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Linear increment in efficiency with the inclusion of surrogate endpoint
Statistics and Probability Letters 96 (2015) 102–108
Contents lists available at ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Linear increment in efficiency with the inclusion of surrogate endpoint Buddhananda Banerjee a , Atanu Biswas b,∗ a
Indian Institute of Science Education and Research, Kolkata, India
b
Indian Statistical Institute, Kolkata, India
article
info
Article history: Received 10 May 2014 Received in revised form 10 September 2014 Accepted 11 September 2014 Available online 27 September 2014
abstract In a two-sample clinical trial, a fixed proportion of true-and-surrogate and the remaining only-surrogate responses are observed. We quantify the increase in efficiency to compare the treatments as a linear function of the proportion of available true responses. © 2014 Elsevier B.V. All rights reserved.
Keywords: Surrogate responses Odds ratio Risk ratio Treatment difference
1. Introduction A surrogate endpoint is chosen as a measure or an indicator of a biological process. Usually it is obtained sooner, at a lesser cost than the true endpoint of health outcome, and is used to arrive at a conclusion about an effect of intervention on the true endpoint. Surrogate endpoints are used with growing interest in medical science. For example, in a trial of a treatment of osteoporosis we might be interested in reduction of the fracture rate, but we measure the bone mineral density (BMD) instead. A change in CD4 cell count in a randomized trial is considered as a surrogate of survival time in the study of HIV affected patients. Again, some damages to the heart muscle due to myocardial infarction can be accurately assessed by an arterioscintography reading. As it is an expensive procedure, the peak cardiac enzyme level in the blood stream, which is more easily obtainable, is used as a surrogate measure of heart vascular damage (see Wittes et al., 1989). Sometimes the observed value of the response variable in the middle of an ongoing experiment is considered as the surrogate endpoint. For example patients with age related macular degeneration (ARMD) progressively loose vision. To compare between placebo and high-dose interferon-α for its treatment, observations are taken after six months and one year. The observation after six months is considered as a surrogate corresponding to the final outcome. Two basic problems are studied in the literature of surrogate responses, namely (a) validation of a surrogate and (b) measurement of gain in inference using the surrogate responses. Prentice (1989) gave validation criteria for a surrogate, which is subsequently discussed by Freedman et al. (1992), Reilly and Pepe (1995), Day and Duffy (1996), Buyse and Molenberghs (1998), Buyse et al. (2000), Molenberghs et al. (2001), and Chen et al. (2007). The use of surrogate endpoints is likely to be beneficial, not only in terms of cost or time, but it gives more accuracy in the estimation of target parametric functions such as treatment difference and odds ratio. For that purpose we first look at the data structure under
∗
Corresponding author. Tel.: +91 33 25752818; fax: +91 33 25773104. E-mail addresses: [email protected] (B. Banerjee), [email protected], [email protected] (A. Biswas).
http://dx.doi.org/10.1016/j.spl.2014.09.015 0167-7152/© 2014 Elsevier B.V. All rights reserved.
B. Banerjee, A. Biswas / Statistics and Probability Letters 96 (2015) 102–108
103
consideration. In this article we are interested to measure the increment of the efficiency with the increase of the proportion of surrogate endpoints where the surrogate is presumed to be validated. Suppose we consider the accumulated data when all of the surrogate responses are known, but only Q % of the true responses are available. Then, if we do not consider the surrogate data, we need to make inference based on Q % true responses only. In the present paper our objective is to use surrogate data efficiently (which consist of Q % bivariate data and (100 − Q )% only univariate surrogate data) to improve the inference. To use (100 − Q )% surrogate data efficiently one need to identify the dependency structure of true and surrogate responses based on the Q % bivariate true-and-surrogate data. A real data example which is appropriate to this situation is described in the next section. Pepe (1992) obtained the distribution of the estimator of regression parameter when the validation sample fraction has a fixed limiting value, ρ (=Q /100), say. Banerjee and Biswas (2011) established that the variance of the estimator of treatment difference is bounded for such a fixed ρ . Lin et al. (1997) measured the extent to which a biological marker is a surrogate endpoint for a clinical event and Wang and Taylor (2002) propose alternative measures of the proportion explained by the surrogate endpoint. Chen (2000) and Begg and Leung (2000) discussed the inferential improvement by the use of surrogate endpoints. Chen et al. (2003) introduced the concept of information recovery from surrogate endpoints by considering linear models for true and surrogate on covariates. Proportion of validation sample, ρ , naturally plays a key role in the gain in associated inference. The validation samples are true and surrogate paired observations, but the rest of the samples are surrogate responses only. In Section 2 we describe the set up in details and the data structure under a general probability model for binary true and binary surrogate responses. In Section 3 we establish that the (inverse of) relative efficiency to estimate the treatment success probability by using surrogate endpoints is a linear function of the validation sample proportion, ρ . As a simple consequence of that we also prove the (inverse of) relative efficiency to estimate treatments difference, log risk ratio and log odds ratio in a two-treatment set up is also linear in ρA and ρB , the validation sample proportions for the two treatments A and B, respectively. In Section 4 we demonstrate our results with data example and conclude. 