Survival data analysis methods are the cornerstone of a wide range of statistical applications. While mean survival time is of utmost relevance, e.g., in health economics (Paltiel et al., 2009) or oncology studies (Zhao et al., 2001), its estimation might be hindered in the presence of censoring, where the time variable is only observed until a certain quantile. In practice, censoring is almost always present, calling for specialised estimation techniques. Several approaches have been considered to overcome this problem, the most used amongst them, the restricted mean, computes mean survival time up to a specific cut-off time point (Irwin, 1949). Estimation of the restricted mean, however, might be heavily affected by the presence of censored observations, which will result in a loss of estimation accuracy. Moreover, clinical relevance and interpretation of restricted mean estimates remains unclear.
We present a novel mean survival measure based on observed quantiles that divides the population in ordered fractions in which the mean survival can be estimated separately. Interpretation of the estimates is straightforward, as they represent mean survival times for the specified fractions of the population. Similarly to the restricted mean, we estimate mean survival up to a specific cut-off point, that we set to the largest observed -th fraction of population to experience the event of interest. Our approach exploits that the distribution of observed and censored events imposes differences in the estimation accuracy of specific quantiles, i.e., those that are close to observed events can be more precisely estimated than those located after the occurrence of censored events. Therefore, estimates for certain fractions can be really precise, which allows quantifying significant mean survival differences across groups, even in scenarios where state-of-the-art methods are unable to detect them.
0.2 Mean survival by ordered fractions
be a non-negative random variable withand let and denote its survival and quantile functions, respectively. An expression for the expectation of in terms of is
Given a grid of proportions with for all , we can divide into separate components as follows
If we now weight each by its corresponding inverse proportion, we obtain
where is the mean survival time for a specific fraction of population delimited by . For example, if we consider , quantifies mean survival time for the first half of the population to experience the event of interest.
In the presence of a censoring variable , when is observed, the decomposition shown in (2) is of utmost convenience because can be set to the largest proportion of observed events, that is, the one corresponding to the last observed quantile. Note that when , the mean survival time for the -th fraction of the population observed to experience the event, does not correspond to the restricted mean computed up to the last observed quantile . Indeed, while
can be easily interpreted in terms of the population under study, the corresponding
does not prove as informative.
0.3 Estimation and simulation results
In the presence of censoring, estimation of is possible via the Kaplan-Meier estimator of the underlying survival function, . Given and the grid of proportions , an estimator for follows easily from equations (1) and (2), with
where denote observed event times such that for all , and and . In this case, we obtain a step-wise constant estimator of the quantile function , in which observed times play the role of estimated quantiles of order .
We tested the performance of in different scenarios, all yielding analogous conclusions. In Table 1 we present results for a simulation study of data sets with samples each, generated from a time variable following a log-logistic distribution with scale and shape
. The censoring variables were sampled independently from a uniform distribution in, yielding an average censoring rate of . Estimated average upper and lower bounds for where computed integrating over equal precision confidence bands for the Kaplan-Meier estimator (Nair, 1984). We observed that our estimates’ precision decreased with increasing (that is, the bands widened with increasing ), which was expected, as the proportion of censored observations also increased with and fewer events were observed. In this sense, one might say that some can be more precisely estimated than others, which proves highly useful in application seetings.
0.4 Application example: Survival after bone marrow transplant in lymphoma patients
We analysed data on 35 patients with lymphoma that received either an allogenic or an autologous bone marrow transplant, that is, they received marrow from either a compatible donor or their own after chemotherapy treatment and cleansing (Avalos et al., 1993). The aim of the study was to find differences between lymphoma-free survival after having received either type of transplant. After 2.5 years of follow-up, 26 patients had died or relapsed and the censoring rate was . The estimated survival curves for both treatments are shown in Figure 1.
While restricted mean survival estimates did not detect any significant difference in mean survival between the allogenic and autologus transplant groups (restricted mean difference of days, with confidence interval )), our approach showed that among earlier failures that difference was actually significant. In particular, considering the weakest of the patients, that is, the first to die or relapse after receiving the transplant, mean survival difference was estimated at days ( CI ) favouring those who received the autologus transplant. In Table 2
we show the results of mean survival time differences after receiving a bone marrow transplant by deciles of population up to theth percentile (last fraction commonly observed in both groups).
Our estimates could detect an improvement on lymphoma-free survival for the autologus transplant group amongst at least the weakest of patients, providing a useful guide for effective decision-making.
0.5 Final remarks
Our approach for quantifying mean survival time takes advantage of the information contained in the data and deals with the censoring hurdle. Our proposed measures are easily interpretable, providing a useful alternative to the restricted mean, which poses interpretation difficulties. By dividing the study population into ordered fractions, the proposed method provides a detailed picture of the underlying probability distribution and can detect mean survival differences across groups that are often undetected by other methods. Results from a simulation study show good performance of our proposed estimation strategy even in the presence of censoring, and support the idea that mean survival can be more accurately estimated in some fractions of the population. In the analysis of survival data after bone marrow transplant, our method detected differences in mean survival between given transplants for certain fractions of population, while those differences were overlooked when using restricted mean estimates instead.
This work was partially supported by the KID funding doctoral grant from Karolinska Institutet
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