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I am running models in R using an OLS as well as using a GLM with a Poisson distribution and log link function.

df <- data.frame(x1=runif(100), x2=runif(100))

df$ols_y <- rnorm(100)
df$glm_y <- rpois(100, lambda=5)


ols_mod <- lm(ols_y ~ x1 + x2, data=df)
glm_mod <- glm(glm_y ~ x1 + x2, data=df, family=poisson)

Now, I can recover the original y values if I use the ols model to predict the training data and add back the residual (rounding to avoid floating-point issues). I can also use fitted.values instead of re-running the model on the training data:

all(round(ols_mod$fitted.values + ols_mod$residuals, 5) == round(df$ols_y, 5))
> TRUE
all(round(predict(ols_mod, df) + ols_mod$residuals, 5) == round(df$ols_y, 5))
> TRUE

But if I do it with my Poisson model, even using type="response", the values don't match.

all(round(predict(glm_mod, df, type="response") + glm_mod$residuals, 5) == round(df$glm_y, 5))
> FALSE
all(round(glm_mod$fitted.values + glm_mod$residuals, 5) == round(df$glm_y, 5))
> FALSE

Assuming this was due to the log-link, I also tried to exponentiate the residuals to get them on the scale of the linear predictor, but this also fails.

all(round(predict(glm_mod, df, type="response") + exp(glm_mod$residuals), 5) == round(df$glm_y, 5))

What am I missing here? Why aren't the residuals on the scale of the response variable? Why isn't $\hat{y}_i + r_i = y_i$ ?

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  • $\begingroup$ Take a look at the help page at ?residuals.glm. What you are looking at are deviance residuals $\endgroup$
    – Knarpie
    Dec 11, 2023 at 10:08
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    $\begingroup$ @Knarpie, the glm$residuals object contains working residuals. It's still correct that you need residuals(glm_mod, type="response"). $\endgroup$
    – PBulls
    Dec 11, 2023 at 10:14
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    $\begingroup$ Mathematical error: exponentiation couldn't possibly retrieve negative residuals, regardless of what you feed it. $\endgroup$
    – Nick Cox
    Dec 11, 2023 at 11:24
  • $\begingroup$ What happens if you exponentiate the absolute value, somehow keeping track of which were negative? $\endgroup$
    – Peter Flom
    Dec 11, 2023 at 11:59

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