Clinical trials with rescue medication applied according to a deterministic rule
aa r X i v : . [ s t a t . M E ] A ug Clinical trials with rescue medicationapplied according to a deterministic rule
Gerd K Rosenkranz ∗ Medical University of Vienna, Vienna, AustriaVersion of August 30, 2016
Abstract
Clinical trials in specific indications require the administration ofrescue medication in case a patient does not sufficiently respond to in-vestigational treatment. The application of additional treatment on anas needed basis causes problems to the analysis and interpretation ofthe results of these studies since the effect of the investigational treat-ment can be confounded by the additional medication. Following-upall patients until study end and capturing all data is not fully address-ing the issue. We present an analysis that takes care of the fact thatrescue is a study outcome and not a covariate when rescue medicationis administered according to a deterministic rule. This approach allowsto clearly define a biological effect. For normally distributed longitudi-nal data a practically unbiased estimator of the biological effect can beobtained. The results are compared to an ITT analysis and an analysison all patients not receiving rescue.
Keywords: rescue medication, potential outcomes, principal strata, biologi-cal effect.
In clinical trials in specific indications it is necessary to provide patients theopportunity to receive an established medication, either alone or in combi-nation with study treatment, in case of insufficient response. This additionaltreatment is called rescue medication, or sometimes reliever medication. Indiabetes trials patients are to receive rescue medication in case blood glu-cose does not decrease over time (see for example [1]). In trials in chronicobstructive pulmonary disease (COPD) [2, 3] rescue may be given to preventexacerbations. ∗ Address: Gerd K Rosenkranz, Center for Medical Statistics, Informatics and Intel-ligent Systems, Medical University of Vienna, Spitalgasse 23, A-1090 Vienna, Austria.Email: [email protected] observed administration of rescue, which is anoutcome affected by the treatment assigned at baseline. Rather they areclassified according to whether they would (or would not) need rescue underinvestigational treatment or control.In the context of rescue medication, one can consider four strata of pa-tients: those who do not need rescue regardless of assigned treatment (ab-breviated 00), those who need rescue when assigned control but not whenassigned treatment (10), those who need rescue when assigned treatmentbut not when assigned control (01) and those in need of rescue no matterhow treated (11). If control treatment is a placebo, one may argue that stra-tum (01) is empty since patients not responding to investigational treatmentare not responding to placebo either since the former is a placebo as well.This approach has been applied under a likelihood [12] as well as a Bayesianparadigm [13] to obtain estimates for the biological effect in subjects thatdo not fail under either treatment option (stratum 00). However, univari-ate methods were considered that would necessitate to define a summaryoutcome for studies with repeated measures.We consider a simple longitudinal setting consisting of 2 visits only andan explicit (deterministic) rule for the administration of rescue. The rulebasically states that if the values at an earlier visit are lower or higher thanacceptable, the subject should receive rescue medication. The endpointto assess an effect of the investigational treatment will be captured at thefinal (second) visit of the trial. The only stratum that provides an non-3onfounded estimator of the treatment effect is patients that do not needrescue under treatment nor control (00). Under plausible distributionalassumptions, the biological effect of treatment over control in the 00 stratumcan be estimated without having to use computationally intensive methodslike EM algorithms or Markov Chain Monte Carlo techniques.After introducing notation and concept of potential outcomes and theirrelationship to observations, a practically unbiased estimator of the biologi-cal effect of interest is presented. Its small sample properties are investigatedby simulation. Some questions for further investigations are discussed.
