J Chron Dis 1976, Vol. 29, pp. 613-624.
Pergamon
Press. Printed
in Great
Britain
DESIGN OF BLOOD PRESSURE SCREENING FOR CLINICAL TRIALS OF HYPERTENSION ANNE GOLDMAN Div. of Biometry,
University (Rwriued
of Minnesota, in revised jbrm
Minneapolis.
MA 55455, U.S.A.
15 April 1976)
Abstract-In clinical trials of hypertension the eligibility of a subject is usually determined at a series of screening visits prior to randomization. Only those subjects whose blood pressures are within certain limits at each of specified visits are eligible. The pressure measurement of any individual has considerable time-to-time variability, therefore subjects who have passed one screen may ‘regress towards the mean’ and not pass subsequent screens. This paper is a statistical examination of the effect of different numbers of screens and pressure limits on the probability of a given subjects passing all screens. the proportion of subjects who pass, and the distribution of their blood pressures. Numerical estimates, using data from the literature, indicate that the best way to maximize yield while controlling the distribution of blood pressures of selected subjects is to have several screens with pressure limits slightly wider than the goal.
INTRODUCTION IN CLINICAL trials involving hypertension, it is often considered necessary to have a series of screening visits where the eligibility of the subjects is determined before they are enrolled in the study. Usually, the blood pressure is taken at each of these visits and only those subjects with diastolic or systolic pressures within a certain range at one or more specified visits are considered eligible. When the subjects with those higher pressures are remeasured at subsequent screening visits, many of them have much lower pressures. This phenomenon is due to two wellknown effects: a ‘regression towards the mean’ and a ‘pressor effect’. The regression effect is a statistical phenomenon that can occur when measurements which show time-to-time random variability are repeated. When selected persons from a population are re-measured because their first blood pressures were much higher (or lower) than the population mean, then those repeat pressures will, on the average, be much closer to the mean. The pressor effect is a postulated psycho-physical effect whereby a person’s blood pressure tends to read higher when taken under the stress of unfamiliar circumstances; when retaken, there is presumably less stress and therefore the pressure is lower. The pressor effect will be ignored in the discussion that follows, but the random variability of repeated measurements and the
613
ANNE GOLDMAN
614
consequent regression effect must be taken into account in the design of pressure screens. Blood pressure screening may be done in several ways. Both the eligibility limits for pressures and the number of screening visits may be varied. For example, in the Multiple Risk Factor Intervention Trial [l], risk eligibility is determined based on three risk factors, one of which is blood pressure, at the first visit only. A second and third visit provides data for exclusion baseline information and for exclusion purposes, but blood pressure is not used for exclusion. Treatment begins after the third visit at which time some subjects, who at first screen had elevated pressure, have normal pressure and do not require treatment. In the NHLI-VA Study of Mild Hypertension [2], the subject’s pressure must be within defined limits on all three screening visits to be eligible. With this screening method, eligible subjects are clearly mild hypertensives at randomization and may be treated. However, the screens are so restrictive that massive numbers of persons must be screened in order to identify sufficient numbers of eligible subjects. Similarly, in the NHLI Hypertension Detection and Follow-up Program [3], the subjects must have pressures above defined limits at each of two screens to be eligible. This paper is a statistical examination of the consequences of different screening designs. By making a few general assumptions, tools are developed for examining the effect of different numbers of screens and pressure limits on the probability of a given subject being eligible for the study, the proportion of subjects who pass the screens, and the distribution of their blood pressures. Numerical estimates are calculated using blood pressure distribution data from the literature. BASIC
DEFINITIONS
AND
ASSUMPTIONS
The screening population is defined as the group of individuals who are screened, and the study population as the group of individuals who pass the screens and are randomized or otherwise followed. The goal of the screening is to select subjects with defined blood pressure characteristics, pressure being the only criterion for eligibility. When an individual’s blood pressure is repeated, it will be somewhat different each time, varying around that individual’s long-term mean. We therefore distinguish between a one time or casual pressure and the long-range mean pressure. We let pi be the mean pressure for individual i and assume that the casual readings will be normally distributed with long term, mean pi and standard deviation pi. In some studies an individual’s blood pressure is measured two or three times at each visit, 5 or 10 min apart, and the average of these readings is used as the pressure for that visit, thus reducing the variability. To avoid confusion the pressure at one visit will be referred to as a ‘casual pressure’ even when it is based on the average of several readings. In past studies of hypertension, the goal of screening has been the identification of persons with one or more casual readings within specified limits. These limits have frequently been based on categories of risk calculated from data of the Framingham Study [4] using one casual blood pressure reading to define the categories. When risk categories are based on one casual reading, it follows that each category will consist of individuals whose mean pressures are distributed over a wide range. The overall risk for the group thus is an average for individuals,
615
Design of Blood Pressure Screening
some of whom are at high risk and some who are at low risk. Contrary to current practice, it is therefore suggested that the goal of screening should be to select a group of subjects who are at homogeneous risk due to blood pressure and that therefore subjects should be selected whose mean pressures are inferred to be within specified limits. It is assumed that there is no ethical difficulty in prescribing the contemplated treatments to persons whose casual pressures occasionally measure slightly higher or lower than the screening goal. Suppose that for a study of mild hypertension, the screening goal is defined as a mean diastolic pressure of 9(r105 mm of mercury. The contemplated treatments are an anti-hypertensive drug and a placebo. All subjects with mean pressures within the goal limits can be considered treatable since the anti-hypertensive drugs may be prescribed to patients with casual diastolic pressures as low as 85 mm, and the placebo to patients with pressures as high as 110 mm. These limits will be called the treatment limits. A screening design consists of a specified number of screens. At each screen j there is a lower pressure limit aj and an upper limit hj. A subject i passes screen visit ,j if the casual pressure measurement for that visit is between Uj and hj.
The main assumptions may be summarized as follows: 1. Screening goal: mean diastolic pressure 90-105 mm. 2. Treatment limits: casual diastolic pressure 85-l 10 mm. 3. Screening limits: aj to bj mm of pressure at screen j. 4. Casual measurements on one individual i are independent and normally distributed with mean Iii and standard deviation pi mm. The results given here are derived from the above assumptions and some elementary probability theory. The mathematical derivations are shown in the Appendix. THE
PROBABILITY
OF A GIVEN
PERSON
PASSING
THE
SCREENS
In order to estimate the probability of a given subject i passing all the screens, we let Sij be the event of subject i passing screen j. Then the probability of Sij given pi and (Ti is:
(1) where Q(x) is the value of the cumulative standard normal up to x. Then the probability of subject i passing three screens is simply the product, or: P(SilPi,ai)
=
Gil
ci2
ci3.
(2)
Probabilities for any number of screens can be similarly derived. For simplicity the probability in (2) will be written P(Slp,o). Table 1 shows the probability in (2) for various values of pi and oi when the screening limits are 9G-105 at each of three visits (Design B). Notice that a person whose mean pressure is 90 mm and any oi has a 12.5% chance or less of being judged eligible. Even a person with mean pressure of 97.5 and 0 = 7 has only a 37% chance of qualifying.
616
ANNE GOLDMAN
TABLE 1. PROBABILITYP(SIp, a) OF PASSINGTHREE SCREENS FOR AN INDIVIDUALWITH GIVEN pi,ui. DIASTOLICPRESSURE LIMITS0~ ALL THREESCREENS 90-105 mm (DESIGN B) eqn (2) Within person mean pressure pi 72.5 71.5 82.5 87.5 90.0 92.5 97.5 102.5 105.5 107.5 112.5 117.5 122.5 127.5 132.5
3
1 0 0 0 0 0.125 0.981 l.OOO 0.981 0.125 0 0 0 0
0 0 0 0.001 0.125 0.715 0.999 0.715 0.125 0.001 0 0 0 0
:
Within person standard deviation oi 6 7 4 5
8
9
10 0 0.001 0.010 0.047 0.081 0.120 0.163 0.120 0.08 1 0.047 0.010 OOOl 0 0 0
0 0.008 0.125 0.508 0.963 0.508 0.125 0.008 0 0 0
0 0 0 0.019 0.125 0.394 0.828 0.394 0.125 0.019 0 0 0
0 0 0 0.029 0.124 0.322 0.650 0.322 0.124 0.029 0 0 0
0 0 0.001 0.038 0.120 0.266 0.491 0.266 0.120 0.038 0.001 0 0
0 0 0.003 0.044 0.113 0.219 0.367 0.219 0,113 0.044 0.003 0 0
0 0 0.005 0.048 0.104 0.179 0.217 0.179 0.104 0.048 0.005 0 0
0 0.001 0.008 0.048 0.092 0.146 0.211 0.146 0.092 0.048 0.008 0.001 0
0
0 O.
O 0
0
0
0
0
Table 2 shows a comparison of various screening designs for c( = 7 (Designs A-H). Screen 1 may be thought of as a prescreening conducted by the study personnel out in the field. In practice, the limits for such a screen are frequently different from those on subsequent screens. Many other designs are obviously possible; only some can be listed here. A within person standard deviation (a) of 7 was chosen for illustration since it seems representative of values reported by different observers (see below for further discussion). For comparison, Design I3 is shown in both tables: in Table 1 for all values TABLE 2.
