glm.poisson.disp: Overdispersed Poisson log-linear models
In dispmod: Modelling Dispersion in GLM
Description
Usage
Arguments
Details
Value
Note
References
See Also
Examples
Description
This function estimates overdispersed Poisson log-linear models using the approach discussed by Breslow N.E. (1984).
Usage
1
glm.poisson.disp(object, maxit = 30, verbose = TRUE)
Arguments
object
an object of class "glm" providing a fitted Poisson log-linear regression model; see glm.
maxit
integer giving the maximal number of iterations for the model fitting procedure.
verbose
logical, if TRUE information are printed during each step of the algorithm.
Details
Breslow (1984) proposed an iterative algorithm for fitting overdispersed Poisson log-linear models. The method is similar to that proposed by Williams (1982) for handling overdispersion in logistic regression models (see glm.binomial.disp).
Suppose we observe n independent responses such that
y_i | λ_i ~ Poisson(λ_i n_i)
for i = 1, …, n.
The response variable y_i may be an event counts variable observed over a period of time (or in the space) of length n_i, whereas λ_i is the rate parameter.
Then,
E(y_i | λ_i) = μ_i = λ_i n_i = exp(log(n_i) + log(λ_i))
where log(n_i) is an offset and log(λ_i) = β'x_i expresses the dependence of the Poisson rate parameter on a set of, say p, predictors. If the periods of time are all of the same length, we can set n_i = 1 for all i so the offset is zero.
The Poisson distribution has E(y_i | λ_i) = V(y_i | λ_i), but it may happen that the actual variance exceeds the nominal variance under the assumed probability model.
Suppose that θ_i = λ_i n_i is a random variable distributed according to
θ_i ~ Gamma (μ_i, 1/φ)
where E(θ_i) = μ_i and V(θ_i) = μ_i^2 φ. Thus, it can be shown that the unconditional mean and variance of y_i are given by
E(y_i) = μ_i
and
V(y_i) = μ_i + μ_i^2 φ = μ_i (1 + μ_iφ)
Hence, for φ > 0 we have overdispersion. It is interesting to note that the same mean and variance arise also if we assume a negative binomial distribution for the response variable.
The method proposed by Breslow uses an iterative algorithm for estimating the dispersion parameter φ and hence the necessary weights 1/(1 + μ_i \hat{φ}) (for details see Breslow, 1984).
Value
The function returns an object of class "glm" with the usual information and the added components:
dispersion
the estimated dispersion parameter.
disp.weights
the final weights used to fit the model.
Note
Based on a similar procedure available in Arc (Cook and Weisberg, http://www.stat.umn.edu/arc)
References
Breslow, N.E. (1984), Extra-Poisson variation in log-linear models,
Applied Statistics, 33, 38–44.
See Also
lm, glm, lm.disp, glm.binomial.disp
Examples
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## Salmonella TA98 data
data(salmonellaTA98)
salmonellaTA98 <- within(salmonellaTA98, logx10 <- log(x+10))
mod <- glm(y ~ logx10 + x, data = salmonellaTA98, family = poisson(log))
summary(mod)
mod.disp <- glm.poisson.disp(mod)
summary(mod.disp)
mod.disp$dispersion
# compute predictions on a grid of x-values...
x0 <- with(salmonellaTA98, seq(min(x), max(x), length=50))
eta0 <- predict(mod, newdata = data.frame(logx10 = log(x0+10), x = x0), se=TRUE)
eta0.disp <- predict(mod.disp, newdata = data.frame(logx10 = log(x0+10), x = x0), se=TRUE)
# ... and plot the mean functions with variability bands
plot(y ~ x, data = salmonellaTA98)
lines(x0, exp(eta0$fit))
lines(x0, exp(eta0$fit+2*eta0$se), lty=2)
lines(x0, exp(eta0$fit-2*eta0$se), lty=2)
lines(x0, exp(eta0.disp$fit), col=3)
lines(x0, exp(eta0.disp$fit+2*eta0.disp$se), lty=2, col=3)
lines(x0, exp(eta0.disp$fit-2*eta0.disp$se), lty=2, col=3)
## Holford's data
data(holford)
mod <- glm(incid ~ offset(log(pop)) + Age + Cohort, data = holford,
family = poisson(log))
summary(mod)
mod.disp <- glm.poisson.disp(mod)
summary(mod.disp)
mod.disp$dispersion
Example output
Call:
glm(formula = y ~ logx10 + x, family = poisson(log), data = salmonellaTA98)
Deviance Residuals:
Min 1Q Median 3Q Max
-2.5173 -1.1731 -0.4032 0.7995 3.7041
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 2.1727730 0.2184269 9.947 < 2e-16 ***
logx10 0.3198250 0.0570014 5.611 2.01e-08 ***
x -0.0010130 0.0002452 -4.131 3.61e-05 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 78.358 on 17 degrees of freedom
Residual deviance: 43.716 on 15 degrees of freedom
AIC: 142.25
Number of Fisher Scoring iterations: 4
Poisson overdispersed log-linear model fitting...