2. Experimental details and data structure We consider a set up of two treatments having binary true endpoints with binary surrogates as well. Begg and Leung (2000) pointed out that for the binary endpoints the probability of concordance is an indicator of association between true and surrogate endpoints. Suppose nA and nB patients are allotted to the treatments A and B, respectively; but we get only mA and mB true endpoints along with all surrogate endpoints within the stipulated time frame or cost limit, where mt ≪ nt , t = A, B. Denote the true and surrogate endpoints for the treatment t by Yt and Wt , where t = A, B. All these endpoints are either 1 or 0 for success or failure, respectively. We denote pt = P (Yt = 1) as the success probability by the true endpoints for treatment t. Furthermore, let us denote P (Wt = 1|Yt = 1) = πt1
and P (Wt = 0|Yt = 0) = πt0 ,
(1)
which are the sensitivity and specificity of the 2 × 2 table for treatment t where the true and surrogate responses are in the two margins. Clearly it is a saturated model with full parameter space. Consequently, the success probabilities by the surrogate responses for the two treatments are rt = P (Wt = 1)
= P (Wt = 1|Yt = 1)P (Yt = 1) + P (Wt = 1|Yt = 0)P (Yt = 0) = πt1 pt + (1 − πt0 )(1 − pt ) = pt (πt1 + πt0 − 1) + (1 − πt0 ). The data corresponding to treatment t can be represented in a table as follows. True Yt = 1 Yt = 0 Total Only surrogate
Surrogate Wt = 1 mt11 mt01 WtT WtS
Wt = 0 mt10 mt00 mt − WtT nt − mt − WtS
Total YtT mt − YtT mt nt − mt
t t t where YtT = i=1 Yti and WtT = i=1 Wti ; also we denote WtS = i=mt +1 Wti for t = A and B. The notation (Yti , Wti ) is specifically used for denoting the response variables corresponding to the ith individual under tth treatment. If any marginal is found to be zero, it is customary to add 0.5 to each of the marginals. As an example/illustration we consider the data set analyzed by Buyse and Molenberghs (1998). This data set is obtained from a randomized clinical trial comparing an experimental treatment interferon-α , with highest dose, 6-million units daily to a corresponding placebo in the treatment of patients with age-related macular degeneration (ARMD). Patients with ARMD progressively lose vision. In the trial, a patient’s visual acuity is assessed at different time points through the ability to read lines of letters on standardized vision charts. It is examined whether the loss of at least two lines of vision at 6 months (denoted as 1, and 0 otherwise) can be used
m
m
n
104
B. Banerjee, A. Biswas / Statistics and Probability Letters 96 (2015) 102–108
as a surrogate for the loss of at least three lines of vision at 1 year (denoted as 1, and 0 otherwise) which is a true endpoint with respect to the effect of interferon-α . A total of 87 patients received interferon-α and 103 received placebo. Treatment procedures are evaluated after the first week, which is considered as surrogate endpoints (Winterferon-α or WPlacebo ) and the second week to assess the helpfulness by true endpoints (Yinterferon-α or YPlacebo ). These data are used in Section 6 for illustration. If we assume that 40% of the patients allocated to the interferon-α drug will turn up with both true and surrogate endpoints, that is ρinterferon-α = 0.4, then the data set for the drug interferon-α and placebo can be represented as mt11 mt10 mt01 mt00 WtS nt − mt − WtS 15 4 4 12 28 24 12 4 4 24 22 47
t Interferon-α Placebo
3. Variance reduction using surrogate responses Our basic objective is to quantify the increase in efficiency or reduction in variance of the estimator of any specified parametric function by properly using the information extracted from the surrogate endpoints. Consider the likelihood for the treatment t, L(ξt ) =
mt
y
ytT
×
mt −ytT
pt tT qt
ytT mt11
mt − ytT mt01
m
πt1t11 (1 − πt1 )ytT −mt11 m
(1 − πt0 )mt01 πt0t00
nt − m t
wtS
wtS
rt
(1 − rt )nt −mt −wtS ,
(2)
where ξt = (pt , πt1 , πt0 ) and qt = 1 − pt . The Fisher information matrix is I(ξt ) and the (1, 1)th element of [I(ξt )]−1 , denoted 1 (S ) by [I(ξt )]− 11 , gives the variance of maximum likelihood estimator pt . Here we define the inverse of efficiency, which is the measure of improvement to estimate pt when surrogate augmented analysis is conducted. So the measure of improvement is
1 [I(ξt )]− 11
. Furthermore, denote mt /nt = ρt ∈ (0, 1]. We note that ρt = 0 is possible only when mt = 0, indicating no true response is available. This is not of any statistical interest. Using mt = ρt nt we get I(ξt ) = nt Iξt (ρt ). Hence the measure of improvement, given by the proportional variance, reduces to pt qt /mt
−1
1 n− Iξt (ρt ) t
Gξt (ρt ) =
11
1 m− t pt qt
=
−1 ρt Iξt (ρt ) 11 pt qt
.
Now we have the following theorem, proof of which is given in Appendix A.1 Theorem 1. (a) Relative gain, denoted by the inverse of efficiency, Gξt (ρt ) by using surrogate endpoints with ρt proportion of available true responses is a linear function of ρt , that is Gξt (ρt ) = Ct + (1 − Ct )ρt = ρt + (1 − ρt )Ct ,
(3)
which is a straight line joining the points (0, Ct ) and (1, 1) with intercept and slope add to unity, with Ct =
qt πt0 (1 − πt0 ) + pt πt1 (1 − πt1 ) rt (1 − rt )
.
(b) Further Ct =
E (Var(Wt |Yt )) Var(Wt )
∈ [0, 1].
