J Clin Epldemiel Vol. 44, No. 8, pp. 831-838, Printed in Great Britain
0895-4356/91 $3.00 + 0.00 Pergamon Press plc
1991
JUDGING THE EFFECTIVENESS OF ANTIHYPERTENSIVE THERAPY IN AN INDIVIDUAL PATIENT B. ROSNJJR’*and H. G. LANGFORD*+ ‘Cbanning Laboratory, Harvard Medical School, Department of Medicine and Br@am and Women’s Hospital, and Department of Biostatistics, Harvard School of Public Health and ZUniversity of Mississippi Medical Center, Mississippi, MS, U.S.A. (Received in revised form
12 February 1991)
physician’s decision as to whether a hypertensive drug has produced satisfactory blood pressure lowering is more difficult than it is usually perceived to be. Variability of blood pressure, regression to the mean and habituation to the taking of blood pressure all conspire to blur the significance of the observed drop. We present algorithms outlining the number of visits before and after the initiation of therapy and the mmHg drop required for determination with acceptable certainty that any response to the hypotensive drug has occurred. Our conclusion is that the variability of blood pressure limits the clinician’s ability to confidently conclude that antihypertensive therapy is effective unless the change in blood pressure exceeds a relatively high threshold and/or the number of visits both before and after the initiation of antihypertensive therapy is large. Because regression to the mean will be minimized, fewer visits and/or a lesser blood pressure fall in the treated patient changing or adding medications will be necessary to determine that a real decline has occurred. Abstract-The
Hypertension
Blood pressure
Screening
INTRODUCTION
It is widely accepted by investigators and clinicians that multiple blood pressure determinations are necessary before the labeling of an individual as hypertensive. Predictive value curves for routine blood pressure measurements have been derived and used as a tool for screening individuals for hypertension based on multiple blood pressure determinations obtained over several visits [l]. This approach has been successfully used in the NHLBI-sponsored Trial of Hypertension Prevention (TOHP) to identify 2164 subjects with high normal diastolic blood *All correspondence should be addressed to: Bernard Rosner. Ph.D. Associate Professor of Medicine (Biostatist&), Harvard Medical School, 180 Lo&ood Avenue, Boston, MA 02115, U.S.A. @Decca& January 1991.
pressure among a screening population of approximately 17,000 subjects and treat them with various non-pharmacologic treatment modalities [2]. Similar issues arise once an individual is identified as hypertensive and is treated. However, no consensus exists as to criteria for determining when a treatment regimen has been effective. Indeed, it is generally agreed that there is a wide individual variation in the blood pressure response to antihypertensive therapy. Many papers classify patients into on the “responders” and “non-responders” basis of one reading, or one set of readings on one occasion. Decision rules are then proposed for changing or adding medication. The purpose of this paper is to provide decision rules for determining whether an agent has actually lowered blood pressure. This problem 831
832
B. ROWERand H. G. LANGFORD
is crucial in the clinical management of hypertensive patients as well as in treatment trials of antihypertensive agents where it is desirable to identify individuals as responders or nonresponders to specific antihypertensive treatment regimens. METHODS
We assume that patients are classified as hypertensive if their observed mean diastolic blood pressure (DBP) over N, pre-treatment visits with k readings per visit exceeds some threshold (c) such as 90 mmHg . We also assume that patients are ascertained after N2 post-treatment visits with k readings per visit after therapy has begun. Let pi, = true underlying pre-treastment level of DBP for the ith person, pi2 = true underlying post-treatment level of DBP for the ith person, where the true underlying level is the average level that would be achieved over a very large number of pre- and post-treatment visits respectively. Under this model, we are implicitly assuming that true mean blood pressure remains stable over the short term during both the pre- and post-treatment periods. We classify therapy as eflective if A = pii - pi2 > 0 and ineffective if A < 0. Thus, therapy is classified as effective if blood pressure has truly declined by any amount. A can be considered as the change in DBP that would occur in a treated patient if a large number of measurements were available both pre- and post-treatment. We will also consider a decision rule whereby therapy is classified as very effective if true blood pressure has declined by a specified amount (AO), where A0 > 0. One of the issues in assessing adequacy of treatment is regression to the mean. Specifically, if the number of pre-treatment measures is small (as is often the case), then based on regression to the mean one would expect some decline in mean blood pressure even if treatment is completely ineffective. Therefore, we cannot simply use the observed mean decline in blood pressure as a guide to the effectiveness of therapy without controlling for the regression to the mean effect. For this purpose, we use a predictive value approach, where predictive value positive is defined as the probability that true blood pressure has declined after treatment (A > 0) given a specified observed decline (we denote the probability that true blood pressure has not declined
given a specified observed decline by predictive value negative). The predictive values depend on
(a) the mean observed pre-treatment
(jr& and post-treatment (j&,) levels of DBP. @I the magnitude of the components of within-person blood pressure variability denoted by 0: for between visit variability and G* for within visit variability, and 04 the prevalence distribution of DBP, which for a specific age-sex-race group is assumed to be normal with mean pp and variance 0 $. Based on these parameters, the formula for predictive value positive is derived in the Appendix and is given in equation (A.@. To use the predictive value approach, we must determine what level of predictive value should be used to identify patients for whom therapy is effective. In a previous study [l], questionnaires were sent to the 15 members of the 1980 Joint National Committee for the Detection, Evaluation and Treatment of High Blood Pressure [3] and to the principal clinical investigator in each center of the Hypertension Detection and Follow-up Program [4] asking: How large should your positive and negative predictive values be, based on the blood pressure measurements you have taken for identifying persons for whom newly initiated antihypertensive drug therapy is warranted? The median predictive value selected by the 30 respondents was approximately 80% for both predictive value positive and negative. If we apply the same criteria for evaluating the effectiveness of antihypertensive therapy, then based on equation (A.6), we can establish a threshold of observed change in mean DBP needed to identify patients for whom the predictive value positive, is at least 80%. This value is given by kRKtivcas defined in equation (A.7). To use this approach, we declare Treatment effective:
Treatment ineffective: Furthermore, a predictive declined by threshold is
if mean pre-treatment DBP - mean post-treatment DBP 2 A&xtive otherwise.
(4)
based on equation (A.7), to insure value of 80% that true DBP has at least 4 mmHg, the critical given by A0+ AdfcEtjyc.
