Table 2 

Microsimulation steps for estimating the cost of productivity losses due to CFS. 

Step 
Equation 


1. Divide the study sample data into two groups: Fatigued (F), N = 3,528, and NonFatigued (NF), N = 3,634. Estimate logistic regressions to obtain the probability that an individual is employed for F, P[W = Y/F], and NF, P[W = Y/NF], subsamples, as a function of human capital characteristics, as displayed in equations 1a and 1b. 

a. P[W = Y/F] = f(age, sex, ethnicity, education, suspected CFS, CFS, history of diseases that exclude respondent from CFS diagnosis) 

b. P[W = Y/NF] = f(age, sex, ethnicity, education, history of diseases that exclude respondent from CFS diagnosis) 

2. Estimate ordinary least squares regressions to predict the natural log of income (I) given employment for the F, E[I/W = Y,F], and NF, E[I/W = Y,NF], subsamples, as a function of human capital characteristics, as displayed in equations 2a and 2b. 

a. E[I/W = Y,F] = f(age, sex, ethnicity, education, suspected CFS, CFS, history of diseases that exclude respondent from CFS diagnosis, occupation) 

b. E[I/W = Y,NF] = f(age, sex, ethnicity, education, history of diseases that exclude respondent from CFS diagnosis, occupation) 

Note: Once these regressions are estimated, only the sample of 43 individuals with CFS are used for the remainder of the microsimulation to estimate mean labor force and household productivity by age and sex in the presence and absence of CFS. 

3. Calculate predicted mean F and NF employment rates by age and sex categories, weighting by sampling weights. Multiply the coefficient estimates from the F (Blf) and NF (Blnf) Logit regressions by the human capital characteristics of the 43 individuals with CFS (X) to obtain their F (P[W = Y/F]) and NF (P[W = Y/NF) employment rates, respectively, as shown in equations 3a and 3b. Then, calculate the mean employment rate across the 43 individuals with CFS for each age and sex category weighting these means to reflect the survey sampling rates. 

a. P[W = Y/F] = {exp(X*Blf)/1+exp(X*Blf) 

b. P[W = Y/NF] = {exp(X*Blnf)/1+exp(X*Blnf) 

4. Calculate predicted mean F and NF income given employment by age and sex categories weighting by sampling weight. Multiply the coefficient estimates from the F (Bolsf) and NF (Bolsnf) OLS income regressions by the human capital characteristics of the 43 individuals with CFS (X) to obtain their F (E[I/W = Y,F]) and NF (E[I/W = Y,NF) income given employment, respectively. Then, apply the smearing adjustment to the exponent of these F and NF products, as shown in equations 4a and 4b, to correct for the "retransformation" bias that arises from estimating impacts using loglinear models and to protect against data issues such as heteroskedasticity^{1}. The smearing factors for the regressions among individuals with F (Sf) and in the absence of F (Snf) are equal to the means of the antilogs of the residuals of the respective income regressions. Calculate predicted F and NF income given employment for each age and sex category weighting by the survey sampling weights, and adjust these means from 1997 to 2002 dollars to account for inflation using the Department of Labor, Bureau of Labor Statistics Consumer Price Index from 1997 to 2002^{2}. Apply an adjustment factor for the difference between mean income in Wichita and the nation based on analysis by the U.S. Department of Commerce^{3 }increasing the estimated losses by 1.3 percent. In addition, to account for fringe benefits, multiply predicted income by a factor of 1.338, which is obtained from the Bureau of Labor Statistics Report on Employer Costs for Employee Compensation – June 2002^{4}. 