Remark. The expression in (3) is amazingly simple in the context of the fairly general joint distribution of the true and surrogate endpoints. So far our knowledge goes, this expression and the following expressions are new in the literature E (Var(Wt |Yt )) and slope is where true and surrogate both are binary endpoints. The intercept of the line in Eq. (3) is Ct = Var(W )
(1 − Ct ) =
Var(E (Wt |Yt )) , Var(Wt )
t
and they add up to 1. It is also evident from Eq. (3) that the value of G is also 1 when ρt = 1, irrespective of the value of Ct . The slope gives the rate of increment of efficiency with the inclusion of true endpoints. If we let the value of ρt to increase or in other words if we let mt to increase and keep nt as fixed then (1 − Ct ) indicates the improvement in efficiency. The intercept gives the lower bound of efficiency gain. Clearly Ct = 1 when true and surrogate endpoints are independent, and Ct = 0 when the conditional distribution of surrogate endpoint given the true is degenerate one, for treatment t. Total variability of Wt is partitioned into two parts. The variability of the regression of Wt on Yt relative to the variability of Wt (i.e. correlation ratio) serves as the slope, and the variability of the unexplained part in the regression related to the total variability is the intercept. Theorem 1 can be effectively used to obtain the gain in efficiencies for the standard measures of treatment difference, like the log odds ratio, difference in success probabilities and relative risk ratio in the two-treatment set up. We present these in Theorem 2 and Corollaries 1–3.
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105
Theorem 2. The inverse of efficiency in variance estimation by using surrogate endpoints to estimate the log odds ratio (OR),
θ = log
pA qB qA pB
, is given by a plane, that is
GOR (ρA , ρB ) =
(mA pA qA )−1 GξA (ρA ) + (mB pB qB )−1 GξB (ρB ) . (mA pA qA )−1 + (mB pB qB )−1
(4)
The proof is immediate with the use of delta method and Theorem 1. Corollary 1. When ρA = ρB = ρ , the inverse of efficiency by using surrogate endpoints to estimate the log odds ratio is a linear function of ρ , that is GOR (ρ, ρ) = G∗OR (ρ) = ρ + (1 − ρ)
(nA pA qA )−1 CA + (nB pB qB )−1 CB . (nA pA qA )−1 + (nB pB qB )−1
Corollary 2. The inverse of efficiency by using surrogate endpoints to estimate θ = pA − pB , the treatment difference (TD), is a plane given by GTD (ρA , ρB ) =
1 −1 m− A pA qA GξA (ρA ) + mB pB qB GξB (ρB ) 1 −1 m− A pA qA + mB pB qB
,
and, for ρA = ρB = ρ we get the line given by
GTD (ρ, ρ) = GTD (ρ) = ρ + (1 − ρ) ∗
1 −1 n− A pA qA CA + nB pB qB CB 1 −1 n− A pA qA + nB pB qB
.
Corollary 3. The inverse of efficiency by using surrogate endpoints to estimate θ = log
pA pB
, the log risk ratio (RR), is a plane
given by GRR (ρA , ρB ) =
(mA pA )−1 qA GξA (ρA ) + (mB pB )−1 qB GξB (ρB ) , (mA pA )−1 qA + (mB pB )−1 qB
and when ρA = ρB = ρ it reduces to a linear function of ρ given by
(nA pA )−1 qA CA + (nB pB )−1 qB CB . GRR (ρ, ρ) = GRR (ρ) = ρ + (1 − ρ) (nA pA )−1 qA + (nB pB )−1 qB ∗
In Fig. 1 we plot GOR (ρA , ρB ), given in (4), against ρA and ρB . We find that, for suitable ρA and ρB , there is considerable gain in estimation of OR. Fig. 2 illustrates the special case of ρA = ρB where GA , GB , G∗OR , G∗TD , G∗RR are all straight lines. In fact, GOR (., .), GTD (., .) and GRR (., .) are convex combinations of GξA and GξB , and belong to in between these two straight lines. We have taken mA = mB = 34 and pA = 0.7, pB = 0.8 in these two figures, for illustration. 4. Data example and discussions We obtain the treatment difference (TD) between interferon-α and placebo alloted to the ARMD patients (see Buyse and Molenberghs, 1998). Estimated treatment difference is 0.162. The observed value of CA = Cinterferon-α = 0.66 and CB = Cplacibo = 0.58. So the equation of G∗TD (ρ) = ρ + 0.62(1 − ρ) where we assume the equal values of ρinterferon-α and ρplacebo , say ρ . For difference values of ρ = {0.2, 0.3, . . . , 0.9} TD is calculated based only on true endpoints and on surrogate augmented data as well. The ratio of their bootstrap variances are plotted along with the line G∗TD (ρ) (see Corollary 2). We find that the observed measures of efficiencies are close to the straight line G∗TD (ρ), see Fig. 3. In this article we discussed the impact of proportion of available true endpoints relative to large number of surrogate endpoints in reduction of variance to estimate treatment success probabilities and related parametric functions like treatment difference, log odds ratio, log risk ratio. We found that the proportion of variance reduction for the estimate of treatment success probability for any treatment is a linear function Gξt (ρt ) of sample proportion of true endpoint (ρt ) when compared to the total (surrogate) endpoints. Proportion of variance for the parametric functions corresponding to two treatments are convex combinations of GξA (ρA ) and GξB (ρB ) and the weights are proportional to the variance of the estimator of the parametric function for individual treatments with true endpoints only. When true and surrogate endpoints are independent that is CA = 1 = CB , there is no reduction in sample size. On the contrary when the conditional distribution of surrogate endpoint given the true is degenerate one, i.e. CA = 0 = CB , the proportion of sample size reduction is (1 − ρ) for ρA = ρB = ρ .
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B. Banerjee, A. Biswas / Statistics and Probability Letters 96 (2015) 102–108
Fig. 1. Proportion of variance against ρA and ρB .
Fig. 2. GξA , GξB , G∗OR , G∗TD , G∗RR against ρ .