PROBABILITYP(Sjp,a)
OF PASSINGALL SCREENS FOR AN INDIVIDUAL WITH GIVEN pi, 0 = 7, eqn (2) Design A
Pressure limits
No. screens Screen 1 Screen 2 Screen 3 Screen 4
62.5 67.5 72.5 77.5 82.5 87.5 92.5 97.5 102.5 107.5 112.5 117.5 122.5 127.5 132.5
B
4 3 9O-120 85-110 9O105 85-110 90-105 85-110 9O105 Probability 0 0 0 0 0 0 0 0 0.007 0.003 0.094 0.044 0.395 0.219 0.680 0.367 0.59 1 0.219 0.249 0.044 0.040 0.003 0.002 0 0 0 0 0 0 0
C
D
3 85-110 855110 85-110
1 90-120 ~ -
P(S 1p, u) 0 0 0 0.00 1 0 0.006 0.003 0.037 0.142 0.047 0.360 0.261 0.618 0.639 0.794 0.857 0.618 0.957 0.957 0.261 0.047 0.857 0.003 0.639 0.360 0 0 0.142 0.037 0
E
F
G
1 2 3 9O-105 9O-120 9O120 ~ 85-110 85-110 ~ 85-110 ~ ~ -
0 0.001 0.006 0.037 0.141 0.354 0.602 0.716 0.602 0.354 0.141 0.037 0.006 OOOl 0
0 0 0 0.005 0.05 1 0.230 0.545 0.794 0.815 0.611 0.309 0.091 0.013 0.001 0
0 0 0 0.001 0.018 0.147 0.464 0.735 0.694 0.390 0.111 0.013 0 0 0
H 4 9O-120 9O-105 9O105 9O-105
0 0
0 0 0 0.016 0.140 0.315 0.209 0.043 0.002 0 0 0 0
Design of Blood Pressure Screening
617
of ci = l-10, and in Table 2 only for (TV = 7. Notice that in designs with three screens, persons with mean pressure within the screening goal (90-105 mm) have 2-6 times the probability of being included in the study when the limits are 85-110 (Design C) than when they are 90-105 (Design B). At the same time, persons with mean pressure outside the goal limits have only a small probability of being included. When there is only one screen, all subjects have a higher probability of being judged eligible, including those with pressures outside the goal limits (Designs D, E). Other values of oi give somewhat different probabilities, but similar trends. SCREENING
YIELD
The probability derived in eqn (2) is the conditional probability of passing the screens given pi and pi. Since it relates to individuals with known mean blood pressure and standard deviation it may not be of practical interest. However, eqn 2 may be used to derive probabilities and distributions which are useful for designing screens. One question of great concern is the percentage yield of a screening design, or the number of persons found eligible for every 100 that must be seen for at least one screening visit. The greater the yield of a design, the lower the cost of screening because fewer subjects need be examined to attain a specified sample size. This unconditional probability can be found by averaging the conditional probability in eqn (2) over the distribution of CLiand (Tiin the screening population. Let P(p) be the probability of having mean pressure p, and P(o) the probability of having a standard deviation (r in the screening population. It is assumed that P(p) and P(o) are independent, that is, the time to time variability of a person’s pressure does not depend on the level of that pressure. P(p) and P(o) may be estimated from blood pressure studies. It is assumed that p and (T are discrete variables since such studies usually report pressure to the nearest 1 or 5 mm. If p and (T are to be continuous, their density functions would need to be known and eqns (3,4) modified appropriately. The probability of subjects with standard deviation e passing the screens is the conditional probability of S given 0. As shown in the Appendix, we have: P(Slo) = CP(SIP, P The yield of the screens, or the unconditional for a population with P(o) known is:
a) . WL). probability
P(S) = c P(S IO) . P(0).
(3) of passing the screens (4)
Frequency tables from studies of blood pressures of a population similar to that being screened may be used to estimate P(p). However, data on individual standard deviations of pressures are not generally available, although estimates of the average G may be possible. If only the average 0 is available, the screening yield, given that CJ,can be calculated from eqn (3). P (~310)is that proportion of the screening population with standard deviation cr which will pass the screens. Equation (4) however, averages that conditional probability over the distribution of CJfound in the population to give a screening yield for the whole population.
618
ANNE GOLDMAN TABLE 3. THE RELATIVEFREQUENCY P(p) OF DIASTOLICBLOOD PRESSURESFORTWO POPULATIONS. DERIVEDFROMDATA OF THENATIONAL HEALTHSURVEY(TABLE 5) Diastolic pressure (mm of mercury)
Midpoint p
Population I
Population II
5G-54 55-59 60-64 6549 70-14 75-79 8S-84 85-89 90-94 95-99 lOCklO 105-109 110-114 115~119 12G-124 125-129 130-134 Mean SD.
52.5 57.5 62.5 67.5 72.5 77.5 82.5 87.5 92.5 97.5 102.5 107.5 112.5 117.5 122.5 127.5 132.5
0.010 0.024 0.061 0.121 0.154 0.192 0.154 0.1 13 0.071 0.045 0.024 0.015 0.006 0.006 0.002 0.002 0.001 79.48 12.17
0 0 0 0 0.005 0.035 0.109 0.202 0.224 0.193 0.1 12 0.07 I 0.026 0.018 0.004 0.001 0 94.05 9.05
Table 3 shows the relative frequency of pressures of two different populations used to estimate P(p). Population I represents the general adult population. The data are taken from the National Health Survey [S] and shows the distribution of diastolic pressure of 11$03 randomly chosen adults age 18-79 yr (pressures of < 50 mm and > 134 mm were ignored). The reported pressures are based on the average of three readings at a single visit and are thus casual pressures rather than the needed distribution of mean pressures. However, the distribution of mean pressures for the general population is not available. It would be a tighter (smaller variance) distribution than that of Population I used here, since the distribution of the mean of a random variable has a smaller variance than that of the random variable itself (casual pressure). Population II represents a population of ‘prescreened’ subjects since frequently a natural source of subjects for screening may be persons who have previously been diagnosed as hypertensives and are then referred to the study. Population II therefore generally has higher pressures than I. The distribution of II was derived from that of population I by applying the method of the next section and will be discussed there. The mean and standard deviations shown in the table are those of the populations, not within individuals. Table 4 shows the relative frequency of within person standard deviatiori used to estimate P(a). Such data are scarce in the literature. The frequencies quoted here were calculated from data of individual blood pressure measurements on 21 subjects [6]. Each subject had supine, diastolic and systolic pressures taken several times daily, except Sunday, over a period of three and a half weeks. All subjects had pressures taken in the morning, repeated about four times at 5-min intervals, and some also had repeated measurements in the evening. The mean
Design of Blood Pressure Screening
6 I9
TABLE 4. THE RELATIVEFREQUENCYP(o) OF WITHIN PERSON STANDARD DEVIATIONOF DIASTOLICPRESSURES. D&RIVED FROM DATA ON 21 SUBJECTS (TABLE 5)
Standard deviation C, mm
123
Number of subjects Relative frequency 00)
0
0
2
4
5
6
7
8
9 10 Total
4
10
2
2
I
0 0
0 0 0.095 0.190 0.476 0.095 0.095 0.048 0
21
0 0.999
Group mean diastolic pressure = 64.14 mm. Average SD. = 5.04mm.
