Supporting Materials for Chapter 2

**T**his appendix provides the supporting technical material for Chapter 2. This information includes the relationship between biological efficacy and effectiveness, approximations for the effects on sample size of incorrect assumptions about HIV incidence and the relative risk of an intervention, and the justification for the approximate formula for determining the relative size of those two factors.

The overall effectiveness of an intervention depends on its biological efficacy and the degree to which individuals adhere to the product’s intended use. Nonadherence can be due to several factors. For example, individuals may never initiate the intervention, or they may be required to discontinue use of a product at some point because of a serious side effect, pregnancy, or other reason.

This section develops a simple approximation to illustrate how imperfect adherence dilutes the effect of an intervention. In this simple case, only a proportion, say *f*, of the individuals assigned to the intervention actually initiate it and use it as intended, while the remaining subjects never initiate the intervention.

Suppose that in the absence of an intervention, the incidence rate of HIV infection is *I*_{0}*.* Suppose also that the biological effect of the intervention, if it was initiated, reduced the incidence rate from *I*_{0} to *I*_{1}. *Biologi-*

Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.

Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.

OCR for page 236

Appendix C
Supporting Materials for Chapter 2
T
his appendix provides the supporting technical material for Chapter
2. This information includes the relationship between biological
efficacy and effectiveness, approximations for the effects on sample
size of incorrect assumptions about HIV incidence and the relative risk
of an intervention, and the justification for the approximate formula for
determining the relative size of those two factors.
RELATIONSHIP OF EFFECTIVENESS TO
BIOLOGICAL EFFICACY AND ADHERENCE
The overall effectiveness of an intervention depends on its biologi-
cal efficacy and the degree to which individuals adhere to the product’s
intended use. Nonadherence can be due to several factors. For example,
individuals may never initiate the intervention, or they may be required to
discontinue use of a product at some point because of a serious side effect,
pregnancy, or other reason.
This section develops a simple approximation to illustrate how imper-
fect adherence dilutes the effect of an intervention. In this simple case, only
a proportion, say f, of the individuals assigned to the intervention actually
initiate it and use it as intended, while the remaining subjects never initiate
the intervention.
Suppose that in the absence of an intervention, the incidence rate of
HIV infection is I0. Suppose also that the biological effect of the interven-
tion, if it was initiated, reduced the incidence rate from I0 to I1. Biologi-

OCR for page 236

APPENDIX C
cal efficacy is defined as the proportionate reduction in the incidence rate
associated with use of the intervention. That is,
I1
Biological efficacy = 1 − = 1 – r,
I0
where r = I1 / I0. The quantity r is sometimes referred to as the relative risk,
denoted RR.
Now consider the effectiveness of the intervention in a population of
individuals where a proportion f use the intervention as intended, and as
a result have incidence rate I1. The remaining (in proportion 1 – f) do not
initiate the intervention, and as a result have incidence rate I0. We refer to f
in this setting as the adherence rate. The incidence rate I in this population
is a weighted average of the incidence rates I0 and I1:
I = frI 0 + (1 − f ) I 0 .
We define the effectiveness of the intervention to be the proportion-
ate reduction in the incidence rate, when accounting for the possibility of
imperfect adherence. That is, the effectiveness of the intervention is defined
as
I
1− R = 1−
I0
= 1 − ( fr + 1 − f )
= (1 − r ) f .
That is,
(1)
Effectiveness = Biological Efficacy × Adherence Rate.
Thus the effectiveness of an intervention depends on both its biological
efficacy and the adherence rate.
IMPACT OF ERRORS IN ASSUMPTIONS ON
REQUIRED SAMPLE SIZE
This section discusses the impact on sample size requirements of errors
in estimating two critical factors: the incidence rate of HIV infection in the
control group, and the relative risk of the intervention.

