Investigating confounders of the association between survival and adjuvant radiation therapy after breast conserving surgery in a sample of elderly breast Cancer patients in Appalachia

Background To explain the association between adjuvant radiation therapy after breast conserving surgery (BCS RT) and overall survival (OS) by quantifying bias due to confounding in a sample of elderly breast cancer beneficiaries in a multi-state region of Appalachia. Methods We used Medicare claims linked registry data for fee-for-service beneficiaries with AJCC stage I-III, treated with BCS, and diagnosed from 2006 to 2008 in Appalachian counties of Kentucky, Ohio, North Carolina, and Pennsylvania. Confounders of BCS RT included age, rurality, regional SES, access to radiation facilities, marital status, Charlson comorbidity, Medicaid dual status, institutionalization, tumor characteristics, and surgical facility characteristics. Adjusted percent change in expected survival by BCS RT was examined using Accelerated Failure Time (AFT) models. Confounding bias was assessed by comparing effects between adjusted and partially adjusted associations using a fully specified structural model. Results The final sample had 2675 beneficiaries with mean age of 75, with 81% 5-year survival from diagnosis. Unadjusted percentage increase in expected survival was 2.75 times greater in the RT group vs. non-RT group, with 5-year survival of 85% vs 60%; fully adjusted percentage increase was 1.70 times greater, with 5-year rates of 83% vs 71%. Quantification of incremental confounding showed age accounted for 71% of the effect reduction, followed by tumor features (12%), comorbidity (10%), dual status(10%), and institutionalization (8%). Adjusting for age and tumor features only resulted in only 4% bias from fully adjusted percent change (70% change vs 66%). Conclusion Quantification of confounding aids in determining covariates to adjust for and in interpreting raw associations. Substantial confounding was present (60% of total association), with age accounting for the largest share (71%); adjusting for age plus tumor features corrected for most of the confounding (4% bias). The direct effect of BCS RT on OS accounted for 40% of the total association.


Quantification of Confounding Bias
Using the well-established potential outcomes framework popularized by Rubin, we seek to estimate a causal effect of treatment on survival time T by comparing expected survival times E[T (0)] and E[T (1)] over the population, where T (x) represents a patient's survival time if treatment is received (x = 1) and not received (x = 0).
Firstly, a structural model for the survival distribution was determined by an AFT model ('fully adjusted') according to the following equation: where βZ is a linear combination of covariates Z 1 , Z 2 , ..., Z n and β 1 is a linear combination of Z, allowing for effect modification of Z. Under the consistency assumption, Under the conditional ignorability assumption that implies no residual confounding is present after accounting for Z, T (x)|Z independent of X|Z. It follows In order to estimate E[T (x)], each observation in the data set was assigned two scores for E[T |X = x, Z 1 , ..., Z n ], setting X to 0 and 1 respectively. These scores were calculated using parameter estimates from the structural model. Scoring procedures are available in several statistical packages. For example, in SAS, proc lifereg output statement can output quantile estimates such as the median survival time for each observation. For log-logistic distributions, the median is proportional to the expected value, where the constant of proportionality depends on σ. Then the following approximation was used: , taking the average across the entire sample. The population effect in terms of percent change, The problem of incremental confounding, which allows for decomposition of the bias into contributions by each confounder, if discussed in [1] and covers two methods. Using the first method, the marginal association removing influence of confounders is estimated, say Z 1 and Z 2 for simplicity (δ * Z1Z2 ). This association has the same formula as δ, except the expected values are based on In order estimate these expected values, all the observations in the sample were scored twice with E[T |X, Z 1 , Z 2 ], setting X to 0 and 1. These expected values were calculated by using partially adjusted models following the following equations: ln T =β 0 + β 1 (Z j )X+βZ j +σ , where Z j is a subset of Z; specifically for the above example, Z j = Z 1 ,Z 2 . Interactions between Z j and β 1 were retained if included in the full model.
Sample averages were then used to approximate E[E[T |X, Z 1 , Z 2 ]] and δ *

Z1Z2
. Incremental confounding was then assessed by the difference δ * Z1Z2 -δ * Z1 and interpreted as the change to confounding bias by accounting jointly for Z 1 , Z 2 compared to Z 1 alone. Overall bias ∆ can be decomposed into a sum of incremental differences, where each difference is derived by successively augmenting the list of confounders (i.e δ * Z1Z2Z3 -δ * Z1Z2 ). Under the second method, the expected conditional association removing the influence of successive confounders is estimated, for example E[δ * Z1Z2 |Z 1 ], where δ * Z1Z2 |Z 1 has the same formula as δ except the expected values are based on ] is compared to the expected conditional association E[δ * Z1 |Z 1 ] to assess the amount of confounding bias removed by Z 2 conditional on Z 1 .
When the effect measure (δ) is a difference between potential outcomes of X =1 and 0 both methods yield equivalent incremental differences [1] . However, in general this is not the case, such as in this presentation, where the effect measure is a ratio. A decomposition of confounding bias using the second method is in terms of incremental differences is not easily derivable in our opinion. For this reason, the first method was applied instead.

Inclusion Criteria
Diagnostically confirmed female breast cancers: N 29,372 Continuously enrolled fee for service 1 year before/after diagnosis in Medicare: N 7,686 First tumor: N 7,470 No multiple or concurrent solid tumors: N 6,592 Survived 1 year after diagnosis: N 6,549 AJCC stage I,II,III: N 4,893 BCS surgery ¡ 6 months from diagnosis: N 2,675