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Segmented
Poisson
models
Enrique Vidal, Roberto Pastor-Barriuso, Marina Pollán, Gonzalo López-Abente
Enviromental and Cancer Epidemiology Unit, National Centre of Epidemiology, Carlos III Institute of Health. Madrid, Spain.
CIBERESP, Spain
background
There are many situations in which threshold effects could be supposed to explain dose-response relationship
@ Diabetes
@ Mortality trends
@ Physics0 2 4 6 8 10
010
2030
Dose
Res
pons
e
We need tools to deal with dose-response analyses
background
Standard dose-response analyses provide flexible tools to describe the overall shape of the relationship
* Categorical Analyses
* Non Parametrical Regressions
* Splines0 2 4 6 8 10
010
2030
Dose
Res
pons
eBUT identification of change points is subjective
We need to test existence & location of possible change points
background
Dose
Res
pons
e
0 2 4 6 8 10
010
2030
Standard dose-response analyses provide flexible tools to describe the overall shape of the relationship
* Categorical Analyses
* Non Parametrical Regressions
* Splines
BUT identification of change points is subjective
We need to test existence & location of possible change points
background
One choice could be linear joinpoint regression
+ It tests existence of joinpoints
+ It’s already implemented
- It assumes an abrupt transition0 2 4 6 8 10
010
2030
Dose
Res
pons
e
May be smooth transitions will be more plausible in many biological settings
Aim
It would be desirable to find a model that
assess changes in response trends related to a dose variable
tests existence and location of change points
allows a gradual transition at the change point
could be implemented in R code
model
We propose a Segmented Poisson Model with
Poisson variance for aggregated counts
Free dispersion parameter for extra variance (small areas)
2 intersecting straight lines for differential dose-response
Hyperbolic transition function for smoothness at the change point
and a log link function
λ+σλ-σ λ
γ = σ
γ = 0,5σ
γ = 0,1σ
β 0+ (β 1
+ β2)(c -λ)
β0 + (β1 - β2)(c - λ)
Hyperbolic Transition
Res
pons
e
2221
0i
)()(
])/(log[
γλβλβ
βΖα
+−+−+
++=
cc
ndE i
Estimation
For change point and transition parameter fixed, the function is lineal in ß so
existence is tested performing a grid search over the dose variable, and applying improved Bonferroni corrections for multiple search to a likelihood ratio test
location is estimated by searching around de ML knot of the above grid. Its CI is approximated by cubic spline interpolation over the knots
Once the existence and location of the change point has been assed, the final model is fitted to obtain the corresponding slopes
function
InputData, as data frame
Outcome variable, as character
Dose variable, as character
Covariates (offset), as formula
OutputChange Point existence test
Change point location point & interval estimates
Slopes below & above change point
examples
[1] Renal cancer mortality
Response: Deaths by municipalities in Spain (1994-2003)
Dose: Distance to he nearest metallurgical facilities (EPER)
Covariables: Expected cases (offset), age, sex, socio-eco. ind.
[2] Breast cancer incidence:
Response: New cases from 16 (of the 50) Spanish registers
Dose: Year of diagnosis (1970-2004)
Covariables: Person-years (offset), register
Results [1] Renal cancer mortality
It does exist a change point (p-value < 0.005), located at 17 Km (CI 95% 0 28 Km) away from the point source
0 20 40 60 80 100
0.5
1.0
1.5
2.0
Distance / Km
Rel
ativ
e R
isk
Renal Cancer Mortality
*
0 25 50 Km
Significant decrease of renal cancer mortality with further distance bellow change point, no trend above it
Results [2] Breast cancer incidence
1970 1975 1980 1985 1990 1995 2000 2005
0.5
12
Yea r
Rat
e ra
tio
B reas t C an cer In c iden ce
It does exist a change point (p-value < 10-10), happening in year 1999 (CI 95% 1996 2001)
Breast cancer incidence increased in Spain (2.8% per year) during the 70s, 80s &90s and levelled in the XXI century
Thank you
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