Using SAS to calculate the Kent and O'Quigley measure of dependence for Cox proportional hazards regression model

Quantifying predictive accuracy in survival models

For time-to-event outcomes in medical research, survival models are the most appropriate to use. Unlike logistic regression models, quantifying the predictive accuracy of these models is not a trivial task. We present the classes of concordance (C) statistics and R 2 statistics often used to assess the predictive ability of these models. The discussion focuses on Harrell's C, Kent and O'Quigley's R 2, and Royston and Sauerbrei's R 2. We present similarities and differences between the statistics, discuss the software options from the most widely used statistical analysis packages, and give a practical example using the Worcester Heart Attack Study dataset.

A novel nonparametric measure of explained variation for survival data with an easy graphical interpretation

INTRODUCTION

For survival data the coefficient of determination cannot be used to describe how good a model fits to the data. Therefore, several measures of explained variation for survival data have been proposed in recent years.

METHODS

We analyse an existing measure of explained variation with regard to minimisation aspects and demonstrate that these are not fulfilled for the measure.

RESULTS

In analogy to the least squares method from linear regression analysis we develop a novel measure for categorical covariates which is based only on the Kaplan-Meier estimator. Hence, the novel measure is a completely nonparametric measure with an easy graphical interpretation. For the novel measure different weighting possibilities are available and a statistical test of significance can be performed. Eventually, we apply the novel measure and further measures of explained variation to a dataset comprising persons with a histopathological papillary thyroid carcinoma.

CONCLUSION

We propose a novel measure of explained variation with a comprehensible derivation as well as a graphical interpretation, which may be used in further analyses with survival data.

Prognostic significance of Ki67 proliferation index, HIF1 alpha index and microvascular density in patients with non-small cell lung cancer brain metastases

BACKGROUND

Survival upon diagnosis of brain metastases (BM) in patients with non-small cell lung cancer (NSCLC) is highly variable and established prognostic scores do not include tissue-based parameters.

METHODS

Patients who underwent neurosurgical resection as first-line therapy for newly diagnosed NSCLC BM were included. Microvascular density (MVD), Ki67 tumor cell proliferation index and hypoxia-inducible factor 1 alpha (HIF-1 alpha) index were determined by immunohistochemistry.

RESULTS

NSCLC BM specimens from 230 patients (151 male, 79 female; median age 56 years; 199 nonsquamous histology) and 53/230 (23.0%) matched primary tumor samples were available. Adjuvant whole-brain radiation therapy (WBRT) was given to 153/230 (66.5%) patients after neurosurgical resection. MVD and HIF-1 alpha indices were significantly higher in BM than in matched primary tumors. In patients treated with adjuvant WBRT, low BM HIF-1 alpha expression was associated with favorable overall survival (OS), while among patients not treated with adjuvant WBRT, BM HIF-1 alpha expression did not correlate with OS. Low diagnosis-specific graded prognostic assessment score (DS-GPA), low Ki67 index, high MVD, low HIF-1 alpha index and administration of adjuvant WBRT were independently associated with favorable OS. Incorporation of tissue-based parameters into the commonly used DS-GPA allowed refined discrimination of prognostic subgroups.

CONCLUSION

Ki67 index, MVD and HIF-1 alpha index have promising prognostic value in BM and should be validated in further studies.

A measure of explained risk in the proportional hazards model

A measure of explained risk is developed for application with the proportional hazards model. The statistic, which is called the estimated explained relative risk, has a simple analytical form and is unaffected by censoring that is independent of survival time conditional on the covariates. An asymptotic confidence interval for the limiting value of the estimated explained relative risk is derived, and the role of individual factors in the computation of its estimate is established. Simulations are performed to compare the results of the estimated explained relative risk to other known explained risk measures with censored data. Prostate cancer data are used to demonstrate an analysis incorporating the proposed approach.

Prognostic and predictive gene signature for adjuvant chemotherapy in resected non-small-cell lung cancer

PURPOSE

The JBR.10 trial demonstrated benefit from adjuvant cisplatin/vinorelbine (ACT) in early-stage non-small-cell lung cancer (NSCLC). We hypothesized that expression profiling may identify stage-independent subgroups who might benefit from ACT.

PATIENTS AND METHODS

Gene expression profiling was conducted on mRNA from 133 frozen JBR.10 tumor samples (62 observation [OBS], 71 ACT). The minimum gene set that was selected for the greatest separation of good and poor prognosis patient subgroups in OBS patients was identified. The prognostic value of this gene signature was tested in four independent published microarray data sets and by quantitative reverse-transcriptase polymerase chain reaction (RT-qPCR).

