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Keywords:

  • breast carcinoma;
  • survival;
  • Cox regression analysis;
  • frailty;
  • prognostic factors

Abstract

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION
  6. REFERENCES

BACKGROUND

Tumor size, lymph node status, and histologic grade are reported to be important predictors of survival in the first 5 years after the diagnosis of invasive breast carcinoma. However, to the authors' knowledge, the effect of these factors in the longer term (> 10 years after diagnosis) is not yet clear.

METHODS

It is now > 20 years since the Swedish Two-County Trial of breast carcinoma screening with mammography was instigated and long-term follow-up is now available to December 1998. In the current study, the authors analyzed the effects of tumor size, lymph node status, and tumor grade on survival to death from breast carcinoma using Cox regression and frailty models that allow the baseline hazard and/or effect of a covariate to vary with time.

RESULTS

The effects of tumor size, lymph node status, and tumor grade were shown to progressively diminish with time from diagnosis. The Cox regression model with time-varying coefficients and a dampening parameter then was fitted to allow for the attenuation of prognostic effects; tumor size, lymph node status, and tumor grade were all found to be highly significant (P < 0.001).

CONCLUSIONS

The results of the current study suggest that long-term survival in women with invasive breast carcinoma could be modelled satisfactorily using either frailty models or Cox regression models with time-varying coefficients. The results also suggest that the value of tumor grade, lymph node status, and tumor size at the time of diagnosis have a lasting influence on subsequent survival, albeit attenuated in later years. The long-term effects of these prognostic factors may explain the fact that the impact of mass screening programs on breast carcinoma mortality rates is still apparent many years later. Cancer 2004;100:1331–6. © 2004 American Cancer Society.

It has been long established that the pathologic variables of tumor size, lymph node status, and histologic tumor grade are significant prognostic indicators in invasive breast carcinoma.1–4 More recently, biomarkers of prognosis have been identified5–7 and a radiologic predictor of survival has been discovered,8 but the value of tumor size, lymph node status, and tumor grade as powerful predictors of survival remains.9

The significance of prognostic factors for the long term (i.e., > 10 years after diagnosis) is not clear. There have been suggestions that tumor grade, lymph node status, and tumor size have no effect on subsequent survival in patients who already have survived for ≥ 10 years.10, 11 These need to be qualified by the observation that tumor series with long-term follow-up, of the order of 20 years, are rare and the majority of such series pertain to a period of diagnosis when tumors typically presented at an advanced stage. If very early disease stage confers a better prognosis in the long term, as well as in the short, long-term follow-up of a substantial number of early-stage cases will be necessary to observe this finding with any precision. In the Swedish Two-County Trial, long-term follow-up is available for 2299 cases of invasive breast carcinoma (including 1371 women with follow-up in excess of 10 years), 1023 of which were diagnosed at the time of screening. In the current study, we use the survival data on these cases to estimate the effects of tumor size, lymph node status, and histologic tumor grade on both long-term and short-term survival.

MATERIALS AND METHODS

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION
  6. REFERENCES

The Swedish Two-County Trial is a cluster-randomized trial of breast carcinoma screening with mammography, in which 77,080 women were randomized to receive regular invitations to mammographic screening (Active Study Population [ASP]) and 55,985 were randomized to receive no invitation (Passive Study Population [PSP]). At the end of 1984, a significant 30% reduction in breast carcinoma mortality was observed in the ASP, the PSP was offered screening, and the trial closed thereafter.12, 13 During the trial, 2468 cases were diagnosed (2299 of which were invasive): 1426 in the ASP and 1042 in the PSP.

The study subjects were followed to December 1998 and all causes of death were reviewed by a committee. Deaths were attributed to breast carcinoma if breast carcinoma was the primary cause or if there was proven residual disease present at the time of death.

We analyzed the effects of tumor size, lymph node status, and tumor grade on survival to death from breast carcinoma (defined as deaths in which breast carcinoma was present) by: 1) fitting five Cox regression14 models (one for the entire period of follow-up and the others relating to each quinquennium after diagnosis) and 2) fitting a Cox regression model with time-varying coefficients.

