Pooled Bayesian Analysis Pooled Bayesian Analysis

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Pooled Bayesian Analysis Pooled Bayesian Analysis

of 28 Studies on Radon Induced of 28 Studies on Radon Induced

Lung Cancers Lung Cancers

Krzysztof Wojciech Fornalski, M.

Krzysztof Wojciech Fornalski, M.Sc Sc., ., Eng Eng. .

The The Andrzej So Andrzej So łtan ł tan Institute

Institute for Nuclear for Nuclear Studies Studies Ś Ś wierk, wierk, Poland Poland

Krzysztof W. Fornalski, Ludwik Dobrzy ń ski

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Overview Overview

  Bayesian Bayesian analysis analysis

  Curve Curve fitting fitting

  Model Model selection selection

  Radon Radon

  Doses Doses from from radon radon

  28 28 papers papers

  Results Results

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Bayesian

Bayesian analysis analysis

methodology

methodology

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Curve

Curve fitting fitting

  Bayesian Bayesian analysis analysis can can be be used used for for fitting fitting any any function function to to experimental experimental points points

  A A probability probability of of getting getting given given point: point:

  The The algorithm algorithm is is insensitive insensitive to to outliers outliers

( ) exp [ ( ) /( 2 ) ] ( )

2

1 2 2

σ σ σ

σ π T E p

E

P = − − ( ) 0 2

σ

σ = σ

p

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Model

Model selection selection

  Let Let two two alternative alternative models models fit well fit well to to the the same set

same set of of experimental experimental points points

  Which Which of of the the models models has has higher higher reliability reliability ? ?

=

=

=

=

 

 

 

 

 −

− −

 

 

 

 

 −

− −

= k

a B

a

B B

m

b B

b

B B

N

i i

i Bi

i Bi

N

i i

i Ai

i Ai

m a a

b b

E T

E T

E T

E W T

1 ( )

) ( min )

( max

1 ( )

) ( min )

( max

1

2 0

2 2

1

2 0

2 2

2 2

2

) exp (

) 1 (

1

2

) exp (

) 1 (

1

π σ

π σ

σ

σ

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Ockham

Ockham ’ ’ s s razor razor

Model

Model selection selection in in bayesian bayesian approach approach is is always

always giving giving preference preference to to simpler simpler models

models , i.e. , i.e. the the ones ones with with a a fewer fewer number number

of of parameters parameters

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Radon Radon

the the data data

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Doses Doses

  Radon Radon concentration concentration : : Bq Bq /m /m 3 3

  Efective Efective dose dose from from radon: radon:

1 1 Bq Bq /m /m 3 3 = 0.179 = 0.179 mSv mSv / / year year

source

source: : (UNSCEAR 2006, Annex E, Table 25) (UNSCEAR 2006, Annex E, Table 25)

  Two Two analysed analysed low low dose dose ranges ranges : :

  Up Up to 70 to 70 mSv mSv / / year year (391 (391 Bq Bq /m /m 3 3 ) )

  Up Up to 150 to 150 mSv/ mSv /year year (838 (838 Bq Bq /m /m 3 3 ) )

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Points

Points from from all all 28 28 studies studies

Uncertainties not shown in order to keep the readability!

However, the scatter of points shows their real values.

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Results

Results

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Results Results

  7 7 mathematical mathematical models models were were tested tested

  Groups Groups of of data: data:

  up up to 70 to 70 and and 150 150 mSv mSv / / year year

  with with and and without without Cohen Cohen ’ ’ s s and and miners miners ’ ’ data data

  Model Model selection selection algorith algorith was was used used to to compare

compare each each of of the the models models and and select select

the the best best one one

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Models

Models used used

* * Model 1 Model 1 – – RR = 1 RR = 1 , ,

* * Model 2 Model 2 – – RR = a RR = a , where , where a a denotes a constant to be fitted, denotes a constant to be fitted,

