From Table 11 in HEI report (p. 28):

All Causes (MSA & DIFF):

Mean estimate: Lower bound estimate:

PM1 = 15 PM1 = 20 PM1 = 15

PM0 = 5 PM0 = 10 PM0 = 5

ln(PM1)) 2.70805 ln(PM1)) 2.995732 ln(PM1)) 2.70805

ln(PM0)) 1.609438 ln(PM0)) 2.302585 ln(PM0)) 1.609438

difference: 1.098612 difference: 0.693147 difference: 1.098612

input: RR = 1.128 RR = 1.079 RR = 1.077

ln(RR) = 0.120446 ln(RR) = 0.076035 ln(RR) = 0.074179

gamma = 0.109635 gamma = 0.109695 gamma = 0.067521

gamma = 0.109665 gamma = 0.06758

1.128 1.077

1.079 1.048

RR from standard Cox log-linear model for delta PM =10: 1.057

RR from random effects log-linear model for delta PM =10: 1.078

CPD (MSA & DIFF):

Mean estimate: Lower bound estimate:

PM1 = 15 PM1 = 20 PM1 = 15

PM0 = 5 PM0 = 10 PM0 = 5

ln(PM1)) 2.70805 ln(PM1)) 2.995732 ln(PM1)) 2.70805

ln(PM0)) 1.609438 ln(PM0)) 2.302585 ln(PM0)) 1.609438

difference: 1.098612 difference: 0.693147 difference: 1.098612

input: RR = 1.208 RR = 1.127 RR = 1.132

ln(RR) = 0.188966 ln(RR) = 0.119559 ln(RR) = 0.123986

gamma = 0.172004 gamma = 0.172488 gamma = 0.112857

gamma = 0.172246 gamma = 0.112612

1.208 1.132

1.127 1.081

RR from standard Cox log-linear model for delta PM =10: 1.138

RR from random effects log-linear model for delta PM =10: 1.128

IHD (MSA & DIFF):

Mean estimate: Lower bound estimate:

PM1 = 15 PM1 = 20 PM1 = 15

PM0 = 5 PM0 = 10 PM0 = 5

ln(PM1)) 2.70805 ln(PM1)) 2.995732 ln(PM1)) 2.70805

ln(PM0)) 1.609438 ln(PM0)) 2.302585 ln(PM0)) 1.609438

difference: 1.098612 difference: 0.693147 difference: 1.098612

input: RR = 1.484 RR = 1.283 RR = 1.311

ln(RR) = 0.394741 ln(RR) = 0.249201 ln(RR) = 0.27079
gamma = 0.359309 gamma = 0.359521 gamma = 0.246484

gamma = 0.359415 gamma = 0.246294

1.484 1.311

1.283 1.186

RR from standard Cox log-linear model for delta PM =10: 1.242

RR from random effects log-linear model for delta PM =10: 1.287

Lung Cancer (MSA & DIFF):

Mean estimate: Lower bound estimate:

PM1 = 15 PM1 = 20 PM1 = 15

PM0 = 5 PM0 = 10 PM0 = 5

ln(PM1)) 2.70805 ln(PM1)) 2.995732 ln(PM1)) 2.70805

ln(PM0)) 1.609438 ln(PM0)) 2.302585 ln(PM0)) 1.609438

difference: 1.098612 difference: 0.693147 difference: 1.098612

input: RR = 1.236 RR = 1.143 RR = 1.114

ln(RR) = 0.21188 ln(RR) = 0.133656 ln(RR) = 0.107957

gamma = 0.192862 gamma = 0.192825 gamma = 0.098267

gamma = 0.192844 gamma = 0.098613

1.236 1.114

1.143 1.071
und estimate: Upper bound estimate:

PM1 = 20 PM1 = 15 PM1 = 20

PM0 = 10 PM0 = 5 PM0 = 10

ln(PM1)) 2.995732 ln(PM1)) 2.70805 ln(PM1)) 2.995732

ln(PM0)) 2.302585 ln(PM0)) 1.609438 ln(PM0)) 2.302585

difference: 0.693147 difference: 1.098612 difference: 0.693147

RR = 1.048 RR = 1.183 RR = 1.112

ln(RR) = 0.046884 ln(RR) = 0.168054 ln(RR) = 0.10616

gamma = 0.067639 gamma = 0.152969 gamma = 0.153157

gamma = 0.153063

1.183

1.112

und estimate: Upper bound estimate:

