Comparing Two Models for Beryllium Benefits 

The purpose of this paper is to compare the results of the quartile model OSHA used in its benefits analysis in the preliminary economic impact analysis to the results of the logistic regression analysis presented in Table 14 of the preliminary risk analysis.  
To achieve this comparison, OSHA separately determines the prevalence of CBD cases under the quartile model and the logistic model, then converts the prevalence rate into an exposure-response incidence rate, as described in more detail in Chapter VII (Benefits), and then applies that incidence rate in the lifetime risk model used to calculate benefits.  
By comparing the different outcomes of the lifetime risk model based on the different incident rate inputs produced by the quartile and logistical models, OSHA provides a further check on the reasonableness of its preferred quartile model.  The summary conclusion for this examination of the logistic model is that even taking into account the uncertainty represented by the 95% confidence interval on the slope coefficient  of the cumulative exposure variable does not change the qualitative result that economic benefits vastly outweigh the costs of the proposed standard.

Background 
All of the models OSHA will examine used the data set from the beryllium machining plant in Cullman, Alabama, to create empirical estimates of beryllium sensitization and CBD prevalence for its exposure response model, which translates beryllium exposure to risk of adverse health endpoints for the purpose of determining the benefits that could be achieved by preventing those adverse health endpoints.  This data set had the advantage of highly accurate historical exposure assessment available for a wide range of cumulative exposures, average exposures, and years of exposure, allowing for greater accuracy in calculating the risk rate for adverse health endpoints, and thus the benefits of preventing them.  
The study contains results for 319 employees, all of whom were exposed for 33 years or less, so even one case would represent a risk in excess of one in a thousand over a forty five year life time, which would satisfy OSHA's legal requirement to show that the health risk is "significant."  Even before a specific exposure-response model is estimated, the raw data show cases of CBD with cumulative exposures that would represent an average exposure level of less than 0.1 ug/m[3] per day if exposed for ten years; show cases of CBD with exposures lasting less than one year; and show cases of CBD with actual average daily exposure of less than 0.1 ug/m[3].  Thus, OSHA has strong reasons to conclude that significant risks would result from exposures at the current and proposed PELs, and benefits could be achieved by reducing employee's exposures from their current levels, without any extrapolation to lower exposure levels than found in the raw data.  Nevertheless, OSHA is aware that the extrapolations may be less reliable, particularly at very low and high exposure levels, because of certain characteristics of the exposure found in the data set.  There are levels below which no cases of CBD were found.  The lowest cumulative exposure that results in cases of CBD is 0.118 ug/m[3]exposure years, equivalent to an average exposure of 0.01 ug/m[3] per day over ten years or an average exposure of 0.003 ug/m[3] per day over 45 years.  There are 70 cases in the Cullman database with aggregate exposure less than this.  Conversely the same database contains very few cases of exposure at very high levels such as might be experienced by a worker with 45-year exposure at 1.0 ug/m[3] per day or more.  The highest exposure resulting in CBD in the data base is 61.86 exposure years, equivalent to an average exposure of approximately 2 ug/m[3] per day over thirty years or 1.37 ug/m[3]over 45 years.
All of the models examined used the version with cumulative exposure as the independent variable.
As explained in the revised Chapter VII, the estimation of the quantitative effects on morbidity and mortality proceed in two stages.  The first stage estimates the numbers of diseases and deaths prevented by comparing the current (baseline) situation to a world in which the proposed PEL is adopted and followed by everyone throughout their lives.  For this stage, OSHA converts the prevalence estimates to incidence estimates and then uses a standard life table to determine the number of cases of beryllium disease that can be expected with and without the proposed rule. The second stage also assumes that the proposed PEL is adopted, but uses the results from the first stage to estimate what would happen under a more realistic scenario in which employees have been exposed for varying periods of time to the baseline situation and will thereafter be exposed to the new PEL.  
In addition, OSHA developed separate estimates of the effects of the ancillary provisions and combined this estimate with the estimates of the benefits of the PEL to develop its estimate of the total benefits of the proposed standard.  In one sector, dental laboratories, OSHA estimated that some employers would substitute away from beryllium alloys.  This effect was placed under airborne- only elements, though this effect is the result of both the PEL and the ancillary provisions. 
