Anscombe Residuals

Fit Analyses

Anscombe Residuals

For nonnormal response distributions in generalized linear models, the distribution of the Pearson residuals is often skewed. Anscombe proposed a residual using a function A(y) in place of y in the residual derivation (Anscombe 1953, McCullagh and Nelder 1989). The function A(y) is chosen to make the distribution of A(y) as normal as possible and is given by

$A(\mu) = \int_{-{\infty}}^{\mu}{V^{-1/3}(t) dt }$

where V(t) is the variance function.

For a binomial distribution with m_i trials in the ith observation, the Anscombe residual is defined as

$r_{Ai} = \sqrt{ m_{i}} \frac{A( y_{i}) -A(\hat{ \mu_{i}})}{{A'}(\hat{ \mu_{i}}) \sqrt{V(\hat{ \mu_{i}})}}$

For other distributions, the Anscombe residual is defined as

$r_{Ai} = \frac{A( y_{i}) -A(\hat{ \mu_{i}})}{{A'}(\hat{ \mu_{i}}) \sqrt{V(\hat{ \mu_{i}})} }$

where ${{A'}(\mu)}$ is the derivative of ${A(\mu)}$ .

For the response distributions used in the fit analysis, Anscombe residuals are

Normal: ${ r_{Ai} = y_{i}-\hat{ \mu_{i}} }$
Inverse Gaussian: ${ r_{Ai} = ( \log( y_{i})- \log(\hat{ \mu_{i}}) ) / \hat{ \mu_{i}}^{1/2}}$
Gamma: ${ r_{Ai} = 3 ( ( y_{i} / \hat{ \mu_{i}})^{1/3} - 1 ) }$
Poisson: ${ r_{Ai} = \frac{3}2 ( y_{i}^{2/3} \hat{ \mu_{i}}^{-1/6} - \hat{ \mu_{i}}^{1/2}) }$
Binomial: $r_{Ai} = \sqrt{ m_{i}} ( B( y_{i},\frac{2}3,\frac{2}3)- B(\hat{ \mu_{i}},\frac{2}3,\frac{2}3) ) (\hat{ \mu_{i}} (1-\hat{ \mu_{i}}))^{-1/6}$

where $B(z,a,b) = \int_{0}^zt^{a-1}(1-t)^{b-1}\, dt$

You can save Anscombe residuals to your data set by using the Output Variables dialog, as shown in Figure 39.5, or the Vars menu, as shown in Figure 39.48. These residuals are stored in variables named RA_yname for each response variable, where yname is the response variable name.

The standardized and studentized Anscombe residuals are

$r_{Asi} = \frac{r_{Ai}}{\sqrt{\hat{\phi} (1- h_{i})} }$

$r_{Ati} = \frac{r_{Ai}}{\sqrt{ \hat{\phi}_{(i)} (1- h_{i})} }$

where ${\hat{ \phi}}$ is the estimate of the dispersion parameter $\phi$ , and ${ \hat{ \phi}_{(i)}}$ is a one-step approximation of $\phi$ after excluding the ith observation.

The standardized Anscombe residuals are stored in variables named RAS_yname and the studentized Anscombe residuals are stored in variables named RAT_yname for each response variable, where yname is the response variable name.

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