pressure and standard deviation was calculated for each individual disregarding when the pressure was taken and tallied in Table 4. The average standard deviation, found by pooling over the 21 subjects is 5.03 mm. This average (T is probably lower than would be found in a study screening since they were: supine pressures, measured several times a day, measured every day. Most study screens use sitting pressures, one measurement per visit (or possibly the average of several), and the visits are weeks or months apart. Current hypertension studies are reporting variabilities of cr = 7-9 [7] and also in a past study of normal subjects [8]. An average 0 = 7, as used in Table 2 for illustration, seems reasonable. Table 5 shows the yields of various screening designs as calculated from eqns (3,4) for population I and II. Generally, broader screening limits give two to three times higher yields (Designs A vs H, C vs B, J vs I); decreasing the number of screens gives a slightly higher yield (compare Designs A, G, F. D). A comparison of screening population I and a prescreened population (Designs B vs I, C vs J) shows a more than three fold increase in yield. TABLE 5. SCREENINGYIELDS
DeWgn No. screens PreWlre hmits
Screen I Screen 2 Screen 3 Screen 4
Population
C?=
I 2 3 4 5 6 7 8 9 IO
The number of subjects enrolled A B C D
for each 100 subjects E F G
4 3 3 I 9&120 90-120 ~ ~ 85-110 90-105 85-110 85-I 10 90105 85-I IO 85-110 90 105 X5-l IO I 1 I I
I 2 3 90-105 Yo- I20 90-120 ~~ 85-I IO 9&105 85-l 10 ~~ ~~ ~ I I I
‘2, Yield 15.5 15.1 14.3 13.1 11.6 10.2 08.8 07.6 06.5 05.6
Unconditional% yield-P(s) 11.6
given o - P(SIo)*lOO eqn (3) 13.x 26.5 16.6 14.0 11.3 23. I 17.0 14.3 OY.3 20.4 17.5 14.7 07.7 18.5 18.0 15.1 06.4 16.7 IS.6 15.5 05.2 15.1 19.4 16.0 04.3 13.6 20.2 16.6 03.6 12.2 21.0 17.1 03.1 10.9 21.9 17.6 02.7 09.8 22.7 IS.0 eqn (4) (Distribution 06.5 16.7 18.7
15.5 15.7 15.6 15.3 14.9 14.5 14.0 13.6 13.2 12.9
15.5 15.4 14.x 14.0 13.0 11.9 10.9 09.9 09.0 ox.3
screened H
I
J
4 .3 3 9&120 ~~ 90-105 9CLlO5 85 110 9&105 9@ 105 85 ~1IO 9OklO5 Y@~lO5 85 I IO I II II
137 10.7 08.4 06.7 05.2 04. I 03.2 02 5 02.0 01.6
of 0 from Table 4 assumed) 15.6 14.8 12.9 05.4
52.3 43.4 35 Y 29.1 24.2 19.5 157 127 IO.4 OX.6
7’) 7 72.4 66.2 60.9 55.3 49.5 43.x 3x.4 33.5 29 I
24.5
54.9
ANNE
620
THE
DISTRIBUTION OF
GOLDMAN
OF MEAN BLOOD ELIGIBLE SUBJECTS
PRESSURES
The goai of the screening was assumed to be the selection of subjects with mean pressures in the interval 9&105 mm. We therefore need to know the distribution of mean blood pressures of subjects who pass all the screens. The higher the proportion of selected subjects who have mean pressures within the screening goal, the more sensitive the design and the more homogeneous the risk. The relative frequency of different blood pressures for the subjects with given 0 who pass the screens is the probability P@IS,a). As shown in the Appendix, we have:
where &!+,a) is from eqn (2) and P(Sl(r) from eqn (3). This probability is the distribution of mean blood pressures in the study population with a given within person standard deviation. If estimates of P(a) are available, the relative frequency of different mean blood pressures in the study population is the probability P@(S). As shown in the Appendix :
WI9 = 1
WI/& 4 . P(p) . P(o)
P(S)
0
Table 6 shows the conditional distribution of blood pressures with D = 7 for various screening designs and population I and II (P(pjS,o) eqn (5)). Population II in Table 4 was actually found by applying (5) to population I, assuming 0 = 7 and one screen with limits 90-120 mm (Design D). Using population II followed by three screens (Designs I, J) is thus equivalent to using population I and having four screens (Designs H, A respectively), when c = 7. The yields of Table 5 (for G = 7) are also related in that, for example (yield Design D) x (yield Design TABLE 6. DISTRIBUTION OF BLOOD PRESSURES(P(~jz,o)) A
D.?Slgtl
No. screens PrcsslJrc limits
Screen1 screen2 Screen3 Screen4
Population
Mean diastolic pW3s"rc (mm) p
70-74 IS-79 SO-84 85-89 90-94 95-99 100-104 105.-109 110-114 115-119 120-124 125-129
B
C
ON d = 7
OF ELIGIISLESUBJECCTS CONDITLONAL D
E
F
G
H
I
J
4 3 3 I I 2 3 4 3 3 9CLl20 -- 9&120 9&105 9&120 9&120 9CklO2 ~ 85%110 9&105 85-110 ~ 85-110 85-110 90-~10590-105 85-110 85.-1109C~105 85-110 ~ .~ 85-110 9G-105 90-105 85SllO 85SllO 9cLlOS 85-110 ~ _ ~- 90 105 90-105 x5-110 1 1 I 1 i 1 I I II II
0 0 0.012 0.120 0.315 0.350 0.158 0.042 0.003 0 0 0
P(pIS.@+qn 0 0 0.008 0 0.010 0.068 0.116 0222 0.355 0 310 0.384 0.256 0.119 0.104 0.015 0029 0003 0 0 0 0 0 0 0
(5) 0.005 0.035 0.109 0.202 0.224 0.193 0.112 0.071 0.026 0.018 0.004 0.001
0.006 0.043 0.132 0.242 0.256 0.196 0.086 a032 0.005 0.001 0 0
0
0.007 0.056 0.186 0.274 0.257 0.137 0 065 0.013 0.004 0 0
0
0.001 0.026 0.153 0.301 0.307 0.151 0.054 0.006 0.001 0 0
0
0
0.0002 0.:2 0.057 0.057 0.312 0.312 0452 0.452 0.156 0156 0.020 0.020 0 0 0 0 0 0 0 0
0
0.092 0.120 0.316 0.349 0.158 0.042 0.003 0 0 0
621
Design of Blood Pressure Screening
TABLE I. THE DISTRIBUTIONOF BLOOD PREWJRESOFTHESTUDY mPm.AnoN. THE DISTRIBUTIONOFSTANDARD DEV~~TION FROM TABLE 4 ISASSUMED Design
A
No. screens
ScreenI Screen2 Screen3 Screen4
PEWWe lilnits
Population
B
C
D
E
F
G
H
I
.I
4 3 3 I I 2 3 4 3 3 _ 9C~i2090.10590-1209@-1209&120 ~~ 9Gl20 85~11090-10585-110 -~ X5-11085-1109CklO59GlO.585 110 XS~IIO9&105 85-~110 ~~ 85 11090~10.5 9% 105 85-110 85.1109&105 85~110 ~~ 90 1059c-10585~110 I I I I 1 I 1 I II I1
mcl9ewKd MWl diastolic pV?Wlre (mm) I’
Average/I S.D. "0 p in range9Gl05mm :,p in range85~~110mm
l&l4 75-79 X&84 85-89 9&94 95-99 100~104 105~109 110~114 115-119 120-124 125-129
0 0 0.003 0.099 0.346 0.343 0.163 0.044 0.001 0 0 0
0 (! 0.002 0.050 0.361 0.459 0.121 0.007 II II (I (I
0 0.001 0.027 0.233 0.339 0.254 0.114 0.013 0.001 0 0 0
0.001 0.012 0.058 0.182 0.263 0.225 0.124 0.079 0030 0.021 0.004 0.001
0.002 0.014 0.069 0.218 0.311 0.251 0.104 0.029 0.003 0 0 0
0 0.002 0.024 0.159 0.309 0.27X 0.146 0.070 0.011 0.002 0 0
0 0 0.009 0.127 0.332 0.314 0.157 0.056
0 0 0 0 0.019 0.308 0.520 0.144
0 0 0 0.073 0.302 0.514 0.151 0.008
o.MM 0.008 0 0 0 0
0 0 0
0 0 0
0 0 0 0.127 0.328 0.329 0.164 0.045
0.00I 0 0 0
95.9 95.X 94.0 95.5 93.4 95.7 95.8 96.5 96.4 95.x 5.9 3.9 5.6 s.5 5.6 5.6 6.3 4.8 5.7 5.3 85.2",, 94.1':" 70.1",, 61.2";, 66.6""73.3",X0.3""97.2:,, 96.7",, 82.1", 99.5:&99.V'~97.1:"873",,91.3",96.2",9X.6"" IOO",,IOO",,99.3",,
I) = (yield Design H). The point here, however, is not the method of prescreening, but to show the effect of using a prescreened population on the outcome of subsequent screens. Table 7 shows the distribution of blood pressures for the study population for the various designs. Equation (6) was applied by assuming a distribution of standard deviation, P(o), as in Table 4. At the bottom of the table is the average mean pressure and standard deviation of the study population, the percentage that has mean pressure within the screening goal (90-105 mm) (specificity), and the percentage within the treatment limits (85-l 10 mm). Both Tables 6 and 7 show that increasing the number of screens and tightening the screening limits tend to decrease the range of blood pressures of the study population having passed the screens, yet has little effect on the average pressure of the group. Notice that designs with a single screen (Designs D and E) show loo/:, of the eligible population outside the treatment limits. The change of limits in screens 2, 3, and 4 from 9&105 to 85-110 mm (Design H and A) causes a decrease in the percentage of population within the screening goal but a trivial decrease in the percentage within the treatment range. At the same time, Tables 4 and 5 indicate that the yield is doubled by the change. COST
ESTIMATES
AND
COMPARISON
OF
DESIGNS
The cost per randomized subject of some of the screening designs can be estimated. Assume that the cost of seeing one subject for one screening visit is C regardless of visit. For design D, 100 subjects must be seen at one visit to enroll 18.7 in to the study (Table 5); hence, the cost/study subject is lOOC/18.7 = 5.3C.