a. E[I/W = Y,F] = {exp(X*Bolsf)*Sf}*1.114*1.013*1.338 

b. E[I/W = Y,NF] = {exp(X*Bolsnf)*Snf}*1.114*1.013*1.338 

5. Calculate predicted household productivity given employment and no employment in absence of F. The value of household productivity by sex, age, and employment status absent F is calculated on the basis of data on the number of hours spent on household chores for the NF sample, given employment (HH hours/W = Y,NF) and no employment (hours/W = N, NF). Value these hours at the average hourly wage for a service industry worker as estimated on the basis of the March Supplement of the Current Population Survey 2002 or $9.20. Similar to employment income, increase the value of the service industry worker wage by a factor of 1.338 to account for the value fringe benefits. This equation is displayed in 5a and 5b. 

a. E[HH/W = Y,NF] = E[HH Hours/W = Y, NF]*$9.20*1.338 

b. E[HH/W = N,NF] = E[HH Hours/W = N, NF]*$9.20*1.338 

6. Calculate predicted household productivity given F. Assume that the percentage reduction in employment related income, given work, is equal to the percentage reduction in household productivity. Apply a reduction factor representing the estimated reduction in employmentrelated income, given work, resulting from CFS to the predicted values of household productivity, given employment and no employment, as displayed in 6a and 6b. Calculate reduction factors separately for males and females. 

a. E[HH/W = Y,F] = E[HH/W = Y,NF] * E[I/W = Y,F]/E[I/W = Y,NF] 

b. E[HH/W = N,F] = E[HH/W = N,NF] * E[I/W = Y,F]/E[I/W = Y,NF] 

7. Calculate predicted mean F and NF total productivity for each CFS individual. Overall, each CFS individual's expected total productivity in the presence or absence of F, E[Y/F] or E[Y/NF] respectively, is equal to the probability that they participate in the labor force, P[W = Y/F] or P[W = Y/NF], times the expected value of their total labor force and household productivity if they participate in the labor force plus the probability they choose not to participate in the labor force, P[W = N/F] or P[W = N/NF], times the expected value of their household productivity when they do not participate in the labor force. Equations 7a and 7b display expected productivity. 

a. E[Y/F] = P[W = Y/F]{E[I/W = Y,F] + E[HH/W = Y,F]} + P(W = N/F) {E[I/W = N,F] + E[HH/W = N,F]} 

b. E[Y/NF] = P[W = Y/NF]{E[I/W = Y,NF] + E[HH/W = Y,NF]} + P(W = N/NF) {E[I/W = N,NF] + E[HH/W = N,NF]} 

8. Calculate estimated number of individuals with CFS nationally by age and sex. Using the Wichita Prevalence Study data, calculate the prevalence of CFS per 100,000 by age and sex cells and then use national population data from the Current Population Survey to calculate the number of individuals in each age and sex category with CFS. 

9. Calculate individual and societal productivity losses due to CFS. Compute the difference between predicted mean total productivity without and with F, (E[Y/NF]E[Y/F]), by age and sex category to estimate the individual loss for each age and sex cell and then multiply these differences for each sex and age cell by the estimated by number of individuals with CFS nationally in each cell and sum across the cells to estimate the total societal cost of lost productivity due to CFS. 



^{1}Duan N. Smearing Estimates: A nonparametric retransformation technique. J Am Stat Assoc 1983,383:605–10. ^{2}Consumer Price Index from 1997 to 2002. Department of Labor, Bureau of Labor Statistics. (http://data.bls.gov/cgibin/surveymost?cu webcite, then select U.S. All items, 198284=100 CUUR0000SA0) ^{3}Per capita net earnings ($) Metro Comparisons. Department of Economics, Iowa State University, Midwest Profiles, Public Resources Online. (http://www.bea.doc.gov/bea/regional/reis/ webcite, then select Personal income and population summary estimates (CA13) plus per capita personal income plus Metropolitan Statistical Areas*)^{4}Employer Costs for Employee Compensation – June 2002. Bureau of Labor Statistics, September 2002. (http://http//data.bls.gov/cgibin/surveymost?cc webcite, then click on Civilian, All workers, Total compensation  CCU110000100000D) 

Reynolds et al. Cost Effectiveness and Resource Allocation 2004 2:4 doi:10.1186/1478754724 