833
Judging the Effectiveness of Antihypertensive Therapy Table 1. Thresholds* for determination of effectiveness of antihyuertensive therapy (N, = 1 pre-treatment visit)t Number of post-treatment visitst
Initial mean pre-treatment DBP 90 95 100 105 110 115
1
2
3
4
5
Expected changej
7.6 8.6 9.6 10.6 11.6 12.6
6.7 7.7 8.7 9.7 10.7 11.7
6.3 7.3 8.4 9.4 10.4 11.4
6.2 7.2 8.2 9.2 10.2 11.2
6.0 7.1 8.1 9.1 10.1 11.1
1.6 2.6 3.6 4.6 5.6 6.6
*Change in mean DBP (pre-post) required to insure that treatment is effective in lowering true mean DBP with 2 80% probability = Aar (see equation A.7). tpr = 82.3 mmHg = mean of prevalence distribution of DBP, a$ = 112.5, IS; = 26.2, u* = 7.0, based on a weighted average of the eight age-race-sex-specific parameter estimates with weights given by the number of screenees in specific age-race-sex groups in the HDFP program [4]. #Three readings per visit for both pre- and post-treatment visits. §Expected change in mean DBP due to regression to the mean in the absence of any treatment effect (see Appendix).
Table 2. Thresholds* for determination of effectiveness of antihvnertensive theranv IN, = 2 me-treatment visit&t Initial mean pre-treatment DBP$
Number of post-treatment visit@ 1
2
3
4
5
Expected change]]
105
6.3 6.8 7.4 7.9
5.2 5.8 6.4 6.9
4.8 5.4 5.9 6.5
4.6 5.2 5.7 6.3
4.5 5.0 5.6 6.2
0.9 1.4 2.0 2.6
110 115
8.5 9.1
7.5 8.0
7.1 7.6
6.9 7.4
6.7 7.3
3.1 3.7
90 95
100
*Change in mean DBP (pre-post) required to insure that treatment is effective in lowering true mean DBP with > 80% probability = & (see equation A.7). tpr = 82.3 mmHg = mean of prevalence distribution of DBP, u: = 112.5, ai = 26.2, u2 = 7.0, based on a weighted average of the eight age-race-sex-specific parameter estimates with weights given by the number of screenees in specific age-race-sex groups in the HDFP program [4]. IBased on six readings, three readings per visit. §Based on three readings per visit. ]]Expected change in mean DBP due to regression to the mean in the absence of any treatment effect (see Appendix).
Therefore, we have computed the cutoff value for Nr = l-3 and 20 (a large number of) preWe note that the critical cutoff value AeEdVe, treatment visits, N2 = l-5 post-treatment visits, depends on k = 3 readings per visit, mean pre-treatment the number of pre-treatment visits (N,) level of DBP = 90-l 15 mmHg. For the variance the number of post-treatment visits (Nr) components and prevalence distribution parthe number of readings per visit (k) ameters, we used a weighted average of the the mean pre-treatment level of DBP (j&) eight age-race-sex-specific values (30-49/50-69; the within-person variance components of male/female; white/black) [ 11,where the weights DBP (a;, a’) used corresponded to the number of persons the prevalence distribution of DBP screened in these age-race-sex groups in the (mean = Pi, variance = a’,) HDFP program [4]. Finally, we used a predicthe predictive value positive (1 - a). :tive value positive of 80%. The cutoff values are presented in tabular form in Tables l-4 and in 7 graphical form in Figs 1-I for 1, 2 and 3 and a large number of pre-treatment visits (N, = 20) !3Number of respectively. We have also computed the Post -Treatment Visits expected decline in DBP due to regression to the -i i2RESULTS
II-
6-
,,,,,,,
0. For example, if one has a mean pre-treatment DBP of 95 mmHg based on two pre-treatment visits and one wishes to insure that true DBP has declined by at least 6 mmHg after beginning therapy with 80% probability, then one would need a threshold of 12.8 mmHg (6.8 + 6.0) based on one post-treatment visit and 11.0 mmHg (5.0 + 6.0) based on five post-treatment visits. Another way of expressing these relationships is in terms of absolute mean level of blood pressure required at post-treatment visits to certify effectiveness of therapy. These data are Table 5. Absolute mean level of post-treatment DBPg required to certify effectiveness of antihypertensive therapy* Initial mean pre-treatment DBPt
Number of post-treatment 1
visits1
2
3
4
5
85.4 86.3 87.2 88.1 88.9
85.5 86.4 87.3 88.2 89.1
90 91 92 93 94
83.7 84.6 85.5 86.4 87.3
84.8 85.7 86.5 87.4 88.3
85.2 86.1 87.0 87.8 88.7
95 96 91 98 99
88.2 89.1 90.0 90.8 91.7
89.2 90.1 91.0 91.9 92.8
89.6 90.5 91.4 92.3 93.2
89.8 90.7 91.6 92.5 93.4
90.0 90.9 91.7 92.6 93.5
100 101 102 103 104
92.6 93.5 94.4 95.3 96.2
93.6 94.5 95.4 96.3 91.2
94.1 94.9 95.8 96.7 97.6
94.3 95.2 96.0 96.9 97.8
94.4 95.3 96.2 97.1 98.0
105 106 107 108 109
97.1 91.9 98.8 99.7 100.6
98.1 99.0 99.9 100.7 101.6
98.5 99.4 100.3 101.2 102.0
98.1 99.6 100.5 101.4 102.3
98.8 99.7 100.6 101.5 102.4
110 111 112 113 114 115
101.5 102.4 103.3 104.2 105.0 105.9
102.5 103.4 104.3 105.2 106.1 107.0
102.9 103.8 104.7 105.6 106.5 107.4
103.1 104.0 104.9 105.8 106.7 107.6
103.3 104.2 105.1 105.9 106.8 107.7
*pp = 82.3 mmHg = mean of prevalence distribution of DBP, a! = 112.5, a? = 26.2. u2 = 7.0. based on a weinhted average’ of the. &ht agerace-sex-specific pa&eter estimates with weights given by the number of screenees in specific age-race-sex groups in the HDPP program
[41. tBased on six readings, three readings per visit. $Based on three readings per visit. $Critical upper threshold of mean post-treatment DBP to insure that treatment is effective in lowering true mean DBP with 380% probability (see equation A.7).
Table 6. Example of use of evaluation algorithm Cumulative Visit (mean)
Sum
No. readings
Mean
288 (96) 282 (94)
288
3
96.0
570
6
95.0
276 (92.0) 266 (88.7) 258 (86.0)
276
3
92.0
542
6
90.3
800
9
88.9
SllUl
Visit
Readings
Pm-treatment 1 96 98 94 2
96 94 92
Post-treatment 94 90 92 1 2
88 88 90
3
88 86 84
presented in Table 5 in the special case of two pre-treatment visits for one to five post-treatment visits and 1 mmHg increments of mean pre-treatment DBP (90-l 15 mmHg). An illustration of the use of this table in clinical practice is provided in Table 6. Suppose one has a patient with a mean pretreatment DBP of 95 mmHg based on six readings over two visits (three readings per visit). Antihypertensive therapy is administered at that time and 3 weekly follow-up visits are scheduled to assess the effectiveness of therapy. Suppose the mean DBP at the first, second and third post-treatment visit is 92.0, 88.7 and 86.0mmHg, respectively, yielding an overall cumulative mean of 88.9 mmHg over the three visits (nine readings). Since 88.9 < 89.6 = upper limit for determining effectiveness of therapy based on three post-treatment visits, we conclude that therapy is effective in lowering DBP. As is illustrated in the previous example, an important issue in implementing an algorithm of this type, is to determine what are an appropriate number of pre- and post-treatment visits so as to efficiently identify patients for whom therapy is or is not effective. To study this question, we have computed the sensitivity of the decision rule for a patient with a given observed mean pre-treatment DBP, given (a) a specific true decline in mean DBP (A) = 0, 2, 5 or lOmmHg, (b) the number of pre-treatment visits (N,) = 1 ,5, 20, (c) the’ * number of post-treatment visits (I$) = 1, . . . ,5, 10 and (d) the number of readings per visit (k) = 3. (e) the level of predictive value positive = 1 -a.