PM1 = 20 PM1 = 15 PM1 = 20

PM0 = 10 PM0 = 5 PM0 = 10

ln(PM1)) 2.995732 ln(PM1)) 2.70805 ln(PM1)) 2.995732

ln(PM0)) 2.302585 ln(PM0)) 1.609438 ln(PM0)) 2.302585

difference: 0.693147 difference: 1.098612 difference: 0.693147

RR = 1.081 RR = 1.29 RR = 1.174

ln(RR) = 0.077887 ln(RR) = 0.254642 ln(RR) = 0.160417

gamma = 0.112367 gamma = 0.231785 gamma = 0.231432

gamma = 0.231609

1.290

1.174

und estimate: Upper bound estimate:

PM1 = 20 PM1 = 15 PM1 = 20

PM0 = 10 PM0 = 5 PM0 = 10

ln(PM1)) 2.995732 ln(PM1)) 2.70805 ln(PM1)) 2.995732

ln(PM0)) 2.302585 ln(PM0)) 1.609438 ln(PM0)) 2.302585

difference: 0.693147 difference: 1.098612 difference: 0.693147

RR = 1.186 RR = 1.68 RR = 1.387

ln(RR) = 0.170586 ln(RR) = 0.518794 ln(RR) = 0.327143
gamma = 0.246104 gamma = 0.472226 gamma = 0.471968

gamma = 0.472097

1.680

1.387

und estimate: Upper bound estimate:

PM1 = 20 PM1 = 15 PM1 = 20

PM0 = 10 PM0 = 5 PM0 = 10

ln(PM1)) 2.995732 ln(PM1)) 2.70805 ln(PM1)) 2.995732

ln(PM0)) 2.302585 ln(PM0)) 1.609438 ln(PM0)) 2.302585

difference: 0.693147 difference: 1.098612 difference: 0.693147

RR = 1.071 RR = 1.372 RR = 1.221

ln(RR) = 0.068593 ln(RR) = 0.31627 ln(RR) = 0.19967

gamma = 0.098958 gamma = 0.287881 gamma = 0.288063

gamma = 0.287972

1.372

1.221
0

0.05

0.1

0.15

0.2

0.25

0.3

0 10 20 30 40 50 60

Estimated All Cause Mortality Associated with PM2.5 Over LML

PM2.5 Concentration

random effects log-linear model

random effects log-log model
Log-log vs. log linear model -- Krewski et al. (2009), later period -- all cause mortality

beta from log-linear model 0.007511 (derived from HR for linear PM from Table 11)

beta from log-log model 0.109665 (derived from HRs for log(PM) from Table 11)