* * Model 3 Model 3 – – RR = a + bD RR = a + bD , where , where a a and and b b are fitting are fitting parameters, and

parameters, and D D denotes the annual dose, denotes the annual dose ,

* * Model 4 Model 4 – – RR = 1 + bD RR = 1 + bD , differs from , differs from Model 3 Model 3 by setting the by setting the parameter

parameter a a to 1, to 1,

* * Model 5 Model 5 – – same as same as Model 4 Model 4 but with the parameter but with the parameter b b

constrained to the positive values (

constrained to the positive values ( LNT model LNT model ), ),

* * Model 6 Model 6 – – RR = a +bD + cD RR = a +bD + cD 2 2 with a, b and c being fitting with a, b and c being fitting parameters,

parameters,

* * Model 7 Model 7 - - RR = 1 +bD + cD RR = 1 +bD + cD 2 2 , , i.e. same as i.e. same as Model 6 Model 6 but with but with the parameter

the parameter a a set to set to 1. 1 .

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Results

Results #1 #1

  Linear Linear models models in in the the dose dose range range up up to 150 to 150 mSv/y mSv /y result

result in in RR RR decreasing decreasing with with increasing increasing dose dose ( ( also also when when Cohen Cohen ’ ’ s s and and miners miners ’ ’ data data are are excluded excluded ) )

  In In the the narrower narrower range range , , up up to 70 to 70 mSv mSv / / year year , , when when Cohen

Cohen ’ ’ s s and and miners miners ’ ’ data data are are excluded excluded , , the the linear linear models

models exhibit exhibit statistically statistically insignificant insignificant increase increase

  Quadratic Quadratic models models result result in in hormetic hormetic effect effect

  However However , , when when Cohen Cohen ’s ’ s and and miners miners ’ ’ data are data are excluded

excluded one arrives one arrives at at inverted inverted parabola. parabola. This This shows

shows the the sensitivity sensitivity of of the the parabolic parabolic dependence dependence

on on the the chosen chosen data set. data set.

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Results

Results #2 #2

  The The most most likely likely is is Model 1, Model 1, where where RR = 1

RR = 1 in in the the whole whole studied studied range range of of doses

doses

  The The second second are are LNT LNT and and other other linear linear models

models . . They They are are , , however however about about two two order

order s s of of magnitude magnitude less less likely likely than than the the Model 1

Model 1

  Quadratic Quadratic models models are are least least likely likely

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Conclusions Conclusions

  7 7 models models were were tested tested in in the the analysis analysis of of the the radon data

radon data coming coming from from 28 28 studies studies

  The The bayesian bayesian methods methods of of statistical statistical analysis analysis were were used used

  The The most most likely likely model model is is dose dose - - independent one independent one ( ( below below 150 mSv 150 mSv/ / year year ) )

  The The data data give give no no base base to to claim claim that that radon radon bears bears

a a risk risk to to health health , , at at least least in in this this range range of of doses doses

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Thank

Thank you you ! !

  krzysztof krzysztof .fornalski@ .fornalski@ gmail gmail . . com com

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Data

Data from from 28 28 studies studies (#1) (#1)

Country/region/group Source Comment

Austria Oberaigner et al, 2002 b

Canada, Winnipeg Letourneau et al., 1994 a

China, Gansu Wang et al., 2002 c

China, Shenyang Blot et al., 1990 a, c

Czech Rep. Tomášek et al., 2001 b

England, south-west Darby et al., 1998 b

Finland I Auvinen et al., 1996 a, b

Finland II Ruosteenoja, 1991 a

Finland III Ruosteenoja et al., 1996 b

a – paper is a part of 8 pooled studies, which were analyzed by (Lubin et al., 1997b; UNSCEAR, 2000)

b – paper is a part of 13 pooled European studies, which were analyzed by (Darby et al., 2004; UNSCEAR, 2006) c – paper is a part of pooled Chinese studies, which were analyzed by (Lubin et al., 2004)

d – paper contains only cancer mortalities, not all incidences

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Data

Data from from 28 28 studies studies (#2) (#2)

a – paper is a part of 8 pooled studies, which were analyzed by (Lubin et al., 1997b; UNSCEAR, 2000)