ANNE GOLDMAN
622
TABLE8. YIELD,COSTAND SENSITIVITY OF SELECTED DESIGNS (C IS THECOSTOF ONLSCREENING VISITFOR ONE SUBJECT)
Design Pressure limits
Yield (Table 5) Cost/study subject Sensitivity (Table 7)
Screen I Screen 2 Screen 3 Screen 4
H
A
B
C
9G-120 90-105 9&105 9&105 5.4%
9LL-120 85-110 85-110 85-l 10 I 1.6%
90-105 90-105 9@105 6.5%
85-l 10 85-t 10 85-l 10 16.1”/
21.7c 97.2%
12.6C 85.2%
19.2C 94.17;
8.9C 70.77;
For design F, if 100 subjects are seen at screen 1, 18.7 will be eligible for screen 2 and 14.8 will be randomized. The cost/study subject is (100 + 18.7)C/14.8 = 8.0 C. Similar estimates can be found for designs F, G, H. Table 8 shows a comparison of selected designs by yield, cost and sensitivity (percentage of study subjects with mean pressure between 90 and 105 mm of pressure). The table shows clearly that designs H and B, although very sensitive, are about twice as costly as designs A and C which have slightly wider limits and slightly less sensitivity. The results in Tables 4-8 show that when only one blood pressure screen is used, some subjects with mean pressures outside the goal limits may be selected and their casual pressures at and after randomization may be outside the treatment limits. On the other hand, when there are several screens and the screening limits are equal to the goal limits, the sensitivity may be very high, but the yield decreases and the cost/study subject increases. The best way to maximize yield and minimize cost while maximizing sensitivity is to have several screens with limits slightly wider than the goal. Decreasing the number of screens has a much less dramatic effect on yield and cost than reasonable modifications in the limits, while a good combination allows control of the distribution of pressures of the study population. The use of a prescreened population, if one is available to the study at low cost, clearly increases the yield dramatically, thereby saving considerable time and money. Unless, however, the nature of the prescreening is known, it may be hard to define the study population accurately. Without a well-defined study population, application of results to the general population may be difficult. DISCUSSION
In applying the tools and results given here for planning a study, some of the assumptions that have been made may have to be modified. The choice of number of screens to be used may be constrained by factors other than those considered here. Eligibility may be determined, not just by blood pressure, but also by the subjects’ willingness to cooperate, absence of other health problems, and ability to tolerate proposed treatments. It may be necessary to determine eligibility according to these other criteria at a series of visits over a period of time. The question then is whether pressure screening is to be done at each of the pre-randomization visits or only at some. If tests other than blood
Design
of Blood
Pressure
Screening
623
pressure are to be done at various visits, the costs/subject visit will vary and the cost calculations for a design modified. Changing the visit when auxilliary testing is done may decrease cost significantly. Generally, the most expensive testing should be done as late in screening as practical when fewer subjects are eligible. In addition, if the other eligibility criteria are very restrictive, they may significantly affect yield and should be taken into account for estimation of cost and possibly sensitivity. The shift of emphasis from casual pressures to inferred mean pressures may present a problem with estimating risk and therefore planning of sample size for a study. Such estimates are based on previous studies such as the Framingham Study [4] which used a single casual pressure. A risk group with a casual pressure in the 90-105 mm range may have mean pressures from 7&114 (design E). Using observations from a single-screen study would therefore result in inflated sample size estimates for a multiple-screen study. It may, however, be possible to adjust estimates to account for the change. From a clinical point of view, results of studies based on single casual pressures seem unsatisfactory. Good practice suggests treatment of the hypertensive patient not begin until the physician is convinced that the elevated pressure is not a transient symptom but is consistently present over time. Thus the shift to consideration of mean pressure has many advantages. The screening phase of a study can be better planned, the study population is better defined, and the inferences drawn have wider application. Acknowl&ement-The problems considered in this paper were suggested by discussions ative Study on Mild Hypertension which is supported by the Veterans Administration Heart and Lung Institute and for which the author is study biostatistician [Z].
of the Cooperand National
REFERENCES 1. Halperin M et al.: Design considerations in the NHLI Multiple Risks Factor Intervention Trial (MRFIT). To be published 2. Protocol: VA-NHLI Cooperative study on anti-hypertensive agents: Mild hypertension. Cooperative Studies Program, Veterans Administration Central Office, Washington, D.C. 3. Remington RD: The hypertension detection and follow-up program (U.S.A.). INSERM 21: 185, 1973 4. Kannel WB, Golden T (Eds): The Framingham Study, DHEW Publication No. (NIH) 74599, 1975 5. Blood pressure of adults by race and area, United States 1960-62. Table 5. p, 14, National Center for Health Statistics, Series II, Number 5, 1964 6. Clark EG, Schweitzer MD, Glock CY, Vought RL: Studies in hypertension-3. Analysis of individual blood pressure changes. J Chron Dis 4: 477, 1956 7. Private Communications from MRFIT, VA cooperative studies of hypertension study 6. and the NHLI-VA Study of mild hypertension 8. Armitage P. Fox W, Rose GA, Tinker CM: The variability of measurements of casual blood pressure-2. Survey experience. J Clin Sci 30: 337, 1966 9. Feller W: An Introduction to Probability Theory and its Applications Chap. V, Vol. 1. New York: Wiley. 1957 APPENDIX Equations (3-6) were derived in the following p = mean blood pressure, (T = within person standard deviation of IL, S = the event of passing all screens. From eqn 2 we have P(SIp, rr), the probability all screens.
way. We have defined,
of a subject
with
given
mean
and
variance
passing
624
ANNE GOLDMAN
From data, we have estimates for the screening population of P(pkThe relative frequency or probability of blood pressure p, P(cbThe relative frequency or probability of standard deviation 6. We also assume that fi and v are independent or P(p and a) = P(p) P(o). By the definition of conditional probability (9) P(S P(S P(p P(S
and and and and
p and u) u) = P(S S) = P(p p and u)
= P(S 1p, a). P(p and u) = P(S 1p, 0). P(p). P(u) a). P(u). S). P(S),
= P@ 1S,a). P(S and a).
Also, Pb and S) = 2 P(p and S and u) 0 P(S and u) = C P(p and S and u) P It follows that, P(S and (7) P(S(0) = -----=c
P(S and p and (7)
0)
,‘
P(u) which is equation
(3)
and, P(S) = 1 P(Slu) d
P(a)
(4)
and.
ptPc(s,o)=
P(S and p and a) P(S and u)
= P(SIP, 0) mo Wlo)
(5)
and finally,
fuls)
=
P(p and S) P(s)
=c
P(SIP, u) . P(p) P(u) .7
P(S)
(6)
Pergamon
Press. Printed
in Great
Britain
DESIGN OF BLOOD PRESSURE SCREENING FOR CLINICAL TRIALS OF HYPERTENSION ANNE GOLDMAN Div. of Biometry,
University (Rwriued
of Minnesota, in revised jbrm
Minneapolis.