836
B. RCSNERand H. G.
Table 7. Sensitivity* of decision rule to identify persons for whom antihypertensive therapy is effective (Nr = l-5, 20; N, = 1-5, 10: A = 0. 2, 5 or 10 mmHd
N,t
A§
1
2
3
4
5
10
1
0 2 5 10
0.200 0.287 0.444 0.711
0.200 0.305 0.493 0.789
0.200 0.313 0.516 0.821
0.200 0.318 0.530 0.839
0.209 0.321 0.539 0.849
0.200 0.328 0.559 0.872
2
0 2 5 10
0.200 0.299 0.476 0.764
0.200 0.325 0.549 0.861
0.209 0.339 0.588 0.900
0.200 0.348 0.612 0.920
0.200 0.354 0.628 0.932
0.200 0.370 0.666 0.955
3
0 2 5 10
0.209 0.304 0.491 0.787
0.200 0.336 0.580 0.893
0.200 0.355 0.629 0.933
0.200 0.368 0.660 0.952
0.200 0.377 0.682 0.963
0.200 0.400 0.735 0.982
4
0 2 5 10
0.200 0.307 0.501 0.801
0.200 0.343 0.599 0.910
0.200 0.366 0.656 0.950
0.200 0.381 0.693 0.968
0.209 0.393 0.719 0.977
0.200 0.423 0.781 0.992
5
0 2 5 10
0.200 0.310 0.507 0.809
0.200 0.349 0.613 0.921
0.200 0.373 0.675 0.960
0.200 0.391 0.716 0.976
0.200 0.405 0.745 0.984
0.200 0.442 0.815 0.9%
20
0 2
0.209 0.318 0.529 0.838
0.209 0.369 0.664 0.954
0.200 0.407 0.750 0.986
0.200 0.438 0.808 0.995
0.200 0.464 0.850 0.998
0.200 0.551 0.943 1.000
1;
*Sensitivity=Q(A/a** -q-J. TN, = number of pre-treatment visits. $N, = number of post-treatment visits. $A = true decline in mean DBP after start of antihypertensive therapy.
The sensitivity is derived in equation (A.8) with results presented in Table 7. We note first that the probability of determining that antihypertensive therapy is effective when it has no true effect (A = 0), i.e. 1 - specificity, is 20% for all parameter combinations. This was planned by design since a = 0.20. Second, we note that the sensitivity is inadequate to enable one to detect a true 2 mmHg change in an individual patient; even with as many as 20 pre- and 10 post-treatment visits, only 55% of patients with a true 2 mmHg decline would be identified by this rule. Third, sensitivity is marginally adequate to allow one to identify individual patients with a true 5 mmHg decline; with A = 5 and 5 pre-treatment visits, one needs as many as 10 post-treatment visits to achieve 80% sensitivity and 4 post-treatment visits to achieve 70% sensitivity. Finally, sensitivity is more than adequate to identify individual patients with a true decline of 10 mmHg; one can achieve 80% sensitivity with as few as 2 pre- and 2 post-treatment visits.
bNGFGRD
DISCUSSION
Most investigators have accepted the necessity of obtaining multiple blood pressure determinations over several visits before labeling an individual as hypertensive. However, there is no consensus as to criteria for determining the effectiveness of antihypertensive therapy once an individual has been identified as a hypertensive. In this paper, specific criteria are provided for determining when treatment is efficacious for an individual patient with high probability based on a threshold of mean blood pressure decline. The required threshold depends on the initial blood pressure level, as well as the number of visits and measurements per visit both before and after therapy. It is clear from the results in this paper that, because of the variability of blood pressure, reliable inferences concerning the efficacy of antihypertensive therapy can only be made based on a large number of visits both before and after therapy. For example, if the initial mean pre-treatment DBP = 95 mmHg and only a single visit is used both before and after therapy to characterize the efficacy of treatment as is often the case in clinical trials [5], then a relatively large threshold is necessary (8.6 mmHg) to identify treatment as effective for an individual patient. Indeed, if initial mean DBP = 95 mmHg and only one pre-treatment visit is used, then the expected decline in mean DBP due to regression to the mean is 2.6 mmHg even if treatment is completely ineffective. Similarly, if the true effect of treatment is relatively small for an individual patient (e.g. 2 mmHg), then it may be impossible to confidently document efficacy with acceptable sensitivity even with a large number of visits. If the true treatment effect is 5 mmHg, then as many as 5 visits before and after treatment are necessary to have approximately 75% sensitivity to document efficacy of treatment in individual patients. The clinician will be aware that some drugs may not reach their full effect for weeks after initiation of therapy. In such cases, the decision on the effectiveness of antihypertensive therapy should be made on the basis of pressures determined after the full effect of the drug has had a chance to be demonstrated. Figure 4 shows that the problem of assessing effectiveness of therapy is considerably less for the important situation produced by the necessity of adding antihypertensive medications. As the patient has been followed in the course of
837
Judging the Effectiveness of Antihypertensive Therapy
determining the necessity of more antihypertensive therapy, a large number of pre-treatment visits are available. Figure 4 shows what must be considered an extreme case, that of 20 pre-treatment visits. However, for 2 post-treatment visits we can now ascertain that a patient with a mean pre-treatment DBP of 90 mmHg who experiences a relatively modest mean observed decline of 3.4 mmHg in DBP will have a probability of 280% of having a true decline in DBP. It would be ideal to use the criteria described in this paper to identify hypertensives who are adequately and inadequately treated and to follow these subjects over time in longitudinal studies to compare cardiovascular disease endpoints in these two subgroups. This is an important goal of future research. The data and discussion presented above have several implications:
preferably spread over some weeks, before and after therapy, is a desirable goal. If this protocol is followed, then based on Table 7, a physician will be able to identify a true treatment effect of 10 mmHg in DBP with 93% sensitivity, while if there is no true change in mean DBP, then the specificity will be 80%. In the case of the currently treated hypertensive, when a second medicine is being substituted or added, an average drop of approximately 3 mmHg with three determinations at three post-treatment visits will offer a sensitivity of 75% and a specificity of 80%.