Quartile Model 
The quartile model groups exposure data into broad groups.  Given the broad grouping of the raw data and the estimates needed, this approach is particularly prone to mis-estimating the effects in two ways.  First, the very low levels of cumulative exposure could result in an overestimate because the entire first quartile is lumped together.  Second, the approach potentially underestimates the effects of very high levels of exposure, since the highest exposures for which estimates are needed are higher than any exposure in the data set.  
OSHA for its primary estimate used a 45 year working life, and the full life risk model includes estimates of how to convert prevalence to incidence data.  To examine the importance of these aspects of the approach, OSHA, for this paper, also examined a simpler model.  Perhaps the simplest approach to estimating the number of cases of CBD prevented when there is full compliance with the PEL would be to look at the Quartile Model and match up the risk projected by the cumulative exposure prevalence estimates to the exposed population.  After 15 years of cumulative exposure, the entire exposed population is estimated to fall in the top two quartiles of the model, having a prevalence rate of 8% or higher, and cause 2,843 cases of CBD.  If one ignores turnover, this results in an average of 190 cases per year.  Using the estimated incidence rate as 10% of the prevalence based on a 10% turnover rate, as is done elsewhere in OSHA's analysis, would result in 284 CBD cases annually.  These estimates bracket OSHA's baseline estimate of 245 cases per year.  (The effect from reducing airborne exposures to the PEL of 0. 2 ug/m[3] alone, including substitution from Beryllium in dental labs, is estimated to be 594 cases total, or between 40 and 59 cases annually, compared to 57 cases in the main analysis of Chapter VII).  
Logistic Regression
A logistic model is often simple to interpret.  Time enters only through any independent variables with a time component.  A standard logistic model gives an estimate of the probability of an outcome, and, applied to a population, estimates a prevalence of being in a state (i.e., having CBD), a stock concept at a point of time.  In order to apply the prevalence model to the lifetime risk model, this prevalence rate must be translated into an incidence rate of the risk of developing CBD at a point in time, a flow concept.  Under steady state conditions, the prevalence rate multiplied by the population turnover rate will equal the incidence rate.  To make this translation, the Agency assumes (both for the quartile model and the logistical model) a 10% turnover rate and hence the estimates of prevalence from the logit model are multiplied by 10% to produce the incidence rates used in the lifetime risk calculation. 
The logistic model, as well as providing point estimates of coefficients, also gives us some measures of the uncertainty of these estimates, of course conditional on the model being the correct specification.  From the CBD logistic model presented in table 14 of the background risk assessment document, OSHA used the coefficient on cumulative exposure, 0.03, as its base model, along with the 95% confidence interval around this estimate (the values of 0.001 and 0.07 for a low and high model)  as a useful measure and range of uncertainty.  OSHA used these parameters and range to implement the life time risk model.  
First, the comparison of the number of CBD cases that occur (and are prevented at various PEL levels) is presented in Table 1 below.  OSHA presents results on CBD cases, for all 4 models, of cases due to airborne only effects (purely due to changes in the PEL), non-airborne included (adding the effects of ancillary benefits), and then the base midpoint estimate.  The quartile model results here are those presented in the PEA Chapter VII revised benefits analysis. 
One thing to keep in mind is that with the complete benefits model there is an offsetting effect from ancillary benefits versus any changes in the slope coefficient on cumulative exposure for the number of cases averted due to a changing PEL, given a constant baseline number of cases. Ancillary programs are assumed to save a certain percentage (45% at the midpoint estimate) of all CBD cases that are not averted by the change in the PEL.  Hence changes in the PEL that leave more CBD cases still occurring post-standard will have a somewhat offsetting increase in cases averted due to ancillary programs.  Comparing the spread in logit model estimates, for airborne only at a PEL option of 0.2, the low estimate is 44% of the mid estimate (1503/3414) while the high estimate is 148% of the mid (5066/3414.)  In contrast, this spread narrows to 74% and 126% in the estimates that include non-airborne effects. The midpoint spread is then between these two.