For estimation we suggested to use MLE in this paper, which is iterative for the problem under consideration. For practical implementation of a surrogate-augmented procedure, Banerjee and Biswas (2011) used EM-based estimates of pA and pB . Alternative estimates based on conditional expectations may be (see Banerjee and Biswas, 2014) (S )
pt
mt11 −1 = Yt /nt = nt YtT + WtS + WtT
mt10 mt − WtT
(nt − mt − WtS ) .
(5)
Here the three terms within brace in the right hand side correspond to the observed number of successes from the true responses, estimate of the true successes out of WtS surrogate successes for which true responses are unobserved, and estimate of true successes out of (nt − mt − WtS ) surrogate failures for which true responses are unobserved. Our detailed simulation studies show that the behavior of the two estimators, EM based and given in (5) are almost similar to the MLE. As an immediate application, these results can be used to obtain the closed form expressions for the allocation proportions of two treatments when the adaptive allocation is used to compare the two treatments. Other studies involving two treatment binary responses like the Cochran–Mantel–Haenszel test can also be modified for the surrogate augmented set up. These
B. Banerjee, A. Biswas / Statistics and Probability Letters 96 (2015) 102–108
107
Fig. 3. G∗TD (ρ) and observed efficiencies against ρ .
expressions are extremely useful to determine the sample to attain a certain level and power of a test when the surrogate data are present. As a far reaching application these results may be extremely helpful when ρA and ρB are random variables depending on time in an ongoing treatment process. It will help to obtain the variance of the estimators of the functions of success probabilities more easily. Acknowledgments The authors wish to thank two anonymous referees for their careful reading and constructive suggestions which led some improvements over the earlier version of the manuscript. Appendix A.1. Proof of Theorem 1 (a) From the likelihood Eq. (2), it is immediate that
nt Iξt (ρt ) ≡ nt 1
where dt1 =
πt0 (1−rt )
ρt dt1 + (1 − ρt )dt2 (1 − ρt )ct1 (1 − ρt )ct2
(1 − ρt )ct1 ρt dt3 + (1 − ρt )dt4 (1 − ρt )ct3
(πt1 +πt0 −1)2 , dt3 rt (1−rt ) rt (πt1 +πt0 −1) 1−π , ct2 = (1−rt1) rt (1−rt ) t
, dt2 =
pt qt t0 − 1−π = rt
= −
pt
, d
(1 − ρt )ct2 (1 − ρt )ct3 , ρt dt5 + (1 − ρt )dt6 =
p2t
, d
t4 t5 πt1 (1−πt1 ) rt (1−rt ) πt1 (1−rt )(πt1 +πt0 −1) =− , ct3 = rt rt (1−rt )
= −
qt
πt0 (1−πt0 ) pt qt . rt (1−rt )
, dt6 =
q2t rt (1−rt )
and ct1 =
Now observing that
2 dt6 dt4 = ct3 2 dt2 dt4 = ct1 2 dt6 dt2 = ct2 2 2 2 dt2 dt4 dt6 + dt1 dt4 dt6 + dt2 dt3 dt6 = dt1 ct3 + dt3 ct2 + dt5 ct1 2 2 2 dt2 dt4 dt6 + 2ct1 ct2 ct3 = dt2 ct3 + dt4 ct2 + dt6 ct1
dt1 dt3 dt5 = dt1 dt4 dt5 + dt2 dt3 dt5 + dt1 dt3 dt6 1
=
πt1 (1 − πt1 )πt0 (1 − πt0 )
.
So we get Ct =
dt6 dt5
+
dt4
=
dt3
qt πt0 (1 − πt0 ) + pt πt1 (1 − πt1 ) rt ( 1 − rt )
,
and 1 n− Iξt (ρt ) t
Gξt (ρt ) =
1 m− t pt qt
−1 11
=
−1 ρt Iξt (ρt ) 11 pt qt
= ρt + (1 − ρt )Ct = Ct + (1 − Ct )ρt .
(6)
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B. Banerjee, A. Biswas / Statistics and Probability Letters 96 (2015) 102–108
For (b) we observe that Var(Wt |Yt = j) = πtj (1 − πtj ) for j = 0, 1 and t = A, B. Hence E (Var(Wt |Yt )) = qt πt0 (1 − πt0 ) + pt πt1 (1 − πt1 ), and consequently we get Ct =
E (Var(Wt |Yt )) Var(Wt )
=
E (Var(Wt |Yt )) E (Var(Wt |Yt )) + Var(E (Wt |Yt ))
∈ [0, 1].