If a minimum of readings are taken prior to and after administration of a new drug, published information on the drug’s hypotensive action may provide a better measure of therapeutic efficacy for an individual patient than a single set of blood pressures. Because of the marked variability of blood pressure from session to session, the frequent practice of describing individuals as “responders” or “non-responders” without taking blood pressure variability and regression to the mean into account is unwarranted. When the drug administered takes weeks to achieve its full effect, the series of blood pressure determinations required to assess efficacy can be deferred until the presumed time of full effect is reached. When patients have been followed in clinic and repeated pressures are obtained under optimal conditions, then efficacy can be demonstrated much more easily than in the case of initial therapy. However, even in the case of a large number of blood pressure determinations before the new therapy is added, one needs approximately a 5 mmHg drop in diastolic blood pressure to determine that there is any efficacy for the new drug, if only one follow-up blood pressure visit is obtained.
Rosner B, Polk BF. Predictive values of routine blood pressure measurements in screening for hypertension. Am J Epidemiol 1983; 117(4): 429-442. 2. Trials of Hypertension Prevention: Phase I Design. AM Epidemiol 1991; (5): 455472. 3. The 1980 Report of the Joint National Committee on Detection, Evaluation and Treatment of High Blood Pressure. Arch Intern Med 1980; 140: 1280-1285. 4. Hypertension Detection and ‘Follow-up Program Cooperative Group. Blood pressure studies in 14 communities: a two-stage screen for hypertension. JAMA 1977; 237: 2385-2391. 5. Dcmanet JC, Paduart P, Fichefet JP et al. Tolerance and ionic effects in hypertensive patients of prolonged administration of an alodsterone antagonist (Amiloride) as compared with a thiaxide (Hydrochlorthiaxide). J CBn Wannacol 1970; lo: 269-273. 6. Rosner B, Polk BF. The instability of blood pressure variability over time. J Chroo Dis 1981; 34: 135-139.
In summary, we recommend that blood pressure be recorded repeatedly, in a standardized and non-stressed fashion, before and after the initiation of antihypertensive therapy. We suggest that three determinations at three visits,
Acknowledgements-The authors wish to acknowledge the support of the National Heart, Lung and Blood Institute (HL 37852 and HL 40619) for assistance in performing the work presented in this manuscript. REFERENCES 1.
APPENDIX Suppose that individual i is seen over N, visits with k readings per visit. To derive predictive values, we characterize the variability of pre-treatment blood pressure by a three-way nested analysis of variance model [l] whereby yti~=lr,+Pi+aU+eV,, i=l
,....,
P;
j=l
,...,
N,;
I=1 ,...,
k
(A.l)
where yp = Ith blood pressure measurement at the jth pretreatment visit from the ith person j,+ = underlying mean blood pressure over the entire age-race-sex group for that individual Pi follows a N(0, ug) distribution and represents between-person variability as follows a N(0, u:) distribution and represents between-visit variability eV, follows o N(0, a2) distribution and represents within-visit variability. We assume that the variance components ui and u2 are the same for all persons in particular age-race-sex groups. The basis for this assumption is given in [6] where 123 adults were ascertained at 2-4 visits 1 week apart at each of two consecutive years and the correlation between variance
838
B. Rm
and H. G. LANGFORD
components of blood pressure over successive years was CO.09. From equation (A.l), it follows tbat the distribution of true pre-treatment level of blood pressure over all persons in a particular age-race4ex group (i.e. the prevalence distribution) is normal with mean or, and variance a& It also follows from equation (A.l) that if j& is the mean blood pressure over N, pre-treatment visits with k readings per visit for the ith person and pi, is the corresponding true level of blood pressure for that person, then ~~~l~,,-N[~,,,~:/N,+~*/(N,~)l
A’YN~,, -A, etiNr +
~2/(Wll
(A3)
Our goal is to make some inference concerning A based on (a) mean observed pre-treatment (y& and post-treatment (j,,) levels of blood pressure (b) the within-person variance components of blood pressure (u:, u2) (c) the parameters of the prevalence distribution of blood pressure for particular age-racesex groups Olrr c:,. We assume a llat prior distribution for A and independence of the prior distributions for p,, and A and wish to derive the conditional distribution of A given jr= and &,,,, . IfJ g and h represent probability density functtons, then it follows that
=
[(2x)“’ cwIcw2+I-’ exp[- (Ym - p,, )2/(2c$ I ~bxP[-(&.a, x ewol,,
11
- piil- 4*/W2)1
- ~~cp)~/@GllhiI
(A.4)
where ui = ui/N,+
d/(N,k),
j = 1,2.
f=u:/uf,.
f(Al~,,,~~,)-N~**,a2++)
Thus, the expected change in the observed mean DBP depends on both the initial mean pm-treatment DBP (j&) as well as the ratio of between-person to within-person variance. The formula for predictive value given in equation (A.6) follows immediately from equation (A.5). Predictive value positive =pr(~>‘Jl&m,&o,,) = WG, - &od/~**l = 1 - Predictive value negative = 1-WA
G Wprm&,st,
64.6)
8, = &&;, u2** =
+ k+J:)/(wt* + l/G) u;* + #J2*
u2* = (l/f& + l/u:)-’ utj= u:/N,+
u2/(Njk),
j = 1,2.
Based on equation (A.6), we can establish the cutoff point AaStiw = the threshold of change in observed mean DBP for identifying patients for whom the predictive value is at least 80% as given in equation (A.7). 4lieCtiK= &prr-pi, + 2, _.u**
(A.7)
where a = 0.20. Similarly, the upper threshold of mean post-treatment DBP needed to insure that treatment is effective is given by j& - &&,. To compute the sensitivity, we have upon standard algebraic manipulation that J& - &MlI up,, 9 A) .., N(j$,, -pi, + A, u***).
Upon integration and straightforward algebraic manipulation, it follows from equation (A.4) that (A.3
where P ** = Pa -~pm,, u*+* = uf + u**,
&re -Pi, = (&We- P(P)/(l +fb
(A.2)
Let &,, be the mean blood pressure over N2 post-treatment visits with k readings per visit. If we assume that the effect of treatment is to decrease the true level of blood pressure by A mmHg for a person, and the within-person variability of blood pressure is the same for both pre- and post-treatment pressure recordings, then from equation (A. 1) we have &&i,,
An interestinginterpretationof equation (A.5) is that p** is our best estimate of the patient’s true decline in blood pressure. after accounting for regression to the mean and the within- and between-person variability of blood pressure. Furthermore, in the absence of any treatment effect, the expected change in observed mean DBP due to regression to the mean is
It follows straightforwardly
that
Sensitivity = Pr(& - ypt > Adid, Ij& , A)
(A.@
= @(A/u** - z, _,). Pi, = @.&:,
+ Prm/W~2,,
u2* = (l/u$ + l/u;)-‘.
+ WX
Correspondingly, the specificity is given by F’r(j& - yporfC L~sti,&,ra~ A=O)=l-a.
