## Appendix A Technical Estimation Details-Free PDF

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Estimation of Breast Cancer Incident Cases and Medical Care Costs Attributable to Alcohol Consumption. Among Insured Women Aged 45 Years in the U S,Ekwueme et al. where denotes the probability that X 0 i e the probability of being a non drinker Since. RR 0 1 and because we can interchange integration and summation operations substituting Eq. A1 2 into Eq A1 1 results in the following alternative formula. 1 0 RR x fX X 0 x X 0 dx 1,1 0 RR x fX X 0 x X 0 dx. Eq A1 3 is almost equivalent to the formula in Rehm et al 1 except that it makes clear that the. continuous portion of the alcohol exposure distribution needs to be weighted by the probability. that X 0 i e 1 so that the overall exposure distribution integrates one. A2 Quadrature, To approximate the integral in Eq A1 3 we used Gauss Legendre quadrature after applying a. change of variables to transform the range of integration from 0 to 1 1 We found this. approach to have better numerical stability than using Gauss Laguerre quadrature after. normalizing by exp x The following change of variables changes the bounds of integration as. Note that as t approaches 1 x approaches infinity Also as t approaches 1 x approaches 0. Accordingly after the change of variables in Eq A2 1 the integral in Eq A1 3 can be. rewritten as follows,1 t 1 t dt,RR x f x dx 2 RR f. 0 1 1 t 1 t 1 t 2,1 1 t 1 t dt,2 1 RR 1 t f 1 t 1 t 2.
American Journal of Preventive Medicine, Estimation of Breast Cancer Incident Cases and Medical Care Costs Attributable to Alcohol Consumption. Among Insured Women Aged 45 Years in the U S,Ekwueme et al. Equation A2 2 can be approximated as follows,1 1 t 1 t dt 1 x 1 x wi. 2 1 RR 1 t f 1 t 1 t 2,2 ni 1 RR 1 xi f 1 xi 1 xi 2. where xi and wi are the nodes and weights for Gauss Legendre quadrature. A3 Accounting for Underreporting, As mentioned in the paper we estimate subpopulation specific AAFs for each subpopulation.
defined by age 18 25 years 26 34 years 35 49 years and 50 years race ethnicity white. non Hispanic black non Hispanic other non Hispanic and Hispanic and insurance. Medicaid private and other no insurance We modeled the alcohol consumption distributions. within each of these subpopulations using the best fitting parametric distribution as determined. by Kolmogorov Smirnov testing, In this section we present additional details on how distributions generalized gamma gamma. Weibull and lognormal were adjusted separately We concluded the section by presenting a. detailed but simplified numerical example of shifting the overall mixture distribution that. utilizes more than one statistical distribution to model alcohol consumption across demographic. sub strata e g females aged 21 25 years 26 34 years etc All of these shifting procedures. depend on estimating a multiplier m which compares the overall survey average alcohol. consumed with an estimate of the per capita volume of alcohol sold in the U S Specifically we. m Per capita sales Average alcohol consumed A3 1,American Journal of Preventive Medicine. Estimation of Breast Cancer Incident Cases and Medical Care Costs Attributable to Alcohol Consumption. Among Insured Women Aged 45 Years in the U S,Ekwueme et al. Generalized Gamma, The PDF for the three parameter generalized gamma distribution can be written as. fX x exp A3 2, where and are shape parameters and is the scale parameter Mittelhammer 2 denotes.
the gamma function which is defined as,0 x 1 e x dx A3 3. We have found an alternative parameterization of the generalized gamma distribution presented. in Manning et al 3 to have smoother convergence properties The parameterization in Manning et. fX x exp z u A3 4, where abs 2 z sign log x and u exp abs z There is no explicit scale. parameter in the parameterization in Eq A3 4 However Manning et al 3 showed that the scale. parameter in Eq A3 2 can be written as a function of the parameters in Eq A3 3 In. particular, Thus one can estimate the parameters of Eq A3 4 and adjust the scale by adding the logarithm. of the multiplier i e Eq A3 1 to the parameter which is equivalent to multiplying by the. multiplier In summary the two steps required to adjust the generalized gamma distribution are. 1 Estimate the generalized gamma parameters in Eq A3 4 using the observed survey. 2 Adjust the parameter by adding the logarithm of the multiplier defined in Eq A3 1. American Journal of Preventive Medicine, Estimation of Breast Cancer Incident Cases and Medical Care Costs Attributable to Alcohol Consumption. Among Insured Women Aged 45 Years in the U S,Ekwueme et al.