                                   Table 1 
 Prevented Mortality and Morbidity by PEL Option  (45-Year Working Life Case)
                                CBD Total Cases
                                 (All Models)

                             Baseline Total Cases
                             PEL Option (ug/m[3])
 
                             Baseline Annual Cases
                             PEL Option (ug/m[3])


Total  Number of Avoided Cases
 

Annual Number of Avoided Cases


                                      0.1
                                      0.2
                                      0.5
                                       
                                       
                                      0.1
                                      0.2
                                      0.5


                                       
                                       
                                       
                                       
                                       
                                       
                                       
                                       


                                       
                                       
                                       
                                       
                                       
                                       
                                       
                                       
Airborne Only Estimates
                                       
                                       
                                       
                                       
                                       
                                       
                                       
                                       


                                       
                                       
                                       
                                       
                                       
                                       
                                       
                                       
Quartile Model
                                    11,017
                                     2,763
                                     2,563
                                     2,463
                                       
                                      245
                                     61.4
                                     56.9
                                     54.7

                                       
                                       
                                       
                                       
                                       
                                       
                                       
                                       
                                       
Logit-Base
                                     9,873
                                     3,493
                                     3,414
                                     3,259
                                       
                                      219
                                     77.6
                                     75.9
                                     72.4
Logit-Low
                                     7,462
                                     1,506
                                     1,503
                                     1,499
                                       
                                      166
                                     33.5
                                     33.4
                                     33.3
Logit-High
                                    12,308
                                     5,282
                                     5,066
                                     4,556
                                       
                                      274
                                     117.4
                                     112.6
                                     101.3

                                       
                                       
                                       
                                       
                                       
                                       
                                       
                                       
                                       

                                       
                                       
                                       
                                       
                                       
                                       
                                       
                                       
                                       
Non-Airborne Included Estimates
                                       
                                       
                                       
                                       
                                       
                                       
                                       


                                       
                                       
                                       
                                       
                                       
                                       
                                       
                                       
Quartile Model
                                    11,017
                                    10,191
                                    10,171
                                     6,312
                                       
                                      245
                                     226.5
                                     226.0
                                     140.3


                                       
                                       
                                       
                                       
                                       
                                       
                                       
                                       
Logit-Base
                                     9,873
                                     9,235
                                     9,227
                                     6,236
                                       
                                      219
                                     205.2
                                     205.1
                                     138.6
Logit-Low
                                     7,462
                                     6,867
                                     6,866
                                     4,183
                                       
                                      166
                                     152.6
                                     152.6
                                     92.9
Logit-High
                                    12,308
                                    11,605
                                    11,583
                                     8,044
                                       
                                      274
                                     257.9
                                     257.4
                                     178.8


                                       
                                       
                                       
                                       
                                       
                                       
                                       
                                       
Midpoint Estimates
                                       
                                       
                                       
                                       
                                       
                                       
                                       
                                       


                                       
                                       
                                       
                                       
                                       
                                       
                                       
                                       
Quartile Model
                                    11,017
                                     6,477
                                     6,367
                                     4,387
                                       
                                      245
                                     143.9
                                     141.5
                                     97.5

                                       
                                       
                                       
                                       
                                       
                                       
                                       
                                       
                                       
Logit-Base
                                     9,873
                                     6,364
                                     6,321
                                     4,747
                                       
                                      219
                                     141.4
                                     140.5
                                     105.5
Logit-Low
                                     7,462
                                     4,186
                                     4,185
                                     2,841
                                       
                                      166
                                     93.0
                                     93.0
                                     63.1
Logit-High
                                    12,308
                                     8,444
                                     8,325
                                     6,300
                                       
                                      274
                                     187.6
                                     185.0
                                     140.0


                                       
                                       
                                       
                                       
                                       
                                       
                                       
                                       
 
 
                                       
                                       
                                       
                                       
                                       
                                       
                                       
                                       


                                       
                                       
                                       
                                       
                                       
                                       
                                       
                                       
Source: Office of Regulatory Analysis, Directorate of Standards and Guidance
                                       
                                       
                                       

