Why glm() can't recover the true parameters?

Question

glm() of the following data gives intercept 0.56916 and slope x .018. But the true slope should be 1/10. Does anybody know why glm() can not recover the true slope? Thanks.

R> tmp = data.frame(x=seq_len(100), y=rpois(100, lambda=seq_len(100)/10))
R> fit = glm(y ~ x, family=poisson, data=tmp)
R> fit

Call:  glm(formula = y ~ x, family = poisson, data = tmp)

Coefficients:
(Intercept)            x  
    0.56916      0.01812  

Degrees of Freedom: 99 Total (i.e. Null);  98 Residual
Null Deviance:      252.6 
Residual Deviance: 125.1    AIC: 442.3
R> library(ggplot2)
R> p=qplot(tmp$x, predict(fit))
R> ggsave(p, file='/tmp/glm_poisson_fit.png')
Saving 7 x 7 in image
R> 

Answer 1

Poisson regression model is

$$ \log (\operatorname{E}(Y\mid\mathbf{x}))=\alpha + \mathbf{\beta}' \mathbf{x} $$

So you are fitting different function to your data, then the one that generated it, so it would have different parameters. Poisson regression uses by default the log as a link function. For your simulation to be in-line with the Poisson regression model, you would need to draw samples from

$$ Y \sim \mathcal{P}(\,\exp[\alpha + \mathbf{\beta}'\mathbf{x} ]\,) $$

Translating this into R code, that gives:

set.seed(123)
n <- 100

x <- seq_len(n)
y <- rpois(n, x/10)
beta <- glm(y ~ x, family='poisson')$coef
plot(x, y)
curve(x/10, min(x), max(x), col='blue', lty=2, lwd=2, add=TRUE)
curve(exp(beta[1] + beta[2] * x), min(x), max(x), col='red', lwd=2, add=TRUE)
title(expression(lambda == x/10))

x <- seq_len(n)
y <- rpois(n, exp(x/10))
beta <- glm(y ~ x, family='poisson')$coef
plot(x, y)
curve(exp(x/10), min(x), max(x), col='blue', lty=2, lwd=2, add=TRUE)
curve(exp(beta[1] + beta[2] * x), min(x), max(x), col='red', lwd=2, add=TRUE)
title(expression(lambda == exp(x/10)))

As you can see on the plots, the true regression line (blue dashed line) in first case differs from the regression line (red line) predicted by the model, while in second case they match so closely, that they are indistinguishable on the plot.