The PDF for the standard gamma distribution can be written as. fX x x 1 exp A3 6, where is a shape parameter and is the scale parameter Since the parameterization in Eq. A3 6 includes a single scale parameter the gamma distribution can be adjusted by multiplying. the scale parameter by the multiplier in Eq A3 1 In summary the two steps required to adjust. the gamma distribution are, 1 Estimate the gamma parameters in Eq A3 6 using the observed survey data. 2 Adjust the parameter by multiplying by the multiplier defined in Eq A3 1. The PDF for the Weibull distribution can be written as. fX x exp A3 7, where is a shape parameter and is the scale parameter Since the parameterization in Eq. A3 7 includes a single scale parameter the Weibull distribution can be adjusted by multiplying. the scale parameter by the multiplier in Eq A3 1 In summary the two steps required to adjust. the Weibull distribution are, 3 Estimate the Weibull parameters in Eq A3 7 using the observed survey data. 4 Adjust the parameter by multiplying by the multiplier defined in Eq A3 1. In the latter three cases we adjust the distribution by multiplying the scale parameter by the. multiplier defined in Eq A3 1 The lognormal distribution is slightly different To see that a. American Journal of Preventive Medicine, Estimation of Breast Cancer Incident Cases and Medical Care Costs Attributable to Alcohol Consumption.
Among Insured Women Aged 45 Years in the U S,Ekwueme et al. lognormally distributed random variable can be thought of as a transformation of a normally. distributed random variable If X is lognormally distributed with parameters and then. X exp Y where Y N 2 A3 9, Accordingly X m exp Y log m for any constant m Thus mis measured and accurately. measured alcohol consumption can be related in terms of a normally distributed random variable. Y as follows,Y log m Y A3 10, Eq A3 10 implies that Y is normally distributed with mean given by log m and variance. given by 2 Combining this with Eq A3 9 it follows that a lognormally distributed random. variable can be adjusted by adding the logarithm of the multiplier defined in Eq A3 1 to the. location parameter rather than the scale parameter as we have done in all cases above In. summary the two steps required to adjust the lognormal distribution are. 1 Estimate the lognormal parameters using the observed survey data. 2 Adjust the parameter by adding the logarithm of the multiplier defined in Eq A3 1. A4 Details Around Monte Carlo Simulations, While Delta Method or bootstrapped SEs could be used here we used a Monte Carlo approach to. assess uncertainty around our point estimates i e AAFs attributable cases and attributable. medical care costs In a Monte Carlo approach one simulates a distribution for all of the. uncertain components within each calculation and then repeats the calculation R times to. simulate a distribution of estimates The 2 5th and 97 5th percentiles provide the lower and upper. bounds of a 95 CI respectively We used R 1 000 Monte Carlo repetitions. American Journal of Preventive Medicine, Estimation of Breast Cancer Incident Cases and Medical Care Costs Attributable to Alcohol Consumption.
Among Insured Women Aged 45 Years in the U S,Ekwueme et al. The uncertain components include, 1 The parameters of the alcohol exposure distribution including the probability of. being a non drinker, 2 The parameters of the dose response function RR x and. 3 The incremental medical care cost estimates, The total number of breast cancer cases are not estimated but represent a complete census of. cases that occurred in 2013 The parameters of the alcohol exposure distribution include and. the parameters of the continuous PDF used to model alcohol exposure among drinkers We used. the proportion of respondents who are non drinkers to estimate The beta distribution is an. appropriate statistical distribution to simulate proportions We used the weighted number of non. drinkers as the first shape parameter and the number of drinkers as the second shape parameter. to generate pseudo random beta variates The parameters of the continuous PDF used to model. alcohol consumption among drinkers are estimated using maximum likelihood models Under. standard assumptions these parameters are asymptotically normal Thus we used estimates of the. distributional parameters and associated variance covariance matrix to generate pseudo random. multivariate normal variates Similarly the dose response function parameter was estimated. using a generalized least squares trend estimator Under standard assumptions the dose response. function parameter is also asymptotically normal Accordingly we used the parameter estimate. and its associated variance estimate to generate pseudo random normal variates Finally cost. estimates are nonnegative and potentially skewed A typical distributional assumption for. medical costs is the gamma distribution Appendix Table 3 summarizes this information. American Journal of Preventive Medicine, Estimation of Breast Cancer Incident Cases and Medical Care Costs Attributable to Alcohol Consumption.