Iter. 1 phi: 0.07178889
Iter. 2 phi: 0.07180372
Iter. 3 phi: 0.07180614
Iter. 4 phi: 0.07180702
Iter. 5 phi: 0.07180731
Iter. 6 phi: 0.07180741
Iter. 7 phi: 0.07180744
Iter. 8 phi: 0.07180745
Iter. 9 phi: 0.07180745
Iter. 10 phi: 0.07180745
Converged after 10 iterations.
Estimated dispersion parameter: 0.07180745
Call:
glm(formula = y ~ logx10 + x, family = poisson(log), data = salmonellaTA98,
weights = disp.weights)
Deviance Residuals:
Min 1Q Median 3Q Max
-1.4419 -0.6272 -0.2374 0.4564 1.9937
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 2.2030681 0.3635996 6.059 1.37e-09 ***
logx10 0.3109623 0.0990523 3.139 0.00169 **
x -0.0009741 0.0004371 -2.228 0.02585 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 25.313 on 17 degrees of freedom
Residual deviance: 14.214 on 15 degrees of freedom
AIC: 50.799
Number of Fisher Scoring iterations: 4
Call:
glm(formula = y ~ logx10 + x, family = poisson(log), data = salmonellaTA98,
weights = disp.weights)
Deviance Residuals:
Min 1Q Median 3Q Max
-1.4419 -0.6272 -0.2374 0.4564 1.9937
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 2.2030681 0.3635996 6.059 1.37e-09 ***
logx10 0.3109623 0.0990523 3.139 0.00169 **
x -0.0009741 0.0004371 -2.228 0.02585 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 25.313 on 17 degrees of freedom
Residual deviance: 14.214 on 15 degrees of freedom
AIC: 50.799
Number of Fisher Scoring iterations: 4
[1] 0.07180745
Call:
glm(formula = incid ~ offset(log(pop)) + Age + Cohort, family = poisson(log),
data = holford)
Deviance Residuals:
Min 1Q Median 3Q Max
-3.214 -1.205 -0.172 1.462 3.670
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -8.62551 0.10123 -85.208 < 2e-16 ***
Age55-59 0.81903 0.02871 28.526 < 2e-16 ***
Age60-64 1.55271 0.02790 55.655 < 2e-16 ***
Age65-69 2.13322 0.02751 77.540 < 2e-16 ***
Age70-74 2.68681 0.02803 95.844 < 2e-16 ***
Age75-79 3.13410 0.02910 107.689 < 2e-16 ***
Age80-84 3.45545 0.03143 109.930 < 2e-16 ***
Cohort1860-64 0.36540 0.10901 3.352 0.000803 ***
Cohort1865-69 0.52338 0.10221 5.120 3.05e-07 ***
Cohort1870-74 0.77303 0.09976 7.749 9.25e-15 ***
Cohort1875-79 1.00959 0.09883 10.216 < 2e-16 ***
Cohort1880-84 1.15061 0.09832 11.702 < 2e-16 ***
Cohort1885-89 1.30700 0.09803 13.333 < 2e-16 ***
Cohort1890-94 1.50956 0.09856 15.317 < 2e-16 ***
Cohort1895-99 1.55288 0.09882 15.713 < 2e-16 ***
Cohort1900-04 1.59181 0.09932 16.027 < 2e-16 ***
Cohort1905-09 1.46105 0.10074 14.503 < 2e-16 ***
Cohort1910-14 1.36926 0.10396 13.171 < 2e-16 ***
Cohort1915-19 1.23725 0.11616 10.651 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 35493.66 on 48 degrees of freedom
Residual deviance: 127.38 on 30 degrees of freedom
AIC: 571.65
Number of Fisher Scoring iterations: 3
Poisson overdispersed log-linear model fitting...