MA 55455, U.S.A.
15 April 1976)
Abstract-In clinical trials of hypertension the eligibility of a subject is usually determined at a series of screening visits prior to randomization. Only those subjects whose blood pressures are within certain limits at each of specified visits are eligible. The pressure measurement of any individual has considerable time-to-time variability, therefore subjects who have passed one screen may ‘regress towards the mean’ and not pass subsequent screens. This paper is a statistical examination of the effect of different numbers of screens and pressure limits on the probability of a given subjects passing all screens. the proportion of subjects who pass, and the distribution of their blood pressures. Numerical estimates, using data from the literature, indicate that the best way to maximize yield while controlling the distribution of blood pressures of selected subjects is to have several screens with pressure limits slightly wider than the goal.
INTRODUCTION IN CLINICAL trials involving hypertension, it is often considered necessary to have a series of screening visits where the eligibility of the subjects is determined before they are enrolled in the study. Usually, the blood pressure is taken at each of these visits and only those subjects with diastolic or systolic pressures within a certain range at one or more specified visits are considered eligible. When the subjects with those higher pressures are remeasured at subsequent screening visits, many of them have much lower pressures. This phenomenon is due to two wellknown effects: a ‘regression towards the mean’ and a ‘pressor effect’. The regression effect is a statistical phenomenon that can occur when measurements which show time-to-time random variability are repeated. When selected persons from a population are re-measured because their first blood pressures were much higher (or lower) than the population mean, then those repeat pressures will, on the average, be much closer to the mean. The pressor effect is a postulated psycho-physical effect whereby a person’s blood pressure tends to read higher when taken under the stress of unfamiliar circumstances; when retaken, there is presumably less stress and therefore the pressure is lower. The pressor effect will be ignored in the discussion that follows, but the random variability of repeated measurements and the
613
ANNE GOLDMAN
614
consequent regression effect must be taken into account in the design of pressure screens. Blood pressure screening may be done in several ways. Both the eligibility limits for pressures and the number of screening visits may be varied. For example, in the Multiple Risk Factor Intervention Trial [l], risk eligibility is determined based on three risk factors, one of which is blood pressure, at the first visit only. A second and third visit provides data for exclusion baseline information and for exclusion purposes, but blood pressure is not used for exclusion. Treatment begins after the third visit at which time some subjects, who at first screen had elevated pressure, have normal pressure and do not require treatment. In the NHLI-VA Study of Mild Hypertension [2], the subject’s pressure must be within defined limits on all three screening visits to be eligible. With this screening method, eligible subjects are clearly mild hypertensives at randomization and may be treated. However, the screens are so restrictive that massive numbers of persons must be screened in order to identify sufficient numbers of eligible subjects. Similarly, in the NHLI Hypertension Detection and Follow-up Program [3], the subjects must have pressures above defined limits at each of two screens to be eligible. This paper is a statistical examination of the consequences of different screening designs. By making a few general assumptions, tools are developed for examining the effect of different numbers of screens and pressure limits on the probability of a given subject being eligible for the study, the proportion of subjects who pass the screens, and the distribution of their blood pressures. Numerical estimates are calculated using blood pressure distribution data from the literature. BASIC
DEFINITIONS
AND
ASSUMPTIONS
The screening population is defined as the group of individuals who are screened, and the study population as the group of individuals who pass the screens and are randomized or otherwise followed. The goal of the screening is to select subjects with defined blood pressure characteristics, pressure being the only criterion for eligibility. When an individual’s blood pressure is repeated, it will be somewhat different each time, varying around that individual’s long-term mean. We therefore distinguish between a one time or casual pressure and the long-range mean pressure. We let pi be the mean pressure for individual i and assume that the casual readings will be normally distributed with long term, mean pi and standard deviation pi. In some studies an individual’s blood pressure is measured two or three times at each visit, 5 or 10 min apart, and the average of these readings is used as the pressure for that visit, thus reducing the variability. To avoid confusion the pressure at one visit will be referred to as a ‘casual pressure’ even when it is based on the average of several readings. In past studies of hypertension, the goal of screening has been the identification of persons with one or more casual readings within specified limits. These limits have frequently been based on categories of risk calculated from data of the Framingham Study [4] using one casual blood pressure reading to define the categories. When risk categories are based on one casual reading, it follows that each category will consist of individuals whose mean pressures are distributed over a wide range. The overall risk for the group thus is an average for individuals,
615
Design of Blood Pressure Screening
some of whom are at high risk and some who are at low risk. Contrary to current practice, it is therefore suggested that the goal of screening should be to select a group of subjects who are at homogeneous risk due to blood pressure and that therefore subjects should be selected whose mean pressures are inferred to be within specified limits. It is assumed that there is no ethical difficulty in prescribing the contemplated treatments to persons whose casual pressures occasionally measure slightly higher or lower than the screening goal. Suppose that for a study of mild hypertension, the screening goal is defined as a mean diastolic pressure of 9(r105 mm of mercury. The contemplated treatments are an anti-hypertensive drug and a placebo. All subjects with mean pressures within the goal limits can be considered treatable since the anti-hypertensive drugs may be prescribed to patients with casual diastolic pressures as low as 85 mm, and the placebo to patients with pressures as high as 110 mm. These limits will be called the treatment limits. A screening design consists of a specified number of screens. At each screen j there is a lower pressure limit aj and an upper limit hj. A subject i passes screen visit ,j if the casual pressure measurement for that visit is between Uj and hj.
The main assumptions may be summarized as follows: 1. Screening goal: mean diastolic pressure 90-105 mm. 2. Treatment limits: casual diastolic pressure 85-l 10 mm. 3. Screening limits: aj to bj mm of pressure at screen j. 4. Casual measurements on one individual i are independent and normally distributed with mean Iii and standard deviation pi mm. The results given here are derived from the above assumptions and some elementary probability theory. The mathematical derivations are shown in the Appendix. THE
PROBABILITY
OF A GIVEN
PERSON
PASSING
THE
SCREENS
In order to estimate the probability of a given subject i passing all the screens, we let Sij be the event of subject i passing screen j. Then the probability of Sij given pi and (Ti is:
(1) where Q(x) is the value of the cumulative standard normal up to x. Then the probability of subject i passing three screens is simply the product, or: P(SilPi,ai)
=
Gil
ci2
ci3.
(2)
Probabilities for any number of screens can be similarly derived. For simplicity the probability in (2) will be written P(Slp,o). Table 1 shows the probability in (2) for various values of pi and oi when the screening limits are 9G-105 at each of three visits (Design B). Notice that a person whose mean pressure is 90 mm and any oi has a 12.5% chance or less of being judged eligible. Even a person with mean pressure of 97.5 and 0 = 7 has only a 37% chance of qualifying.
616
ANNE GOLDMAN
TABLE 1. PROBABILITYP(SIp, a) OF PASSINGTHREE SCREENS FOR AN INDIVIDUALWITH GIVEN pi,ui. DIASTOLICPRESSURE LIMITS0~ ALL THREESCREENS 90-105 mm (DESIGN B) eqn (2) Within person mean pressure pi 72.5 71.5 82.5 87.5 90.0 92.5 97.5 102.5 105.5 107.5 112.5 117.5 122.5 127.5 132.5
3
1 0 0 0 0 0.125 0.981 l.OOO 0.981 0.125 0 0 0 0
0 0 0 0.001 0.125 0.715 0.999 0.715 0.125 0.001 0 0 0 0
:
Within person standard deviation oi 6 7 4 5
8
9
10 0 0.001 0.010 0.047 0.081 0.120 0.163 0.120 0.08 1 0.047 0.010 OOOl 0 0 0
0 0.008 0.125 0.508 0.963 0.508 0.125 0.008 0 0 0
0 0 0 0.019 0.125 0.394 0.828 0.394 0.125 0.019 0 0 0
0 0 0 0.029 0.124 0.322 0.650 0.322 0.124 0.029 0 0 0
0 0 0.001 0.038 0.120 0.266 0.491 0.266 0.120 0.038 0.001 0 0
0 0 0.003 0.044 0.113 0.219 0.367 0.219 0,113 0.044 0.003 0 0
0 0 0.005 0.048 0.104 0.179 0.217 0.179 0.104 0.048 0.005 0 0
0 0.001 0.008 0.048 0.092 0.146 0.211 0.146 0.092 0.048 0.008 0.001 0
0
0 O.
O 0
0
0
0
0
Table 2 shows a comparison of various screening designs for c( = 7 (Designs A-H). Screen 1 may be thought of as a prescreening conducted by the study personnel out in the field. In practice, the limits for such a screen are frequently different from those on subsequent screens. Many other designs are obviously possible; only some can be listed here. A within person standard deviation (a) of 7 was chosen for illustration since it seems representative of values reported by different observers (see below for further discussion). For comparison, Design I3 is shown in both tables: in Table 1 for all values TABLE 2.
