Variables with time-varying effects and the Cox model: some statistical concepts illustrated with a prognostic factor study in breast cancer. BMC Med Res Methodol 10(1):20

Department of Clinical Epidemiology and Clinical Research, Institut Bergonié, Regional Comprehensive Cancer Centre, Bordeaux, France.
BMC Medical Research Methodology (Impact Factor: 2.27). 03/2010; 10(1):20. DOI: 10.1186/1471-2288-10-20
Source: PubMed


The Cox model relies on the proportional hazards (PH) assumption, implying that the factors investigated have a constant impact on the hazard - or risk - over time. We emphasize the importance of this assumption and the misleading conclusions that can be inferred if it is violated; this is particularly essential in the presence of long follow-ups.
We illustrate our discussion by analyzing prognostic factors of metastases in 979 women treated for breast cancer with surgery. Age, tumour size and grade, lymph node involvement, peritumoral vascular invasion (PVI), status of hormone receptors (HRec), Her2, and Mib1 were considered.
Median follow-up was 14 years; 264 women developed metastases. The conventional Cox model suggested that all factors but HRec, Her2, and Mib1 status were strong prognostic factors of metastases. Additional tests indicated that the PH assumption was not satisfied for some variables of the model. Tumour grade had a significant time-varying effect, but although its effect diminished over time, it remained strong. Interestingly, while the conventional Cox model did not show any significant effect of the HRec status, tests provided strong evidence that this variable had a non-constant effect over time. Negative HRec status increased the risk of metastases early but became protective thereafter. This reversal of effect may explain non-significant hazard ratios provided by previous conventional Cox analyses in studies with long follow-ups.
Investigating time-varying effects should be an integral part of Cox survival analyses. Detecting and accounting for time-varying effects provide insights on some specific time patterns, and on valuable biological information that could be missed otherwise.