RESULTS

A 15-gene signature separated OBS patients into high-risk and low-risk subgroups with significantly different survival (hazard ratio [HR], 15.02; 95% CI, 5.12 to 44.04; P < .001; stage I HR, 13.31; P < .001; stage II HR, 13.47; P < .001). The prognostic effect was verified in the same 62 OBS patients where gene expression was assessed by qPCR. Furthermore, it was validated consistently in four separate microarray data sets (total 356 stage IB to II patients without adjuvant treatment) and additional JBR.10 OBS patients by qPCR (n = 19). The signature was also predictive of improved survival after ACT in JBR.10 high-risk patients (HR, 0.33; 95% CI, 0.17 to 0.63; P = .0005), but not in low-risk patients (HR, 3.67; 95% CI, 1.22 to 11.06; P = .0133; interaction P < .001). Significant interaction between risk groups and ACT was verified by qPCR.

CONCLUSION

This 15-gene expression signature is an independent prognostic marker in early-stage, completely resected NSCLC, and to our knowledge, is the first signature that has demonstrated the potential to select patients with stage IB to II NSCLC most likely to benefit from adjuvant chemotherapy with cisplatin/vinorelbine.

Prognostic indices for Budd-Chiari syndrome: valid for clinical studies but insufficient for individual management

OBJECTIVES

Several prognostic indices (PIs) have been proposed for Budd-Chiari syndrome (BCS). However, patient characteristics, causal factors, and treatment outcomes have changed since these indices have been elaborated. Validation in a recent patient population and comparison of predictive accuracy between these PIs are needed.

METHODS

A database of 96 BCS patients diagnosed between 1995 and 2005 was analyzed. Cox survival models were fitted with time to liver transplantation or death, and time to invasive therapy or death, as end points. The prognostic values of known indices (Child-Pugh score, model for end-stage liver disease (MELD), Clichy, Rotterdam BCS index, New Clichy, and BCS-TIPS (transjugular intrahepatic portosystemic shunt)) at diagnosis were assessed in Cox models using the chi-square test, the Kent and O'Quigley measure of dependence, and unrestricted bootstrapping analysis. Areas under receiver operating characteristic curves (AUROCs) were built for both end points and compared.

RESULTS

All prognostic indices, except BCS-TIPS, were significant predictors of transplant-free and invasive therapy-free survival. However, only 31 and 37% of the variance in transplant-free and invasive therapy-free survival, respectively, were explained by the best performing indices. For transplant-free survival, AUROCs were < 0.70. For invasive therapy-free survival, AUROCs were < 0.80. For both end points, BCS-TIPS PI AUROCs were significantly lower than others.

CONCLUSIONS

Most PIs are valid for transplant-free survival and invasive therapy-free survival in a population of current BCS patients, and thus can be used for stratification in clinical studies. However, predictive accuracy is insufficient to be used for individual patients.

Explained randomness in proportional hazards models

A coefficient of explained randomness, analogous to explained variation but for non-linear models, was presented by Kent. The construct hinges upon the notion of Kullback-Leibler information gain. Kent and O'Quigley developed these ideas, obtaining simple, multiple and partial coefficients for the situation of proportional hazards regression. Their approach was based upon the idea of transforming a general proportional hazards model to a specific one of Weibull form. Xu and O'Quigley developed a more direct approach, more in harmony with the semi-parametric nature of the proportional hazards model thereby simplifying inference and allowing, for instance, the use of time dependent covariates. A potential drawback to the coefficient of Xu and O'Quigley is its interpretation as explained randomness in the covariate given time. An investigator might feel that the interpretation of the Kent and O'Quigley coefficient, as a proportion of explained randomness of time given the covariate, is preferable. One purpose of this note is to indicate that, under an independent censoring assumption, the two population coefficients coincide. Thus the simpler inferential setting for Xu and O'Quigley can also be applied to the coefficient of Kent and O'Quigley. Our second purpose is to point out that a sample-based coefficient in common use in the SAS statistical package can be interpreted as an estimate of explained randomness when there is no censoring. When there is censoring the SAS coefficient would not seem satisfactory in that its population counterpart depends on an independent censoring mechanism. However there is a quick fix and we argue in favour of its use.