We see at the proposed PEL of 0.2 that the quartile model for the baseline number of cases (the scenario under the current regime with no change in the current PEL) predicts 9,563 cases, while the logit model at its estimated cumulative exposure coefficient is somewhat higher at 11,017 cases.  With the range estimated using the 95% confidence level around this coefficient we get 7,462 and 12,308 cases. To the right are total cases and annual cases averted as the PEL is lowered.  We see even for the low estimate of the logit model that the number of cases averted is 33.4 per year at a proposed PEL of 0.2.
Next in Table 2, OSHA shows how various daily levels of exposure map into cumulative exposure years given various numbers of years of exposure.  Then in Table 3, OSHA presents the estimates of prevalence at different cumulative exposure levels.  At 10 years at the proposed PEL of 0.2, the quartile model has already reached its highest point of 8.8%.  The logit model instead has a continuous function and for the high model at a large number of years reaches very high percentages.  However, as noted above, the values for these high levels in the logistic regression model are extrapolated well beyond the available data.
Table 2.  Cumulative Exposure by Daily Exposure Level and Years of Exposure 
Cumulative Exposure
                             Daily Exposure Level


                                                                            0.1
                                                                            0.2
                                                                            0.5
                                                                            1.0
                                                                            2.0
Years of exposure





                                                                             10

                                                                            1.0
                                                                            2.0
                                                                            5.0
                                                                           10.0
                                                                           20.0
                                                                             20

                                                                            2.0
                                                                            4.0
                                                                           10.0
                                                                           20.0
                                                                           40.0
                                                                             30

                                                                            3.0
                                                                            6.0
                                                                           15.0
                                                                           30.0
                                                                           60.0
                                                                             40

                                                                            4.0
                                                                            8.0
                                                                           20.0
                                                                           40.0
                                                                           80.0
                                                                             45

                                                                            4.5
                                                                            9.0
                                                                           22.5
                                                                           45.0
                                                                           90.0
















             Table 3.  Prevalence by Model and Years of Exposure 
                                       







Quartile Model







                                Exposure Level
Years of exposure
                                                                            0.1
                                                                            0.2
                                                                            0.5
                                                                            1.0
                                                                            2.0
                                                                             10

                                                                          2.50%
                                                                          8.00%
                                                                          8.00%
                                                                          8.80%
                                                                          8.80%
                                                                             20

                                                                          8.00%
                                                                          8.00%
                                                                          8.80%
                                                                          8.80%
                                                                          8.80%
                                                                             30

                                                                          8.00%
                                                                          8.00%
                                                                          8.80%
                                                                          8.80%
                                                                          8.80%
                                                                             40

                                                                          8.00%
                                                                          8.80%
                                                                          8.80%
                                                                          8.80%
                                                                          8.80%
                                                                             45

                                                                          8.00%
                                                                          8.80%
                                                                          8.80%
                                                                          8.80%
                                                                          8.80%














Logit Model -Base







                                Exposure Level
Years of exposure
                                                                            0.1
                                                                            0.2
                                                                            0.5
                                                                            1.0
                                                                            2.0
                                                                             10

                                                                          4.70%
                                                                          4.83%
                                                                          5.27%
                                                                          6.07%
                                                                          8.02%
                                                                             20

                                                                          4.83%
                                                                          5.12%
                                                                          6.07%
                                                                          8.02%
                                                                         13.71%
                                                                             30

                                                                          4.97%
                                                                          5.42%
                                                                          6.98%
                                                                         10.53%
                                                                         22.44%
                                                                             40

                                                                          5.12%
                                                                          5.73%
                                                                          8.02%
                                                                         13.71%
                                                                         34.52%
                                                                             45

                                                                          5.19%
                                                                          5.90%
                                                                          8.59%
                                                                         15.58%
                                                                         41.58%














Logit Model - High







                                Exposure Level
Years of exposure
                                                                            0.1
                                                                            0.2
                                                                            0.5
                                                                            1.0
                                                                            2.0
                                                                             10

                                                                          4.88%
                                                                          5.22%
                                                                          6.36%
                                                                          8.79%
                                                                         16.25%
                                                                             20

                                                                          5.22%
                                                                          5.95%
                                                                          8.79%
                                                                         16.25%
                                                                         44.03%
                                                                             30

                                                                          5.57%
                                                                          6.79%
                                                                         12.03%
                                                                         28.09%
                                                                         76.13%
                                                                             40

