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Statistical Monitoring of Clinical Trials

Chapter 6 Looking Forward - Conditional Power.pdf 220KB

Chapter 3 Probability Tools for Monitoring Rules.pdf 226KB

Chapter 2 The Basis of Statistical Reasoning in Medicine.pdf 149KB

Back Matter.pdf 237KB

Chapter 7 Safety and Futility.pdf 197KB

Chapter 8 Bayesian Statistical Monitoring.pdf 191KB

Chapter 5 Group Sequential Analysis Procedures.pdf 270KB

Chapter 4 Issues and Intuition in Path Analysis.pdf 285KB

Chapter 1 Here, There be dragons.pdf 186KB

front-matter.pdf 111KB

Issues and Intuition in Path Analysis

4.1 Pondering the Meaning of It All

Monitoring is an ethical requirement of modern clinical research, and its application

must balance the need for the study to end when clear evidence demonstrates the

direction and magnitude of the treatment effect with the requisite for precise and

persuasive estimates of that treatment effect. In addition, difficulties generalizing

from a sample to a population are compounded by the new problem of generalizing

from only a fraction of the sample to the larger population.

In addition, the investigator is beset by the vexing problem of drawing

conclusions from a stream of unpredictable data observed for only a short period of

time. Observing a random process for a relatively brief time during which one

hopes to see its long-term target presents its own problems, as the following illus-

tration, adapted from [1] demonstrates.

In a rush to get to the hospital, the university, the airport, or the

institute, you get caught in a traffic jam that stretches out for

miles in front of you There are two lines of slowly moving traf-

fic, and you slide into one, nudging in behind the car in front of

you. As it turns out, I am in the car immediately behind you, and

because you chose one lane, I choose the other. There we are,

side by side in two different lanes of traffic, sitting still more

than we are moving. You have a statistician in your car as a pas-

senger, and nudging him, you point to my car, saying,

“You’re an expert in these things. How likely is it that

we will pull ahead of Lem?”

Answering with smooth assuredness, he says, “You will

definitely pull far ahead of him. Just be patient.”

However, as you watch, I slowly pull ahead of you.

First I am one car length ahead of you, then two, and then three

full car lengths. A few minutes later, I am out of sight. You are

befuddled, but your statistician still appears remarkably confi-

dent. Turning to him, you remark, “I thought that you said I

would pull ahead of him?”

“Yes, I did. And I will say it again. You will pull far

ahead.”

80 4. Issues and Intuition in Path Analysis

“Well, just when was that going to happen?”

“You have to wait an infinite amount of time.”

It just takes you a moment to compose your answer,

“So do you. Get out!”

After your passenger steps out, now himself looking be-

fuddled, you are left alone (still stuck in traffic) to ponder the

meaning of it all.

The trends appearing in research data that are collected, analyzed, and then

sent to a DMC can appear to be like the relative movement of traffic in the preced-

ing example. First a pattern appears, suggesting one result. This pattern then disap-

pears, replaced by another ephemeral trend. One reason that results from random

data are so challenging is because their content can frequently appear to be unreli-

able. We want to believe what they “say”, but if we wait a little longer they “say”

something else. Perhaps it is not quite fair to conclude that this incoming data is

misleading, but it can be fickle.

4.2 Generalization Complexities

In Chapter Two, we described the types of misleading conclusions from samples

that are the product of sample-to-sample variability or sampling error. Because

sampling error is embedded in the data, actually insinuating itself into the study

results, a careful and discerning eye is required to separate the signal that reflects a

true population effect from the background noise.

The circumstance is not less, but more, perilous when scientists contem-

plate discontinuing a study early. Consider an investigator who is designing a study

of 400 patients in order to examine the effect of a new therapy on the reduction of

systolic blood pressure (SBP) in patients with isolated systolic hypertension (ISH).

