Assessing maximally allowed concentration; application to the regulation of PNOS

Transcription

1 Assessng maxmally allowed concentraton; alcaton to the regulaton of PNOS G. SALANTI, K. ULM Insttut für Medznsche Statstk und Edemologe der TU München Klnkum rechts der Isar Ismannger Str Muenchen Deutschland Introducton A threshold value, n Germany called MAK value (maxmum worklace concentraton) s defned as the maxmum concentraton of a chemcal substance not have known adverse effects on the health of the emloyee nor cause unreasonable annoyance even when the erson s reeatedly exosed durng long erods, gven a 40-hour workng week. As a rule, the MAK value refers to the average concentraton obtaned by ntegratng the concentratons determned durng a erod of u to one workng day or shft (DFG, 2002). In assessng a threshold value (a MAK value) the most senstve endont has to be taken nto account. Regardng the exosure to dust, whch can cause unsecfc effects on the resratory organs, chronc bronchtc reacton (= CBR) was used. The decson whether there was such a reacton or not was based on a classfcaton tree combnng an anamnestc questonnare and lung functon tests. In order to assess a threshold value n ths stuaton and to adjust for the effect of other mortant factors, lke age, a statstcal model s necessary.

2 Statstcal methods General consderatons To analyse the mact of a contnuous varable on a defned resonse a model has to be used, n whch the form of the relatonsh s secfed. The general model usng the varables dust concentraton c and tme snce frst exosure t was as follows: logt = α + f() c + f2() t () The data of the CBR-study were dvded nto several subsamles wth resect to lant and smokng. Several models were aled to analyze the DFG-Chronc Bronchts study. The models dffer wth resect to the choce of the functons f (c) and f 2 (t). Logstc regresson and threshold value estmaton The smlest model s the lnear logstc regresson that assumes lnear relatonshs. That means both functons n formula () are lnear (e. g. f (c) = ß c c f 2 (t) = ß t t). Ths functon can be extended for estmatng a threshold value τ f ( c) 0 c τ = βc ( c τ) c > τ (2) In case of a threshold value there s the restrcton that the regresson lne has to be constant u to the break ont. Bascally the dea s to ntroduce an addtonal arameter nto the regresson equaton, the threshold value. A more formal descrton of the statstcal method s gven by Ulm (99). One roblem s related to the outcome. The relatonsh can result n a local decrease n the rsk wth ncreasng exosure. Such a result could be seen n some subsamles n the DFG Bronchts Study where the rsk of the dsease s lower than the baselne rsk over some range of the lower dust concentraton. Ths henomenon was also observed n other studes. There are several exlanatons to observe a decrease n the rsk. Frst the workers wth zero or very low exosure mght be dfferent from the exosed workers, or there s a selecton bas. Second ths result may be exlaned by chance. 2

3 Isotonc regresson The sotonc transformaton and ts features Alternatve to the logstc model a nonarametrc aroach, sotonc regresson, has been used (Robertson, Wrght & Dykstra, 988). Isotonc regresson has some advantages comared to arametrc models. No secfc assumtons for the form of the dose-resonse relatonsh aart from monotoncty are requred. Isotonc regresson s ftted by the Polled Adjacent Volators Algorthm (PAVA), an algorthm wdely used and easy to aly. Isotonc framework has been well establshed by Robertson et al. and ganed further attenton snce then. Isotonc regresson also rovdes a test for trend whch s owerful, stable and ndeendent of any monotonc transformaton of the redctors (Chuang-Sten et al., 997), a feature that s not rovded by commonly used test (as the Cochran- Armtage test) or the logstc regresson. Techncal detals of the PAVA and sotonc test for trend are resented n the aendx. Isotonc regresson and threshold estmaton Isotonc regresson roceeds by slttng the redctors n constant rsk grous and yelds a small set of cutonts. These cutonts are canddate threshold locatons (Morton-Jones et al., 2000). Note that sotonc regresson s a tool for modelng and testng, that can be aled n dfferent forms: as unvarate regresson, as surface-estmator or to extend generalzed addtve models. Regardng alcatons already resented for assessng a threshold value for dust, sotonc regresson has been ftted n two dmensons alyng an sotonc-surfaces model. In ast aers (Ulm 999) the followng model has been used: = f(,) c t (3) wth f c t f c2 t2 (, ) (, ) for c c or 2 t t2. For estmatng f (,) ct the data had to be groued n order to reduce the dmensonalty. In lack of an arorate test rocedure for assessng threshold values, a rsk ncrease of 5% 3