PROBABILITYP(Sjp,a)
OF PASSINGALL SCREENS FOR AN INDIVIDUAL WITH GIVEN pi, 0 = 7, eqn (2) Design A
Pressure limits
No. screens Screen 1 Screen 2 Screen 3 Screen 4
62.5 67.5 72.5 77.5 82.5 87.5 92.5 97.5 102.5 107.5 112.5 117.5 122.5 127.5 132.5
B
4 3 9O-120 85-110 9O105 85-110 90-105 85-110 9O105 Probability 0 0 0 0 0 0 0 0 0.007 0.003 0.094 0.044 0.395 0.219 0.680 0.367 0.59 1 0.219 0.249 0.044 0.040 0.003 0.002 0 0 0 0 0 0 0
C
D
3 85-110 855110 85-110
1 90-120 ~ -
P(S 1p, u) 0 0 0 0.00 1 0 0.006 0.003 0.037 0.142 0.047 0.360 0.261 0.618 0.639 0.794 0.857 0.618 0.957 0.957 0.261 0.047 0.857 0.003 0.639 0.360 0 0 0.142 0.037 0
E
F
G
1 2 3 9O-105 9O-120 9O120 ~ 85-110 85-110 ~ 85-110 ~ ~ -
0 0.001 0.006 0.037 0.141 0.354 0.602 0.716 0.602 0.354 0.141 0.037 0.006 OOOl 0
0 0 0 0.005 0.05 1 0.230 0.545 0.794 0.815 0.611 0.309 0.091 0.013 0.001 0
0 0 0 0.001 0.018 0.147 0.464 0.735 0.694 0.390 0.111 0.013 0 0 0
H 4 9O-120 9O-105 9O105 9O-105
0 0
0 0 0 0.016 0.140 0.315 0.209 0.043 0.002 0 0 0 0
Design of Blood Pressure Screening
617
of ci = l-10, and in Table 2 only for (TV = 7. Notice that in designs with three screens, persons with mean pressure within the screening goal (90-105 mm) have 2-6 times the probability of being included in the study when the limits are 85-110 (Design C) than when they are 90-105 (Design B). At the same time, persons with mean pressure outside the goal limits have only a small probability of being included. When there is only one screen, all subjects have a higher probability of being judged eligible, including those with pressures outside the goal limits (Designs D, E). Other values of oi give somewhat different probabilities, but similar trends. SCREENING
YIELD
The probability derived in eqn (2) is the conditional probability of passing the screens given pi and pi. Since it relates to individuals with known mean blood pressure and standard deviation it may not be of practical interest. However, eqn 2 may be used to derive probabilities and distributions which are useful for designing screens. One question of great concern is the percentage yield of a screening design, or the number of persons found eligible for every 100 that must be seen for at least one screening visit. The greater the yield of a design, the lower the cost of screening because fewer subjects need be examined to attain a specified sample size. This unconditional probability can be found by averaging the conditional probability in eqn (2) over the distribution of CLiand (Tiin the screening population. Let P(p) be the probability of having mean pressure p, and P(o) the probability of having a standard deviation (r in the screening population. It is assumed that P(p) and P(o) are independent, that is, the time to time variability of a person’s pressure does not depend on the level of that pressure. P(p) and P(o) may be estimated from blood pressure studies. It is assumed that p and (T are discrete variables since such studies usually report pressure to the nearest 1 or 5 mm. If p and (T are to be continuous, their density functions would need to be known and eqns (3,4) modified appropriately. The probability of subjects with standard deviation e passing the screens is the conditional probability of S given 0. As shown in the Appendix, we have: P(Slo) = CP(SIP, P The yield of the screens, or the unconditional for a population with P(o) known is:
a) . WL). probability
P(S) = c P(S IO) . P(0).
(3) of passing the screens (4)
Frequency tables from studies of blood pressures of a population similar to that being screened may be used to estimate P(p). However, data on individual standard deviations of pressures are not generally available, although estimates of the average G may be possible. If only the average 0 is available, the screening yield, given that CJ,can be calculated from eqn (3). P (~310)is that proportion of the screening population with standard deviation cr which will pass the screens. Equation (4) however, averages that conditional probability over the distribution of CJfound in the population to give a screening yield for the whole population.
618
ANNE GOLDMAN TABLE 3. THE RELATIVEFREQUENCY P(p) OF DIASTOLICBLOOD PRESSURESFORTWO POPULATIONS. DERIVEDFROMDATA OF THENATIONAL HEALTHSURVEY(TABLE 5) Diastolic pressure (mm of mercury)
Midpoint p
Population I
Population II
5G-54 55-59 60-64 6549 70-14 75-79 8S-84 85-89 90-94 95-99 lOCklO 105-109 110-114 115~119 12G-124 125-129 130-134 Mean SD.
52.5 57.5 62.5 67.5 72.5 77.5 82.5 87.5 92.5 97.5 102.5 107.5 112.5 117.5 122.5 127.5 132.5
0.010 0.024 0.061 0.121 0.154 0.192 0.154 0.1 13 0.071 0.045 0.024 0.015 0.006 0.006 0.002 0.002 0.001 79.48 12.17
0 0 0 0 0.005 0.035 0.109 0.202 0.224 0.193 0.1 12 0.07 I 0.026 0.018 0.004 0.001 0 94.05 9.05
Table 3 shows the relative frequency of pressures of two different populations used to estimate P(p). Population I represents the general adult population. The data are taken from the National Health Survey [S] and shows the distribution of diastolic pressure of 11$03 randomly chosen adults age 18-79 yr (pressures of < 50 mm and > 134 mm were ignored). The reported pressures are based on the average of three readings at a single visit and are thus casual pressures rather than the needed distribution of mean pressures. However, the distribution of mean pressures for the general population is not available. It would be a tighter (smaller variance) distribution than that of Population I used here, since the distribution of the mean of a random variable has a smaller variance than that of the random variable itself (casual pressure). Population II represents a population of ‘prescreened’ subjects since frequently a natural source of subjects for screening may be persons who have previously been diagnosed as hypertensives and are then referred to the study. Population II therefore generally has higher pressures than I. The distribution of II was derived from that of population I by applying the method of the next section and will be discussed there. The mean and standard deviations shown in the table are those of the populations, not within individuals. Table 4 shows the relative frequency of within person standard deviatiori used to estimate P(a). Such data are scarce in the literature. The frequencies quoted here were calculated from data of individual blood pressure measurements on 21 subjects [6]. Each subject had supine, diastolic and systolic pressures taken several times daily, except Sunday, over a period of three and a half weeks. All subjects had pressures taken in the morning, repeated about four times at 5-min intervals, and some also had repeated measurements in the evening. The mean
Design of Blood Pressure Screening
6 I9
TABLE 4. THE RELATIVEFREQUENCYP(o) OF WITHIN PERSON STANDARD DEVIATIONOF DIASTOLICPRESSURES. D&RIVED FROM DATA ON 21 SUBJECTS (TABLE 5)
Standard deviation C, mm
123
Number of subjects Relative frequency 00)
0
0
2
4
5
6
7
8
9 10 Total
4
10
2
2
I
0 0
0 0 0.095 0.190 0.476 0.095 0.095 0.048 0
21
0 0.999
Group mean diastolic pressure = 64.14 mm. Average SD. = 5.04mm.