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Available from: Simone Mathoulin-Pelissier, Mar 09, 2014
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    • "This assumption implies that the factors investigated have a constant impact on the hazard over time. Savignoni et al (2012) and Bellera et al (2010) have recently emphasised that this assumption in long-term breast cancer prognostic analysis may be misleading and a significant effect of varying prognostic value of the markers in the early or late follow-up period may be missed. "
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    ABSTRACT: Recent studies have indicated the prognostic value of tumour subtype and pathological complete response (pCR) after neoadjuvant chemotherapy (NAC). However these results were reported after a short follow-up and using a standard Cox model which could be unsatisfactory for time-dependent factors. In the present study, we identified the prognostic factors for long-term outcome after NAC, considering that they could have an inconstant impact over time. Prognostic factors from 956 consecutive breast cancer patients treated with NAC were identified and associated with long-term outcomes. We estimated survival by a time function multivariate Cox model regression and stratified by follow-up length. The prognostic value of tumour histological grade and hormone receptors status varied as distant recurrence-free interval (DRFI) increased. The multivariate analysis identified the following significant prognostic factors: tumour size, N stage, clinical and pathological response to NAC, hormone receptors (HR) status and histological tumour grade. The 'prognostic benefit' of low-grade and positive-HR status decreased over the years. Thus, in the early years after cancer diagnosis, the hazard ratio of distant recurrences in patients with positive-HR status increased from 0.26 (95% CI 0.1-0.4) at 6 months to 2.2 (95% CI 1.3-3.7) at 120 months. The histological tumour grade followed a similar trend. The hazard ratio of grade III patients compared with grade I was 1.83 (95% CI 1.1-2.8) at 36 months and diminished over time to 0.70 (95% CI 0.4-1.3) at 120 months. This indicates that the risk of recurrence for positive-HR patients was 74% lower at 6 months compared with the negative-hormone receptor group, but 30% higher at 5 years and more than double at 10 years. High-grade tumours presented a risk of 83% in the earlier years decreasing to 30% at 10 years versus the low-grade group. From the present study, we conclude the importance of identifying time-dependent prognostic factors. Distant recurrence-free interval within women who receive NAC are influenced by achieving pCR and breast cancer subtype. Tumours with more aggressive biology have poorer survival during the first 5 years, but if they exceed this point their prognostic impact is no longer significant. Conversely, positive-HR patients remain at risk for distant recurrence for many years.British Journal of Cancer advance online publication, 16 June 2015; doi:10.1038/bjc.2015.174
    British Journal of Cancer 06/2015; 113(1). DOI:10.1038/bjc.2015.174 · 4.84 Impact Factor
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    • "The potential time-dependency of prognostic and predictive variables is well-known [11,24,25], but it is rarely studied purposefully. In this paper we examined the time-dependent interaction of proliferation [17] and ER-related gene expression [18,26] with the risk of early versus late recurrence in ER-positive early-stage breast cancer. "
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    ABSTRACT: We examined if a combination of proliferation markers and estrogen receptor (ER) activity could predict early versus late relapses in ER-positive breast cancer and inform the choice and length of adjuvant endocrine therapy. Baseline affymetrix gene-expression profiles from ER-positive patients who received no systemic therapy (n = 559) or adjuvant tamoxifen for 5 years (cohort-1: n = 683, cohort-2: n = 282) and from 58 patients treated with neoadjuvant letrozole for 3 months (gene-expression available at baseline, 14 and 90 days) were analyzed. A proliferation score based on the expression of mitotic kinases (MKS) and an ER-related score (ERS) adopted from Oncotype DX(R) were calculated. The same analysis was performed using the Genomic Grade Index as proliferation marker and the luminal gene score from the PAM50 classifier as measure of estrogen-related genes. Median values were used to define low and high marker groups and four combinations were created. Relapses were grouped into time cohorts of 0--2.5, 0--5, >5-10 years. In the overall 10 years period, the proportional hazards assumption was violated for several biomarker groups indicating time-dependent effects. In tamoxifen-treated patients Low-MKS/Low-ERS cancers had continuously increasing risk of relapse that was higher after 5 years than Low-MKS/High-ERS cancers [0 to 10 year, HR 3.36; p = 0.013]. High-MKS/High-ERS cancers had low risk of early relapse [0--2.5 years HR 0.13; p = 0.0006], but high risk of late relapse which was higher than in the High-MKS/Low-ERS group [after 5 years HR 3.86; p = 0.007]. The High-MKS/Low-ERS subset had most of the early relapses [0 to 2.5 years, HR 6.53; p < 0.0001] especially in node negative tumors and showed minimal response to neoadjuvant letrozole. These findings were qualitatively confirmed in a smaller independent cohort of tamoxifen-treated patients. Using different biomarkers provided similar results. Early relapses are highest in highly proliferative/low-ERS cancers, in particular in node negative tumors. Relapses occurring after 5 years of adjuvant tamoxifen are highest among the highly-proliferative/high-ERS tumors although their risk of recurrence is modest in the first 5 years on tamoxifen. These tumors could be the best candidates for extended endocrine therapy.
    Breast cancer research: BCR 09/2013; 15(5):R86. DOI:10.1186/bcr3481 · 5.49 Impact Factor
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    • "Similarly, in observational studies, it is often important to determine whether a factor is associated with a constantly higher or lower risk of the outcome over time. For example, Bellera et al. [9] showed that the prognostic relevance of tumor grade for breast cancer metastases diminished over time and negative hormone receptor status was associated with an increased risk of metastases early but became protective thereafter. "
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    ABSTRACT: Case-cohort studies have become common in epidemiological studies of rare disease, with Cox regression models the principal method used in their analysis. However, no appropriate procedures to assess the assumption of proportional hazards of case-cohort Cox models have been proposed. We extended the correlation test based on Schoenfeld residuals, an approach used to evaluate the proportionality of hazards in standard Cox models. Specifically, pseudolikelihood functions were used to define "case-cohort Schoenfeld residuals", and then the correlation of these residuals with each of three functions of event time (i.e., the event time itself, rank order, Kaplan-Meier estimates) was determined. The performances of the proposed tests were examined using simulation studies. We then applied these methods to data from a previously published case-cohort investigation of the insulin/IGF-axis and colorectal cancer. Simulation studies showed that each of the three correlation tests accurately detected non-proportionality. Application of the proposed tests to the example case-cohort investigation dataset showed that the Cox proportional hazards assumption was not satisfied for certain exposure variables in that study, an issue we addressed through use of available, alternative analytical approaches. The proposed correlation tests provide a simple and accurate approach for testing the proportional hazards assumption of Cox models in case-cohort analysis. Evaluation of the proportional hazards assumption is essential since its violation raises questions regarding the validity of Cox model results which, if unrecognized, could result in the publication of erroneous scientific findings.
    BMC Medical Research Methodology 07/2013; 13(1):88. DOI:10.1186/1471-2288-13-88 · 2.27 Impact Factor
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