Understanding that any attempt to generalize the results of a 400 patient study to a

population of millions of patients might be hazardous, she proceeds carefully. She

chooses a sample size that was computed based on her best estimate of (1) the ef-

fect of her therapy on SBP, (2) the variability of SBP, and (3) acceptable levels of

type I and type II error rates. In addition, she selects her subjects as randomly as

possible in order to reduce the problems that she would have in extending her find-

ings.

Finally, she also knows that her assumptions about the study findings

might be wrong. For example, the intervention might be far more effective in reduc-

ing SBP than she anticipated. Alternatively, a new and unanticipated adverse effect

may appear in these subjects that would require her to terminate the study early. In

recognition of these possible outcomes, she begins the thought process that would

lead to the justifiable early termination of the study.

It is relatively easy to understand the ethical difficulties that would ensue

if premature study termination were not possible. However, early termination is not

without its own set of difficulties. A principal problem involves generalizability.

Another point of view is that we are just too impatient. If we could wait an infinite amount

of time, the clear solution would be readily apparent.

4.3. Why Are Special Tools Necessary? 81

The investigator’s initial plans required her to recruit, randomize, and follow 400

patients in order to provide assurances to the medical and regulatory community

that sampling error had been adequately controlled (Figure 4.1 Panel 1). Now, she

is contemplating the possibility that she may have to end the study before that goal

is achieved. Thus, in the case of early termination, there are two levels of generali-

zation. The first involves extending the results of the analysis that was carried out

on only a fraction of the data to the entire sample; the second is to generalize from

the sample to the entire population (Figure 4.1Panel 2)

Panel 1

Panel 2

Population

Sample

Figure 4.1. Comparing the task of generalizing a sample result to the population

(Panel 1), to generalizing from a fraction of the sample to the population (Panel 2).

Sample

All sample data is evaluated before generalization

is attempted.

Only a fraction of the sample data is evaluated, leading to

prediction of the result of the sample, and then the final

generalization to the population.

Thus, in circumstances where there will be early termination of a research effort,

there must be especially tight control over the type I error rate if the result is to sur-

vive this two-step generalization process. In addition, we must ensure that the esti-

mates of effect sizes are especially trustworthy in order to successfully guide the

investigator through these multiple levels of generalization to a clear view of the

population findings.

4.3. Why Are Special Tools Necessary?

We have provided ample motivation for the need to monitor clinical studies; how-

ever, we have not yet justified the use of any particular procedure to carry out this

monitoring. In fact, the investigator who has observed the wealth of tests already in

82 4. Issues and Intuition in Path Analysis

service

may be forgiven for asking the question, “Why is yet one more test neces-

sary?” However, there are three new issues in clinical trial monitoring that the clas-

sic hypothesis testing procedures do not directly address. They are (1) repeated test-

ing of data, (2) variability, and (3) the notion of dependency.

4.3.1 Repeated Testing

We have established that one of the most important issues in generalizing results

from a sample to the population is the effect of sampling variability. One notewor-

thy way in which sampling error can mislead an investigator is by the construction

(through the random aggregation of subjects in a sample) of a result that can appear

to be clinically relevant but at the same time is not representative of the population.

The two types of sampling errors that clinical researchers measure are the type I or

alpha error rate and the type II or beta error rate, as discussed in Chapter One. The

medical and regulatory communities are comfortable with drawing conclusions

from concordantly executed studies

†

when these rates are kept at acceptably low

levels.

However, these rates can grow to unacceptably high levels when statistical

testing is carried out repeatedly in the same research effort. The monitoring of a

clinical study during the course of its follow-up period is a clear example of this

phenomenon. Other illustrations are multiple endpoint evaluation, subgroup ana-

lyzes, and the evaluation of different contrasts between the arms of a clinical trial

with more than two treatment groups. Difficulties with these analyzes have been

well elucidated in the literature [2,3,4]. The particular problems induced in the in-

terim monitoring setting have also been elaborated [5].