0895-4356/91 $3.00 + 0.00 Pergamon Press plc
1991
JUDGING THE EFFECTIVENESS OF ANTIHYPERTENSIVE THERAPY IN AN INDIVIDUAL PATIENT B. ROSNJJR’*and H. G. LANGFORD*+ ‘Cbanning Laboratory, Harvard Medical School, Department of Medicine and Br@am and Women’s Hospital, and Department of Biostatistics, Harvard School of Public Health and ZUniversity of Mississippi Medical Center, Mississippi, MS, U.S.A. (Received in revised form
12 February 1991)
physician’s decision as to whether a hypertensive drug has produced satisfactory blood pressure lowering is more difficult than it is usually perceived to be. Variability of blood pressure, regression to the mean and habituation to the taking of blood pressure all conspire to blur the significance of the observed drop. We present algorithms outlining the number of visits before and after the initiation of therapy and the mmHg drop required for determination with acceptable certainty that any response to the hypotensive drug has occurred. Our conclusion is that the variability of blood pressure limits the clinician’s ability to confidently conclude that antihypertensive therapy is effective unless the change in blood pressure exceeds a relatively high threshold and/or the number of visits both before and after the initiation of antihypertensive therapy is large. Because regression to the mean will be minimized, fewer visits and/or a lesser blood pressure fall in the treated patient changing or adding medications will be necessary to determine that a real decline has occurred. Abstract-The
Hypertension
Blood pressure
Screening
INTRODUCTION
It is widely accepted by investigators and clinicians that multiple blood pressure determinations are necessary before the labeling of an individual as hypertensive. Predictive value curves for routine blood pressure measurements have been derived and used as a tool for screening individuals for hypertension based on multiple blood pressure determinations obtained over several visits [l]. This approach has been successfully used in the NHLBI-sponsored Trial of Hypertension Prevention (TOHP) to identify 2164 subjects with high normal diastolic blood *All correspondence should be addressed to: Bernard Rosner. Ph.D. Associate Professor of Medicine (Biostatist&), Harvard Medical School, 180 Lo&ood Avenue, Boston, MA 02115, U.S.A. @Decca& January 1991.
pressure among a screening population of approximately 17,000 subjects and treat them with various non-pharmacologic treatment modalities [2]. Similar issues arise once an individual is identified as hypertensive and is treated. However, no consensus exists as to criteria for determining when a treatment regimen has been effective. Indeed, it is generally agreed that there is a wide individual variation in the blood pressure response to antihypertensive therapy. Many papers classify patients into on the “responders” and “non-responders” basis of one reading, or one set of readings on one occasion. Decision rules are then proposed for changing or adding medication. The purpose of this paper is to provide decision rules for determining whether an agent has actually lowered blood pressure. This problem 831
832
B. ROWERand H. G. LANGFORD
is crucial in the clinical management of hypertensive patients as well as in treatment trials of antihypertensive agents where it is desirable to identify individuals as responders or nonresponders to specific antihypertensive treatment regimens. METHODS
We assume that patients are classified as hypertensive if their observed mean diastolic blood pressure (DBP) over N, pre-treatment visits with k readings per visit exceeds some threshold (c) such as 90 mmHg . We also assume that patients are ascertained after N2 post-treatment visits with k readings per visit after therapy has begun. Let pi, = true underlying pre-treastment level of DBP for the ith person, pi2 = true underlying post-treatment level of DBP for the ith person, where the true underlying level is the average level that would be achieved over a very large number of pre- and post-treatment visits respectively. Under this model, we are implicitly assuming that true mean blood pressure remains stable over the short term during both the pre- and post-treatment periods. We classify therapy as eflective if A = pii - pi2 > 0 and ineffective if A < 0. Thus, therapy is classified as effective if blood pressure has truly declined by any amount. A can be considered as the change in DBP that would occur in a treated patient if a large number of measurements were available both pre- and post-treatment. We will also consider a decision rule whereby therapy is classified as very effective if true blood pressure has declined by a specified amount (AO), where A0 > 0. One of the issues in assessing adequacy of treatment is regression to the mean. Specifically, if the number of pre-treatment measures is small (as is often the case), then based on regression to the mean one would expect some decline in mean blood pressure even if treatment is completely ineffective. Therefore, we cannot simply use the observed mean decline in blood pressure as a guide to the effectiveness of therapy without controlling for the regression to the mean effect. For this purpose, we use a predictive value approach, where predictive value positive is defined as the probability that true blood pressure has declined after treatment (A > 0) given a specified observed decline (we denote the probability that true blood pressure has not declined
given a specified observed decline by predictive value negative). The predictive values depend on
(a) the mean observed pre-treatment
(jr& and post-treatment (j&,) levels of DBP. @I the magnitude of the components of within-person blood pressure variability denoted by 0: for between visit variability and G* for within visit variability, and 04 the prevalence distribution of DBP, which for a specific age-sex-race group is assumed to be normal with mean pp and variance 0 $. Based on these parameters, the formula for predictive value positive is derived in the Appendix and is given in equation (A.@. To use the predictive value approach, we must determine what level of predictive value should be used to identify patients for whom therapy is effective. In a previous study [l], questionnaires were sent to the 15 members of the 1980 Joint National Committee for the Detection, Evaluation and Treatment of High Blood Pressure [3] and to the principal clinical investigator in each center of the Hypertension Detection and Follow-up Program [4] asking: How large should your positive and negative predictive values be, based on the blood pressure measurements you have taken for identifying persons for whom newly initiated antihypertensive drug therapy is warranted? The median predictive value selected by the 30 respondents was approximately 80% for both predictive value positive and negative. If we apply the same criteria for evaluating the effectiveness of antihypertensive therapy, then based on equation (A.6), we can establish a threshold of observed change in mean DBP needed to identify patients for whom the predictive value positive, is at least 80%. This value is given by kRKtivcas defined in equation (A.7). To use this approach, we declare Treatment effective:
Treatment ineffective: Furthermore, a predictive declined by threshold is
if mean pre-treatment DBP - mean post-treatment DBP 2 A&xtive otherwise.
(4)
based on equation (A.7), to insure value of 80% that true DBP has at least 4 mmHg, the critical given by A0+ AdfcEtjyc.