PM LN(PM) LML LN(LML) PM - LML LN(PM) - LN(LML) log-linear log-log

6 1.791759 5.8 1.757858 0.2 0.033901552 0.001501 0.003711

7 1.94591 1.2 0.188052232 0.008972 0.020412

8 2.079442 2.2 0.321583624 0.016388 0.034652

9 2.197225 3.2 0.43936666 0.023748 0.047041

10 2.302585 4.2 0.544727175 0.031053 0.057988

11 2.397895 5.2 0.640037355 0.038303 0.067783

12 2.484907 6.2 0.727048732 0.045499 0.076636

13 2.564949 7.2 0.80709144 0.052641 0.084706

14 2.639057 8.2 0.881199412 0.05973 0.092114

15 2.70805 9.2 0.950192284 0.066766 0.098957

16 2.772589 10.2 1.014730805 0.073749 0.105312

17 2.833213 11.2 1.075355427 0.080679 0.111241

18 2.890372 12.2 1.13251384 0.087558 0.116794

19 2.944439 13.2 1.186581062 0.094386 0.122015

20 2.995732 14.2 1.237874356 0.101162 0.12694

21 3.044522 15.2 1.28666452 0.107888 0.131599

22 3.091042 16.2 1.333184536 0.114563 0.136018

23 3.135494 17.2 1.377636298 0.121189 0.14022

24 3.178054 18.2 1.420195913 0.127764 0.144223

25 3.218876 19.2 1.461017907 0.134291 0.148046

26 3.258097 20.2 1.50023862 0.140769 0.151702

27 3.295837 21.2 1.537978948 0.147198 0.155206

28 3.332205 22.2 1.574346593 0.153579 0.158568

29 3.367296 23.2 1.609437912 0.159913 0.1618

30 3.401197 24.2 1.643339464 0.166199 0.164911

31 3.433987 25.2 1.676129287 0.172438 0.167908

32 3.465736 26.2 1.707877985 0.17863 0.1708

33 3.496508 27.2 1.738649644 0.184776 0.173594

34 3.526361 28.2 1.768502607 0.190876 0.176295

35 3.555348 29.2 1.797490144 0.19693 0.178909

36 3.583519 30.2 1.825661021 0.202939 0.181442

37 3.610918 31.2 1.853059995 0.208903 0.183898

38 3.637586 32.2 1.879728242 0.214823 0.186281

39 3.663562 33.2 1.905703729 0.220698 0.188596

40 3.688879 34.2 1.931021537 0.226529 0.190845

41 3.713572 35.2 1.955714149 0.232317 0.193033

42 3.73767 36.2 1.979811701 0.238061 0.195163

43 3.7612 37.2 2.003342198 0.243762 0.197237

44 3.78419 38.2 2.026331716 0.249421 0.199259

45 3.806662 39.2 2.048804572 0.255037 0.20123

46 3.828641 40.2 2.070783479 0.260612 0.203153

47 3.850148 41.2 2.092289684 0.266144 0.20503

48 3.871201 42.2 2.113343093 0.271635 0.206863
49 3.89182 43.2 2.133962381 0.277085 0.208655

50 3.912023 44.2 2.154165088 0.282495 0.210406
0

0.05

0.1

0.15

0.2

0.25

0.3

0 10 20 30 40 50 60

Estimated All Cause Mortality Associated with PM2.5 Over LML

PM2.5 Concentration

standard log-linear model

random effects log-log model
Log-log vs. log linear model -- Krewski et al. (2009), later period -- all cause mortality

beta from log-linear model 0.005543 (derived from HR for standard Cox proportional hazard model from T

beta from log-log model 0.109665 (derived from HRs for log(PM) from Table 11)

PM LN(PM) LML LN(LML) PM - LML LN(PM) - LN(LML) log-linear log-log

6 1.791759 5.8 1.757858 0.2 0.033901552 0.001108 0.003711

7 1.94591 1.2 0.188052232 0.00663 0.020412

8 2.079442 2.2 0.321583624 0.012122 0.034652

9 2.197225 3.2 0.43936666 0.017583 0.047041

10 2.302585 4.2 0.544727175 0.023014 0.057988

11 2.397895 5.2 0.640037355 0.028415 0.067783

12 2.484907 6.2 0.727048732 0.033786 0.076636

13 2.564949 7.2 0.80709144 0.039127 0.084706

14 2.639057 8.2 0.881199412 0.044439 0.092114

15 2.70805 9.2 0.950192284 0.049721 0.098957

16 2.772589 10.2 1.014730805 0.054975 0.105312

17 2.833213 11.2 1.075355427 0.060199 0.111241

18 2.890372 12.2 1.13251384 0.065394 0.116794

19 2.944439 13.2 1.186581062 0.070561 0.122015

20 2.995732 14.2 1.237874356 0.075699 0.12694

21 3.044522 15.2 1.28666452 0.080808 0.131599

22 3.091042 16.2 1.333184536 0.08589 0.136018

23 3.135494 17.2 1.377636298 0.090943 0.14022

24 3.178054 18.2 1.420195913 0.095969 0.144223

25 3.218876 19.2 1.461017907 0.100966 0.148046

26 3.258097 20.2 1.50023862 0.105936 0.151702

27 3.295837 21.2 1.537978948 0.110879 0.155206

28 3.332205 22.2 1.574346593 0.115794 0.158568

29 3.367296 23.2 1.609437912 0.120682 0.1618

30 3.401197 24.2 1.643339464 0.125543 0.164911

31 3.433987 25.2 1.676129287 0.130377 0.167908

32 3.465736 26.2 1.707877985 0.135184 0.1708

33 3.496508 27.2 1.738649644 0.139965 0.173594

34 3.526361 28.2 1.768502607 0.14472 0.176295

35 3.555348 29.2 1.797490144 0.149448 0.178909

36 3.583519 30.2 1.825661021 0.15415 0.181442

37 3.610918 31.2 1.853059995 0.158826 0.183898

38 3.637586 32.2 1.879728242 0.163476 0.186281

39 3.663562 33.2 1.905703729 0.1681 0.188596

40 3.688879 34.2 1.931021537 0.172699 0.190845

41 3.713572 35.2 1.955714149 0.177272 0.193033

42 3.73767 36.2 1.979811701 0.181821 0.195163

43 3.7612 37.2 2.003342198 0.186344 0.197237

44 3.78419 38.2 2.026331716 0.190842 0.199259

45 3.806662 39.2 2.048804572 0.195315 0.20123

46 3.828641 40.2 2.070783479 0.199763 0.203153

47 3.850148 41.2 2.092289684 0.204187 0.20503

48 3.871201 42.2 2.113343093 0.208586 0.206863
49 3.89182 43.2 2.133962381 0.212961 0.208655

50 3.912023 44.2 2.154165088 0.217312 0.210406
ional hazard model from Table 33)