The principal difficulty with multiple analyzes is that the overall false

positive error rate or alpha error rate increases with the number of tests that are exe-

cuted. Thus, although each test provides the same level of protection, the integrity

of the overall process degrades. This is easily demonstrated. Consider an example

of a clinical study that assesses the ability of a therapy to reduce the fatal stroke rate.

At the conclusion of the research, the investigator plans to construct a test statistic

that will produce a type I error rate of 0.05. However, the investigator intends to

have the study monitored every year until the five-year study has concluded. At

each monitoring point, he is looking for an early demonstration of the same effect

that he hopes will be demonstrated at the conclusion of the five-year study. There-

fore, a treatment effect finding resulting in a p-value  0.05 for any of the five ana-

lyzes would be sufficient for him.

At first appearance, this collection of tests may appear to offer substantial

protection against the occurrence of an alpha error or false positive results. After all,

this 0.05 level of protection was satisfactory for drawing conclusions at the end of

the research effort. If it will be adequate when applied at the end of the study, why

wouldn’t it afford adequate protection during the interim monitoring times?

T-tests, chi-square tests, tests of equality of proportions, life table analyzes, and Bayes pro-

cedures are but a few of the many types of test statistics brought to bear in the evaluation of

clinical research data.

†

A concordantly executed study is one that is follows its prospectively written protocol.

4.3. Why Are Special Tools Necessary? 83

The problem is produced by the change in the fundamental event of inter-

est to the investigator. When there was only one evaluation at the conclusion of the

study, the focus of his attention was the occurrence of a type I error at the end of the

study. With the institution of multiple monitoring, the question is now, “What is the

probability that the study produces a false positive result either at the end of the

study or during the monitoring procedures?” Because the study could be stopped at

every monitoring point, we now need the probability that it meets the criteria for

early termination at any one of these points.

Assuming for the moment that these tests (designed to determine the supe-

riority of treatment) are independent of one another,

this probability is easy to

identify. Let ȟ be the probability that at least one type I error occurs. Then, when

there is only one hypothesis test at the end of the study, ȟ = 0.05. However, when

there are five such tests, we must find the probability of at least one type I error. We

begin by finding the probability of no type I errors. The

P[no type I error on the

first test ] = 1 – 0.05 = 0.95. Thus

P[no type I error on all five tests] = (0.95)

P[at least one type I error on the five tests] = 1 – (0.95)

= 0.226.

The likelihood of a type I error has increased from 0.05 to 0.226, representing more

than a fourfold increase. A more complete examination reveals the rapid increase in

the overall type I error rate (sometimes called the familywise error rate or FWER [6,

7]) as a function of the number of tests (Figure 4.2.)Armitage et al. [8] took this one

step further by computing the probability that a test statistic that follows the stan-

dard normal distribution will exceed any threshold under the null hypothesis. This

demonstration revealed that, if an investigator evaluated his data five times during

the course of a study, the actual type I error level is 2.5 to 4 times as large as he had

planned.

†

These evaluations are the quantitative justification for the use of the pejo-

rative term “testing to significance”.

The underlying reason driving this phenomenon is the random sample. As

the sample grows, the data randomly aggregates and disaggregates in different pat-

terns, sometimes making sense, other times not, but always random. As the data set

increases, the inclusion of additional data provides continued opportunity for the

data points to randomly arrange themselves into the pattern for which the investiga-

tor was looking. Just as searching the clouds long enough will eventually reveal the

ephemeral shape of an elephant, or a ship moving backwards, the continued inter-

rogation of data, itself randomly aggregated, will reveal the interesting pattern that

the investigator anticipated. As Miles pointed out, “If you torture a dataset long

enough, it will tell you want you want” [9].

We will see shortly that they are not independent at all, but this scenario is provided solely

for illustration.

†

This is a basis for requiring smaller type I errors during the interim monitoring procedure.

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