Iter. 1 phi: 0.004150272
Iter. 2 phi: 0.003747225
Iter. 3 phi: 0.003651023
Iter. 4 phi: 0.003626604
Iter. 5 phi: 0.003620276
Iter. 6 phi: 0.003618627
Iter. 7 phi: 0.003618197
Iter. 8 phi: 0.003618085
Iter. 9 phi: 0.003618056
Iter. 10 phi: 0.003618048
Iter. 11 phi: 0.003618046
Iter. 12 phi: 0.003618045
Iter. 13 phi: 0.003618045
Converged after 13 iterations.
Estimated dispersion parameter: 0.003618045
Call:
glm(formula = incid ~ offset(log(pop)) + Age + Cohort, family = poisson(log),
data = holford, weights = disp.weights)
Deviance Residuals:
Min 1Q Median 3Q Max
-1.77737 -0.50352 -0.08971 0.58891 1.74713
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -8.64453 0.12539 -68.940 < 2e-16 ***
Age55-59 0.82267 0.04461 18.443 < 2e-16 ***
Age60-64 1.54904 0.04520 34.269 < 2e-16 ***
Age65-69 2.12764 0.04619 46.066 < 2e-16 ***
Age70-74 2.69624 0.04785 56.348 < 2e-16 ***
Age75-79 3.17241 0.05003 63.408 < 2e-16 ***
Age80-84 3.47447 0.05335 65.129 < 2e-16 ***
Cohort1860-64 0.35468 0.13296 2.668 0.00764 **
Cohort1865-69 0.51920 0.12542 4.140 3.48e-05 ***
Cohort1870-74 0.77428 0.12250 6.321 2.60e-10 ***
Cohort1875-79 1.01212 0.12128 8.345 < 2e-16 ***
Cohort1880-84 1.15071 0.12066 9.536 < 2e-16 ***
Cohort1885-89 1.29932 0.12035 10.796 < 2e-16 ***
Cohort1890-94 1.54559 0.12230 12.638 < 2e-16 ***
Cohort1895-99 1.57541 0.12344 12.763 < 2e-16 ***
Cohort1900-04 1.62779 0.12502 13.021 < 2e-16 ***
Cohort1905-09 1.46431 0.12792 11.447 < 2e-16 ***
Cohort1910-14 1.37192 0.13347 10.279 < 2e-16 ***
Cohort1915-19 1.25628 0.15029 8.359 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 9139.332 on 48 degrees of freedom
Residual deviance: 29.984 on 30 degrees of freedom
AIC: 193.63
Number of Fisher Scoring iterations: 3
Call:
glm(formula = incid ~ offset(log(pop)) + Age + Cohort, family = poisson(log),
data = holford, weights = disp.weights)
Deviance Residuals:
Min 1Q Median 3Q Max
-1.77737 -0.50352 -0.08971 0.58891 1.74713
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -8.64453 0.12539 -68.940 < 2e-16 ***
Age55-59 0.82267 0.04461 18.443 < 2e-16 ***
Age60-64 1.54904 0.04520 34.269 < 2e-16 ***
Age65-69 2.12764 0.04619 46.066 < 2e-16 ***
Age70-74 2.69624 0.04785 56.348 < 2e-16 ***
Age75-79 3.17241 0.05003 63.408 < 2e-16 ***
Age80-84 3.47447 0.05335 65.129 < 2e-16 ***
Cohort1860-64 0.35468 0.13296 2.668 0.00764 **
Cohort1865-69 0.51920 0.12542 4.140 3.48e-05 ***
Cohort1870-74 0.77428 0.12250 6.321 2.60e-10 ***
Cohort1875-79 1.01212 0.12128 8.345 < 2e-16 ***
Cohort1880-84 1.15071 0.12066 9.536 < 2e-16 ***
Cohort1885-89 1.29932 0.12035 10.796 < 2e-16 ***
Cohort1890-94 1.54559 0.12230 12.638 < 2e-16 ***
Cohort1895-99 1.57541 0.12344 12.763 < 2e-16 ***
Cohort1900-04 1.62779 0.12502 13.021 < 2e-16 ***
Cohort1905-09 1.46431 0.12792 11.447 < 2e-16 ***
Cohort1910-14 1.37192 0.13347 10.279 < 2e-16 ***
Cohort1915-19 1.25628 0.15029 8.359 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 9139.332 on 48 degrees of freedom
Residual deviance: 29.984 on 30 degrees of freedom
AIC: 193.63
Number of Fisher Scoring iterations: 3
[1] 0.003618045
dispmod documentation built on May 2, 2019, 2:48 p.m.