We consider a parallel group trial where experimental treatment ( Z = 1)is to be compared with control ( Z = 0) treatment. The treatment is to berandomly assigned to 1 ≤ i ≤ n j , ( j = 0 ,
1) patients who provide obser-vations ( Y i , Y i ) from two visits. For the discussion we assume that highvalues are beneficial and low values indicate lack of efficacy. The rule forapplying rescue is that if the observed values are too small at the first visitafter treatment initiation, i.e., Y i ≤ c for some predefined critical value c ,the subject has to receive rescue medication, denoted by R i = 1. R i = 0means that no rescue is necessary. As a consequence, the outcome at thesecond visit will be affected not only by the treatment assigned at baseline,but also by the rescue medication administered after the first visit. Notto complicate matters further, we assume complete data at each visit, i.e.,( Y i , Y i ) are observed for all patients and investigators and patients adhereto the rules for administration of rescue medication.In the following we use the notion of potential outcomes. We drop theindex accounting for the subject number whenever possible to ease notation.Let Y ( z ) be the potential outcome of a subject at visit 1 had treatment z been assigned at baseline. Let R ( z ) denote the potential rescue medication ofa subject assigned to treatment z . Likewise, let Y ( z, r ) denote the potentialoutcome of a subject at visit 2 assigned treatment z at baseline and havingreceived rescue medication r at visit 1. The biological effect of treatment z on the outcomes at visit 1 is given by E = E [ Y (1)] − E [ Y (0)], andthe biological effect of treatment z at visit 2 in stratum ( i, j ) is E ( i, j ) = E [ Y (1 , i )] − E [ Y (0 , j )]. Specifically in the stratum of subjects who do notneed rescue under either treatment it is E (0 ,
0) = E [ Y (1 , − E [ Y (0 , R (0) E [ Y (0 , R (0))] R (1) E [ Y (1 , R (1))]60 00 0 0 0 1 110 01 0 0 1 3 310 10 1 1 0 1 020 11 1 1 1 3 2Table 1: Example of expected outcomes in principal strata for a trial requir-ing rescue medicationtreatment beneficial for 80% of subjects. The ITT effect is 0 . × . × . × . E (0 , Y , Y , R has to be established.Obviously, Y = ZY (1) + (1 − Z ) Y (0) R = ZR (1) + (1 − Z ) R (0) Y = ZY (1 , R (1)) + (1 − Z ) Y (0 , R (0)) Y Y = ZY (1) Y (1 , R (1)) + (1 − Z ) Y (0) Y (0 , R (0))The last equation follows from Z (1 − Z ) = 0, Z = Z and (1 − Z ) = 1 − Z .Under the assumption that Z is assigned randomly to subjects, Y ( z ) and R ( z ) are independent of Z and therefore E [ Y | Z = z ] = E [ Y ( z )] E [ R | Z = z ] = E [ R ( z )] . This can be seen for R from E [ R | Z = 1] = E [ ZR (1) + (1 − Z ) R (0) | Z = 1] = E [ R (1) | Z = 1] = E [ R (1)] . As a result, the conditional expectation of the observations Y and R given Z = z equals the expectation of their respective counterfactual outcomes.This is no longer true for Y , since Y is confounded by R . In fact E [ Y | Z = 1 , R = 0]= E [ ZY (1 , R (1)) + (1 − Z ) Y (0 , R (0)) | Z = 1 , R = 0]= E [ Y (1 , R (1)) | Z = 1 , R = 0]= E [ Y (1 , R (1)) | R (1) = 0]= E [ Y (1 , | R (1) = 0]= E [ Y (1 , | Y (1) > c ] 5imilarly one can show E [ Y | Z = 0 , R = 0] = E [ Y (0 , | Y (1) > c ], andtherefore E [ Y | Z = z, R = 0] = E [ Y ( z, | Y ( z ) > c ] (1)confirming that an estimator of E [ Y | Z = z, R = 0] is not appropriate as anestimator of E [ Y ( z, To obtain an estimator of E [ Y ( z, E [ Y | Z = z, R = 0], a correctionis necessary. It is shown in this section that such a correction can be ob-tained for normally distributed variables. Assume that ( Y ( z ) , Y ( z, r )) hasa bivariate normal distribution with means µ ( z ) = E [ Y ( z )] = α + β zµ ( z, r ) = E [ Y ( z, r )] = α + β z + γr + δzr and covariance matrixΣ( z, r ) = σ ( z ) σ ( z, r ) σ ( z, r ) σ ( z, r ) ! It should be noted that the decision on rescue at visit 1 can be based on anendpoint different from the one on which the efficacy assessment is based onat visit 2.Under the model above, β = µ (1 , − µ (0 ,