Clearly Ct = 1 when true and surrogate endpoints are independent, and Ct = 0 when the conditional distribution of surrogate endpoint given the true is degenerate. References Banerjee, B., Biswas, A., 2011. Estimating treatment difference for binary responses in the presence of surrogate end points. Stat. Med. 30, 186–196. Banerjee, B., Biswas, A., 2014. Odds ratio for 2 × 2 tables: Mantel–Haenszel estimator, profile likelihood and presence of surrogate responses. J. Biopharm. Statist. 24 (3), 649–659. Begg, C.B., Leung, D.H.Y., 2000. On the use of surrogate endpoints in randomized trials. J. Roy. Statist. Soc. Ser. A 163, 15–28. Buyse, M., Molenberghs, G., 1998. Criteria for the validation of surrogate endpoints in randomized experiments. Biometrics 54, 1014–1029. Buyse, M., Molenberghs, G., Burzykowski, T., Renard, D., Geys, H., 2000. The validation of surrogate endpoints in meta-analysis of randomized experiments. Biometrics 56, 49–67. Chen, Y.H., 2000. A robust imputation method for surrogate outcome data. Biometrika 87, 711–716. Chen, H., Geng, Z., Jia, J., 2007. Criteria for surrogate endpoints. J. R. Stat. Soc. Ser. B 69, 919–932. Chen, S.X., Leung, D.H.Y., Qin, J., 2003. Information recovery in a study with surrogate endpoints. J. Amer. Statist. Assoc. 98, 1052–1062. Day, N.E., Duffy, S.W., 1996. Trial design based on surrogate endpoints—applications to comparison of different breast screening frequencies. J. Roy. Statist. Soc. Ser. A 159, 49–60. Freedman, L.S., Graubard, B.I., Schatzkin, A., 1992. Statistical validation of intermediate endpoints for chronic diseases. Stat. Med. 11, 167–178. Lin, D.Y., Fleming, T.R., Gruttola, V.D., 1997. Estimating the proportion of treatment effect explained by a surrogate marker. Stat. Med. 16, 1515–1527. Molenberghs, G., Geys, H., Buyse, M., 2001. Evaluation of surrogate endpoints in randomized experiments with mixed discrete and continuous outcomes. Stat. Med. 20, 3023–3038. Pepe, M., 1992. Inference using surrogate outcome data and validation sample. Biometrica 79, 355–365. Prentice, R.L., 1989. Surrogate endpoints in clinical trials: definition and operational criteria. Stat. Med. 8, 431–440. Reilly, M., Pepe, M.S., 1995. A mean score method for missing and auxiliary covariate data in regression models. Biometrika 82, 299–314. Wang, Y., Taylor, J.M.G., 2002. A measure of the proportion of treatment effect explained by a surrogate marker. Biometrics 58, 803–812. Wittes, J., Lakatos, E., Probstfield, J., 1989. Surrogate endpoints in clinical trials: cardiovascular disease. Stat. Med. 8, 415–425.
Contents lists available at ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Linear increment in efficiency with the inclusion of surrogate endpoint Buddhananda Banerjee a , Atanu Biswas b,∗ a
Indian Institute of Science Education and Research, Kolkata, India
b
Indian Statistical Institute, Kolkata, India
article
info
Article history: Received 10 May 2014 Received in revised form 10 September 2014 Accepted 11 September 2014 Available online 27 September 2014
abstract In a two-sample clinical trial, a fixed proportion of true-and-surrogate and the remaining only-surrogate responses are observed. We quantify the increase in efficiency to compare the treatments as a linear function of the proportion of available true responses. © 2014 Elsevier B.V. All rights reserved.
Keywords: Surrogate responses Odds ratio Risk ratio Treatment difference
1. Introduction A surrogate endpoint is chosen as a measure or an indicator of a biological process. Usually it is obtained sooner, at a lesser cost than the true endpoint of health outcome, and is used to arrive at a conclusion about an effect of intervention on the true endpoint. Surrogate endpoints are used with growing interest in medical science. For example, in a trial of a treatment of osteoporosis we might be interested in reduction of the fracture rate, but we measure the bone mineral density (BMD) instead. A change in CD4 cell count in a randomized trial is considered as a surrogate of survival time in the study of HIV affected patients. Again, some damages to the heart muscle due to myocardial infarction can be accurately assessed by an arterioscintography reading. As it is an expensive procedure, the peak cardiac enzyme level in the blood stream, which is more easily obtainable, is used as a surrogate measure of heart vascular damage (see Wittes et al., 1989). Sometimes the observed value of the response variable in the middle of an ongoing experiment is considered as the surrogate endpoint. For example patients with age related macular degeneration (ARMD) progressively loose vision. To compare between placebo and high-dose interferon-α for its treatment, observations are taken after six months and one year. The observation after six months is considered as a surrogate corresponding to the final outcome. Two basic problems are studied in the literature of surrogate responses, namely (a) validation of a surrogate and (b) measurement of gain in inference using the surrogate responses. Prentice (1989) gave validation criteria for a surrogate, which is subsequently discussed by Freedman et al. (1992), Reilly and Pepe (1995), Day and Duffy (1996), Buyse and Molenberghs (1998), Buyse et al. (2000), Molenberghs et al. (2001), and Chen et al. (2007). The use of surrogate endpoints is likely to be beneficial, not only in terms of cost or time, but it gives more accuracy in the estimation of target parametric functions such as treatment difference and odds ratio. For that purpose we first look at the data structure under
∗
Corresponding author. Tel.: +91 33 25752818; fax: +91 33 25773104. E-mail addresses: [email protected] (B. Banerjee), [email protected], [email protected] (A. Biswas).
http://dx.doi.org/10.1016/j.spl.2014.09.015 0167-7152/© 2014 Elsevier B.V. All rights reserved.