833
Judging the Effectiveness of Antihypertensive Therapy Table 1. Thresholds* for determination of effectiveness of antihyuertensive therapy (N, = 1 pre-treatment visit)t Number of post-treatment visitst
Initial mean pre-treatment DBP 90 95 100 105 110 115
1
2
3
4
5
Expected changej
7.6 8.6 9.6 10.6 11.6 12.6
6.7 7.7 8.7 9.7 10.7 11.7
6.3 7.3 8.4 9.4 10.4 11.4
6.2 7.2 8.2 9.2 10.2 11.2
6.0 7.1 8.1 9.1 10.1 11.1
1.6 2.6 3.6 4.6 5.6 6.6
*Change in mean DBP (pre-post) required to insure that treatment is effective in lowering true mean DBP with 2 80% probability = Aar (see equation A.7). tpr = 82.3 mmHg = mean of prevalence distribution of DBP, a$ = 112.5, IS; = 26.2, u* = 7.0, based on a weighted average of the eight age-race-sex-specific parameter estimates with weights given by the number of screenees in specific age-race-sex groups in the HDFP program [4]. #Three readings per visit for both pre- and post-treatment visits. §Expected change in mean DBP due to regression to the mean in the absence of any treatment effect (see Appendix).
Table 2. Thresholds* for determination of effectiveness of antihvnertensive theranv IN, = 2 me-treatment visit&t Initial mean pre-treatment DBP$
Number of post-treatment visit@ 1
2
3
4
5
Expected change]]
105
6.3 6.8 7.4 7.9
5.2 5.8 6.4 6.9
4.8 5.4 5.9 6.5
4.6 5.2 5.7 6.3
4.5 5.0 5.6 6.2
0.9 1.4 2.0 2.6
110 115
8.5 9.1
7.5 8.0
7.1 7.6
6.9 7.4
6.7 7.3
3.1 3.7
90 95
100
*Change in mean DBP (pre-post) required to insure that treatment is effective in lowering true mean DBP with > 80% probability = & (see equation A.7). tpr = 82.3 mmHg = mean of prevalence distribution of DBP, u: = 112.5, ai = 26.2, u2 = 7.0, based on a weighted average of the eight age-race-sex-specific parameter estimates with weights given by the number of screenees in specific age-race-sex groups in the HDFP program [4]. IBased on six readings, three readings per visit. §Based on three readings per visit. ]]Expected change in mean DBP due to regression to the mean in the absence of any treatment effect (see Appendix).
Therefore, we have computed the cutoff value for Nr = l-3 and 20 (a large number of) preWe note that the critical cutoff value AeEdVe, treatment visits, N2 = l-5 post-treatment visits, depends on k = 3 readings per visit, mean pre-treatment the number of pre-treatment visits (N,) level of DBP = 90-l 15 mmHg. For the variance the number of post-treatment visits (Nr) components and prevalence distribution parthe number of readings per visit (k) ameters, we used a weighted average of the the mean pre-treatment level of DBP (j&) eight age-race-sex-specific values (30-49/50-69; the within-person variance components of male/female; white/black) [ 11,where the weights DBP (a;, a’) used corresponded to the number of persons the prevalence distribution of DBP screened in these age-race-sex groups in the (mean = Pi, variance = a’,) HDFP program [4]. Finally, we used a predicthe predictive value positive (1 - a). :tive value positive of 80%. The cutoff values are presented in tabular form in Tables l-4 and in 7 graphical form in Figs 1-I for 1, 2 and 3 and a large number of pre-treatment visits (N, = 20) !3Number of respectively. We have also computed the Post -Treatment Visits expected decline in DBP due to regression to the -i i2RESULTS
II-
6-
,,,,,,,
0. For example, if one has a mean pre-treatment DBP of 95 mmHg based on two pre-treatment visits and one wishes to insure that true DBP has declined by at least 6 mmHg after beginning therapy with 80% probability, then one would need a threshold of 12.8 mmHg (6.8 + 6.0) based on one post-treatment visit and 11.0 mmHg (5.0 + 6.0) based on five post-treatment visits. Another way of expressing these relationships is in terms of absolute mean level of blood pressure required at post-treatment visits to certify effectiveness of therapy. These data are Table 5. Absolute mean level of post-treatment DBPg required to certify effectiveness of antihypertensive therapy* Initial mean pre-treatment DBPt
Number of post-treatment 1
visits1
2
3
4
5
85.4 86.3 87.2 88.1 88.9
85.5 86.4 87.3 88.2 89.1
90 91 92 93 94
83.7 84.6 85.5 86.4 87.3
84.8 85.7 86.5 87.4 88.3
85.2 86.1 87.0 87.8 88.7
95 96 91 98 99
88.2 89.1 90.0 90.8 91.7
89.2 90.1 91.0 91.9 92.8
89.6 90.5 91.4 92.3 93.2
89.8 90.7 91.6 92.5 93.4
90.0 90.9 91.7 92.6 93.5
100 101 102 103 104
92.6 93.5 94.4 95.3 96.2
93.6 94.5 95.4 96.3 91.2
94.1 94.9 95.8 96.7 97.6
94.3 95.2 96.0 96.9 97.8
94.4 95.3 96.2 97.1 98.0
105 106 107 108 109
97.1 91.9 98.8 99.7 100.6
98.1 99.0 99.9 100.7 101.6
98.5 99.4 100.3 101.2 102.0
98.1 99.6 100.5 101.4 102.3
98.8 99.7 100.6 101.5 102.4
110 111 112 113 114 115
101.5 102.4 103.3 104.2 105.0 105.9
102.5 103.4 104.3 105.2 106.1 107.0
102.9 103.8 104.7 105.6 106.5 107.4
103.1 104.0 104.9 105.8 106.7 107.6
103.3 104.2 105.1 105.9 106.8 107.7
*pp = 82.3 mmHg = mean of prevalence distribution of DBP, a! = 112.5, a? = 26.2. u2 = 7.0. based on a weinhted average’ of the. &ht agerace-sex-specific pa&eter estimates with weights given by the number of screenees in specific age-race-sex groups in the HDPP program
[41. tBased on six readings, three readings per visit. $Based on three readings per visit. $Critical upper threshold of mean post-treatment DBP to insure that treatment is effective in lowering true mean DBP with 380% probability (see equation A.7).
Table 6. Example of use of evaluation algorithm Cumulative Visit (mean)
Sum
No. readings
Mean
288 (96) 282 (94)
288
3
96.0
570
6
95.0
276 (92.0) 266 (88.7) 258 (86.0)
276
3
92.0
542
6
90.3
800
9
88.9
SllUl
Visit
Readings
Pm-treatment 1 96 98 94 2
96 94 92
Post-treatment 94 90 92 1 2
88 88 90
3
88 86 84
presented in Table 5 in the special case of two pre-treatment visits for one to five post-treatment visits and 1 mmHg increments of mean pre-treatment DBP (90-l 15 mmHg). An illustration of the use of this table in clinical practice is provided in Table 6. Suppose one has a patient with a mean pretreatment DBP of 95 mmHg based on six readings over two visits (three readings per visit). Antihypertensive therapy is administered at that time and 3 weekly follow-up visits are scheduled to assess the effectiveness of therapy. Suppose the mean DBP at the first, second and third post-treatment visit is 92.0, 88.7 and 86.0mmHg, respectively, yielding an overall cumulative mean of 88.9 mmHg over the three visits (nine readings). Since 88.9 < 89.6 = upper limit for determining effectiveness of therapy based on three post-treatment visits, we conclude that therapy is effective in lowering DBP. As is illustrated in the previous example, an important issue in implementing an algorithm of this type, is to determine what are an appropriate number of pre- and post-treatment visits so as to efficiently identify patients for whom therapy is or is not effective. To study this question, we have computed the sensitivity of the decision rule for a patient with a given observed mean pre-treatment DBP, given (a) a specific true decline in mean DBP (A) = 0, 2, 5 or lOmmHg, (b) the number of pre-treatment visits (N,) = 1 ,5, 20, (c) the’ * number of post-treatment visits (I$) = 1, . . . ,5, 10 and (d) the number of readings per visit (k) = 3. (e) the level of predictive value positive = 1 -a.