Country/region/group Source Comment

France Baysson et al., 2004 b

Germany Wichmann et al., 2005 b

Germany, Saxony Conrady & Martin, 1996 cited in Becker, 2003

Germany, Schneeberg Conrady et al., 2002

Germany, western Kreienbrock et al., 2001 b

Italy, Mediterranean Bochicchio et al., 2005 b

Italy, Alps Pisa et al., 2001

Japan, Misasa Sobue et al., 2000

Uranium miners Lubin et al., 1997a cited in UNSCEAR, 2000 d

Spain Barros-Dios et al., 2002 b

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Data

Data from from 28 28 studies studies (#3) (#3)

a – paper is a part of 8 pooled studies, which were analyzed by (Lubin et al., 1997b; UNSCEAR, 2000)

b – paper is a part of 13 pooled European studies, which were analyzed by (Darby et al., 2004; UNSCEAR, 2006) c – paper is a part of pooled Chinese studies, which were analyzed by (Lubin et al., 2004)

d – paper contains only cancer mortalities, not all incidences

Country/region/group Source Comment

Sweden I Lagarde et al., 2001 b

Sweden II Pershagen et al., 1992 a, b

Sweden III Pershagen et al., 1994 a, b

USA Cohen, 1995 d

USA, Iowa Field et al., 2000

USA, Missouri I Alavanja et al., 1994 a

USA, Missouri II Alavanja et al., 1999

USA, New Jersey Schoenberg et al., 1990 a

USA, Worcester Thompson et al., 2008

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UNSCEAR 2000

UNSCEAR 2000

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8 8 studies studies

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Selected

Selected W W m m values values

  M1/M2 = 400 M1/M2 = 400

  M1/M5 = 100 M1/M5 = 100

  M1/M6 = 1200 M1/M6 = 1200

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Results of pooled Bayesian analysis of 28 radon studies up to 150 mSv (838 Bq m

-3

) per year.

Type of data Model 2

(Constant)

Model 3 (Linear 1)

Model 4 (Linear 2)

Model 6 (Quadratic 1)

Model 7 (Quadratic 2)