pressure and standard deviation was calculated for each individual disregarding when the pressure was taken and tallied in Table 4. The average standard deviation, found by pooling over the 21 subjects is 5.03 mm. This average (T is probably lower than would be found in a study screening since they were: supine pressures, measured several times a day, measured every day. Most study screens use sitting pressures, one measurement per visit (or possibly the average of several), and the visits are weeks or months apart. Current hypertension studies are reporting variabilities of cr = 7-9 [7] and also in a past study of normal subjects [8]. An average 0 = 7, as used in Table 2 for illustration, seems reasonable. Table 5 shows the yields of various screening designs as calculated from eqns (3,4) for population I and II. Generally, broader screening limits give two to three times higher yields (Designs A vs H, C vs B, J vs I); decreasing the number of screens gives a slightly higher yield (compare Designs A, G, F. D). A comparison of screening population I and a prescreened population (Designs B vs I, C vs J) shows a more than three fold increase in yield. TABLE 5. SCREENINGYIELDS
DeWgn No. screens PreWlre hmits
Screen I Screen 2 Screen 3 Screen 4
Population
C?=
I 2 3 4 5 6 7 8 9 IO
The number of subjects enrolled A B C D
for each 100 subjects E F G
4 3 3 I 9&120 90-120 ~ ~ 85-110 90-105 85-110 85-I 10 90105 85-I IO 85-110 90 105 X5-l IO I 1 I I
I 2 3 90-105 Yo- I20 90-120 ~~ 85-I IO 9&105 85-l 10 ~~ ~~ ~ I I I
‘2, Yield 15.5 15.1 14.3 13.1 11.6 10.2 08.8 07.6 06.5 05.6
Unconditional% yield-P(s) 11.6
given o - P(SIo)*lOO eqn (3) 13.x 26.5 16.6 14.0 11.3 23. I 17.0 14.3 OY.3 20.4 17.5 14.7 07.7 18.5 18.0 15.1 06.4 16.7 IS.6 15.5 05.2 15.1 19.4 16.0 04.3 13.6 20.2 16.6 03.6 12.2 21.0 17.1 03.1 10.9 21.9 17.6 02.7 09.8 22.7 IS.0 eqn (4) (Distribution 06.5 16.7 18.7
15.5 15.7 15.6 15.3 14.9 14.5 14.0 13.6 13.2 12.9
15.5 15.4 14.x 14.0 13.0 11.9 10.9 09.9 09.0 ox.3
screened H
I
J
4 .3 3 9&120 ~~ 90-105 9CLlO5 85 110 9&105 9@ 105 85 ~1IO 9OklO5 Y@~lO5 85 I IO I II II
137 10.7 08.4 06.7 05.2 04. I 03.2 02 5 02.0 01.6
of 0 from Table 4 assumed) 15.6 14.8 12.9 05.4
52.3 43.4 35 Y 29.1 24.2 19.5 157 127 IO.4 OX.6
7’) 7 72.4 66.2 60.9 55.3 49.5 43.x 3x.4 33.5 29 I
24.5
54.9
ANNE
620
THE
DISTRIBUTION OF
GOLDMAN
OF MEAN BLOOD ELIGIBLE SUBJECTS
PRESSURES
The goai of the screening was assumed to be the selection of subjects with mean pressures in the interval 9&105 mm. We therefore need to know the distribution of mean blood pressures of subjects who pass all the screens. The higher the proportion of selected subjects who have mean pressures within the screening goal, the more sensitive the design and the more homogeneous the risk. The relative frequency of different blood pressures for the subjects with given 0 who pass the screens is the probability P@IS,a). As shown in the Appendix, we have:
where &!+,a) is from eqn (2) and P(Sl(r) from eqn (3). This probability is the distribution of mean blood pressures in the study population with a given within person standard deviation. If estimates of P(a) are available, the relative frequency of different mean blood pressures in the study population is the probability P@(S). As shown in the Appendix :
WI9 = 1
WI/& 4 . P(p) . P(o)
P(S)
0
Table 6 shows the conditional distribution of blood pressures with D = 7 for various screening designs and population I and II (P(pjS,o) eqn (5)). Population II in Table 4 was actually found by applying (5) to population I, assuming 0 = 7 and one screen with limits 90-120 mm (Design D). Using population II followed by three screens (Designs I, J) is thus equivalent to using population I and having four screens (Designs H, A respectively), when c = 7. The yields of Table 5 (for G = 7) are also related in that, for example (yield Design D) x (yield Design TABLE 6. DISTRIBUTION OF BLOOD PRESSURES(P(~jz,o)) A
D.?Slgtl
No. screens PrcsslJrc limits
Screen1 screen2 Screen3 Screen4
Population
Mean diastolic pW3s"rc (mm) p
70-74 IS-79 SO-84 85-89 90-94 95-99 100-104 105.-109 110-114 115-119 120-124 125-129
B
C
ON d = 7
OF ELIGIISLESUBJECCTS CONDITLONAL D
E
F
G
H
I
J
4 3 3 I I 2 3 4 3 3 9CLl20 -- 9&120 9&105 9&120 9&120 9CklO2 ~ 85%110 9&105 85-110 ~ 85-110 85-110 90-~10590-105 85-110 85.-1109C~105 85-110 ~ .~ 85-110 9G-105 90-105 85SllO 85SllO 9cLlOS 85-110 ~ _ ~- 90 105 90-105 x5-110 1 1 I 1 i 1 I I II II
0 0 0.012 0.120 0.315 0.350 0.158 0.042 0.003 0 0 0
P(pIS.@+qn 0 0 0.008 0 0.010 0.068 0.116 0222 0.355 0 310 0.384 0.256 0.119 0.104 0.015 0029 0003 0 0 0 0 0 0 0
(5) 0.005 0.035 0.109 0.202 0.224 0.193 0.112 0.071 0.026 0.018 0.004 0.001
0.006 0.043 0.132 0.242 0.256 0.196 0.086 a032 0.005 0.001 0 0
0
0.007 0.056 0.186 0.274 0.257 0.137 0 065 0.013 0.004 0 0
0
0.001 0.026 0.153 0.301 0.307 0.151 0.054 0.006 0.001 0 0
0
0
0.0002 0.:2 0.057 0.057 0.312 0.312 0452 0.452 0.156 0156 0.020 0.020 0 0 0 0 0 0 0 0
0
0.092 0.120 0.316 0.349 0.158 0.042 0.003 0 0 0
621
Design of Blood Pressure Screening
TABLE I. THE DISTRIBUTIONOF BLOOD PREWJRESOFTHESTUDY mPm.AnoN. THE DISTRIBUTIONOFSTANDARD DEV~~TION FROM TABLE 4 ISASSUMED Design
A
No. screens
ScreenI Screen2 Screen3 Screen4
PEWWe lilnits
Population
B
C
D
E
F
G
H
I
.I
4 3 3 I I 2 3 4 3 3 _ 9C~i2090.10590-1209@-1209&120 ~~ 9Gl20 85~11090-10585-110 -~ X5-11085-1109CklO59GlO.585 110 XS~IIO9&105 85-~110 ~~ 85 11090~10.5 9% 105 85-110 85.1109&105 85~110 ~~ 90 1059c-10585~110 I I I I 1 I 1 I II I1
mcl9ewKd MWl diastolic pV?Wlre (mm) I’
Average/I S.D. "0 p in range9Gl05mm :,p in range85~~110mm
l&l4 75-79 X&84 85-89 9&94 95-99 100~104 105~109 110~114 115-119 120-124 125-129
0 0 0.003 0.099 0.346 0.343 0.163 0.044 0.001 0 0 0
0 (! 0.002 0.050 0.361 0.459 0.121 0.007 II II (I (I
0 0.001 0.027 0.233 0.339 0.254 0.114 0.013 0.001 0 0 0
0.001 0.012 0.058 0.182 0.263 0.225 0.124 0.079 0030 0.021 0.004 0.001
0.002 0.014 0.069 0.218 0.311 0.251 0.104 0.029 0.003 0 0 0
0 0.002 0.024 0.159 0.309 0.27X 0.146 0.070 0.011 0.002 0 0
0 0 0.009 0.127 0.332 0.314 0.157 0.056
0 0 0 0 0.019 0.308 0.520 0.144
0 0 0 0.073 0.302 0.514 0.151 0.008
o.MM 0.008 0 0 0 0
0 0 0
0 0 0
0 0 0 0.127 0.328 0.329 0.164 0.045
0.00I 0 0 0
95.9 95.X 94.0 95.5 93.4 95.7 95.8 96.5 96.4 95.x 5.9 3.9 5.6 s.5 5.6 5.6 6.3 4.8 5.7 5.3 85.2",, 94.1':" 70.1",, 61.2";, 66.6""73.3",X0.3""97.2:,, 96.7",, 82.1", 99.5:&99.V'~97.1:"873",,91.3",96.2",9X.6"" IOO",,IOO",,99.3",,
I) = (yield Design H). The point here, however, is not the method of prescreening, but to show the effect of using a prescreened population on the outcome of subsequent screens. Table 7 shows the distribution of blood pressures for the study population for the various designs. Equation (6) was applied by assuming a distribution of standard deviation, P(o), as in Table 4. At the bottom of the table is the average mean pressure and standard deviation of the study population, the percentage that has mean pressure within the screening goal (90-105 mm) (specificity), and the percentage within the treatment limits (85-l 10 mm). Both Tables 6 and 7 show that increasing the number of screens and tightening the screening limits tend to decrease the range of blood pressures of the study population having passed the screens, yet has little effect on the average pressure of the group. Notice that designs with a single screen (Designs D and E) show loo/:, of the eligible population outside the treatment limits. The change of limits in screens 2, 3, and 4 from 9&105 to 85-110 mm (Design H and A) causes a decrease in the percentage of population within the screening goal but a trivial decrease in the percentage within the treatment range. At the same time, Tables 4 and 5 indicate that the yield is doubled by the change. COST
ESTIMATES
AND
COMPARISON
OF
DESIGNS
The cost per randomized subject of some of the screening designs can be estimated. Assume that the cost of seeing one subject for one screening visit is C regardless of visit. For design D, 100 subjects must be seen at one visit to enroll 18.7 in to the study (Table 5); hence, the cost/study subject is lOOC/18.7 = 5.3C.