OCR for page 236

METHODOLOGICAL CHALLENGES IN HIV PREVENTION TRIALS
Impact of the HIV Incidence Rate on Required Sample Size
Consider a two-arm randomized intervention trial with equal numbers
of subjects on each arm. Suppose that the number of events (infections),
X1, in the control group has a Poisson distribution with expectation I T,
where I is the HIV incidence rate and T is the total person-years of follow-
up in the control group. Similarly, assume that the number of events in the
intervention group has a Poisson distribution with mean (I R T), where R
is the relative risk of the intervention.
We assume that the trial is designed with equal sample sizes and follow-
up in the two groups, so that person-years are the same in each group. We
are interested in testing the null hypothesis of no effect of the intervention:
that is, Ho:R = . Investigators typically calculate sample size proceed by
assuming a plausible a priori value for R, and then size the trial to have
adequate power to detect that relative risk. The value of R is usually of
sufficient public health significance to warrant adoption of the intervention
in the community if the trial demonstrates efficacy.
As discussed by Cox and Hinkley (1974), it is appropriate to condition
on the total number of events (D = X + X). Conditional on D, we have
X~binomial(D, p), where p = 1/(1 + R). Thus, the problem of determining
the sample size for the two-arm trial is essentially equivalent to finding the
sample size in a one-sample binomial situation.
Accordingly, the number of required events D depends only on the
Type 1 error rate α, and on the power β to detect a specified relative risk
R (see Breslow and Day, 1987, p. 282). We denote D, the required number
of events, by the function g (R, α, β).
Suppose we initially assume that the incidence in the control group is
I1. Then, because the expected number of events is equal to the person-time
per group (T) multiplied by I1 ( + R), the person-time of follow-up that
is needed is
g (α , β , R)
.
I1 (1 + R)
However, suppose that the true HIV incidence rate in the control group
is I2 and not I1. Then the person-time per group that would be required to
have the power β to detect the relative risk R is actually
g (α , β , R)
.
I 2 (1 + R)
Taking the ratio of these last two equations, we see that the factor g (α,

OCR for page 236

APPENDIX C
β, R)/(1 + R) cancels. Thus the ratio of person-time that is actually needed
to that which was originally planned is I/I. That is, the person-time that
is actually required to achieve the desired power is (1/I2) times larger than
originally planned. If the duration of follow-up is unchanged, then the
required sample size is also (I1/I2) times larger than originally planned.
We illustrate this result with an example. Suppose the incidence rate is
initially assumed to be 5 percent, but that it is actually only 4 percent. Then
the study would require 25 percent more person-years (1/I2 = 1.25) than
originally planned to achieve the desired power. The increase in person-
years can be achieved by increasing the sample size by 25 percent, increas-
ing follow-up time by 25 percent, or through a combination of increases in
follow-up duration and sample size.
For example, a 25 percent increase in person-years can be achieved by
a 10 percent increase in sample size together with a 14 percent increase
in duration of follow-up for each participant (that follows because 1.10 ×
1.14 = 1.25). As shown in Chapter 2, the consequence of not modifying
the sample size or planned duration of a trial in this setting is a reduction
in the power of the trial to detect an intervention effect.
Impact of the Relative Risk on Required Sample Size
In this section we develop some simple approximations for the impact
on the sample size of changes in the assumed relative risk. The required
total number of events (D) for a two-sided test at level α, with power β to
detect a relative risk of R is to a first approximation (see Breslow and Day,
1987, p. 282), is
(z ) (1 + R)
2 2
+ z1− β (2)
α /2
D≈ ,
(1 − R)
2
where zα is the α critical value of a standard normal.
Suppose a study is designed with an assumed HIV incidence rate of I
and power β to detect a relative risk R1. Then the number of person-years
T1 (per group) needed to yield the expected number of events is obtained
by dividing equation (2) by I1 (1+R).
However, suppose in fact that the true incidence rate and relative risk
are I and R, respectively. Then the ratio of the required person-time T2
(per group) based on the correct specifications (I2, R2) to the person-time
T1 (per group) originally planned based on the incorrect assumptions (I1,
R1) is
T2 I1 (1 − R1 ) (1 + R2 )
2
≈ .
T1 I 2 (1 − R )2 (1 + R ) (3)
2 1

