Parametric and Semi-parametric Approaches to Estimation of Survival Distributions of Treatment Strategies in Sequential Multiple Assignment Randomized Trials
Open AccessBackground: Dynamic treatment regimens (DTRs) are a sequence of treatments tailoredto individual patients to achieve optimal outcomes. DTRs are also known as adaptive treatmentstrategies/policies. They are sequences of decision rules that adjust dynamically to thetime-varying patients conditions at multiple stages. DTRs have gaind popularity in chronicconditions such as cancer and HIV. In this dissertation we are particualarly interested inone type of DTR: Sequential Multiple Assignment Randomized trial (SMART), in whichpatients are sequentially randomized to multiple treatments with the assignment dependenton responses in previous stages. Because of this relationship between treatment stages, it’sof interest to estimate the overall outcome of the entire treatment strategy instead of eachsingle stage. Nonparametric inverse probability weighting methods have been developedand studied in recent years. These methods are unbiased and consistent but not as efficientas their parametric counterparts. We propose parametric and semiparametric methods toestimate survival probabilities for two-stage two-treatment SMARTs.Purpose: To develop parsimonious and efficient methods for estimation of survival probabilitiesin SMART. When the data distribution is correctly specified, parametric and nonparametricmodels not only improve efficiency of estimation but also facilitate understandingand help generalization. We’d like to study appropriate models for various types of dataand explore application of our methods to identify optimal treatment strategies.Methods: We developed two parametric and two semiparametric approaches. Let Ajbk bea stragegy in which patients receive Aj first, then at time the responses are evaluated,responders then receive Bk and nonresponders are not further treated; where j=1,2 andk=1,2. We define ! as the proportion of deaths before . Among those who survived past , we define p as the probability of responding. The probability of death at time t (t )for strategy Ajbk is:FAjbk(t) = ! + (1 !)fpFAjBk(t ) + (1 p)FAjN(t )gvHere FAjBk is the probability of death for patients who received AjBk and FAjN is theprobability of death for those received Aj and didn’t respond. To estimate FAjBk and FAjNwe used exponential, Weibull and Cox proportional hazard models (CoxI). We developeda second Cox model, which we call the CoxII. Insead of using ! to estimate the death ratebefore , we used a time-dependent indicators Z(t) for each treatment assignment.The variance-covariance matrix for FAjbk(t) is derived using maximum likelihood approachesand -method. We also derived sandwich estimator of variance for the exponentialmodel. The estimated variances from exponential, Weibull, CoxI and CoxII models arecompared to the variance of estimator from inverse probability weighting approach (IPW).Relative efficiency(RE), which is variance of our estimator devided by variance of the IPWestimator, is used to measure approvement in efficiency. RE greater than one means ourestimator is more efficient.We used the CoxII model as an example to briefly explore the application of our methodsin identification of optimal strategy based on patient characteristics. We added onebinary variable X for patient characteristics and the interaction of X and treatment in theCoxII model. The difference given X=0 is denoted as 0 and 1 given X=1, the nullhypothesis is H0 : 0 = 1. We studied three scenarios: null interaction, quantitativeinteraction and qualitative interaction. For each of the three scenarios the relationship between0 and 1 are 0 = 1, 0 6= 1 with equal signs and 0 6= 1 with differentsigns. Wald test was used to test for interaction.We conducted simulations and analyzed real data to compare our methods to IPW.Results: We simulated data from exponential distribution and compared variance of theexponential estimator to variance of the IPW estimator. We conducted simulation studiesfor = 0 and 0 and set the probability of response p to be 0.1, 0.5 and 0.9 for each . We calculated relative efficiency from 6 months to five years in the increment of 0.5years. Regardless of prespecified values of and p, the exponential model is always moreefficient and the efficiency improves from 50% to 4 folds depending on the time pointsof evaluation. For each scenario we calculated both the model based variance and thesandwich variance, as expected the two are always close to each other with the sandwichvariance being slightly larger.To further study the sandwich estimator under misspecification of data