B. Banerjee, A. Biswas / Statistics and Probability Letters 96 (2015) 102–108
103
consideration. In this article we are interested to measure the increment of the efficiency with the increase of the proportion of surrogate endpoints where the surrogate is presumed to be validated. Suppose we consider the accumulated data when all of the surrogate responses are known, but only Q % of the true responses are available. Then, if we do not consider the surrogate data, we need to make inference based on Q % true responses only. In the present paper our objective is to use surrogate data efficiently (which consist of Q % bivariate data and (100 − Q )% only univariate surrogate data) to improve the inference. To use (100 − Q )% surrogate data efficiently one need to identify the dependency structure of true and surrogate responses based on the Q % bivariate true-and-surrogate data. A real data example which is appropriate to this situation is described in the next section. Pepe (1992) obtained the distribution of the estimator of regression parameter when the validation sample fraction has a fixed limiting value, ρ (=Q /100), say. Banerjee and Biswas (2011) established that the variance of the estimator of treatment difference is bounded for such a fixed ρ . Lin et al. (1997) measured the extent to which a biological marker is a surrogate endpoint for a clinical event and Wang and Taylor (2002) propose alternative measures of the proportion explained by the surrogate endpoint. Chen (2000) and Begg and Leung (2000) discussed the inferential improvement by the use of surrogate endpoints. Chen et al. (2003) introduced the concept of information recovery from surrogate endpoints by considering linear models for true and surrogate on covariates. Proportion of validation sample, ρ , naturally plays a key role in the gain in associated inference. The validation samples are true and surrogate paired observations, but the rest of the samples are surrogate responses only. In Section 2 we describe the set up in details and the data structure under a general probability model for binary true and binary surrogate responses. In Section 3 we establish that the (inverse of) relative efficiency to estimate the treatment success probability by using surrogate endpoints is a linear function of the validation sample proportion, ρ . As a simple consequence of that we also prove the (inverse of) relative efficiency to estimate treatments difference, log risk ratio and log odds ratio in a two-treatment set up is also linear in ρA and ρB , the validation sample proportions for the two treatments A and B, respectively. In Section 4 we demonstrate our results with data example and conclude. 2. Experimental details and data structure We consider a set up of two treatments having binary true endpoints with binary surrogates as well. Begg and Leung (2000) pointed out that for the binary endpoints the probability of concordance is an indicator of association between true and surrogate endpoints. Suppose nA and nB patients are allotted to the treatments A and B, respectively; but we get only mA and mB true endpoints along with all surrogate endpoints within the stipulated time frame or cost limit, where mt ≪ nt , t = A, B. Denote the true and surrogate endpoints for the treatment t by Yt and Wt , where t = A, B. All these endpoints are either 1 or 0 for success or failure, respectively. We denote pt = P (Yt = 1) as the success probability by the true endpoints for treatment t. Furthermore, let us denote P (Wt = 1|Yt = 1) = πt1
and P (Wt = 0|Yt = 0) = πt0 ,
(1)
which are the sensitivity and specificity of the 2 × 2 table for treatment t where the true and surrogate responses are in the two margins. Clearly it is a saturated model with full parameter space. Consequently, the success probabilities by the surrogate responses for the two treatments are rt = P (Wt = 1)
= P (Wt = 1|Yt = 1)P (Yt = 1) + P (Wt = 1|Yt = 0)P (Yt = 0) = πt1 pt + (1 − πt0 )(1 − pt ) = pt (πt1 + πt0 − 1) + (1 − πt0 ). The data corresponding to treatment t can be represented in a table as follows. True Yt = 1 Yt = 0 Total Only surrogate
Surrogate Wt = 1 mt11 mt01 WtT WtS
Wt = 0 mt10 mt00 mt − WtT nt − mt − WtS
Total YtT mt − YtT mt nt − mt
t t t where YtT = i=1 Yti and WtT = i=1 Wti ; also we denote WtS = i=mt +1 Wti for t = A and B. The notation (Yti , Wti ) is specifically used for denoting the response variables corresponding to the ith individual under tth treatment. If any marginal is found to be zero, it is customary to add 0.5 to each of the marginals. As an example/illustration we consider the data set analyzed by Buyse and Molenberghs (1998). This data set is obtained from a randomized clinical trial comparing an experimental treatment interferon-α , with highest dose, 6-million units daily to a corresponding placebo in the treatment of patients with age-related macular degeneration (ARMD). Patients with ARMD progressively lose vision. In the trial, a patient’s visual acuity is assessed at different time points through the ability to read lines of letters on standardized vision charts. It is examined whether the loss of at least two lines of vision at 6 months (denoted as 1, and 0 otherwise) can be used
m
m
n
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as a surrogate for the loss of at least three lines of vision at 1 year (denoted as 1, and 0 otherwise) which is a true endpoint with respect to the effect of interferon-α . A total of 87 patients received interferon-α and 103 received placebo. Treatment procedures are evaluated after the first week, which is considered as surrogate endpoints (Winterferon-α or WPlacebo ) and the second week to assess the helpfulness by true endpoints (Yinterferon-α or YPlacebo ). These data are used in Section 6 for illustration. If we assume that 40% of the patients allocated to the interferon-α drug will turn up with both true and surrogate endpoints, that is ρinterferon-α = 0.4, then the data set for the drug interferon-α and placebo can be represented as mt11 mt10 mt01 mt00 WtS nt − mt − WtS 15 4 4 12 28 24 12 4 4 24 22 47
t Interferon-α Placebo
3. Variance reduction using surrogate responses Our basic objective is to quantify the increase in efficiency or reduction in variance of the estimator of any specified parametric function by properly using the information extracted from the surrogate endpoints. Consider the likelihood for the treatment t, L(ξt ) =
mt
y
ytT
×
mt −ytT
pt tT qt
ytT mt11
mt − ytT mt01
m
πt1t11 (1 − πt1 )ytT −mt11 m
(1 − πt0 )mt01 πt0t00
nt − m t
wtS
wtS
rt
(1 − rt )nt −mt −wtS ,
(2)
where ξt = (pt , πt1 , πt0 ) and qt = 1 − pt . The Fisher information matrix is I(ξt ) and the (1, 1)th element of [I(ξt )]−1 , denoted 1 (S ) by [I(ξt )]− 11 , gives the variance of maximum likelihood estimator pt . Here we define the inverse of efficiency, which is the measure of improvement to estimate pt when surrogate augmented analysis is conducted. So the measure of improvement is
1 [I(ξt )]− 11
. Furthermore, denote mt /nt = ρt ∈ (0, 1]. We note that ρt = 0 is possible only when mt = 0, indicating no true response is available. This is not of any statistical interest. Using mt = ρt nt we get I(ξt ) = nt Iξt (ρt ). Hence the measure of improvement, given by the proportional variance, reduces to pt qt /mt
−1
1 n− Iξt (ρt ) t
Gξt (ρt ) =
11
1 m− t pt qt
=
−1 ρt Iξt (ρt ) 11 pt qt
.