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Description Usage Arguments Details Value Note References See Also Examples
Description
This function estimates overdispersed Poisson log-linear models using the approach discussed by Breslow N.E. (1984).
Usage
1 | glm.poisson.disp(object, maxit = 30, verbose = TRUE)
|
Arguments
object |
an object of class |
maxit |
integer giving the maximal number of iterations for the model fitting procedure. |
verbose |
logical, if |
Details
Breslow (1984) proposed an iterative algorithm for fitting overdispersed Poisson log-linear models. The method is similar to that proposed by Williams (1982) for handling overdispersion in logistic regression models (see glm.binomial.disp).
Suppose we observe n independent responses such that
y_i | λ_i ~ Poisson(λ_i n_i)
for i = 1, …, n. The response variable y_i may be an event counts variable observed over a period of time (or in the space) of length n_i, whereas λ_i is the rate parameter. Then,
E(y_i | λ_i) = μ_i = λ_i n_i = exp(log(n_i) + log(λ_i))
where log(n_i) is an offset and log(λ_i) = β'x_i expresses the dependence of the Poisson rate parameter on a set of, say p, predictors. If the periods of time are all of the same length, we can set n_i = 1 for all i so the offset is zero.
The Poisson distribution has E(y_i | λ_i) = V(y_i | λ_i), but it may happen that the actual variance exceeds the nominal variance under the assumed probability model.
Suppose that θ_i = λ_i n_i is a random variable distributed according to
θ_i ~ Gamma (μ_i, 1/φ)
where E(θ_i) = μ_i and V(θ_i) = μ_i^2 φ. Thus, it can be shown that the unconditional mean and variance of y_i are given by
E(y_i) = μ_i
and
V(y_i) = μ_i + μ_i^2 φ = μ_i (1 + μ_iφ)
Hence, for φ > 0 we have overdispersion. It is interesting to note that the same mean and variance arise also if we assume a negative binomial distribution for the response variable.
The method proposed by Breslow uses an iterative algorithm for estimating the dispersion parameter φ and hence the necessary weights 1/(1 + μ_i \hat{φ}) (for details see Breslow, 1984).
Value
The function returns an object of class "glm" with the usual information and the added components:
dispersion |
the estimated dispersion parameter. |
disp.weights |
the final weights used to fit the model. |
Note
Based on a similar procedure available in Arc (Cook and Weisberg, http://www.stat.umn.edu/arc)
References
Breslow, N.E. (1984), Extra-Poisson variation in log-linear models, Applied Statistics, 33, 38–44.
See Also
lm, glm, lm.disp, glm.binomial.disp
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 | ## Salmonella TA98 data
data(salmonellaTA98)
salmonellaTA98 <- within(salmonellaTA98, logx10 <- log(x+10))
mod <- glm(y ~ logx10 + x, data = salmonellaTA98, family = poisson(log))
summary(mod)
mod.disp <- glm.poisson.disp(mod)
summary(mod.disp)
mod.disp$dispersion
# compute predictions on a grid of x-values...
x0 <- with(salmonellaTA98, seq(min(x), max(x), length=50))
eta0 <- predict(mod, newdata = data.frame(logx10 = log(x0+10), x = x0), se=TRUE)
eta0.disp <- predict(mod.disp, newdata = data.frame(logx10 = log(x0+10), x = x0), se=TRUE)
# ... and plot the mean functions with variability bands
plot(y ~ x, data = salmonellaTA98)
lines(x0, exp(eta0$fit))
lines(x0, exp(eta0$fit+2*eta0$se), lty=2)
lines(x0, exp(eta0$fit-2*eta0$se), lty=2)
lines(x0, exp(eta0.disp$fit), col=3)
lines(x0, exp(eta0.disp$fit+2*eta0.disp$se), lty=2, col=3)
lines(x0, exp(eta0.disp$fit-2*eta0.disp$se), lty=2, col=3)
## Holford's data
data(holford)
mod <- glm(incid ~ offset(log(pop)) + Age + Cohort, data = holford,
family = poisson(log))
summary(mod)
mod.disp <- glm.poisson.disp(mod)
summary(mod.disp)
mod.disp$dispersion
|
Example output
Call:
glm(formula = y ~ logx10 + x, family = poisson(log), data = salmonellaTA98)
Deviance Residuals:
Min 1Q Median 3Q Max
-2.5173 -1.1731 -0.4032 0.7995 3.7041
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 2.1727730 0.2184269 9.947 < 2e-16 ***
logx10 0.3198250 0.0570014 5.611 2.01e-08 ***
x -0.0010130 0.0002452 -4.131 3.61e-05 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 78.358 on 17 degrees of freedom
Residual deviance: 43.716 on 15 degrees of freedom
AIC: 142.25
Number of Fisher Scoring iterations: 4
Poisson overdispersed log-linear model fitting...
