Disease-specific survival for limited-stage small-cell lung cancer affected by statistical method of assessment

The Cox proportional hazards model [1] has been the standard tool for multivariate assessments of the influence of prognostic/predictive factors for censored survival data. It may encounter serious difficulties with departures from the proportional hazards assumption, even when the departures are not readily detected by commonly used statistical analytic methods [2].

As well, it has been questioned whether a standard survival analysis framework is appropriate if there is an identifiable mixture of "cured" and "non-cured" patients [3–5], for whom there may be distinguishably different factor effects [4, 6]. Various models have been used in the context of censored patient data to assess the cured fraction, e.g. Boag log-normal with a cure-fraction parameter [7], modified-Boag model with covariates [6], Cox model for grouped survival data [3], logistic mixture model [4], or a Box-Cox transformation on the population survival function that includes both the mixture cure model and the biologic promotion time cure model [5]. Generalizability of survival analysis may be accomplished with flexible models that comprehensively span a range of options, such as the transformation cure model [5], or the Pettit model [8] which encompasses both Cox and Generalized F models. However, increases in model complexity may affect study conclusions, leading to non-identifiability or instability of results [4, 5, 9]. Higher level statistical (research) assessment is operationally essential for complex models.

Meanwhile, there is a clinical practice movement to the use of real-time medical decision making tools like Adjuvant! Online [10]. Users input prognostic/predictive factor data, and are quickly provided with estimated survival, without the need to consult a statistician or consider underlying modeling, which is provided within the software. Modeling is operationally becoming a new de facto staging system, which is iteratively updated by a provider with accumulating literature-based evidence.

Our goal here was to compare disease-specific survival (DSS) obtained by modeling of prognostic/predictive effects for a few simple model systems, to set the stage for a re-examination of commonly available clinical factors for limited-stage small-cell lung cancer, where there is a need to develop a new staging system. Standard Kaplan-Meier and Cox modeling were used because of their prevalence in clinical practice, along with two different types of log-normal models, the Boag cure-rate [7] and log-normal survival analysis [11].

Boag demonstrated more than five decades ago that a log-normal model was appropriate for breast cancer patients [7]. Royston [11] found that the prognosis for breast cancer patients differed by up to a year, depending on whether one utilized a Cox or log-normal assessment. Tai et al. validated the Boag log-normal model for the estimation of survival in patients with limited-stage small-cell lung cancer (SCLC) [12]. Overduin used follow-up to events in Tai's data to examine the goodness of fit for the Weibull, gamma, and lognormal models [13]. The fit was poor for both the Weibull and gamma (p < 0.001, in each instance), while acceptable for the log-normal (p = 0.37). Tai's updated long-term Saskatchewan patient cohort data were used here for the model comparisons. We hypothesized that there might be substantive differences in DSS for the same limited-stage SCLC patients, if Boag log-normal (cure-rate model) [7], Kaplan-Meier [14], Cox [1], and log-normal survival analysis [15, 16] were used.

The median follow-up of those alive was 9.5 years. One hundred and eighty-four (75.7%) of the 243 patients died from lung cancer: by 1 year, 42 of 238 (17.6%); by 3 years, 173 of 230 (75.2%); and by 5 years, 186 of 223 (83.4%). There was no substantive evidence against the assumption of proportional hazards. The times of lung cancer deaths were reasonably consistent with the log-normal model both in the Q-Q plot and by the chi-squared test (p = 0.42).

Four factors were found to significantly affect DSS in both multivariate Cox and log-normal survival analyses: surgical resection (p = 0.01, with Cox; p = 0.001, with log-normal), hemoglobin level (p = 0.02, with Cox; p = 0.005, with log-normal), lymph node involvement (p = 0.02, with Cox; p = 0.04, with log-normal), and superior vena cava (SVC) obstruction (p = 0.06, with Cox; p = 0.10, with log-normal). These factors then became the focus for characterizing patients and reporting DSS.

(Please note: A hemoglobin cut-point of 100 is a medical intervention level for both males and females. It is a value that transfusion is generally recommended at during treatment, irrespective of sex. Also, "incomplete or no resection" for groups B and C includes patients with no surgery, or partial resection, who had residual disease after surgery, and were treated with a combination of chemotherapy and radiotherapy.)

The log-normal survivor plot (Figure 1) contains a smooth modeling of DSS. The discontinuities observed in the Cox plot (Figure 1) arise because DSS is adjusted at the time of events.

These subgroup effects for the four factors indicated by multivariate analyses, with both Cox and log-normal survival analyses, could not be considered here in a Kaplan-Meier framework. The subgroups contained a total of 8 patients and 4 events.

Patient management decisions are made on the basis of prognosis, or prediction about tumor responsiveness to particular therapeutic regimens. Long-term evidence has supported the relevance of TNM staging or other readily ascertainable clinical characteristics. Kaplan-Meier plots and Cox modeling have been standard assessment tools for decades. Infrequently, other models like the Boag log-normal or log-normal survival analysis have been utilized. SCLC is a disease site for which it might be advantageous to develop a new staging system since the requirement of complete excision to define elements of TNM staging is frequently not met; 94 percent of the patients in this cohort did not have complete resection. We presented here a case study that illustrates differences in DSS obtained using the same patients and the above four methods, each of which has underlying assumptions which may be imperfectly met in any particular data.

