Despite the versatility of generalized linear mixed models in handling complex experimental designs, they often suffer from misspecification and convergence problems. This makes inference on the values of coefficients problematic. In addition, the researcher’s choice of random and fixed effects directly affects statistical inference correctness. To address these challenges, we propose a robust extension of the “two-stage summary statistics” approach using sign-flipping transformations of the score statistic in the second stage. Our approach efficiently handles within-variance structure and heteroscedasticity, ensuring accurate regression coefficient testing for 2-level hierarchical data structures. The approach is illustrated by analyzing the reduction of health issues over time for newly adopted children. The model is characterized by a binomial response with unbalanced frequencies and several categorical and continuous predictors. The proposed approach efficiently deals with critical problems related to longitudinal nonlinear models, surpassing common statistical approaches such as generalized estimating equations and generalized linear mixed models.

Maximum likelihood estimation of generalized linear mixed models (GLMMs) is difficult due to marginalization of the random effects. Derivative computations of a fitted GLMM’s likelihood are also difficult, especially because the derivatives are not by-products of popular estimation algorithms. In this paper, we first describe theoretical results related to GLMM derivatives along with a quadrature method to efficiently compute the derivatives, focusing on fitted lme4 models with a single clustering variable. We describe how psychometric results related to item response models are helpful for obtaining the derivatives, as well as for verifying the derivatives’ accuracies. We then provide a tutorial on the many possible uses of these derivatives, including robust standard errors, score tests of fixed effect parameters, and likelihood ratio tests of non-nested models. The derivative computation methods and applications described in the paper are all available in easily obtained R packages.

Major depression is a severe mental disorder that is associated with strongly increased mortality. The quantification of its prevalence on regional levels represents an important indicator for public health reporting. In addition to that, it marks a crucial basis for further explorative studies regarding environmental determinants of the condition. However, assessing the distribution of major depression in the population is challenging. The topic is highly sensitive, and national statistical institutions rarely have administrative records on this matter. Published prevalence figures as well as available auxiliary data are typically derived from survey estimates. These are often subject to high uncertainty due to large sampling variances and do not allow for sound regional analysis. We propose a new area-level Poisson mixed model that accounts for measurement errors in auxiliary data to close this gap. We derive the empirical best predictor under the model and present a parametric bootstrap estimator for the mean squared error. A method of moments algorithm for consistent model parameter estimation is developed. Simulation experiments are conducted to show the effectiveness of the approach. The methodology is applied to estimate the major depression prevalence in Germany on regional levels crossed by sex and age groups.

Composite links and exploded likelihoods are powerful yet simple tools for specifying a wide range of latent variable models. Applications considered include survival or duration models, models for rankings, small area estimation with census information, models for ordinal responses, item response models with guessing, randomized response models, unfolding models, latent class models with random effects, multilevel latent class models, models with log-normal latent variables, and zero-inflated Poisson models with random effects. Some of the ideas are illustrated by estimating an unfolding model for attitudes to female work participation.

Despite the prevalence of catch per unit effort (CPUE) as a key metric in fisheries assessments it can be fraught with inherent problems that often cause its use as an index of abundance to become contentious. This is particularly the case with abalone, a sedentary shellfish targeted by commercial dive fishers around the globe. It is common practice to standardize CPUE to at least partly address issues about how well it reflects the actual abundance of a stock. Differences between standardized and unstandardized trends may lead to controversy between scientists and stakeholders when standardized trends provide a less optimistic picture of stock status. It is within this context that we applied Linear Mixed Model (LMM) and Generalized Linear Mixed Model (GLMM) methods to standardize CPUE for the Western Zone blacklip abalone fishery in Victoria, Australia. This fishery was chosen for our evaluation because it included substantial population losses from a disease shock during the middle of the time series. The effects of diver, reef location, month and their interactions with year were included as random effects in these models and the results compared with nominal geometric means. The two standardization methods provided similar standardized CPUE trends and clearly demonstrated that a large proportion of the variance could be attributed to diver and spatial effects. The GLMM seemed to explain more variability in the data and produced better precision for standardized CPUEs than LMM. The temporal trend in variability attributed to divers and spatial scales reveals the impact of disease as well as any homo/heterogeneity effect. The CPUE trends responded to the impact of disease against a backdrop of declining stock, however when compared with the inter-annual pattern in nominal CPUE, the standardized trends showed that the decline immediately following the onset of disease was less precipitous. In contrast to what appeared to be an increase in the nominal series during the more recent post-disease period, there was only a slight non-significant increase observable in the standardized trends.