Among Insured Women Aged 45 Years in the U S,Ekwueme et al. Appendix Table 1 Average Number of Breast Cancer Cases From the U S Cancer Statistics. Database by SEER Summary Stage and Age 2012 2013a b. SEER summary stagec, Breast cancer incident cases Localized Regional Distant Overall. 18 44 years 11 448 8 952 1 281 21 681,45 64 years 62 964 30 373 5 786 99 123. Total 74 412 39 325 7 067 120 802, Data are from selected statewide and metropolitan area cancer registries that meet the data. quality criteria for all invasive cancer sites combined. Source 2012 data U S Cancer Statistics Working Group United States cancer statistics 1999. 2012 Incidence and mortality web based report Atlanta GA U S DHHS CDC National. Cancer Institute 2015 2013 data U S Cancer Statistics Working Group United States cancer. statistics 1999 2013 Incidence and mortality web based report Atlanta GA U S DHHS. CDC National Cancer Institute 2016,http seer cancer gov tools ssm.
Appendix Table 2 Incremental Annual Medical Care Costs by Stage of Cancer at 12 months. for Medicaid Beneficiaries and for Women With Private Health Insurancea. Younger women aged 18 44 Older women aged 45 64 years. Insurance Incremental 95 CI Incremental 95 CI,type breast cost cost. cancer stage estimate estimate,12 months 12 months. Localized 46 616 43 394 49 837 28 674 27 122 30 226. Regional 59 431 56 603 62 260 45 288 43 265 47 311. Distant 93 471 83 203 103 739 62 868 57 464 68 271. All stages 66 596 63 551 69 641 45 914 57 464 68 271. Private insurance, Localized 79 432 74 885 83 980 59 719 57 910 61 529. Regional 115 416 110 034 120 798 104 749 101 431 108 067. Distant 142 797 117 509 168 084 126 691 111 529 141 853. All stages 97 299 93 443 101 155 75 667 73 893 77 441. Medical care costs were adjusted to 2014 dollars using a gross domestic product deflator 4. Appendix Table 3 Uncertain Parameters and the Statistical Distribution Assumed in the Monte. Carlo Analysis of Uncertainty,Uncertain parameter s Statistical distribution. Probability of being a non drinker Beta, Prevalence distribution parameters Multivariate normal.
Dose response parameter Normal,Incremental medical care cost estimates Gamma. American Journal of Preventive Medicine, Estimation of Breast Cancer Incident Cases and Medical Care Costs Attributable to Alcohol Consumption. Among Insured Women Aged 45 Years in the U S,Ekwueme et al. Supplementary Tables, In this section we present supplementary result tables used as inputs into our calculations. Specifically we present 1 the dose response modeling results 2 results determining the best. fitting distribution for alcohol consumption among each modeled subpopulation as well as model. predicted average daily consumption before and after adjusting for underreporting 3 substrata. specific AAFs 4 the total number of breast cancer incident cases obtained from the U S. Cancer Statistics database and 5 estimated incremental medical care costs by stage of cancer at. 12 months for women who are Medicaid beneficiaries and those with private health insurance. The dose response modeling results can be summarized as follows. log 0 0097784,American Journal of Preventive Medicine.
Estimation of Breast Cancer Incident Cases and Medical Care Costs Attributable to Alcohol Consumption. Among Insured Women Aged 45 Years in the U S,Ekwueme et al. Appendix Table 4 Best fitting Statistical Distributions for Alcohol Consumption Model by. Race Ethnicity Age Group and Insurance Status,predicted Model. average predicted,before average after,Stratum Model choice adjustment a adjustment a. Female white non Hispanic 18 25 years Medicaid Generalized gamma 10 82 24 85. fraction AAF of female breast cancer cases in which we separate drinkers from non drinkers in the probability distribution over alcohol consumption Second we describe the quadrature routine that was used to approximate the integrals in our AAF formula Third we