distribution, wevisimulated data from distributions of lognormal, Weibull with shape parameter = 1:05and Weibull with = 2. In each simulation we estimated FAjbk using the exponentialmodel and calculated both the model based and sandwich variances. The results showedthat the sandwich estimator of variance gives slightly better coverage probability than themodel based variance but does not protect the estimator of survival probability againstmodel misspecification.We set p = 0:5 and conducted simulation studies using Weibull distribution with = 0:5; 1; 1:5 to represent situations when the failure rate decreases, holds constant andincreases with time. In these Weibull simulations the hazard functions of different treatmentgroups are proportional. We compared all of the 4 proposed approaches to IPW. Theexponential model is always the most efficient but biased when 6= 1; the Weibull modelis the second most efficient with RE ranging from 1.6 to 8 and always unbiased; The CoxIand CoxII models are very close to each other, both of them are slightly less efficient thanthe Weibull model with RE ranging from 1.5 to 5. The two Cox models are unbiased aswell.We simulated data from Weibull distributions where the hazard functions of differentgroups are not proportional. In this case the Weibull model is still more efficient than theIPW with RE ranging from 1.3 to 8.4. The other 3 models are all biased therefore notdiscussed.We used CoxII model to explore the application of our model in identification of optimalstrategy. We assumed a binary patient characteristic variable X and simulated datafrom Weibull distributions with = 0:75. Data sets for X=0 and X=1 were generatedseparately according to prespecified survival probability at 2 years. From 5,000 simulations,the rejection rate of null hypothesis is 5% for null interaction, from 20% to 100% forquantitative interaction and above 90% for qualitative interaction. The numerical value ofrejection rate is positively associated with the magnitude of difference between strategies.All 4 methods were applied to the CALGB 8923 study, which was a double-blind, twostage,randomized trial that enrolled 388 elderly patients with acute myelogenous leukemia(AML). In terms of survival function, most of the estimated probabilities from the Coxmodels were closer to the IPW approach than the other two. The Weibull is closer to theIPW approach than the exponential model. A likelihood ratio tests showed that theWeibullmodel fits the data significantly better(p-value=3.83e-15) than the exponential. All fourviimethods outperform IPW in efficiency and the RE went up to as high as 3.2. Among thefour methods, exponential is the least efficient and its RE to the IPW approach droppedbelow 1 at some time points. In early follow-up times, the Weibull method is the most efficientwhile as the follow-up time gets longer the Cox models have better efficiency gains.Equality of the four strategies were tested at days of 25%, 50% and 80% mortality of thecohort and no difference were found. We then used the ECOG score, which is a standardmeasure to quantify cancer patient’s general well being, to test for interaction withtreatment groups using the CoxII model. When ECOG score=1, A2b1 has higher survivalprobability at 1 year compared to A1b1(p-value=0.0485).Conclusions: We have proposed two parametric models and two semiparametric modelsto estimate the survival probability of treatment strategy. When the underlying distributionof data is correctly specified, all of them are significantly more efficient than the IPWapproach. Fewer number of parameters need to be estimated is associated with higherrelative efficiency. The efficiency gain can be as high as 6 folds. Although semiparametric,the two Cox models are only slightly less efficient than the Weibull model while holdingfewer assumptions about baseline hazard. Therefore we prefer the Cox models. Whenearly death is few and is small, CoxI is accurate and easier to conduct. Otherwise CoxIIis preferred.When the assumption of proportional hazards does not hold, the Cox models are notappropriate. In our brief exploration, the Weibull model that assign each treatment groupits own can still be as much as 7 times more efficient than the IPW approach. This willbe the start point to further research of other more complex parametric models.Our models are capable of identifying interactions between covariates and treatmentgroup so that they can be applied to find optimal strategy based on patient characteristics.
- All rights reserved
Notice to Authors
If you are the author of this work and you have any questions about the information on this page, please use the Contact form to get in touch with us.