Generalized estimating equations for ordinal categorical data: arbitrary patterns of missing responses and missingness in a key covariate

We propose methods for regression analysis of repeatedly measured ordinal categorical data when there is nonmonotone missingness in these responses and when a key covariate is missing depending on observables. The methods use ordinal regression models in conjunction with generalized estimating equations (GEEs). We extend the GEE methodology to accommodate arbitrary patterns of missingness in the responses when this missingness is independent of the unobserved responses. We further extend the methodology to provide correction for possible bias when missingness in knowledge of a key covariate may depend on observables. The approach is illustrated with the analysis of data from a study in diagnostic oncology in which multiple correlated receiver operating characteristic curves are estimated and corrected for possible verification bias when the true disease status is missing depending on observables.     

A tale of two defectors: the importance of standing for evolution of indirect reciprocity

Indirect reciprocity occurs when the cooperative behavior between two individuals is contingent on their previous behavior toward others. Previous theoretical analysis indicates that indirect reciprocity can evolve if individuals use an image-scoring strategy. In this paper, we show that, when errors are added, indirect reciprocity cannot be based on an image-scoring strategy. However, if individuals use a standing strategy, then cooperation through indirect reciprocity is evolutionarily stable. These two strategies differ with respect to the information to which they attend. While image-scoring strategies only need attend to the actions of others, standing strategies also require information about intent. We speculate that this difference may shed light on the evolvability of indirect reciprocity. Additionally, we show that systems of indirect reciprocity are highly sensitive to the availability of information. Finally, we present a model which shows that if indirect reciprocity were to evolve, selection should also favor trusting behavior in relations between strangers.     

Information gain for genetic parameter estimation with incorporation of marker data

Genetic marker data has been increasingly incorporated into segregation analysis, as combined segregation and linkage analysis has been performed more frequently. In this article, we study the extent of information gains with incorporation of marker data in segregation analysis, a topic that has not been investigated rigorously. Specifically, the current study is to investigate the influence of marker data on genetic model parameter estimation. A variance matrix criterion (as the inverse of the Fisher information matrix) and a relative entropy criterion (a measure of flatness of expected log-likelihood surface) are used to quantify the information gains. Our results indicate that substantial information gain can be achieved with the incorporation of marker data. The amount of variance reduction increases as the heterozygosity of the linked marker increases and as the trait gets closer to the linked marker(s). Incorporation of marker data in larger pedigrees also yields greater information gains based on both criteria. The effect of pedigree structure is also studied.     

Exact log-rank tests for unequal follow-up

The asymptotic log-rank and generalized Wilcoxon tests are the standard procedures for comparing samples of possibly censored survival times. For comparison of samples of very different sizes, an exact test is available that is based on a complete permutation of log-rank or Wilcoxon scores. While the asymptotic tests do not keep their nominal sizes if sample sizes differ substantially, the exact complete permutation test requires equal follow-up of the samples. Therefore, we have developed and present two new exact tests also suitable for unequal follow-up. The first of these is an exact analogue of the asymptotic log-rank test and conditions on observed risk sets, whereas the second approach permutes survival times while conditioning on the realized follow-up in each group. In an empirical study, we compare the new procedures with the asymptotic log-rank test, the exact complete permutation test, and an earlier proposed approach that equalizes the follow-up distributions using artificial censoring. Results confirm highly satisfactory performance of the exact procedure conditioning on realized follow-up, particularly in case of unequal follow-up. The advantage of this test over other options of analysis is finally exemplified in the analysis of a breast cancer study.     

A comparison of methods for predicting vegetation type

Predictive modeling of vegetation patterns has wide application in vegetation science. In this paper I discuss three methods of predictive modeling using data from the alpine treeline ecotone as a case study. The study area is a portion of Glacier National Park, Montana. Parametric general linear models (GLM), artificial neural networks (ANN) and classification tree (CT) methods of predicting vegetation type are compared to determine the relative strength of each predictive approach and how they may be used in concert to increase understanding of important vegetation – environment relations. For each predictive method, vegetation type within the alpine treeline ecotone is predicted using a suite of environmental indicator variables including elevation, moisture potential, solar radiation potential, snow potential index, and disturbance history. Results from each of the predictive methods are compared against the real vegetation types to determine the relative accuracy of the methods.When the entire data field is examined (i.e., not evaluated by smaller spatial aggregates of data) the ANN procedure produces the most accurate predictions (=0.571); the CT predictions are the least accurate (=0.351). The predicted patterns of vegetation on the landscape are considerably different using the three methods. The GLM and CT methods produce large contiguous swaths of vegetation types throughout the study area, whereas the ANN method produces patterns with much more heterogeneity and smaller patches.When predictions are compared to reality at catchment scale, it becomes evident that the accuracy of each method varies depending upon the specific situation. The ANN procedure remains the most accurate method in the majority of the catchments, but both the GLM and PCT produce the most accurate classifications in at least one basin each.The variability in predictive ability of the three methods tested here indicates that there may not be a single best predictive method. Rather it may be important to use a suite of predictive models to help understand the environment – vegetation relationships. The ability to use multiple predictive methods to determine which spatial subunits of a landscape are outliers is important when identifying locations useful for climate change monitoring studies.     