0) is the causal effect of z at visit 2 in the principal stratum of subjects that do not require rescueunder any assigned treatment. Let γ ( z, r ) = σ ( z, r ) σ − ( z ) and recallthat for bivariate normal variables E [ Y ( z, r ) | Y ( z )] = µ ( z, r ) + γ ( z, r ) { Y ( z ) − µ ( z ) } holds. Let φ , Φ, and λ denote the pdf, cdf and hazard function of a standardnormal variable, respectively. With η ( z ) = [ c − µ ( z )] /σ ( z ), it follows that E [ Y ( z ) | Y ( z ) > c ] = µ ( z ) + σ ( z ) φ ( η ( z ))[1 − Φ( η ( z ))] − = µ ( z ) + σ ( z ) λ ( η ( z ))Then E [ Y ( z, | Y ( z ) > c ]= E [ E [ Y ( z, | Y ( z )] | Y ( z ) > c ]= E [ µ ( z,
0) + γ ( z, { Y ( z ) − µ ( z ) }| Y ( z ) > c ]= µ ( z,
0) + γ ( z, { E [ Y ( z ) | Y ( z ) > c ] − µ ( z ) } = µ ( z,
0) + γ ( z, σ ( z ) λ ( η ( z )) (2)6stimator of β α β α β γ δ ITT Conditional Corrected1 0 0 0 0 0 0.001 (0.201) 0.001 (0.200) 0.001 (0.200)0 1 0 0 0 0 -.003 (0.195) -.226 (0.205) -.003 (0.205)0 0 1 0 0 0 -.002 (0.205) -.001 (0.221) -.001 (0.221)0 0 0 1 0 0 1.001 (0.201) 1.000 (0.220) 1.000 (0.220)0 0 0 0 1 0 0.001 (0.179) 0.001 (0.219) 0.001 (0.219)0 0 0 0 0 1 0.308 (0.191) -.002 (0.219) -.002 (0.219)0 0 0 0 1 1 0.307 (0.192) -.000 (0.219) -.000 (0.219)0 1 0 1 0 0 1.003 (0.199) 0.781 (0.209) 1.003 (0.209)0 1 0 1 1 0 0.759 (0.183) 0.777 (0.208) 0.999 (0.208)0 1 0 1 1 1 0.825 (0.187) 0.779 (0.209) 1.001 (0.209)0 0 0 1 1 1 1.309 (0.190) 0.999 (0.218) 0.999 (0.218)Table 2: Means and standard deviations of the estimates of the effect oftreatment at visit 2 from an ITT analysis, an estimator of E [ Y | Z = 1 , R =0] − E [ Y | Z = 0 , R = 0] and the corrected estimator (3) for n = n = 50, σ ( z ) = σ ( z ) = 1, σ ( z, r ) = 0 . c = − . µ ( z,
0) = E [ Y | Z = z, R = 0] − γ ( z, σ ( z ) λ ( η ( z )) . (3)The conditional expectation of the observed value equals the expectationof the potential outcome if σ ( z,
0) = 0, in which case observations fromdifferent visits are independent and the probability of rescue medicationdepends only on the randomly assigned treatment z and is thus independentof the potential outcome. In this case, the causal effect β equals E [ Y | Z =1 , R = 0] − E [ Y | Z = 0 , R = 0]. The latter is also the case if σ ( z, λ ( η ( z ))does not depend on the assigned treatment z which is a somewhat artificialcondition.To appreciate the discrepancies between different estimators of a treat-ment effect of z in the absence of rescue, Table 2 presents the effect estimatorfrom an ITT analysis, an estimator of E [ Y | Z = 1 , R = 0] − E [ Y | Z = 0 , R =0] and the estimator corrected according to (3). The entries are obtainedfrom 10000 simulations per row. Only the corrected estimator is estimatingthe causal effect β appropriately.To make the correction practically useful, the parameters it comprisesneed to be estimated from the data. Most of them are indeed estimablefrom observations at visit 1 with the exception of σ ( z, E [ Y ( z ) | Y ( z ) > c ] = µ ( z ) + σ ( z ) { σ ( z ) + [ c + µ ( z )] λ ( η ( z )) } one obtains using similar arguments as in the previous section E [ Y Y | Z = z, R = 0] 7 β α β γ δ ˆ β ˆ σ (0) ˆ σ (1)1 0 0 0 0 0 0.001 (0.215) 0.592 (0.149) 0.591 (0.151)0 1 0 0 0 0 -.006 (0.252) 0.596 (0.228) 0.591 (0.149)0 0 1 0 0 0 -.003 (0.289) 0.560 (0.279) 0.561 (0.273)0 0 0 1 0 0 1.019 (0.297) 0.597 (0.229) 0.555 (0.282)0 0 0 0 1 0 0.002 (0.301) 0.597 (0.229) 0.596 (0.226)0 0 0 0 0 1 -.005 (0.299) 0.594 (0.228) 0.597 (0.226)0 0 0 0 1 1 0.002 (0.300) 0.596 (0.227) 0.592 (0.229)0 1 0 1 0 0 1.002 (0.257) 0.596 (0.226) 0.588 (0.136)0 1 0 1 1 0 0.997 (0.257) 0.595 (0.227) 0.584 (0.140)0 1 0 1 1 1 1.001 (0.259) 0.599 (0.226) 0.586 (0.138)0 0 0 1 1 1 1.017 (0.292) 0.597 (0.227) 0.557 (0.279)Table 3: Means and standard deviations of the estimates of the effect oftreatment at visit 2 and the correlation of Y ( z ) and Y ( z,
0) for n = n =50, σ ( z ) = σ ( z ) = 1, σ ( z, r ) = 0 . c = − . E [ Y ( z ) Y ( z, | Y ( z ) > c ]= E [ Y ( z ) E [ Y ( z, | Y ( z )] | Y ( z ) > c ]= E [ Y ( z ) { µ ( z,