Original data – set 1

major data* a = 0.976 ± 0.003

a = 0.988 ± 0.005 b = (-7.3 ± 3.0) · 10-

4

a = 1 b = (-10.2 ± 1.6) ·

10-4

a = 1.031 ± 0.002 b = (-4.4 ± 0.2) · 10-3

c = (0.3 ± 0.1) · 10-4

a = 1 b = (-2.4 ± 0.2) · 10-3

c = (0.1 ± 0.1) · 10-4

plus reference pointsx a = 0.988 ± 0.003

a = 0.998 ± 0.005 b = (-8.5 ± 3.8) · 10-

4

a = 1 b = (-9.1 ± 2.1) · 10-4

a = 1.018 ± 0.003 b = (-3.1 ± 0.2) · 10-3

c = (0.2 ± 0.1) · 10-4

a = 1 b = (-1.8 ± 0.2) · 10-3

c = (0.1 ± 0.1) · 10-4

without Cohen’s and

miners’ datay a = 1.060 ± 0.028

a = 1.069 ± 0.056 b = (-3.4 ± 5.1) · 10-

4

a = 1 b = (10.6 ± 2.1) · 10-

4

a = 0.932 ± 0.037 b = (8.5 ± 1.4) · 10-3 c = (-0.6 ± 0.2) · 10-4

a = 1 b = (5.5 ± 1.1) · 10-3 c = (-0.4 ± 0.2) · 10-4 Partly pooled dataz– set 2

major dataz a = 0.981 ± 0.003

a = 0.998 ± 0.005 b = (-10.5 ± 3.7) ·

10-4

a = 1 b = (-11.0 ± 2.1) ·

10-4

a = 1.053 ± 0.003 b = (-5.9 ± 0.2) · 10-3

c = (0.4 ± 0.1) · 10-4

a = 1 b = (-2.2 ± 0.2) · 10-3

c = (0.1 ± 0.1) · 10-4

plus reference

pointsx a = 0.982 ± 0.003

a = 1.000 ± 0.005 b = (-12.0 ± 3.4) ·

10-4

a = 1 b = (-12.1 ± 2.0) ·

10-4

a = 1.050 ± 0.003 b = (-5.9 ± 0.2) · 10-3

c = (0.4 ± 0.1) · 10-4

a = 1 b = (-2.4 ± 0.2) · 10-3

c = (0.1 ± 0.1) · 10-4

without Cohen’s and

miners’ datay a = 1.060 ± 0.023

a = 1.074 ± 0.033 b = (-5.6 ± 4.3) · 10-

4

a = 1 b = (8.6 ± 2.1) · 10-4

a = 0.975 ± 0.030 b = (6.8 ± 1.3) · 10-3 c = (-0.5 ± 0.3) · 10-4

a = 1 b = (5.6 ± 1.1) · 10-3 c = (-0.4 ± 0.3) · 10-4

* - data as in Fig. 1

x – reference points set as RR = 1 for the lowest doses

y – Cohen’s data (Cohen, 1995) and miner’s data (Lubin et al., 1997a; UNSCEAR, 2000) contains only cancer mortalities

z – papers labeled “13 European studies”, “8 studies” and “Chinese studies” are substituted with pooled studies of (Darby et al., 2004; Lubin et

al., 1997b, 2004; UNSCEAR, 2000, 2006)

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Results of pooled Bayesian analysis of 28 radon studies up to 70 mSv per year (391 Bq m

-3

).

Type of data Model 2

(constant)

Model 3 (Linear 1)

Model 4 (Linear 2)

Model 6 (quadratic 1)

Model 7 (quadratic 2)

Original data – set 1

major data* a = 0.976 ± 0.003

a = 1.014 ± 0.005 b = (-26.8 ± 2.9) · 10-

4

a = 1 b = (-19.9 ± 1.5) · 10-

4

a = 1.180 ± 0.004 b = (-23.9 ± 0.3) · 10-3

c = (4.7 ± 0.2) · 10-4

a = 1 b = (-6.6 ± 0.2) · 10-3

c = (1.5 ± 0.1) · 10-4

without Cohen’s

and miners’ datax a = 1.065 ± 0.045

a = 0.985 ± 0.146 b = (38.1 ± 40.8) · 10-

4

a = 1 b = (33.2 ± 12.8) · 10-

4

a = 0.931 ± 0.038 b = (9.3 ± 1.3) · 10-3 c = (-0.9 ± 0.4) · 10-4

a = 1 b = (3.4 ± 1.3) · 10-3 c = (-0.1 ± 0.4) · 10-4

Partly pooled datay– set 2

major data a = 0.981 ± 0.003

a = 1.036 ± 0.005 b = (-40.0 ± 3.8) · 10-

4

a = 1 b = (-19.8 ± 2.2) · 10-

4

a = 1.176 ± 0.004 b = (-22.1 ± 0.4) · 10-3

c = (4.1 ± 0.2) · 10-4

a = 1 b = (-5.0 ± 0.3) · 10-3

c = (1.1 ± 0.2) · 10-4

without Cohen’s

and miners’ datax a = 1.066 ± 0.028

a = 1.019 ± 0.116 b = (25.3 ± 40.7) · 10-

4

a = 1 b = (31.7 ± 11.9) · 10-

4

a = 1.011 ± 0.034 b = (3.4 ± 1.2) · 10-3 c = (-0.1 ± 0.4) · 10-4

a = 1 b = (4.4 ± 1.2) · 10-3 c = (-0.3 ± 0.4) · 10-4

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Analysis for the model 2, which assumes that the incidence of lu

Analysis for the model 2, which assumes that the incidence of lung cancer is ng cancer is independent on the dose, gives the average of

independent on the dose, gives the average of RR of 97.6 ± RR of 97.6 ± 0.3%. 0.3%. For the linear For the linear models 3 and 4, in all studied cases the risk decreases with inc

models 3 and 4, in all studied cases the risk decreases with increasing dose. reasing dose. However, if However, if one forces in the model 5 the LNT assumption the value of the s

one forces in the model 5 the LNT assumption the value of the slope equals lope equals b b = = 0.0011

