Deviance residual ($r_i^D=\mathrm{sign}(y_i-{\hat{\mu }}_i)\sqrt{d_i}$)

A deviance residual is the signed square root of an observation's contribution $d_i$ to fitted-model deviance. For a Poisson regression, $d_i=2[y_i\log (y_i/{\hat{\mu }}_i)-(y_i-{\hat{\mu }}_i)]$, with $0\log 0=0$. Its squared sum is the Poisson deviance. Leverage and estimated dispersion can further standardize these residuals; a normal quantile-quantile plot is a diagnostic approximation, not a requirement that count errors have a normal distribution.