The Cox model (also known as the proportional hazards model) provides estimates of the probability of surviving to time t given a particular set of prognostic values and may be used to assess the joint and independent effects of such factors on survival. The model assumes a baseline hazard (risk of death) that is common to all the individuals in the data set and then adds additional risk on an individual basis, as determined by the individuals' prognostic information. The distribution of the baseline hazard is not specified but strong assumptions are made regarding its behavior nevertheless, with the key assumption being that the relative risk of death (relative hazard) is constant over time. That is to say that for a prognostic variable with two levels, the risk of death for individuals in each group may vary over time but the ratio of the risk in one group to the other remains the same throughout the period of study. In the short term, when considering factors affecting survival in the early postdiagnosis period, such an assumption is not unreasonable. However, when long-term follow-up is available (as it is with the Swedish Two-County data), this requirement may be inappropriate because it appears unrealistic to expect the relative hazards to remain unchanged during such a lengthy study period. Therefore, the data were also modelled using an extension to the classic Cox regression model that relaxes the proportionality constraint and allows the relative hazards to vary over time. Again, a common baseline hazard is assumed for all individuals, but with the additional risk added now dependent on time since diagnosis as well as the prognostic information. The prognostic effects in the model are assumed to diminish over time, a feature achieved by multiplying the relevant coefficients in the model by the factor e−λt in which t is time since diagnosis and λ > 0 controls the rate of attenuation. The larger the value of λ, the more rapidly the prognostic effect diminishes. This approach enables one to investigate whether there is some point (in time) at which the risk of death for an individual who had a poor prognosis initially (e.g., a tumor grade of 3) becomes the same, or nearly the same, as that for an individual who had Grade 1 disease on entry to the study, an issue of obvious clinical interest.

The mathematical forms of the models used, with that for a frailty model, are as follows.

Notation

In the following models t is used to represent time as a continuous variable, j indexes intervals of time in the discrete case, h0(t) is the baseline hazard function, hi(t) is the hazard function for the ith individual, βT is a vector of coefficients, and xi is the vector of covariate values relating to the ith individual.

  • 1
    Cox regression model
    • equation image
  • 2
    Cox regression model with time-varying coefficients
    • equation image
  • 3
    Complementary log-log model with Gamma frailty
    • equation image
    in which λi ∼ Gamma(1,σ2) and γj=the baseline hazard in the jth time interval.
  • 4
    Complementary log-log model with time-varying coefficients and Gamma frailty
    • equation image
    in which εi ∼ Gamma(1,σ2) and γj=the baseline hazard in the jth time interval.

The complementary log-log model is the discrete time equivalent of the Cox regression model.15 To use this model, the continuous survival times are rendered discrete by splitting them into yearly intervals. This results in some loss of information, particularly in relation to those patients with the shortest survival times, but has the advantage of being easily adapted to allow for changes in covariate values and/or their effects over time. Introducing frailty gives a more flexible model that allows nonproportional hazards (relative hazards that change over time) but has the disadvantage of being more difficult to interpret. This is because frailty is neither observed nor observable at the individual level. The resulting model reduces the estimate of the baseline hazard and correspondingly increases covariate effects and therefore is useful when identifying attenuated covariate effects. It allows the baseline hazard to vary with time but keeps covariate effects constant.

RESULTS

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION
  6. REFERENCES

Table 1 shows the numbers of cases and breast carcinoma deaths, with the Cox regression results (relative hazard ratios and 95% confidence intervals) from analysis of the entire follow-up period. The effects of tumor size, lymph node status, and histologic tumor grade all were found to be highly significant, with particularly intense hazards noted for patients with large tumors and those with distant metastases at the time of diagnosis.

Table 1. Estimated Hazard Ratios from the Univariate and Multivariate Cox Regression Models for the Entire Follow-Up Period
Factor/categoryCasesDeathsUnivariate HRMultivariate HR95% CIP value for trend
  1. HR: hazard ratio; 95% CI: 95% confidence interval.

Grade      
 1454491.001.00 
 27421712.381.62(1.17–2.24) 
 37923285.142.53(1.85–3.46)< 0.001
Lymph node status      
 Negative13072001.001.00 
 Positive6162843.972.40(1.98–2.92) 
 Metastases656453.2422.73(16.05–32.18)< 0.001
Size (mm)      
 1–9298251.001.00 
 10–14442611.651.21(0.76–1.94) 
 15–19407872.781.66(1.05–2.62) 
 20–294701584.962.48(1.59–3.85) 
 30–492541309.423.83(2.43–6.03) 
 50+1178718.824.64(2.85–7.57)< 0.001

Table 2 shows the univariate effects of dichotomized random variables by 5-year epochs. Here, one can see the effects progressively attenuating with time from the time of diagnosis. However, even 15 years after diagnosis, these factors still appeared to have some effect. The largest differences were noted between the first 5-year epoch and the subsequent epochs. This suggests an element of frailty in that a subgroup of individuals (including those with large, poorly differentiated tumors that have metastasized) have a particularly high death rate within the first few years. Thereafter, the change in the estimated hazard ratio is less pronounced but nonetheless suggests that the covariate effects are not constant over time.