0.0011 ± ± ± ± ± ± ± ± 0.0003 and increases to b 0.0003 and increases to b = 0.0019 = 0.0019 ± ± ± ± ± ± ± ± 0.0003 when Cohen 0.0003 when Cohen ’s and miner s and miner’ ’s s data are excluded

data are excluded. For quadratic models, 6 and 7, inclusion of Cohen . For quadratic models, 6 and 7, inclusion of Cohen’ ’s (Cohen, 1995) s (Cohen, 1995) and miners

and miners ’s (Lubin et al., 1997a) data results in ’ s (Lubin et al., 1997a) data results in hormetic curve with the NOAEL point hormetic curve with the NOAEL point ( ( No Observed Adverse Effect Level No Observed Adverse Effect Level , see Calabrese and Baldwin, (1993)) at 140 mSv , see Calabrese and Baldwin, (1993)) at 140 mSv (782 Bq m

(782 Bq m - -3 3 ) and the maximal reduction (13± ) and the maximal reduction (13 ± 7)% of lung cancer incidences at 73 mSv 7)% of lung cancer incidences at 73 mSv per year (408 Bq m

per year (408 Bq m - - 3 3 ). ). No significant increase of risk is observed below 8 mSv per year No significant increase of risk is observed below 8 mSv per yea r (45 Bq m

(45 Bq m - -3 3 ). This result heavily relies on the Cohen’ ). This result heavily relies on the Cohen ’s and miners s and miners’ ’ data. Exclusion of data. Exclusion of these data from the analysis leads to non

these data from the analysis leads to non- -physical behavior presented by an inverted physical behavior presented by an inverted parabola. This result can be expected if one notes that for Cohe

parabola. This result can be expected if one notes that for Cohen n ’s and miners ’ s and miners ’ ’ set of set of data, model 3 (

data, model 3 (“ “linear 1 linear 1” ”) produces RR > 100%. Because the risk is decreasing with the ) produces RR > 100%. Because the risk is decreasing with the dose, final result using quadratic models (6 and 7) cannot be di

dose, final result using quadratic models (6 and 7) cannot be different. Obviously, fferent. Obviously, inverted parabola is just the mathematical result and has no phy

inverted parabola is just the mathematical result and has no physical meaning. sical meaning.

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The Bayesian analysis of the LNT model (model 5, Appendix) produ

The Bayesian analysis of the LNT model (model 5, Appendix) produces ces results of the

results of the same order of magnitude as presented by UNSCEAR same order of magnitude as presented by UNSCEAR (2006, (2006, Annex E), where the slope, recalculated from the Fig. 18 in p. 2

Annex E), where the slope, recalculated from the Fig. 18 in p. 2 91, is 91, is b b

≈ ≈ 0.0047. 0.0047. Our analysis of model 5 for the annual dose range below Our analysis of model 5 for the annual dose range below 150 mSv, gives

150 mSv, gives b b = 0.0011 = 0.0011 ± ± ± ± ± ± ± ± 0.0003 for the data set 1, and 0.0003 for the data set 1, and b b = 0.0019 = 0.0019

± ±

± ±

±

± ±

± 0.0003 when Cohen 0.0003 when Cohen s and miner s and miner s data are excluded. Taking s data are excluded. Taking narrower range of annual doses, up to 70 mSv, the slope values a

narrower range of annual doses, up to 70 mSv, the slope values a re re b b

= 0.0013

= 0.0013 ± ± ± ± ± ± ± ± 0.0003 and 0.0003 and b b = 0.0043 = 0.0043 ± ± ± ± ± ± ± ± 0.0016, respectively. The latter 0.0016, respectively . The latter value is practically the same as UNSCEAR estimate. The negative

value is practically the same as UNSCEAR estimate. The negative values values obtained for model 4 (linear 2) changed to the positive ones for

obtained for model 4 (linear 2) changed to the positive ones for model 5. model 5.

This is natural consequence of forcing mathematical analysis to

This is natural consequence of forcing mathematical analysis to accept in accept in model 4 the positive slope only. Such an assumption should not b model 4 the positive slope only. Such an assumption should not b e e

accepted in objective science.

accepted in objective science.

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