind of the old way that we would do
this is we would enroll three patients.

166
00:11:42,334 --> 00:11:46,472
If zero out of three patients have these dose
limiting toxicities, we'd escalate

167
00:11:46,472 --> 00:11:47,740
to the new dose.

168
00:11:47,740 --> 00:11:51,877
If you saw a dose limiting toxicity
in one of the three patients,

169
00:11:51,877 --> 00:11:53,779
we expanded the cohort to six.

170
00:11:53,779 --> 00:11:58,584
So, we added three more people to it,
because maybe it was a fluke.

171
00:11:58,584 --> 00:12:02,722
It may not be related to this drug
that we're giving them.

172
00:12:02,722 --> 00:12:08,260
And we would escalate to a new dose
if zero of the next three new patients

173
00:12:08,260 --> 00:12:09,628
didn't develop that DLT.

174
00:12:09,628 --> 00:12:13,766
In other words,
only one out of three at a given --

175
00:12:13,766 --> 00:12:18,738
or one out of six at a given dose
developed a dose limiting toxicity.

176
00:12:18,738 --> 00:12:24,443
The idea is that you are trying to work
your way to what's called a maximum

177
00:12:24,443 --> 00:12:26,412
tolerated dose or MTD.

178
00:12:26,412 --> 00:12:30,983
This is the dose level immediately
below the level at which two

179
00:12:30,983 --> 00:12:36,655
or more patients in a cohort of three or six
experienced that dose limiting toxicity.

180
00:12:36,655 --> 00:12:39,692
We're trying to go for a safe dose.

181
00:12:39,692 --> 00:12:43,129
Typically, the maximum dosage
is pre-specified in the protocol.

182
00:12:43,129 --> 00:12:47,867
So, we're going to stop at the MTD
or the maximum dosage.

183
00:12:47,867 --> 00:12:49,969
This is the big picture.

184
00:12:49,969 --> 00:12:53,005
We're going to go through it in steps.

185
00:12:54,373 --> 00:12:56,442
So, first, I enroll three people.

186
00:12:56,442 --> 00:13:00,579
One of two to three or three to four
things can happen.

187
00:13:00,579 --> 00:13:03,682
Maybe none of them have a dose
limiting toxicity.

188
00:13:03,682 --> 00:13:07,820
One of them or two or three of them
have a dose

189
00:13:07,820 --> 00:13:09,188
limiting toxicity.

190
00:13:13,826 --> 00:13:14,894
Of course,

191
00:13:14,894 --> 00:13:19,064
I drink water or my voice gets worse.

192
00:13:19,064 --> 00:13:24,837
So, nobody has a DLT,
I escalate to the new dose.

193
00:13:24,837 --> 00:13:33,212
One out of three have a DLT,
I enroll three more people at the same dose.

194
00:13:33,212 --> 00:13:39,518
And if none of those
new three people have it, I escalate.

195
00:13:39,518 --> 00:13:43,689
If one or more in this new group

196
00:13:43,689 --> 00:13:49,461
have the DLT, depending on my risk tolerance,
I may stop.

197
00:13:49,461 --> 00:13:55,768
But let's say
I only have three people on that lower dose.

198
00:13:56,468 --> 00:13:59,138
I don't really know that much.

199
00:13:59,138 --> 00:14:05,811
So, what I might do is in fact
drop down a dose and start over.

200
00:14:05,811 --> 00:14:13,385
So, what about two or three people out of my
first three have a dose limiting toxicity?

201
00:14:13,385 --> 00:14:18,257
You may stop,
or I might drop down a dose, fill

202
00:14:18,257 --> 00:14:24,029
in three more people at that lower dose,
make sure it's actually okay.

203
00:14:24,029 --> 00:14:26,265
And I might start over.

204
00:14:26,265 --> 00:14:29,435
I might say, "Well, you know,
it was flu season.

205
00:14:29,435 --> 00:14:32,905
I did have my flu shot."
But it was flu season.

206
00:14:32,905 --> 00:14:35,441
These people actually
where they were already immunosuppressed,

207
00:14:35,441 --> 00:14:40,512
and it seemed like they died of influenza,
so we kind of want to restart this.

