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We present generalized additive latent and mixed models (GALAMMs) for analysis of clustered data with responses and latent variables depending smoothly on observed variables. A scalable maximum likelihood estimation algorithm is proposed, utilizing the Laplace approximation, sparse matrix computation, and automatic differentiation. Mixed response types, heteroscedasticity, and crossed random effects are naturally incorporated into the framework. The models developed were motivated by applications in cognitive neuroscience, and two case studies are presented. First, we show how GALAMMs can jointly model the complex lifespan trajectories of episodic memory, working memory, and speed/executive function, measured by the California Verbal Learning Test (CVLT), digit span tests, and Stroop tests, respectively. Next, we study the effect of socioeconomic status on brain structure, using data on education and income together with hippocampal volumes estimated by magnetic resonance imaging. By combining semiparametric estimation with latent variable modeling, GALAMMs allow a more realistic representation of how brain and cognition vary across the lifespan, while simultaneously estimating latent traits from measured items. Simulation experiments suggest that model estimates are accurate even with moderate sample sizes.

We address the problem of estimating how different parts of the brain develop and change throughout the lifespan, and how these trajectories are affected by genetic and environmental factors. Estimation of these lifespan trajectories is statistically challenging, since their shapes are typically highly nonlinear, and although true change can only be quantified by longitudinal examinations, as follow-up intervals in neuroimaging studies typically cover less than 10 \% of the lifespan, use of cross-sectional information is necessary. Linear mixed models (LMMs) and structural equation models (SEMs) commonly used in longitudinal analysis rely on assumptions which are typically not met with lifespan data, in particular when the data consist of observations combined from multiple studies. While LMMs require a priori specification of a polynomial functional form, SEMs do not easily handle data with unstructured time intervals between measurements. Generalized additive mixed models (GAMMs) offer an attractive alternative, and in this paper we propose various ways of formulating GAMMs for estimation of lifespan trajectories of 12 brain regions, using a large longitudinal dataset and realistic simulation experiments. We show that GAMMs are able to more accurately fit lifespan trajectories, distinguish longitudinal and cross-sectional effects, and estimate effects of genetic and environmental exposures. Finally, we discuss and contrast questions related to lifespan research which strictly require repeated measures data and questions which can be answered with a single measurement per participant, and in the latter case, which simplifying assumptions that need to be made. The examples are accompanied with R code, providing a tutorial for researchers interested in using GAMMs.

Analyzing data from multiple neuroimaging studies has great potential in terms of increasing statistical power, enabling detection of effects of smaller magnitude than would be possible when analyzing each study separately and also allowing to systematically investigate between-study differences. Restrictions due to privacy or proprietary data as well as more practical concerns can make it hard to share neuroimaging datasets, such that analyzing all data in a common location might be impractical or impossible. Meta-analytic methods provide a way to overcome this issue, by combining aggregated quantities like model parameters or risk ratios. Most meta-analytic tools focus on parametric statistical models, and methods for meta-analyzing semi-parametric models like generalized additive models have not been well developed. Parametric models are often not appropriate in neuroimaging, where for instance age-brain relationships may take forms that are difficult to accurately describe using such models. In this paper we introduce meta-GAM, a method for meta-analysis of generalized additive models which does not require individual participant data, and hence is suitable for increasing statistical power while upholding privacy and other regulatory concerns. We extend previous works by enabling the analysis of multiple model terms as well as multivariate smooth functions. In addition, we show how meta-analytic $p$-values can be computed for smooth terms. The proposed methods are shown to perform well in simulation experiments, and are demonstrated in a real data analysis on hippocampal volume and self-reported sleep quality data from the Lifebrain consortium. We argue that application of meta-GAM is especially beneficial in lifespan neuroscience and imaging genetics. The methods are implemented in an accompanying R package \verb!metagam!, which is also demonstrated.

BayesMallows is an R package for analyzing data in the form of rankings or preferences with the Mallows rank model, and its finite mixture extension, in a Bayesian probabilistic framework. The Mallows model is a well-known model, grounded on the idea that the probability density of an observed ranking decreases exponentially fast as its distance to the location parameter increases. Despite the model being quite popular, this is the first Bayesian implementation that allows a wide choice of distances, and that works well with a large amount of items to be ranked. BayesMallows supports footrule, Spearman, Kendall, Cayley, Hamming and Ulam distances, allowing full use of the rich expressiveness of the Mallows model. This is possible thanks to the implementation of fast algorithms for approximating the partition function of the model under various distances. Although developed for being used in computing the posterior distribution of the model, these algorithms may be of interest in their own right. BayesMallows handles non-standard data: partial rankings and pairwise comparisons, even in cases including non-transitive preference patterns. The advantage of the Bayesian paradigm in this context comes from its ability to coherently quantify posterior uncertainties of estimates of any quantity of interest. These posteriors are fully available to the user, and the package comes with convienient tools for summarizing and visualizing the posterior distributions.

In many problems involving generalized linear models, the covariates are subject to measurement error. When the number of covariates p exceeds the sample size n, regularized methods like the lasso or Dantzig selector are required. Several recent papers have studied methods which correct for measurement error in the lasso or Dantzig selector for linear models in the p>n setting. We study a correction for generalized linear models based on Rosenbaum and Tsybakov's matrix uncertainty selector. By not requiring an estimate of the measurement error covariance matrix, this generalized matrix uncertainty selector has a great practical advantage in problems involving high-dimensional data. We further derive an alternative method based on the lasso, and develop efficient algorithms for both methods. In our simulation studies of logistic and Poisson regression with measurement error, the proposed methods outperform the standard lasso and Dantzig selector with respect to covariate selection, by reducing the number of false positives considerably. We also consider classification of patients on the basis of gene expression data with noisy measurements.

Ranking and comparing items is crucial for collecting information about preferences in many areas, from marketing to politics. The Mallows rank model is among the most successful approaches to analyse rank data, but its computational complexity has limited its use to a particular form based on Kendall distance. We develop new computationally tractable methods for Bayesian inference in Mallows models that work with any right-invariant distance. Our method performs inference on the consensus ranking of the items, also when based on partial rankings, such as top-k items or pairwise comparisons. We prove that items that none of the assessors has ranked do not influence the maximum a posteriori consensus ranking, and can therefore be ignored. When assessors are many or heterogeneous, we propose a mixture model for clustering them in homogeneous subgroups, with cluster-specific consensus rankings. We develop approximate stochastic algorithms that allow a fully probabilistic analysis, leading to coherent quantifications of uncertainties. We make probabilistic predictions on the class membership of assessors based on their ranking of just some items, and predict missing individual preferences, as needed in recommendation systems. We test our approach using several experimental and benchmark datasets.

Regression with the lasso penalty is a popular tool for performing dimension reduction when the number of covariates is large. In many applications of the lasso, like in genomics, covariates are subject to measurement error. We study the impact of measurement error on linear regression with the lasso penalty, both analytically and in simulation experiments. A simple method of correction for measurement error in the lasso is then considered. In the large sample limit, the corrected lasso yields sign consistent covariate selection under conditions very similar to the lasso with perfect measurements, whereas the uncorrected lasso requires much more stringent conditions on the covariance structure of the data. Finally, we suggest methods to correct for measurement error in generalized linear models with the lasso penalty, which we study empirically in simulation experiments with logistic regression, and also apply to a classification problem with microarray data. We see that the corrected lasso selects less false positives than the standard lasso, at a similar level of true positives. The corrected lasso can therefore be used to obtain more conservative covariate selection in genomic analysis.