4 comared to baselne was used, whch s of course somewhat arbtrary. In contrast to that, n recent aers (Ulm 999 and Salant & Ulm 200) sotonc regresson was extended and new rocedures were ntroduced n order to assess a threshold value. Here we wll resent how an addtve sotonc model can be used to assess threshold values. It has been frst ntroduced by Bacchett (989) and mroved by Morton-Jones (2000). The model takes the usual addtve form descrbed earler (formula ()). logt = α + f ( c) + f ( t) () 2 where now f (c) and f 2 (t) are sotonc transformaton functons (ste functons). Here, the data need no longer to be groued nto certan categores. The effect of dust concentraton can be tested by comarng the values of the lkelhood functons of model () wth and wthout the dust n the model ( f ( c ) = 0). There s so far no dstrbutons theory avalable for the corresondng lkelhood rato test, so condtonal ermutatons as descrbed by Salant & Ulm (200) need to be used to assess sgnfcance (see aendx). Note that ths test s a multvarate test for trend for dust (adjusted for varable tme snce frst exosure), and therefore rovdes an mortant tool n establshng monotonc doseresonse relatonsh. As result of the estmaton rocedure, f( c ) amalgamates certan levels of dust concentraton n order to gve one rsk estmate for each category. Note that ths estmaton as well as the ermutaton test are adjusted for other redctors (e. g. tme) ncluded n model. Once the model s establshed and a dose-resonse relatonsh s roven, -.e. the ermutatons test s sgnfcant, a threshold can be assessed. The rocedure can be descrbed as follows:. The categores defned for dust from the artal ft n model ().e. f ( c ) are lumed together startng from the two lowest categores 2. The change n the goodness of ft ( = devance) s estmated 4

5 3. If the loss n the ft s sgnfcant (one-sded χ 2 -dstrbuted) a threshold s assessed. Otherwse the rocedure contnues (go to ste ) untl a sgnfcant change n the devance s reached. Results Data from 5578 workers of three dfferent lants (Moers, Munch and Saarbrücken) were avalable. A detaled descrton of the data can be found n the monograh of the study (DFG-reort 978) or n Ulm et al. (996). The three lants had a mxture of dust, manly from ron, steel, foundry and engneerng. Logstc regresson and sotonc-surfaces models Wthn ths aer we restrct ourselves to the assessment of the threshold for nhalable dust. The results of ths aroach conssted the bass for assessng the MAK-values and are resented n table. The estmaton of the threshold regardng sotonc regresson has been erformed usng a 5% excess n the rsk. The threshold values are varyng between 3.8 and 20.6 mg/m 3 (Ulm et al., 996). Table : Results used for assessng the threshold value for total dust Non-smokers samle tye of regresson Moers Munch Saarbrücken logstc Isotonc-surfaces Smokers samle tye of regresson Moers Munch Saarbrücken logstc Isotonc-surfaces Whch of these estmates should be taken as MAK-values? The deends also on the olcy of the MAK-commsson. Gven an estmated nterval wthn whch the threshold les, the lowest 5