0) + γ ( z, Y ( z ) − µ ( z )) }| Y ( z ) > c ]= { µ ( z, − γ ( z, µ ( z ) } E [ Y ( z ) | Y ( z ) > c ]+ γ ( z, E [ Y ( z ) | Y ( z ) > c ]= [ µ ( z, − γ ( z, µ ( z )][ µ ( z ) + σ ( z ) λ ( η ( z ))]+ γ ( z, { µ ( z ) + σ ( z )[ σ ( z ) + ( c + µ ( z )) λ ( η ( z ))] } = µ ( z, µ ( z ) + σ ( z ) λ ( η ( z ))] + γ ( z, σ [ σ ( z ) + cλ ( η ( z ))] . Substituting µ ( z,
0) from equation (3) yields σ ( z,
0) = E [ Y Y | Z = z, R = 0] − E [ Y | Z = z, R = 0][ µ ( z ) + σ ( z ) λ ( η ( z ))]1 + λ ( η ( z ))[ η ( z ) − λ ( η ( z ))] (4)The parameters on the right hand side of (4) are consistently estimable fromthe data, hence a consistent estimator of σ ( z,
0) can be obtained. Datafrom all patients are used for estimating µ ( z ), σ and λ ( η ( z )). Only datafrom patients that did not require rescue medication during the trial areused for E [ Y Y | Z = z, R = 0] and E [ Y | Z = z, R = 0].We run a small simulation study to get an idea of the performance ofa plug-in estimator obtained by replacing the parameters on the right-handside of the preceding equation by their estimates. The results are displayedin Table 3. As can be seen, the proposed estimate works quite well in thegiven scenario, however, there seems to be a tendency to underestimate the8ovariance in particular when the proportion of rescue is increasing. Whenwe re-run the simulation with a higher sample size of 500 in each treatmentgroup this phenomenon practically disappeared. In any case this suggeststhat the estimate may become inaccurate for small sample sizes. An estimator of the biological effect of an investigational treatment overcontrol has been proposed for clinical trials where patients are administeredrescue medication if the assigned treatment fails. The main assumptionsare a bivariate normal model for the potential outcomes at visits 1 and 2,and a deterministic rule for administration of rescue medication based onobservations obtained at visit 1. It should be noted that the decision onrescue at visit 1 can be based on an endpoint different from the one onwhich the efficacy assessment is based on at visit 2. A generalization tomore visits should be feasible.When estimating the biological effect at visit 2, data from all subjectsfrom visit 1 are used in the analysis. However, for visit 2, only observationsfrom subjects without rescue medication are analyzed. The smaller effec-tive sample size results in a somewhat larger variability of the estimate ascompared to the ITT estimate which uses the information of the full sam-ple. However, though the latter is using more data, it is not an appropriatebiological effect estimator from an accuracy perspective.As a recommendation for the analysis of clinical trials requiring rescuemedication one should conduct an ITT analysis to obtain an estimator oftreatment effectiveness, an analysis of the amount and/or timing of rescuemedication, and an analysis of the biological effect. If an investigationaltreatment plus rescue has no or little advantage over control plus rescue,one may wonder what could constitute an additional benefit of the newtreatment. Likewise, the efficacy of the investigational treatment is ques-tionable if more rescue is taken there as compared to control. Neverthelessin such a case investigational treatment may still do better than control inpatients that would not need rescue under either treatment. To figure thatout, an estimate of the biological effect would be helpful. In short there isnot one analysis that covers all aspects of the issue. An analysis based onsubjects that did not need rescue in there respective groups without anycorrection is discouraged.The standard error of the biological estimates can be obtained by re-sampling. More importantly the smaller sample size in principal stratum00 has to be taken into consideration during the planning phase of a trial.Furthermore, we did not account for missing data in this paper but ratherassumed that all data needed to obtain estimates of biological effects areavailable. 9 eferences [1] Bailey CJ, Gross JL, Pieters A, Bastien A, List JF. Effect of da-pagliflozin in patients with tpye 2 diabetes who have inadequate gly-caemic control with metformin: a randomised, double-blind, placebo-controlled trial.
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