OCR for page 236

0 METHODOLOGICAL CHALLENGES IN HIV PREVENTION TRIALS
This equation can be further simplified if we ignore the last factor,
yielding the following approximation:
T2 I1 (1 − R1 )
2
(4)
≈ .
T1 I 2 (1 − R )2
2
Thus, the ratio of the person-years that are actually required to those
originally planned approximately varies inversely with the product of the
ratio of incidence rates and the ratio of the square of the effectiveness
parameters. The approximations leading to equations (3) and (4) are most
accurate when the relative risks are near 1.
We illustrate these results with an example. Suppose the study was orig-
inally designed to detect an effectiveness of 0.3 (corresponding to a relative
risk of 0.70), but that in fact the effectiveness is actually 0.2 (corresponding
to a relative risk of 0.80). How much larger would the study need to be to
detect this more modest effect with the same power as originally planned?
If all other factors remained unchanged, the person-years required
would need to be at least about 2.25 times larger than originally planned
(because (0.3/0.2)2 = 2.25). (This calculation is based on equation 4. If a
more precise calculation is desired, equation 3 could be used instead, which
gives a factor of 2.38.) If no change is made to the duration of follow-up,
then the sample size would need to be inflated by the factor 2.25 to achieve
the needed increase in person-years.
Impact of Imperfect Adherence on Required Sample Size
The results in the preceding section can be used to illustrate the impact
of imperfect adherence on the size of trials. As shown by equation 1, non-
adherence dilutes the effectiveness of an intervention.
Suppose we design a trial to evaluate an intervention that is believed to
have a biological efficacy (1 – r). Suppose we also believe that the adher-
ence rate can be expected to be about f1. We design the trial to detect an
effectiveness of f1 (1 – r) with a specified power. We then ask how sensi-
tive the required size of the trial is to the assumed value of the adherence
rate, all other things being equal (such as HIV incidence, study power, and
biological efficacy).
That is, suppose we assume an adherence rate of f instead of f1. It
follows from equation 4 that the ratio of person-years, if we assume an
adherence rate of f instead of f1, is
2
T2 f1
≈ .
T1 f2

OCR for page 236

APPENDIX C
Thus, the ratio of required person-years varies inversely with the square
of the ratio of the adherence rates.
For example, suppose a trial is designed with the expectation that the
adherence rate for an intervention (such as a microbicide) is f1 = 0.90. We
ask how much larger the trial would have to be if the adherence rate is
actually only f = 0.5. We find that the person-years required would need
to be 3.24 times larger (because (0.9/0.5)2 = 3.24).
IMPACT OF MULTIPLE FACTORS
The required sample size for a trial can be sensitive to a small differ-
ence in a single key design parameter, such as the HIV incidence rate or the
relative risk, as the preceding sections show. If there are small differences
in both the actual incidence rate and the actual relative risk compared with
the assumed rates, then these differences compound, and can result in large
differences in the required sample sizes.
For example, suppose a trial is designed to detect an effectiveness of 0.30
based on an annual incidence rate of 0.05. If instead we assume an effective-
ness of 0.25 based on an incidence rate of 0.04, we find that the trial would
need to be approximately 1.8 times larger (because (0.05/0.04)(3/0.2)2 = 1.8).
Such sensitivity analyses of the trial size to multiple design parameters
are important in assessing the feasibility of a trial and the likelihood of
success.
JUSTIFICATION FOR THE APPROXIMATE FORMULA FOR
DETERMINING THE RELATIVE SIZE OF
EFFECTIVENESS AND EFFICACY TRIALS
As noted in Chapter 2, because of the lack of a validated surrogate
endpoint for HIV infection for non-vaccine HIV prevention trials, a tradi-
tional efficacy trial and and an effectiveness trial differ in duration, the HIV
incidence rate in the control group, and the relative risk of the intervention.
For example, a trial that focuses on evaluating the biological efficacy of an
intervention may follow a highly compliant population for a short period
of time, whereas an effectiveness trial may follow a more heterogeneous
population for a longer period of time.
The degree to which the risk-taking behavior of subjects changes dur-
ing either trial may determine the HIV incidence rate in the control group.
And the underlying incidence rate in the more compliant population could
well be different from that in the more general population—either lower
or higher.
The third factor, relative risk, may largely reflect adherence. Investiga-
tors might expect a population used in an efficacy trial to be more adherent