Further, SCLC is a particularly good cancer site in which to consider a cure rate framework as very few events are expected after 5 years. The long accrual period (approx. 18 years) may be a limitation; however, the study population is representative of patients treated in this era. With a median follow-up of 9.5 years for these patients, the 5-year DSS estimates should be stable, and the estimates by the four methods differed by only 1%, to be 19 or 20%. This is a survival rate that would be anticipated from the literature.

The survival rates and confidence limits are those which could be reported for the four model-types, although reports usually consider at most 1 or 2 methods. Formal direct statistical tests are not possible between frameworks, e.g. between the semi-parametric Cox and two parametric log-normal models, nor even between Boag log-normal cure rate and log-normal survival analysis. However, 95% confidence limits are provided for DSS obtained by each method. DSS at 3 years was 38% with log-normal survival analysis, 10–12% higher than with other methods. Univariate 95% confidence limits for log-normal survival analysis did not overlap with those obtained for any of the other methods. In the breast cancer setting, Royston [11] observed that prognosis by model-type differed by up to a year depending on whether a Cox or log-normal model was used, and Chapman, et al [16] found up to 8% absolute difference by model-type. Large differences in survival estimates appear possible according to whether one uses a Cox or log-normal model.

The much narrower range in confidence limits for the Boag model reflects that the DSS estimates are for those who have died, while the other three methods incorporate the uncertainty from censored survival times. Further, the Boag model fits an extra parameter, C, for the proportion cured. The estimate of proportion cured, at a time when few or no events are expected, is derived directly with the survivor function for the Cox and log-normal survival analysis models.

The assessments in this cohort for both the Boag and Kaplan-Meier were limited to those with the full patient group due to the size of the SCLC patient cohort which precluded subgrouping with the four factors indicated by multivariate analyses, while more extensive modeling was possible for specific patient characteristics with both Cox and log-normal survival analysis. Apparent differences in DSS of up to 26% at 3-years were seen between these latter two model-types, although all the multivariate 95% confidence limits for the 2 methods overlap. The variability will reflect patient heterogeneity, the imprecise nature of the particular clinical factors investigated, inadequacy of the Cox and log-normal modeling, and sample size. More precise molecular or genomic factors would eventually provide better precision, but could still involve the same assessment methods. Again, large differences between apparent prognosis with the Cox and log-normal survival models have been seen by others: up to a year's difference, by Royston [11]; and up to 8% absolute difference, by Chapman et al. [16]. Absolute differences at 5-years varied less: respectively, 5, 0, 12, and 6%, for patient Groups A-D.

The Cox plot presents discontinuous estimates of DSS, as would the Kaplan-Meier, as DSS is adjusted at events. The Boag model requires the estimate of one more parameter, the cure fraction, which while reasonable in a lung cancer population might not be appropriate for many other cancers with short or medium follow-up. Royston [11] described the log-normal model as a pragmatic tool that provides a continuous estimate. Where the model-type is concordant with the data, smooth log-normal survival analysis may be advantageous over the semi-parametric proportional hazards model. The Cox adjustments at events may unduly alter estimates of DSS at the end of the study period when there are few patients left. In this instance, with few lung cancer deaths expected after 5 years, the DSS estimates should approximate the proportion of patients who are likely 'cured'.

In this study, Groups A to D represent examples of combinations of clinical characteristics for which DSS was determined based on modeling for all patients, rather than subgroup analyses. The main objective of the study was to illustrate the use of different statistical methods to analyze the effects of prognostic/predictive factors on DSS at differing time points. The data used for these investigations accrued from an eighteen year Provinicial cohort of data, recognized externally for its excellence. Unfortuneately, only 244 of the 1417 patients had limited stage disease, and were considered to have received curative management, so they were eligible for these DSS investigations. These investigations are hypothesis generating, requiring external validation in a much larger series. However, follow-up to 5 years was complete for 223/243 (92%) of patients. The reason for considering estimates of 5 year rates was that this would provide the best estimate of the "cured fraction".

Different factors may have been described from additional studies because of the exclusion of factors, small numbers of patients, different patient population, extent of disease and follow-up. Newer factors mentioned in studies include: neuron-specific enolase (NSE) [21], Cyfra21-1 [22], integrin beta1 [23, 24], p53 [21], and cytoplasmic MAPK [25].

In conclusion, among different factors tested, surgical resection, hemoglobin level, lymph node involvement, and SVC obstruction were found to have significant effects on DSS, regardless of model-type. When one attempts to re-evaluate available data aiming towards the formation of a new staging system, different statistical methods are available. Estimation of DSS for patients was influenced by the statistical assessment method. Parametric models should be considered more frequently for survival analysis to assess prognostic and predictive effects; like Royston [11] and Chapman et al. [16], we found here that the log-normal was an appropriate parametric model choice. Multiple methods may be clinically relevant in the development or improvement of clinical management strategies; a software tool might provide estimates with several survival models.