836
B. RCSNERand H. G.
Table 7. Sensitivity* of decision rule to identify persons for whom antihypertensive therapy is effective (Nr = l-5, 20; N, = 1-5, 10: A = 0. 2, 5 or 10 mmHd
N,t
A§
1
2
3
4
5
10
1
0 2 5 10
0.200 0.287 0.444 0.711
0.200 0.305 0.493 0.789
0.200 0.313 0.516 0.821
0.200 0.318 0.530 0.839
0.209 0.321 0.539 0.849
0.200 0.328 0.559 0.872
2
0 2 5 10
0.200 0.299 0.476 0.764
0.200 0.325 0.549 0.861
0.209 0.339 0.588 0.900
0.200 0.348 0.612 0.920
0.200 0.354 0.628 0.932
0.200 0.370 0.666 0.955
3
0 2 5 10
0.209 0.304 0.491 0.787
0.200 0.336 0.580 0.893
0.200 0.355 0.629 0.933
0.200 0.368 0.660 0.952
0.200 0.377 0.682 0.963
0.200 0.400 0.735 0.982
4
0 2 5 10
0.200 0.307 0.501 0.801
0.200 0.343 0.599 0.910
0.200 0.366 0.656 0.950
0.200 0.381 0.693 0.968
0.209 0.393 0.719 0.977
0.200 0.423 0.781 0.992
5
0 2 5 10
0.200 0.310 0.507 0.809
0.200 0.349 0.613 0.921
0.200 0.373 0.675 0.960
0.200 0.391 0.716 0.976
0.200 0.405 0.745 0.984
0.200 0.442 0.815 0.9%
20
0 2
0.209 0.318 0.529 0.838
0.209 0.369 0.664 0.954
0.200 0.407 0.750 0.986
0.200 0.438 0.808 0.995
0.200 0.464 0.850 0.998
0.200 0.551 0.943 1.000
1;
*Sensitivity=Q(A/a** -q-J. TN, = number of pre-treatment visits. $N, = number of post-treatment visits. $A = true decline in mean DBP after start of antihypertensive therapy.
The sensitivity is derived in equation (A.8) with results presented in Table 7. We note first that the probability of determining that antihypertensive therapy is effective when it has no true effect (A = 0), i.e. 1 - specificity, is 20% for all parameter combinations. This was planned by design since a = 0.20. Second, we note that the sensitivity is inadequate to enable one to detect a true 2 mmHg change in an individual patient; even with as many as 20 pre- and 10 post-treatment visits, only 55% of patients with a true 2 mmHg decline would be identified by this rule. Third, sensitivity is marginally adequate to allow one to identify individual patients with a true 5 mmHg decline; with A = 5 and 5 pre-treatment visits, one needs as many as 10 post-treatment visits to achieve 80% sensitivity and 4 post-treatment visits to achieve 70% sensitivity. Finally, sensitivity is more than adequate to identify individual patients with a true decline of 10 mmHg; one can achieve 80% sensitivity with as few as 2 pre- and 2 post-treatment visits.
bNGFGRD
DISCUSSION
Most investigators have accepted the necessity of obtaining multiple blood pressure determinations over several visits before labeling an individual as hypertensive. However, there is no consensus as to criteria for determining the effectiveness of antihypertensive therapy once an individual has been identified as a hypertensive. In this paper, specific criteria are provided for determining when treatment is efficacious for an individual patient with high probability based on a threshold of mean blood pressure decline. The required threshold depends on the initial blood pressure level, as well as the number of visits and measurements per visit both before and after therapy. It is clear from the results in this paper that, because of the variability of blood pressure, reliable inferences concerning the efficacy of antihypertensive therapy can only be made based on a large number of visits both before and after therapy. For example, if the initial mean pre-treatment DBP = 95 mmHg and only a single visit is used both before and after therapy to characterize the efficacy of treatment as is often the case in clinical trials [5], then a relatively large threshold is necessary (8.6 mmHg) to identify treatment as effective for an individual patient. Indeed, if initial mean DBP = 95 mmHg and only one pre-treatment visit is used, then the expected decline in mean DBP due to regression to the mean is 2.6 mmHg even if treatment is completely ineffective. Similarly, if the true effect of treatment is relatively small for an individual patient (e.g. 2 mmHg), then it may be impossible to confidently document efficacy with acceptable sensitivity even with a large number of visits. If the true treatment effect is 5 mmHg, then as many as 5 visits before and after treatment are necessary to have approximately 75% sensitivity to document efficacy of treatment in individual patients. The clinician will be aware that some drugs may not reach their full effect for weeks after initiation of therapy. In such cases, the decision on the effectiveness of antihypertensive therapy should be made on the basis of pressures determined after the full effect of the drug has had a chance to be demonstrated. Figure 4 shows that the problem of assessing effectiveness of therapy is considerably less for the important situation produced by the necessity of adding antihypertensive medications. As the patient has been followed in the course of
837
Judging the Effectiveness of Antihypertensive Therapy
determining the necessity of more antihypertensive therapy, a large number of pre-treatment visits are available. Figure 4 shows what must be considered an extreme case, that of 20 pre-treatment visits. However, for 2 post-treatment visits we can now ascertain that a patient with a mean pre-treatment DBP of 90 mmHg who experiences a relatively modest mean observed decline of 3.4 mmHg in DBP will have a probability of 280% of having a true decline in DBP. It would be ideal to use the criteria described in this paper to identify hypertensives who are adequately and inadequately treated and to follow these subjects over time in longitudinal studies to compare cardiovascular disease endpoints in these two subgroups. This is an important goal of future research. The data and discussion presented above have several implications:
preferably spread over some weeks, before and after therapy, is a desirable goal. If this protocol is followed, then based on Table 7, a physician will be able to identify a true treatment effect of 10 mmHg in DBP with 93% sensitivity, while if there is no true change in mean DBP, then the specificity will be 80%. In the case of the currently treated hypertensive, when a second medicine is being substituted or added, an average drop of approximately 3 mmHg with three determinations at three post-treatment visits will offer a sensitivity of 75% and a specificity of 80%.