Iter. 1 phi: 0.07178889
Iter. 2 phi: 0.07180372
Iter. 3 phi: 0.07180614
Iter. 4 phi: 0.07180702
Iter. 5 phi: 0.07180731
Iter. 6 phi: 0.07180741
Iter. 7 phi: 0.07180744
Iter. 8 phi: 0.07180745
Iter. 9 phi: 0.07180745
Iter. 10 phi: 0.07180745
Converged after 10 iterations.
Estimated dispersion parameter: 0.07180745
Call:
glm(formula = y ~ logx10 + x, family = poisson(log), data = salmonellaTA98,
weights = disp.weights)
Deviance Residuals:
Min 1Q Median 3Q Max
-1.4419 -0.6272 -0.2374 0.4564 1.9937
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 2.2030681 0.3635996 6.059 1.37e-09 ***
logx10 0.3109623 0.0990523 3.139 0.00169 **
x -0.0009741 0.0004371 -2.228 0.02585 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 25.313 on 17 degrees of freedom
Residual deviance: 14.214 on 15 degrees of freedom
AIC: 50.799
Number of Fisher Scoring iterations: 4
Call:
glm(formula = y ~ logx10 + x, family = poisson(log), data = salmonellaTA98,
weights = disp.weights)
Deviance Residuals:
Min 1Q Median 3Q Max
-1.4419 -0.6272 -0.2374 0.4564 1.9937
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 2.2030681 0.3635996 6.059 1.37e-09 ***
logx10 0.3109623 0.0990523 3.139 0.00169 **
x -0.0009741 0.0004371 -2.228 0.02585 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 25.313 on 17 degrees of freedom
Residual deviance: 14.214 on 15 degrees of freedom
AIC: 50.799
Number of Fisher Scoring iterations: 4
[1] 0.07180745
Call:
glm(formula = incid ~ offset(log(pop)) + Age + Cohort, family = poisson(log),
data = holford)
Deviance Residuals:
Min 1Q Median 3Q Max
-3.214 -1.205 -0.172 1.462 3.670
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -8.62551 0.10123 -85.208 < 2e-16 ***
Age55-59 0.81903 0.02871 28.526 < 2e-16 ***
Age60-64 1.55271 0.02790 55.655 < 2e-16 ***
Age65-69 2.13322 0.02751 77.540 < 2e-16 ***
Age70-74 2.68681 0.02803 95.844 < 2e-16 ***
Age75-79 3.13410 0.02910 107.689 < 2e-16 ***
Age80-84 3.45545 0.03143 109.930 < 2e-16 ***
Cohort1860-64 0.36540 0.10901 3.352 0.000803 ***
Cohort1865-69 0.52338 0.10221 5.120 3.05e-07 ***
Cohort1870-74 0.77303 0.09976 7.749 9.25e-15 ***
Cohort1875-79 1.00959 0.09883 10.216 < 2e-16 ***
Cohort1880-84 1.15061 0.09832 11.702 < 2e-16 ***
Cohort1885-89 1.30700 0.09803 13.333 < 2e-16 ***
Cohort1890-94 1.50956 0.09856 15.317 < 2e-16 ***
Cohort1895-99 1.55288 0.09882 15.713 < 2e-16 ***
Cohort1900-04 1.59181 0.09932 16.027 < 2e-16 ***
Cohort1905-09 1.46105 0.10074 14.503 < 2e-16 ***
Cohort1910-14 1.36926 0.10396 13.171 < 2e-16 ***
Cohort1915-19 1.23725 0.11616 10.651 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 35493.66 on 48 degrees of freedom
Residual deviance: 127.38 on 30 degrees of freedom
AIC: 571.65
Number of Fisher Scoring iterations: 3
Poisson overdispersed log-linear model fitting...
