Poisson Regression with Rates
Traffic Deaths in Finland on
Friday the 13th and Other Fridays
1971-1997
Simo Näyhä (2002). “Traffic Deaths and Superstion on Friday the 13th,”
American Journal of Psychiatry, Vol. 159, #12, pp. 2110-21111
Data
• Traffic Deaths and Exposure by Gender
on Friday the 13th and Other Fridays in
Finland for years 1971-1997
• Traffic Deaths are Given as Counts by
Friday type and gender
• Exposure is person days (100,000s)
Group(i)
1
2
3
4
sum
Gender
M(X1=0)
M(X1=0)
F(X1=1)
F(X1=1)
Friday
13th(X2=1)
Other(X2=0)
13th(X2=1)
Other(X2=0)
deaths(Y)
82
2423
41
789
3335
exposure(t)
79.9
2483.7
86.5
2687.1
5337.2
Poisson Model for Rates
Sample Rate :
Yi
ti
 Yi  i
Expected Value : E   
 ti  ti
 
Loglinear Model : log  i    0  1 X 1i   2 X 2i   3 X 1i X 2i
 ti 
 log i   log ti    0  1 X 1i   2 X 2i   3 X 1i X 2i
 log i   log ti    0  1 X 1i   2 X 2i   3 X 1i X 2i
 i  ti e  0  1 X 1i   2 X 2 i  3 X 1i X 2 i
e  i i  i
Poisson Probabilit y Mass Function : PYi  yi  
yi !
y
yi  0,1,... i  1,2,3,4
Likelihood and ML Estimates - I
4
4
L   p  yi   
i 1
4

e ti e
xi 'β
t e 
x i 'β yi
i
yi !
4
l  ln( L)   ti e
i 1
  ti e
i 1
x i 'β
x i 'β
t e

 0  1 X 1i   2 X 2 i   3 X 1i X 2 i yi
i
yi !
i 1
i 1
4
e
ti e 0  1X1i  2 X 2 i  3 X1i X 2 i
where : x i '  1 X 1i
4

  yi ln ti e
x i 'β
i 1
4
4
i 1
i 1
X 2i
X 1i X 2i 
0 
 
β   1
 2 
 
 3 
  ln  y ! 
4
i 1
i
  yi ln ti   x i ' β   ln  yi !
 1 
X 
4
4
4
l
x i 'β
  ti x i e   yi x i    yi  i x i    yi  i  1i 
 X 2i 

i 1
i 1
i 1
 
 X 3i 
Likelihood and ML Estimates - II
 1 
X 
4
4
4
l
xi ' 
  ti xi e   yi xi    yi  i xi    yi  i  1i 
 X 2i 
β
i 1
i 1
i 1
 
 X 3i 
 1 
X 
l
Setting
 0    yi  i  1i   0  X' (Y  μ)  0
 X 2i 
β
 
 X 3i 
'
'
 2l


  x i e x iβ   yi x i   x i e x iβ x'i   X' WX where W  diag μ 
ββ' β'
Setting : G  X' WX and g   X' (Y  μ) leads to the the estimate of β via Newton - Raphson algorithm :

^ New
β

   yi  

ln 
1
   ti 
^ Old   ^ Old  
^0
 ^ Old 

 β  G  β  g β  with a reasonable starting vector of β  
0


 
 

0




0


with approximat e large - sample estimated variance - covariance matrix :
^
^


1
V  β   G   X' W X 
 


^
1
Friday the 13th Results – Regression Coefficients
Group(i)
1
2
3
4
sum
betahat
-0.02474
-1.20071
0.05069
0.42819
Diff(Beta)
Variable
Intercept
Female
Frdy13
F*Frdy13
Gender
M(X1=0)
M(X1=0)
F(X1=1)
F(X1=1)
Friday
13th(X2=1)
Other(X2=0)
13th(X2=1)
Other(X2=0)
start
-0.20422
0.00000
0.00000
0.00000
betahat
-0.0247
-1.2007
0.0507
0.4282
deaths(Y)
82
2423
41
789
3335
exposure(t)
79.9
2483.7
86.5
2687.1
5337.2
X
1
1
1
1
0
0
1
1
1
0
1
0
0
0
1
0
iteration1 iteration2 iteration3 iteration4 iteration5
-0.00763
-0.02460
-0.02474
-0.02474
-0.02474
-0.83644
-1.13655
-1.19868
-1.20071
-1.20071
0.06221
0.05095
0.05069
0.05069
0.05069
0.15901
0.37098
0.42620
0.42819
0.42819
0.767432 0.13541497 0.00690897 8.0656E-06 8.3718E-12
SE(beta)
0.0203
0.0410
0.1123
0.1956
Z(beta)
-1.2179
-29.2931
0.4514
2.1889
P-value
0.2232
0.0000
0.6517
0.0286
V(Beta)
0.000413 -0.000413 -0.000413 0.000413
-0.000413 0.001680 0.000413 -0.001680
-0.000413 0.000413 0.012608 -0.012608
0.000413 -0.001680 -0.012608 0.038266
Predicted Values, Residuals, Risk Ratios
^
^
i
Predicted Values : Y i  ti e  ti e
^
^
(Note for ' saturated' model, Yi  Y i )
x i 'β
^
Risk Ratios (Group i relative to j ) :
i
^
j
^

e
x i 'β
^
e
^
e
^
x i 'β  x j 'β
x j 'β
^
Pearson Residual : ei 
Yi  Y i
^
(Note for ' saturated' model, ei  0)
Yi
Note for Risk Ratios :
Males : Friday 13/Other Friday RR M  exp  0   2    0   e  2
Females : Friday 13/Other Friday RR F  exp  0  1   2   3    0  1   e  2   3
For Confidence Intervals, first obtain CI for function of  s , then exponentia te :
^ 
Males :  2  1.96 * SE   2 
 
^
^
^

^

Females :   2   3   1.96 * SE   2   3 




^
^
^
^

^

^ 
^ 
^ ^ 
where SE   2   3   V   2   3   V   2   V   3   2COV   2 ,  3 




 
 


Friday the 13th Results
Group(i)
1
2
3
4
sum
Gender
M(X1=0)
M(X1=0)
F(X1=1)
F(X1=1)
Friday
13th(X2=1)
Other(X2=0)
13th(X2=1)
Other(X2=0)
deaths(Y)
82
2423
41
789
3335
exposure(t)
79.9
2483.7
86.5
2687.1
5337.2
mu-hat
0.025943
-0.02474
-0.74657
-1.22545
Y-hat
82
2423
41
789
TDR (exp(mu))
1.0263
0.9756
0.4740
0.2936
TDR=Traffic Death Rate (per 100,000 exposed)
Male Risk Ratio
Female Risk Ratio
Parameter
2
23
Parameter
Estimate
0.0507
0.4789
SE
0.1123
0.1602
Parameter Parameter
Lower
Upper
-0.1694
0.2708
0.1649
0.7929
Risk Ratio
Estimate
1.0520
1.6143
Risk Ratio Risk Ratio
Lower
Upper
0.8442
1.3110
1.1793
2.2097
Males tend to have higher accident death rates than females. The
Friday the 13th effect is not significant for males (Risk Ratio interval
contains 1). The Friday the 13th Effect is significant for females (Risk
Ratio interval is entirely above one)
1.2
Traffic Death Rates per 100,000
1.0263
1
Red line
is
1.052x
higher
0.9756
0.8
fridayother
0.6
friday13
0.4740
0.4
Red line
is
1.614x
higher
0.2
0.2936
Male
0
Female