Table 2. Estimated Hazard Ratios and 95% CIs from the Multivariate Cox Regression Models for the Entire Follow-Up Period Split into 5-Year Epochs
Factor/categoryQuantityEpoch (yrs)
0–4.95–9.910–14.915+
  • 95% CIs: 95% confidence intervals; HR: hazard ratio.

  • a

    Includes distant metastases.

Grade 3HR4.46 (3.53–5.63)1.79 (1.30–2.47)1.60 (1.00–2.55)1.82 (0.79–4.21)
 P value< 0.001< 0.0010.0480.163
Lymph node positiveaHR7.88 (6.12–10.16)3.47 (2.55–4.71)2.40 (1.49–3.89)1.84 (0.74–4.56)
 P value< 0.001< 0.001< 0.0010.188
Size ≥ 20 mmHR5.88 (4.58–7.55)3.41 (2.50–4.64)2.20 (1.41–3.42)2.05 (0.88–4.79)
 P value< 0.001< 0.0010.0010.098

Table 3 shows the results for the Cox regression model with the time-varying regression coefficients. In this model, the effects of tumor grade and lymph node status were allowed to attenuate over time at the rate of e-λt with λ = 0.15. The effect of size was maintained as a constant in this model, partly because the results in Table 2 showed a smaller proportional reduction in the size effect in the first 5 years than was observed for lymph node status and tumor grade, and partly to avoid the problem of overcorrection for dampening because of the collinearity of the three factors: tumor size, tumor grade, and lymph node status. The decision to set λ equal to 0.15 is arbitrary, but was guided by a combination of clinical and statistical criteria. Of primary clinical concern was the question of whether the prognostic factors, which are highly influential in the first few years after diagnosis, continue to matter after long-term survival (e.g., > 10 years) has been achieved. Potentially appropriate choices for λ were identified by tabulating the function e-λt across a range of values (λ = 0.1, 0.15, 0.20, 0.25, 0.5, and 0 ≤ t ≤ 20) and selecting those that give a steady attenuation over the first 10 years with a small effect thereafter as suggested by the literature.10, 11 This gave a suggested range of 0.1–0.5. Cox regression models then were fitted for several values of λ in this region and the final value of λ was chosen to maximize the profile likelihood. The hazard ratios relating to tumor grade and lymph node status (shown in Table 3) relate to time 0; the respective hazard ratios at time t are given by the expression exp[log{HR at time 0}e-λt]. Thus, the relative hazard for an individual with Grade 3 disease at the time of diagnosis (compared with a similar individual with ICDO Grade 1 disease at the time of diagnosis) was reduced from 9.93 postoperatively to just 1.67 after 10 years.

Table 3. Estimated Hazard Ratios from the Cox Regression Model with Time-Varying Coefficients and Dampening Parameter, λ = 0.15
Factor/categoryMultivariate HR (at time 0)95% CIP value for trend
  1. HR: hazard ratio; 95% CI: 95% confidence interval.

Grade   
 11.00 
 23.88(1.78–8.46) 
 39.93(4.64–21.23)< 0.001
Lymph node status   
 Negative1.00 
 Positive6.28(4.25–9.28) 
 Metastases79.60(47.36–133.80)< 0.001
Tumor size (mm)   
 0–91.00 
 10–141.25(0.78–1.99) 
 15–191.77(1.13–2.78) 
 20–292.58(1.67–3.99) 
 30–493.92(2.50–6.14) 
 50+4.69(2.89–7.61)< 0.001

DISCUSSION

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION
  6. REFERENCES

The results of the current study suggest very strongly that the effects of the main prognostic factors in women with breast carcinoma change over time. It therefore is clear that, when analyzing long-term survival, a model that allows nonproportional hazards should be chosen over classic Cox regression time-stratified analysis. The introduction of time-varying coefficients, or the use of frailty terms in the Cox regression model (or its discrete time equivalent, the complementary log-log model) are approaches that can be used to relax the proportionality assumption.