6 value can be used n order to fulfl the defnton of the reamble (s. Grem at al., 2002). However, n the fnal reort, the MAK commsson decded to set the threshold at 4 mg/cm³. Isotonc regresson The mroved verson of sotonc regresson was used to estmate a threshold value. In three of the subsamles sotonc regresson leads to a sgnfcant effect for dust concentraton (s. Table 2). In the estmaton of a threshold value for dust we roceed as descrbed n the revous secton. The estmated threshold values are between 2.09 and.09 mg/m³. In three of the subsamles a sgnfcant effect of dust was to be seen. In two of the three subsamles wth a sgnfcant dust effect (the smokers of Munch and Moers) the old and the new threshold values are close. Table 2: Samles Results from the addtve sotonc model for assessng threshold values (new results) -value for the effect of dust threshold value for total dust Moers smokers 0.03* 2.09 non-smokers 0.28 Munch smokers 0.0* 4.96 non-smokers 0.0 Saarbrücken smokers 0.35 non-smokers 0.049*.09 * statstcally sgnfcant: <

7 Aendx A. PAVA rocedure Focusng on bnary resonse the PAVA for ncreasng trend can be descrbed as follows: Consder the stuaton of k dose grous where the dose d, =,... k s n ncreasng order and the endont s the robablty of an event (here: chronc bronchtc reacton). We wsh to have n non decreasng order, gven that d d +. If there s somewhere a volator such that d > d + for some, then the sotonc estmator of both values s needed to be assessed. That s rovded by ther weghted mean n + n = * + +, + n + n+. Now the observatons,+ form a block. Ths rocess s reeated usng the udated robabltes untl an sotonc set of * s obtaned. The algorthm assumng decreasng trend s smlar. B. Test for trend Ths test s known as the sotonc lkelhood rato test. We defne the followng hyothess: H0: = 2 =... = k = 0 aganst the alternatve H: 2... k wth at least one strct nequalty. Then the sotonc lkelhood rato test has the followng form: T = 2 [ n ln( ) + n ( )ln( )] 0 k * * = 0 0 Ths test follows asymtotcally a weghted ch-square dstrbuton. However, ths aroxmaton does not always hold. Thus a condtonal ermutatons test s roosed. Based on the observed number of atents er tme and dust and the total number of events, a large number of randomzatons s analyzed. Each worker s characterzed by four elements ( d, t, s, St) wth t denotng tme, d dust, s smokng habts, and St dsease status (0 absence, occurrence). For the ermutaton test dust d s consdered ndeendent from ( t, s, St ) so that the events occur n random allocaton. Wthn each ermutaton H0 s consdered, the sotonc model s obtaned and test T 0 s assessed. If the observed test value exceeds the 95th 7

8 ercentle, the observed allocaton of events can not be exlaned by chance and H 0 s to be rejected. The exact -value s estmated as the robablty that the result of a ermutaton s equal or greater to the observed. References Baccett P, (989) Addtve Isotonc Models. JASA 84: Chuang-Sten C, Agrest A (997) Tutoral n Bostatstcs: A revew of tests for detectng a monotone dose-resonse relatonsh wth ordnal resonse data, Stat Med 6: DFG (978) Chronc Bronchts and Occuatonal Dust Exosure. Harald Boldt Verlag, Board DFG (2002) Lst of MAK and BAT Values Wley-VCH, Wenhem Grem H et al. (2002) Offcal Answer of the German MAK-commsson to the revew by McLaughln et al. Int Arch Occ Env Health Morton-Jones T, Dggle P, Parker L (2000) Addtve Isotonc Regresson Models n Edemology. Stat Med 9: Robertson T, Wrght FT, Dykstra RL (988) Order Restrcted Statstcal Inference. J. Wley, New York Salant G, Ulm K (200) Multdmensonal Isotonc Regresson and Estmaton of the Threshold Value. Dscusson Paer No. 234, Sonderforschungsberech 386 der Ludwgs- Maxmlan-Unverstät München Ulm K (99) A Statstcal Method for Assessng a Threshold n Edemologcal Studes. Stat Med 0: Ulm K, Dannegger F, Saner M (996) Bercht über de Auswertung der Daten der Stude Chronsche Bronchts. Dscusson Paer No. 39, Sonderforschungsberech 386 der Ludwgs-Maxmlan-Unverstät München Ulm K (999) Nonarametrc Analyss of Dose-Resonse Relatonshs. Ann NY Acad Sc 895: