Goodness‐of‐Fit Methods for Generalized Linear Mixed Models

Abstract

We develop graphical and numerical methods for checking the adequacy of generalized linear mixed models (GLMMs). These methods are based on the cumulative sums of residuals over covariates or predicted values of the response variable. Under the assumed model, the asymptotic distributions of these stochastic processes can be approximated by certain zero-mean Gaussian processes, whose realizations can be generated through Monte Carlo simulation. Each observed process can then be compared, both visually and analytically, to a number of realizations simulated from the null distribution. These comparisons enable one to assess objectively whether the observed residual patterns reflect model misspecification or random variation. The proposed methods are particularly useful for checking the functional form of a covariate or the link function. Extensive simulation studies show that the proposed goodness-of-fit tests have proper sizes and are sensitive to model misspecification. Applications to two medical studies lead to improved models.

Unilateral upper-limb loss: Satisfaction and prosthetic-device use in veterans and servicemembers from Vietnam and OIF/OEF conflicts

“…Potential interactions were assessed using the log-likelihood ratio. Goodness of fit of the final model was assessed using diagnostic plots of residual errors, and outliers were investigated [27]. Due to the low frequency of limb loss in the wrist, elbow, and shoulder levels, only individuals with transfemoral and transradial limb-loss levels were included in the models.…”

Cumulative Sums of Residuals in GLMM and Its Implementation

“…This type of test statistic was firstly studied in GLMs by Lin et al (2002), and later extended to GLMMs by Pan and Lin (2005). We have a target to implement this testing procedure in a freely accessible package.…”

“…But we have some difficulty to confirm the model misspecification or random fluctuation from a simple plot of residuals. After Pan and Lin (2005) proposed the objective This work was supported by a 2-year Research Grant of Pusan National University.and informative procedures using residuals, the GOF test based on CUSUM has a widespread use as a complementary method for the overall GOF tests such as Pearson's chi-squared statistic and deviance statistic.…”

“…Recently, Pan and Lin (2005) extended this testing procedure to the generalized linear mixed models(GLMM) having random effects, in which we encounter difficulties in computing the marginal likelihood that is expressed as an integral of random effects distribution. The Gaussian quadrature algorithm is commonly used to approximate the marginal likelihood.…”

“…Some kinds of GOF tests using cumulative sums of residuals(CUSUM) are so popular that they can be performed by common statistical packages, for example, PROC GENMOD in SAS, or gof package in R. They provide the empirical p-value with the graphical plot of CUSUM statistic. The asymptotic processes of cumulative sums(CUSUM) of residuals in various models have been studied by many researchers among which we may refer to Pierce and Schafer (1986), Su and Wei (1991), Cook and Weisberg (1994), Stute (1997), Lin et al (2002), Pan and Lin (2005).…”