OCR for page 236

METHODOLOGICAL CHALLENGES IN HIV PREVENTION TRIALS
than a population used in a longer effectiveness trial, though this may not
always be the case, as Chapter 2 notes.
Table C-1 gives the approximate ratios in sample size of efficacy and
effectiveness trials for different HIV incidence rates (I1 and I2) in the control
group, relative risks (RR1 and RR2), and duration of the effectiveness trial
(D2). For all settings, the efficacy trial is of six months duration. In almost
all settings, the sample size needed for the efficacy trial exceeds that of the
effectiveness trial, in many cases by a factor of two or more. The efficacy
trial is smaller only when its control-group incidence rate is substantially
higher and the intervention effect is substantially stronger in the efficacy
trial than in the effectiveness trial.
Let I1 and I2 be the HIV incidence rates in the control group for the
two designs, and let R1 and R2 denote the corresponding relative risk of
intervention: control. The ratio of the required person-years in the second
trial compared with the first trial is approximately
2
T2 I1 1 − R1
= .
T1 I 2 1 − R2
Suppose the duration of follow-up of each participant in these trials is
denoted d1 for the first design and d2 for the second design. Then the ratio
of required sample sizes (n) for the two trial designs is approximately
2
n2 I1 1 − R1 d1
= .
n1 I 2 1 − R2 d2
Thus, the relative sample sizes approximately depend multiplicatively
on the ratio of incidence rates, the ratio of the square of the effectiveness,
and the relative duration of follow-up.
To illustrate, suppose investigators are considering two designs to eval-
uate a particular microbicide. The adherence rates in the first and second
designs are 0.90 and 0.50, respectively. Assume that the biological efficacy
of the microbicide is the same in the two designs. If follow-up (per person)
were twice as long in the second design as in the first, the first design would
require a larger sample size if I1/I2 <0.62 (because (0.5/0.9)2 × 2 = 0.62).
Thus, in this example, if the HIV incidence rate in the control arm
of the first design is smaller than that in the second design—in particular,
less than 62 percent that in the second design—then the first design would
require a larger sample size.

OCR for page 236

APPENDIX C
TABLE C-1 Relative Sample Sizes of Efficacy Trials to Effectiveness
Trials for Varying Control Group HIV Incidence Rates, Relative Risks
(RR), and Participant Follow-Up (in years)
Efficacy trial size Effectiveness trial Sample
Pt FU Incidence RR Pt FU Incidence RR Ratio
0.5 2% 0.4 4 4% 0.6 7.11
0.5 3 5.33
0.5 2 3.56
0.5 0.3 4 0.6 5.22
0.5 3 3.92
0.5 2 3.56
0.5 3% 0.4 4 4% 0.6 4.74
0.5 3 3.56
0.5 2 2.37
0.5 0.3 4 0.6 3.48
0.5 3 2.61
0.5 2 1.74
0.5 4% 0.3 4 4% 0.6 2.61
0.5 3 1.96
0.5 2 1.30
0.5 0.2 4 0.6 2.00
0.5 3 1.50
0.5 2 1.00
0.5 4% 0.4 4 4% 0.6 3.56
0.5 3 2.67
0.5 2 1.78
0.5 0.3 4 0.6 2.61
0.5 3 1.96
0.5 2 1.31
0.5 4% 0.4 4 3% 0.6 2.67
0.5 3 2.00
0.5 2 1.33
0.5 0.3 4 0.6 1.96
0.5 3 1.47
0.5 2 0.98
0.5 4% 0.3 4 2% 0.6 1.31
0.5 3 0.98
0.5 2 0.65
0.5 0.2 4 0.6 1.00
0.5 3 0.75
0.5 2 0.50
NOTE: Pt FU = participant follow-up.

OCR for page 236

METHODOLOGICAL CHALLENGES IN HIV PREVENTION TRIALS
REFERENCES
Breslow, N. E., and N. E. Day. 1987. Statistical methods in cancer research. International
Agency for Research on Cancer workshop, May 25–27, 1983. IARC Scientific Publica-
tions (82):1-406.
Cox, D. R., and D. V. Hinkley. 1974. Theoretical statistics. Toronto: Chapman & Hall.