If a minimum of readings are taken prior to and after administration of a new drug, published information on the drug’s hypotensive action may provide a better measure of therapeutic efficacy for an individual patient than a single set of blood pressures. Because of the marked variability of blood pressure from session to session, the frequent practice of describing individuals as “responders” or “non-responders” without taking blood pressure variability and regression to the mean into account is unwarranted. When the drug administered takes weeks to achieve its full effect, the series of blood pressure determinations required to assess efficacy can be deferred until the presumed time of full effect is reached. When patients have been followed in clinic and repeated pressures are obtained under optimal conditions, then efficacy can be demonstrated much more easily than in the case of initial therapy. However, even in the case of a large number of blood pressure determinations before the new therapy is added, one needs approximately a 5 mmHg drop in diastolic blood pressure to determine that there is any efficacy for the new drug, if only one follow-up blood pressure visit is obtained.
Rosner B, Polk BF. Predictive values of routine blood pressure measurements in screening for hypertension. Am J Epidemiol 1983; 117(4): 429-442. 2. Trials of Hypertension Prevention: Phase I Design. AM Epidemiol 1991; (5): 455472. 3. The 1980 Report of the Joint National Committee on Detection, Evaluation and Treatment of High Blood Pressure. Arch Intern Med 1980; 140: 1280-1285. 4. Hypertension Detection and ‘Follow-up Program Cooperative Group. Blood pressure studies in 14 communities: a two-stage screen for hypertension. JAMA 1977; 237: 2385-2391. 5. Dcmanet JC, Paduart P, Fichefet JP et al. Tolerance and ionic effects in hypertensive patients of prolonged administration of an alodsterone antagonist (Amiloride) as compared with a thiaxide (Hydrochlorthiaxide). J CBn Wannacol 1970; lo: 269-273. 6. Rosner B, Polk BF. The instability of blood pressure variability over time. J Chroo Dis 1981; 34: 135-139.
In summary, we recommend that blood pressure be recorded repeatedly, in a standardized and non-stressed fashion, before and after the initiation of antihypertensive therapy. We suggest that three determinations at three visits,
Acknowledgements-The authors wish to acknowledge the support of the National Heart, Lung and Blood Institute (HL 37852 and HL 40619) for assistance in performing the work presented in this manuscript. REFERENCES 1.
APPENDIX Suppose that individual i is seen over N, visits with k readings per visit. To derive predictive values, we characterize the variability of pre-treatment blood pressure by a three-way nested analysis of variance model [l] whereby yti~=lr,+Pi+aU+eV,, i=l
,....,
P;
j=l
,...,
N,;
I=1 ,...,
k
(A.l)
where yp = Ith blood pressure measurement at the jth pretreatment visit from the ith person j,+ = underlying mean blood pressure over the entire age-race-sex group for that individual Pi follows a N(0, ug) distribution and represents between-person variability as follows a N(0, u:) distribution and represents between-visit variability eV, follows o N(0, a2) distribution and represents within-visit variability. We assume that the variance components ui and u2 are the same for all persons in particular age-race-sex groups. The basis for this assumption is given in [6] where 123 adults were ascertained at 2-4 visits 1 week apart at each of two consecutive years and the correlation between variance
838
B. Rm
and H. G. LANGFORD
components of blood pressure over successive years was CO.09. From equation (A.l), it follows tbat the distribution of true pre-treatment level of blood pressure over all persons in a particular age-race4ex group (i.e. the prevalence distribution) is normal with mean or, and variance a& It also follows from equation (A.l) that if j& is the mean blood pressure over N, pre-treatment visits with k readings per visit for the ith person and pi, is the corresponding true level of blood pressure for that person, then ~~~l~,,-N[~,,,~:/N,+~*/(N,~)l
A’YN~,, -A, etiNr +
~2/(Wll
(A3)
Our goal is to make some inference concerning A based on (a) mean observed pre-treatment (y& and post-treatment (j,,) levels of blood pressure (b) the within-person variance components of blood pressure (u:, u2) (c) the parameters of the prevalence distribution of blood pressure for particular age-racesex groups Olrr c:,. We assume a llat prior distribution for A and independence of the prior distributions for p,, and A and wish to derive the conditional distribution of A given jr= and &,,,, . IfJ g and h represent probability density functtons, then it follows that
=
[(2x)“’ cwIcw2+I-’ exp[- (Ym - p,, )2/(2c$ I ~bxP[-(&.a, x ewol,,
11
- piil- 4*/W2)1
- ~~cp)~/@GllhiI
(A.4)
where ui = ui/N,+
d/(N,k),
j = 1,2.
f=u:/uf,.
f(Al~,,,~~,)-N~**,a2++)
Thus, the expected change in the observed mean DBP depends on both the initial mean pm-treatment DBP (j&) as well as the ratio of between-person to within-person variance. The formula for predictive value given in equation (A.6) follows immediately from equation (A.5). Predictive value positive =pr(~>‘Jl&m,&o,,) = WG, - &od/~**l = 1 - Predictive value negative = 1-WA
G Wprm&,st,
64.6)
8, = &&;, u2** =
+ k+J:)/(wt* + l/G) u;* + #J2*
u2* = (l/f& + l/u:)-’ utj= u:/N,+
u2/(Njk),
j = 1,2.
Based on equation (A.6), we can establish the cutoff point AaStiw = the threshold of change in observed mean DBP for identifying patients for whom the predictive value is at least 80% as given in equation (A.7). 4lieCtiK= &prr-pi, + 2, _.u**
(A.7)
where a = 0.20. Similarly, the upper threshold of mean post-treatment DBP needed to insure that treatment is effective is given by j& - &&,. To compute the sensitivity, we have upon standard algebraic manipulation that J& - &MlI up,, 9 A) .., N(j$,, -pi, + A, u***).
Upon integration and straightforward algebraic manipulation, it follows from equation (A.4) that (A.3
where P ** = Pa -~pm,, u*+* = uf + u**,
&re -Pi, = (&We- P(P)/(l +fb
(A.2)
Let &,, be the mean blood pressure over N2 post-treatment visits with k readings per visit. If we assume that the effect of treatment is to decrease the true level of blood pressure by A mmHg for a person, and the within-person variability of blood pressure is the same for both pre- and post-treatment pressure recordings, then from equation (A. 1) we have &&i,,
An interestinginterpretationof equation (A.5) is that p** is our best estimate of the patient’s true decline in blood pressure. after accounting for regression to the mean and the within- and between-person variability of blood pressure. Furthermore, in the absence of any treatment effect, the expected change in observed mean DBP due to regression to the mean is
It follows straightforwardly
that
Sensitivity = Pr(& - ypt > Adid, Ij& , A)
(A.@
= @(A/u** - z, _,). Pi, = @.&:,
+ Prm/W~2,,
u2* = (l/u$ + l/u;)-‘.
+ WX
Correspondingly, the specificity is given by F’r(j& - yporfC L~sti,&,ra~ A=O)=l-a.