Now we have the following theorem, proof of which is given in Appendix A.1 Theorem 1. (a) Relative gain, denoted by the inverse of efficiency, Gξt (ρt ) by using surrogate endpoints with ρt proportion of available true responses is a linear function of ρt , that is Gξt (ρt ) = Ct + (1 − Ct )ρt = ρt + (1 − ρt )Ct ,
(3)
which is a straight line joining the points (0, Ct ) and (1, 1) with intercept and slope add to unity, with Ct =
qt πt0 (1 − πt0 ) + pt πt1 (1 − πt1 ) rt (1 − rt )
.
(b) Further Ct =
E (Var(Wt |Yt )) Var(Wt )
∈ [0, 1].
Remark. The expression in (3) is amazingly simple in the context of the fairly general joint distribution of the true and surrogate endpoints. So far our knowledge goes, this expression and the following expressions are new in the literature E (Var(Wt |Yt )) and slope is where true and surrogate both are binary endpoints. The intercept of the line in Eq. (3) is Ct = Var(W )
(1 − Ct ) =
Var(E (Wt |Yt )) , Var(Wt )
t
and they add up to 1. It is also evident from Eq. (3) that the value of G is also 1 when ρt = 1, irrespective of the value of Ct . The slope gives the rate of increment of efficiency with the inclusion of true endpoints. If we let the value of ρt to increase or in other words if we let mt to increase and keep nt as fixed then (1 − Ct ) indicates the improvement in efficiency. The intercept gives the lower bound of efficiency gain. Clearly Ct = 1 when true and surrogate endpoints are independent, and Ct = 0 when the conditional distribution of surrogate endpoint given the true is degenerate one, for treatment t. Total variability of Wt is partitioned into two parts. The variability of the regression of Wt on Yt relative to the variability of Wt (i.e. correlation ratio) serves as the slope, and the variability of the unexplained part in the regression related to the total variability is the intercept. Theorem 1 can be effectively used to obtain the gain in efficiencies for the standard measures of treatment difference, like the log odds ratio, difference in success probabilities and relative risk ratio in the two-treatment set up. We present these in Theorem 2 and Corollaries 1–3.
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105
Theorem 2. The inverse of efficiency in variance estimation by using surrogate endpoints to estimate the log odds ratio (OR),
θ = log
pA qB qA pB
, is given by a plane, that is
GOR (ρA , ρB ) =
(mA pA qA )−1 GξA (ρA ) + (mB pB qB )−1 GξB (ρB ) . (mA pA qA )−1 + (mB pB qB )−1
(4)
The proof is immediate with the use of delta method and Theorem 1. Corollary 1. When ρA = ρB = ρ , the inverse of efficiency by using surrogate endpoints to estimate the log odds ratio is a linear function of ρ , that is GOR (ρ, ρ) = G∗OR (ρ) = ρ + (1 − ρ)
(nA pA qA )−1 CA + (nB pB qB )−1 CB . (nA pA qA )−1 + (nB pB qB )−1
Corollary 2. The inverse of efficiency by using surrogate endpoints to estimate θ = pA − pB , the treatment difference (TD), is a plane given by GTD (ρA , ρB ) =
1 −1 m− A pA qA GξA (ρA ) + mB pB qB GξB (ρB ) 1 −1 m− A pA qA + mB pB qB
,
and, for ρA = ρB = ρ we get the line given by
GTD (ρ, ρ) = GTD (ρ) = ρ + (1 − ρ) ∗
1 −1 n− A pA qA CA + nB pB qB CB 1 −1 n− A pA qA + nB pB qB
.
Corollary 3. The inverse of efficiency by using surrogate endpoints to estimate θ = log
pA pB
, the log risk ratio (RR), is a plane
given by GRR (ρA , ρB ) =
(mA pA )−1 qA GξA (ρA ) + (mB pB )−1 qB GξB (ρB ) , (mA pA )−1 qA + (mB pB )−1 qB
and when ρA = ρB = ρ it reduces to a linear function of ρ given by
(nA pA )−1 qA CA + (nB pB )−1 qB CB . GRR (ρ, ρ) = GRR (ρ) = ρ + (1 − ρ) (nA pA )−1 qA + (nB pB )−1 qB ∗
In Fig. 1 we plot GOR (ρA , ρB ), given in (4), against ρA and ρB . We find that, for suitable ρA and ρB , there is considerable gain in estimation of OR. Fig. 2 illustrates the special case of ρA = ρB where GA , GB , G∗OR , G∗TD , G∗RR are all straight lines. In fact, GOR (., .), GTD (., .) and GRR (., .) are convex combinations of GξA and GξB , and belong to in between these two straight lines. We have taken mA = mB = 34 and pA = 0.7, pB = 0.8 in these two figures, for illustration. 4. Data example and discussions We obtain the treatment difference (TD) between interferon-α and placebo alloted to the ARMD patients (see Buyse and Molenberghs, 1998). Estimated treatment difference is 0.162. The observed value of CA = Cinterferon-α = 0.66 and CB = Cplacibo = 0.58. So the equation of G∗TD (ρ) = ρ + 0.62(1 − ρ) where we assume the equal values of ρinterferon-α and ρplacebo , say ρ . For difference values of ρ = {0.2, 0.3, . . . , 0.9} TD is calculated based only on true endpoints and on surrogate augmented data as well. The ratio of their bootstrap variances are plotted along with the line G∗TD (ρ) (see Corollary 2). We find that the observed measures of efficiencies are close to the straight line G∗TD (ρ), see Fig. 3. In this article we discussed the impact of proportion of available true endpoints relative to large number of surrogate endpoints in reduction of variance to estimate treatment success probabilities and related parametric functions like treatment difference, log odds ratio, log risk ratio. We found that the proportion of variance reduction for the estimate of treatment success probability for any treatment is a linear function Gξt (ρt ) of sample proportion of true endpoint (ρt ) when compared to the total (surrogate) endpoints. Proportion of variance for the parametric functions corresponding to two treatments are convex combinations of GξA (ρA ) and GξB (ρB ) and the weights are proportional to the variance of the estimator of the parametric function for individual treatments with true endpoints only. When true and surrogate endpoints are independent that is CA = 1 = CB , there is no reduction in sample size. On the contrary when the conditional distribution of surrogate endpoint given the true is degenerate one, i.e. CA = 0 = CB , the proportion of sample size reduction is (1 − ρ) for ρA = ρB = ρ .