Iter. 1 phi: 0.004150272
Iter. 2 phi: 0.003747225
Iter. 3 phi: 0.003651023
Iter. 4 phi: 0.003626604
Iter. 5 phi: 0.003620276
Iter. 6 phi: 0.003618627
Iter. 7 phi: 0.003618197
Iter. 8 phi: 0.003618085
Iter. 9 phi: 0.003618056
Iter. 10 phi: 0.003618048
Iter. 11 phi: 0.003618046
Iter. 12 phi: 0.003618045
Iter. 13 phi: 0.003618045
Converged after 13 iterations.
Estimated dispersion parameter: 0.003618045
Call:
glm(formula = incid ~ offset(log(pop)) + Age + Cohort, family = poisson(log),
data = holford, weights = disp.weights)
Deviance Residuals:
Min 1Q Median 3Q Max
-1.77737 -0.50352 -0.08971 0.58891 1.74713
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -8.64453 0.12539 -68.940 < 2e-16 ***
Age55-59 0.82267 0.04461 18.443 < 2e-16 ***
Age60-64 1.54904 0.04520 34.269 < 2e-16 ***
Age65-69 2.12764 0.04619 46.066 < 2e-16 ***
Age70-74 2.69624 0.04785 56.348 < 2e-16 ***
Age75-79 3.17241 0.05003 63.408 < 2e-16 ***
Age80-84 3.47447 0.05335 65.129 < 2e-16 ***
Cohort1860-64 0.35468 0.13296 2.668 0.00764 **
Cohort1865-69 0.51920 0.12542 4.140 3.48e-05 ***
Cohort1870-74 0.77428 0.12250 6.321 2.60e-10 ***
Cohort1875-79 1.01212 0.12128 8.345 < 2e-16 ***
Cohort1880-84 1.15071 0.12066 9.536 < 2e-16 ***
Cohort1885-89 1.29932 0.12035 10.796 < 2e-16 ***
Cohort1890-94 1.54559 0.12230 12.638 < 2e-16 ***
Cohort1895-99 1.57541 0.12344 12.763 < 2e-16 ***
Cohort1900-04 1.62779 0.12502 13.021 < 2e-16 ***
Cohort1905-09 1.46431 0.12792 11.447 < 2e-16 ***
Cohort1910-14 1.37192 0.13347 10.279 < 2e-16 ***
Cohort1915-19 1.25628 0.15029 8.359 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 9139.332 on 48 degrees of freedom
Residual deviance: 29.984 on 30 degrees of freedom
AIC: 193.63
Number of Fisher Scoring iterations: 3
Call:
glm(formula = incid ~ offset(log(pop)) + Age + Cohort, family = poisson(log),
data = holford, weights = disp.weights)
Deviance Residuals:
Min 1Q Median 3Q Max
-1.77737 -0.50352 -0.08971 0.58891 1.74713
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -8.64453 0.12539 -68.940 < 2e-16 ***
Age55-59 0.82267 0.04461 18.443 < 2e-16 ***
Age60-64 1.54904 0.04520 34.269 < 2e-16 ***
Age65-69 2.12764 0.04619 46.066 < 2e-16 ***
Age70-74 2.69624 0.04785 56.348 < 2e-16 ***
Age75-79 3.17241 0.05003 63.408 < 2e-16 ***
Age80-84 3.47447 0.05335 65.129 < 2e-16 ***
Cohort1860-64 0.35468 0.13296 2.668 0.00764 **
Cohort1865-69 0.51920 0.12542 4.140 3.48e-05 ***
Cohort1870-74 0.77428 0.12250 6.321 2.60e-10 ***
Cohort1875-79 1.01212 0.12128 8.345 < 2e-16 ***
Cohort1880-84 1.15071 0.12066 9.536 < 2e-16 ***
Cohort1885-89 1.29932 0.12035 10.796 < 2e-16 ***
Cohort1890-94 1.54559 0.12230 12.638 < 2e-16 ***
Cohort1895-99 1.57541 0.12344 12.763 < 2e-16 ***
Cohort1900-04 1.62779 0.12502 13.021 < 2e-16 ***
Cohort1905-09 1.46431 0.12792 11.447 < 2e-16 ***
Cohort1910-14 1.37192 0.13347 10.279 < 2e-16 ***
Cohort1915-19 1.25628 0.15029 8.359 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 9139.332 on 48 degrees of freedom
Residual deviance: 29.984 on 30 degrees of freedom
AIC: 193.63
Number of Fisher Scoring iterations: 3
[1] 0.003618045
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