Time-dependent effects take on importance when long-term survival is studied because a natural selection process is taking place; that is to say that the subgroup of patients who achieves long-term survival is by definition different from the group initially studied at the time of diagnosis. The estimated relative hazard ratios from the Cox models change over the 5-year epochs at least in part because the group of individuals on which the estimate is based changes. This may be because of interactions with unobserved covariates such as vascular involvement or DNA ploidy. It also is likely to be related to residual effects within categories of the known covariates. For example, those patients with lymph node involvement who die earlier may be for the most part patients with large numbers of positive lymph nodes. After these subjects have died, the average number of lymph nodes involved in the lymph node-positive cases will be smaller, and hence the observed effect of being lymph node positive on the hazard may be similarly reduced.

To model such heterogeneity more directly, we fitted explicit frailty models using a complementary log-log function with gamma-distributed frailty. Our results demonstrated a significant heterogeneity (substantial variation around the “average” hazards by tumor size, lymph node status, and tumor grade) when no dampening parameter was included. However, when the time-varying covariate model was fitted, with a parameter of 0.15, the heterogeneity was no longer significant. This suggests that the time-varying model adequately describes the heterogeneity in survival.

In the context of long-term survival from breast carcinoma, it is clear that the results obtained from the Cox model with time-varying coefficients should be of the most interest. Diagnostic plots of the Cox-Snell residuals for Models 1 and 2 are shown in Figure 1. The 45-degree reference line shows what this plot would look like if the model fitted the data perfectly. The line relating to the model with time-varying coefficients (which allows the effect of tumor grade and lymph node status to diminish with time) clearly fits better than the model in which the relevant coefficients are independent of time. Utilizing this model, the estimated hazard ratios for tumor grade and lymph node status at the time of diagnosis and at 5-year intervals thereafter are presented in Table 4. The related 95% confidence intervals may be calculated by applying the formula provided earlier to the hazard ratios at time 0 as reported in Table 3. For clarity, only the 95% confidence intervals at 20 years are reported. By this time, the relative hazards are only a little larger than 1, but in all cases the 95% confidence intervals exclude 1, so the supposition that tumor grade and lymph node status no longer influence survival at this time is clearly refuted.

thumbnail image

Figure 1. Diagnostic plots of Cox-Snell residuals versus the cumulative hazard function for the Cox regression models. Solid line: 45° reference line, representing the perfect model fit; dashed line: classic Cox regression model; thickly dashed line: Cox regression model with time-varying coefficients.

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Table 4. Estimated Hazard Ratios (at the Time of Diagnosis and at 5-Year Intervals Thereafter) from the Multivariate Cox Regression Model with Time-Varying Coefficients
Factor/categoryTime (yrs)
Diagnosis5101520
HRHRHRHRHR (95% CI)
  1. HR: hazard ratio; 95% CI: 95% confidence interval.

Grade     
 11.001.001.001.001.00
 23.881.901.351.151.07 (1.03–1.11)
 39.932.961.671.271.12 (1.08–1.16)
Lymph node status     
 Negative1.001.001.001.001.00
 Positive6.282.381.511.211.10 (1.07–1.12)
 Metastases79.607.902.661.581.25 (1.21–1.28)

In a similar study of long-term survival in breast carcinoma patients in Norway, Zahl and Tretli16 used an excess hazards model with time-varying coefficients to demonstrate that age and clinical disease stage were significant long-term prognostic factors. Clinical disease stage may be viewed as a surrogate for the combined information captured by lymph node status and tumor size so there is no conflict with the results of the current study. The difference with regard to the importance of age, which did not appear to be significant in the current study results, is less easily explained but may be due to different epochs of diagnosis and therefore different treatment practices.

The results of the current study indicate that the value of tumor grade, lymph node status, and tumor size at the time of diagnosis have a lasting influence on subsequent survival, albeit attenuated in later years. Twenty years after diagnosis, a woman who had Grade 1 disease at the time of diagnosis still has a significantly lower risk of dying from breast carcinoma compared with her counterpart who was diagnosed with Grade 2 disease. This suggests two tentative conclusions: first that the pathologic features of the tumor at the time of diagnosis may have implications for long-term follow-up and second that the long-term effects of the prognostic factors may explain the finding that the impact of mass screening programs on breast carcinoma mortality rates remains apparent many years later.17

REFERENCES

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION
  6. REFERENCES