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Fig. 1. Proportion of variance against ρA and ρB .
Fig. 2. GξA , GξB , G∗OR , G∗TD , G∗RR against ρ .
For estimation we suggested to use MLE in this paper, which is iterative for the problem under consideration. For practical implementation of a surrogate-augmented procedure, Banerjee and Biswas (2011) used EM-based estimates of pA and pB . Alternative estimates based on conditional expectations may be (see Banerjee and Biswas, 2014) (S )
pt
mt11 −1 = Yt /nt = nt YtT + WtS + WtT
mt10 mt − WtT
(nt − mt − WtS ) .
(5)
Here the three terms within brace in the right hand side correspond to the observed number of successes from the true responses, estimate of the true successes out of WtS surrogate successes for which true responses are unobserved, and estimate of true successes out of (nt − mt − WtS ) surrogate failures for which true responses are unobserved. Our detailed simulation studies show that the behavior of the two estimators, EM based and given in (5) are almost similar to the MLE. As an immediate application, these results can be used to obtain the closed form expressions for the allocation proportions of two treatments when the adaptive allocation is used to compare the two treatments. Other studies involving two treatment binary responses like the Cochran–Mantel–Haenszel test can also be modified for the surrogate augmented set up. These
B. Banerjee, A. Biswas / Statistics and Probability Letters 96 (2015) 102–108
107
Fig. 3. G∗TD (ρ) and observed efficiencies against ρ .
expressions are extremely useful to determine the sample to attain a certain level and power of a test when the surrogate data are present. As a far reaching application these results may be extremely helpful when ρA and ρB are random variables depending on time in an ongoing treatment process. It will help to obtain the variance of the estimators of the functions of success probabilities more easily. Acknowledgments The authors wish to thank two anonymous referees for their careful reading and constructive suggestions which led some improvements over the earlier version of the manuscript. Appendix A.1. Proof of Theorem 1 (a) From the likelihood Eq. (2), it is immediate that
nt Iξt (ρt ) ≡ nt 1
where dt1 =
πt0 (1−rt )
ρt dt1 + (1 − ρt )dt2 (1 − ρt )ct1 (1 − ρt )ct2
(1 − ρt )ct1 ρt dt3 + (1 − ρt )dt4 (1 − ρt )ct3
(πt1 +πt0 −1)2 , dt3 rt (1−rt ) rt (πt1 +πt0 −1) 1−π , ct2 = (1−rt1) rt (1−rt ) t
, dt2 =
pt qt t0 − 1−π = rt
= −
pt
, d
(1 − ρt )ct2 (1 − ρt )ct3 , ρt dt5 + (1 − ρt )dt6 =
p2t
, d
t4 t5 πt1 (1−πt1 ) rt (1−rt ) πt1 (1−rt )(πt1 +πt0 −1) =− , ct3 = rt rt (1−rt )
= −
qt
πt0 (1−πt0 ) pt qt . rt (1−rt )
, dt6 =
q2t rt (1−rt )
and ct1 =
Now observing that
2 dt6 dt4 = ct3 2 dt2 dt4 = ct1 2 dt6 dt2 = ct2 2 2 2 dt2 dt4 dt6 + dt1 dt4 dt6 + dt2 dt3 dt6 = dt1 ct3 + dt3 ct2 + dt5 ct1 2 2 2 dt2 dt4 dt6 + 2ct1 ct2 ct3 = dt2 ct3 + dt4 ct2 + dt6 ct1
dt1 dt3 dt5 = dt1 dt4 dt5 + dt2 dt3 dt5 + dt1 dt3 dt6 1
=
πt1 (1 − πt1 )πt0 (1 − πt0 )
.
So we get Ct =
dt6 dt5
+
dt4
=
dt3
qt πt0 (1 − πt0 ) + pt πt1 (1 − πt1 ) rt ( 1 − rt )
,
and 1 n− Iξt (ρt ) t
Gξt (ρt ) =
1 m− t pt qt
−1 11
=
−1 ρt Iξt (ρt ) 11 pt qt
= ρt + (1 − ρt )Ct = Ct + (1 − Ct )ρt .
(6)
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B. Banerjee, A. Biswas / Statistics and Probability Letters 96 (2015) 102–108
For (b) we observe that Var(Wt |Yt = j) = πtj (1 − πtj ) for j = 0, 1 and t = A, B. Hence E (Var(Wt |Yt )) = qt πt0 (1 − πt0 ) + pt πt1 (1 − πt1 ), and consequently we get Ct =
E (Var(Wt |Yt )) Var(Wt )
=
E (Var(Wt |Yt )) E (Var(Wt |Yt )) + Var(E (Wt |Yt ))
∈ [0, 1].
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