208
00:14:40,512 --> 00:14:44,950
We're not so sure how much of this
is really due to our therapy.

209
00:14:44,950 --> 00:14:46,485
So, you have rules.

210
00:14:46,485 --> 00:14:50,356
I don't want you to talk yourself
out of your rules.

211
00:14:50,356 --> 00:14:54,593
You need to think to have reasonable rules
in the first place.

212
00:14:54,593 --> 00:14:56,695
But sometimes something weird has happened.

213
00:14:56,695 --> 00:15:00,232
And you went to kind of plan
these different starts.

214
00:15:00,232 --> 00:15:04,436
I will also tell you,
sometimes this happens on your first dose.

215
00:15:04,436 --> 00:15:09,375
You should always have a backup plan
to drop down even from your lowest

216
00:15:09,375 --> 00:15:10,976
-- supposedly lowest dose.

217
00:15:10,976 --> 00:15:16,382
Here it is in a nice chart for those of you
who prefer the charts.

218
00:15:16,382 --> 00:15:20,920
This is actually taken out of something
that Peter Thall at MD Anderson

219
00:15:20,920 --> 00:15:25,090
did, which is interesting because Peter
Thall really doesn't like this design.

220
00:15:25,090 --> 00:15:31,030
But the basic idea is how many patients do
you have with a DLT and the decision

221
00:15:31,030 --> 00:15:33,098
you should make. A few thoughts.

222
00:15:33,098 --> 00:15:39,371
So, if you have zero in three and zero
in three, you really had to get here through

223
00:15:39,371 --> 00:15:43,909
some de-escalation rule that had to have been
applied at a higher dose.

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00:15:43,909 --> 00:15:49,481
Again, sometimes if I had one in three
and then zero in three, I might escalate

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unless it was a de-escalation rule
that got me to this zero in three.

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But you have to

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decide what's your risk of a DLT
that you're willing to take.

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Is it, you know, 16 to 17 percent,
or is it 33 percent?

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So, what's the risk of having that
severe adverse event?

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It may or may not be that severe depending on
how you set your toxicity levels.

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What percentage of patients
are you willing to have that?

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When you're working with so few patients
here, the confidence interval around

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it is really huge.

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This is the reason a lot of statisticians
now don't use these.

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They use these different adaptive designs to
enroll more people and get more information.

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But it is important to figure out like,
if there is nothing available except death,

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you might be willing to tolerate
a lot of toxicity, or you may say no.

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So, it's important
to figure out what percent of toxicity

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and what toxicities
should be your dividing line.

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So, those stars in that previous table,
again, you're implicitly targeting

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a dose with the probability of toxicity
less than or equal to 17 percent.

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But if you're going to look at the two
out of six, then you're saying you're willing

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to have an MTD with a probability of toxicity
at about 33 percent.

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Key point in these trials, patients
do not go on to the next dose level

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until all the patients at the previous dose
level are beyond

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the time frame
where you plan to be looking for toxicity.

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Do not tell me
you need to keep enrolling them, "But

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they're here." You have to make sure
it's safe to put them on the next level.

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Your patients have to be there.

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You have to finish analyzing
-- you have to finish following them.

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You have to clean their data.

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Analyze their data. And make a decision,
do we escalate?

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Yes or no?

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If you do not have the data to decide
if you can escalate,

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you cannot ethically escalate.

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Again,
not a power or sample size calculation issue.

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There
are lots and lots and lots of new methods.

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Almost all of them are better than the method

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I just told you,
but they are also all based on that method.

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Sometimes in Phase I now I'm
randomizing to multiple different arms.

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I may have various control arms, might be
placebo, might be active control. Why?

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Because when I'm expecting a lot of toxicity,

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again, I need to have this comparison
to figure out, is this additional toxicity

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because of my new therapy, or is this
my general background rate of toxicity?

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Especially when I'm dealing with diseases

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that maybe
I don't want to have a lot of toxicity,

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I want to make sure that they're safer,
I might put six to 15 plus people per arm.

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Also, if I'm expecting a wide heterogeneity
in the effects and the toxicities,

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I may want to include more people
in this Phase I layer of work.

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But there are CRM, Bayesian methods, and lots
of different methods you can use here.