ANNE GOLDMAN
622
TABLE8. YIELD,COSTAND SENSITIVITY OF SELECTED DESIGNS (C IS THECOSTOF ONLSCREENING VISITFOR ONE SUBJECT)
Design Pressure limits
Yield (Table 5) Cost/study subject Sensitivity (Table 7)
Screen I Screen 2 Screen 3 Screen 4
H
A
B
C
9G-120 90-105 9&105 9&105 5.4%
9LL-120 85-110 85-110 85-l 10 I 1.6%
90-105 90-105 9@105 6.5%
85-l 10 85-t 10 85-l 10 16.1”/
21.7c 97.2%
12.6C 85.2%
19.2C 94.17;
8.9C 70.77;
For design F, if 100 subjects are seen at screen 1, 18.7 will be eligible for screen 2 and 14.8 will be randomized. The cost/study subject is (100 + 18.7)C/14.8 = 8.0 C. Similar estimates can be found for designs F, G, H. Table 8 shows a comparison of selected designs by yield, cost and sensitivity (percentage of study subjects with mean pressure between 90 and 105 mm of pressure). The table shows clearly that designs H and B, although very sensitive, are about twice as costly as designs A and C which have slightly wider limits and slightly less sensitivity. The results in Tables 4-8 show that when only one blood pressure screen is used, some subjects with mean pressures outside the goal limits may be selected and their casual pressures at and after randomization may be outside the treatment limits. On the other hand, when there are several screens and the screening limits are equal to the goal limits, the sensitivity may be very high, but the yield decreases and the cost/study subject increases. The best way to maximize yield and minimize cost while maximizing sensitivity is to have several screens with limits slightly wider than the goal. Decreasing the number of screens has a much less dramatic effect on yield and cost than reasonable modifications in the limits, while a good combination allows control of the distribution of pressures of the study population. The use of a prescreened population, if one is available to the study at low cost, clearly increases the yield dramatically, thereby saving considerable time and money. Unless, however, the nature of the prescreening is known, it may be hard to define the study population accurately. Without a well-defined study population, application of results to the general population may be difficult. DISCUSSION
In applying the tools and results given here for planning a study, some of the assumptions that have been made may have to be modified. The choice of number of screens to be used may be constrained by factors other than those considered here. Eligibility may be determined, not just by blood pressure, but also by the subjects’ willingness to cooperate, absence of other health problems, and ability to tolerate proposed treatments. It may be necessary to determine eligibility according to these other criteria at a series of visits over a period of time. The question then is whether pressure screening is to be done at each of the pre-randomization visits or only at some. If tests other than blood
Design
of Blood
Pressure
Screening
623
pressure are to be done at various visits, the costs/subject visit will vary and the cost calculations for a design modified. Changing the visit when auxilliary testing is done may decrease cost significantly. Generally, the most expensive testing should be done as late in screening as practical when fewer subjects are eligible. In addition, if the other eligibility criteria are very restrictive, they may significantly affect yield and should be taken into account for estimation of cost and possibly sensitivity. The shift of emphasis from casual pressures to inferred mean pressures may present a problem with estimating risk and therefore planning of sample size for a study. Such estimates are based on previous studies such as the Framingham Study [4] which used a single casual pressure. A risk group with a casual pressure in the 90-105 mm range may have mean pressures from 7&114 (design E). Using observations from a single-screen study would therefore result in inflated sample size estimates for a multiple-screen study. It may, however, be possible to adjust estimates to account for the change. From a clinical point of view, results of studies based on single casual pressures seem unsatisfactory. Good practice suggests treatment of the hypertensive patient not begin until the physician is convinced that the elevated pressure is not a transient symptom but is consistently present over time. Thus the shift to consideration of mean pressure has many advantages. The screening phase of a study can be better planned, the study population is better defined, and the inferences drawn have wider application. Acknowl&ement-The problems considered in this paper were suggested by discussions ative Study on Mild Hypertension which is supported by the Veterans Administration Heart and Lung Institute and for which the author is study biostatistician [Z].
of the Cooperand National
REFERENCES 1. Halperin M et al.: Design considerations in the NHLI Multiple Risks Factor Intervention Trial (MRFIT). To be published 2. Protocol: VA-NHLI Cooperative study on anti-hypertensive agents: Mild hypertension. Cooperative Studies Program, Veterans Administration Central Office, Washington, D.C. 3. Remington RD: The hypertension detection and follow-up program (U.S.A.). INSERM 21: 185, 1973 4. Kannel WB, Golden T (Eds): The Framingham Study, DHEW Publication No. (NIH) 74599, 1975 5. Blood pressure of adults by race and area, United States 1960-62. Table 5. p, 14, National Center for Health Statistics, Series II, Number 5, 1964 6. Clark EG, Schweitzer MD, Glock CY, Vought RL: Studies in hypertension-3. Analysis of individual blood pressure changes. J Chron Dis 4: 477, 1956 7. Private Communications from MRFIT, VA cooperative studies of hypertension study 6. and the NHLI-VA Study of mild hypertension 8. Armitage P. Fox W, Rose GA, Tinker CM: The variability of measurements of casual blood pressure-2. Survey experience. J Clin Sci 30: 337, 1966 9. Feller W: An Introduction to Probability Theory and its Applications Chap. V, Vol. 1. New York: Wiley. 1957 APPENDIX Equations (3-6) were derived in the following p = mean blood pressure, (T = within person standard deviation of IL, S = the event of passing all screens. From eqn 2 we have P(SIp, rr), the probability all screens.
way. We have defined,
of a subject
with
given
mean
and
variance
passing
624
ANNE GOLDMAN
From data, we have estimates for the screening population of P(pkThe relative frequency or probability of blood pressure p, P(cbThe relative frequency or probability of standard deviation 6. We also assume that fi and v are independent or P(p and a) = P(p) P(o). By the definition of conditional probability (9) P(S P(S P(p P(S
and and and and
p and u) u) = P(S S) = P(p p and u)
= P(S 1p, a). P(p and u) = P(S 1p, 0). P(p). P(u) a). P(u). S). P(S),
= P@ 1S,a). P(S and a).
Also, Pb and S) = 2 P(p and S and u) 0 P(S and u) = C P(p and S and u) P It follows that, P(S and (7) P(S(0) = -----=c
P(S and p and (7)
0)
,‘
P(u) which is equation
(3)
and, P(S) = 1 P(Slu) d
P(a)
(4)
and.
ptPc(s,o)=
P(S and p and a) P(S and u)
= P(SIP, 0) mo Wlo)
(5)
and finally,
fuls)
=
P(p and S) P(s)
=c
P(SIP, u) . P(p) P(u) .7
P(S)
(6)