Despite well-recognized heterogeneity in malaria transmission, key parameters such as the force of infection (FOI) are generally estimated ignoring the intrinsic variability in individual infection risks. Given the potential impact of heterogeneity on the estimation of the FOI, we estimate this quantity accounting for both observed and unobserved heterogeneity. We used cohort data of children aged 0·5–10 years evaluated for the presence of malaria parasites at three sites in Uganda. Assuming a Susceptible–Infected–Susceptible model, we show how the FOI relates to the point prevalence, enabling the estimation of the FOI by modelling the prevalence using a generalized linear mixed model. We derive bounds for varying parasite clearance distributions. The resulting FOI varies significantly with age and is estimated to be highest among children aged 5–10 years in areas of high and medium malaria transmission and highest in children aged below 1 year in a low transmission setting. Heterogeneity is greater between than within households and it increases with decreasing risk of malaria infection. This suggests that next to the individual's age, heterogeneity in malaria FOI may be attributed to household conditions. When estimating the FOI, accounting for both observed and unobserved heterogeneity in malaria acquisition is important for refining malaria spread models.

Giant ragweed is a competitive, allergenic weed that persists in agricultural fields and early successional sites. Field experiments were conducted to determine the effects of seed size and seed burial depth on giant ragweed emergence and seed demise. In a seedling emergence experiment, small (< 4.8 mm in diameter) and large (> 6.6 mm in diameter) seeds were buried 0, 5, 10, and 20 cm in fall 1997, and weed emergence was monitored over the next seven growing seasons. A generalized linear mixed model fit to the cumulative emergence data showed that maximum emergence for both seed sizes occurred at the 5-cm burial depth, where probability of emergence was 19% for small seeds and 49% for large seeds. Emergence probability at the 10-cm burial depth was 9% for small seeds and 30% for large seeds, and no seedlings emerged from the 20-cm burial depth. The model predicted that ≥ 98% of total cumulative emergence was completed after four growing seasons for large seeds buried 5 cm, five growing seasons for small seeds buried 5 cm and large seeds buried 10 cm, and seven growing seasons for small seeds buried 10 cm. Seed size and burial treatment effects on seed demise were tested in a second experiment using seed packets. Rates of seed demise were inversely proportional to burial depth, and the percentage of viable seeds remaining after 4 yr ranged from 0% on the soil surface to 19% at the 20-cm burial depth. Some seeds recovered from the 20-cm burial depth were viable after 9 yr of burial. These results, coupled with previous research, suggest that seed size polymorphism facilitates giant ragweed adaptation across habitats and that a combination of no-tillage cropping practices, habitat modification, and timely weed control measures can reduce its active seed bank in agricultural fields by 90% or more after 4 yr.

Genetics affects not only the weight of piglets at birth but also the variability of birth weight within litter. Previous studies on this topic assigned the sample standard deviation of piglet birth weights within litter as an observation to the sow. However, the contribution of the difference in mean birth weight per sex on the within-litter variance has been neglected so far. This work deals with the genetic effect on within-litter variance when different statistical models with different distributional assumptions are used and considers the sex effect and appropriate weights per trait. Traits were formed from the pooled sample variance of male and female birth weights within litter. A linear model approach was fitted to the logarithmized within-litter variance and the sample standard deviation. A generalized linear model with gamma-distributed residuals and log-link function was applied to the untransformed sample variance. Models were compared by analysing data from 9439 litters from Landrace and Large White of a commercial breeding programme. The estimates of heritability for different traits ranged from 7% to 11%. Although the generalized linear mixed model is preferred from a mathematical view, the rank correlations between breeding values of the linear mixed models and the generalized linear mixed model were relatively high, i.e. 94% to 98%. By residual diagnostics, a linear mixed model using the weighted and pooled within-litter standard deviation was identified as most suitable.

The statistical aggregation of parasites among hosts is often described empirically by the negative binomial (Poisson-gamma) distribution. Alternatively, the Poisson-lognormal model can be used. This has the advantage that it can be fitted as a generalized linear mixed model, thereby quantifying the sources of aggregation in terms of both fixed and random effects. We give a worked example, assigning aggregation in the distribution of sheep ticks Ixodes ricinus on red grouse Lagopus lagopus scoticus chicks to temporal (year), spatial (altitude and location), brood and individual effects. Apparent aggregation among random individuals in random broods fell 8-fold when spatial and temporal effects had been accounted for.