                                                                          5.95%
                                                                          7.73%
                                                                         16.25%
                                                                         44.03%
                                                                         92.82%
                                                                             45

                                                                          6.15%
                                                                          8.24%
                                                                         18.77%
                                                                         52.75%
                                                                         96.30%














Logit Model - Low







                                Exposure Level
Years of exposure
                                                                            0.1
                                                                            0.2
                                                                            0.5
                                                                            1.0
                                                                            2.0
                                                                             10

                                                                          4.57%
                                                                          4.57%
                                                                          4.59%
                                                                          4.61%
                                                                          4.65%
                                                                             20

                                                                          4.57%
                                                                          4.58%
                                                                          4.61%
                                                                          4.65%
                                                                          4.74%
                                                                             30

                                                                          4.58%
                                                                          4.59%
                                                                          4.63%
                                                                          4.70%
                                                                          4.83%
                                                                             40

                                                                          4.58%
                                                                          4.60%
                                                                          4.65%
                                                                          4.74%
                                                                          4.93%
                                                                             45

                                                                          4.58%
                                                                          4.60%
                                                                          4.66%
                                                                          4.77%
                                                                          4.97%







Source: Office of Regulatory Analysis, Directorate of Standards and Guidance

Finally, in Table 4 OSHA presents results on end point annualized monetary benefits, for all models at a 3% discount rate, with the same breakout as in Table 1.  These monetized benefit results include the lung cancer estimates, which were unaffected by this analysis.  At the midpoint estimate the quartile model is $575.8 million while the Logit-Mid model is $571.9 million.  Even at the Logit-Low estimate we have sizeable benefits of $388.5 million.  Again we see the offsetting effect of ancillary programs versus the airborne only cases.  For the 0.2 PEL, Airborne only estimate, the Logit-Low estimate is 50.4% of the base estimate, while the high estimate is 142.9% of the base estimate.  Then for the maximum Non-Airborne Included estimates we see this compression, 75.2% and 124.8%.  Table 3 also includes the base estimate and the values for the high, low, and midpoint for each of those estimates.

                                    Table 4
               Total Annualized Monetized Benefits  ($Millions) 
                        (All Models, Discount Rate 3%)
                             PEL Option (ug/m[3])
                                                                               
                                                                               
                                      0.1
                                      0.2
                                      0.5





Airborne Only Estimates





                                       
                                       
                                       

Quartile Model
                                    $170.3
                                    $158.0
                                    $150.7


                                       
                                       
                                       

Logit-Base
                                    $208.8
                                    $202.8
                                    $192.8

Logit-Low
                                    $103.9
                                    $102.1
                                     $99.9

Logit-High
                                    $303.3
                                    $289.9
                                    $261.3


                                       
                                       
                                       
Non-Airborne Included Estimates
                                       
                                       


                                       
                                       
                                       

Quartile Model
                                   $1,250.0
                                   $1,245.2
                                    $782.8


                                       
                                       
                                       

Logit-Base
                                   $1,136.3
                                   $1,133.0
                                    $773.7

Logit-Low
                                    $854.5
                                    $852.1
                                    $529.8

Logit-High
                                   $1,418.2
                                   $1,413.3
                                    $988.8


                                       
                                       
                                       
Midpoint Estimates
                                       
                                       
                                       


                                       
                                       
                                       

Quartile Model
                                    $587.3
                                    $575.8
                                    $403.1


                                       
                                       
                                       

Logit-Base
                                    $577.6
                                    $571.9
                                    $434.0

Logit-Low
                                    $390.6
                                    $388.5
                                    $270.4

Logit-High
                                    $756.0
                                    $743.8
                                    $567.3





Source: Office of Regulatory Analysis, Directorate of Standards and Guidance

In summary, examination of the alternative logistic regression model changes no essential conclusion of the Agency's analysis.  Both the airborne only estimates and the total estimates for the base estimate for the logistic regression are little different from the preferred quartile model.  Even if one uses the airborne only estimate for the low logistic regression estimate, benefits exceed costs.  There is nothing in the comparison between the quartile and logistic model, including the range of uncertainty in the logistic model that change either OSHA's conclusions about significant risk or the results of the benefit cost analysis. 