Involvement of glycogen synthase kinase-3beta and tau phosphorylation in neuronal Golgi disassembly

The dissociation of the neuronal Golgi complex is a classical feature observed in neurodegenerative disorders including Alzheimer's disease. The goal of this study is to determine if the phosphorylation of tau protein is involved in neuronal Golgi disassembly. Primary cortical cultures were exposed to two Golgi toxins, brefeldin A (BFA) or nordihydroguaiaretic acid (NDGA). Immunocytochemical studies using the anti58 k antibody revealed that Golgi disassembly started in exposed neurons a few minutes after treatment. BFA and NDGA induced a rapid and transient increase in tau phosphorylation in a site-specific manner on immunoblots. In addition, the increase in tau phosphorylation directly correlated with a transient dissociation of tau from the cytoskeleton and a decrease of the acetylated tubulin. Furthermore, the activity of glycogen synthase kinase-3beta (GSK-3beta) increased transiently, as demonstrated by the kinase activity assay and by immunoblottings of serine-9 and tyrosine-216 phosphorylated of GSK-3beta. A decrease of the Akt phosphorylated form was also shown. The increase in tau phosphorylation was inhibited by the GSK-3beta inhibitor, lithium. Finally, morphometric studies showed that lithium partially blocked the Golgi disassembly caused by BFA or NDGA. Together these findings indicate that GSK-3beta activity and tau phosphorylation state are involved in the maintenance of the neuronal Golgi organization.     

Dephosphorylation of tau protein by calcineurin triturated into neural living cells

Alzheimer disease and related dementia are characterized by the presence of hyperphosphorylated tau aggregated into filaments. The role of tau phosphorylation in the fibrillogenesis has not yet been unraveled. Therefore, it is important to know which phosphatases can dephosphorylate tau protein in vivo. The effect of recombinant purified calcineurin (CN(PP2B)) and several calcineurin mutants on tau phosphorylation was studied in two neuronal like cell lines PC12 and SH-SY5Y. The modulation of tau phosphorylation at Ser199/Ser202, Ser396/Ser404, Ser262/Ser356, and Thr181 sites was examined in these cell lines using the phosphorylation state-dependent antitau antibodies Tau 1, PHF1, 12E8, and AT270. The results have shown that CN directly dephosphorylates all of those sites of tau protein. Recombinant calcineurin introduced into cells that have previously been treated with okadaic acid and cyclosporin A, which are inhibitors of phosphatases (PP1/PP2A and PP2B), has a direct effect on the phosphorylation status on all phosphorylation sites studied. We conclude that calcineurin is (besides PP2A) a important modulator of tau phosphorylation in vivo.     

MAP2 and tau bind longitudinally along the outer ridges of microtubule protofilaments

MAP2 and tau exhibit microtubule-stabilizing activities that are implicated in the development and maintenance of neuronal axons and dendrites. The proteins share a homologous COOH-terminal domain, composed of three or four microtubule binding repeats separated by inter-repeats (IRs). To investigate how MAP2 and tau stabilize microtubules, we calculated 3D maps of microtubules fully decorated with MAP2c or tau using cryo-EM and helical image analysis. Comparing these maps with an undecorated microtubule map revealed additional densities along protofilament ridges on the microtubule exterior, indicating that MAP2c and tau form an ordered structure when they bind microtubules. Localization of undecagold attached to the second IR of MAP2c showed that IRs also lie along the ridges, not between protofilaments. The densities attributable to the microtubule-associated proteins lie in close proximity to helices 11 and 12 and the COOH terminus of tubulin. Our data further suggest that the evolutionarily maintained differences observed in the repeat domain may be important for the specific targeting of different repeats to either alpha or beta tubulin. These results provide strong evidence suggesting that MAP2c and tau stabilize microtubules by binding along individual protofilaments, possibly by bridging the tubulin interfaces.     

Generalized linear mixture models for handling nonignorable dropouts in longitudinal studies

This paper presents a method for analysing longitudinal data when there are dropouts. In particular, we develop a simple method based on generalized linear mixture models for handling nonignorable dropouts for a variety of discrete and continuous outcomes. Statistical inference for the model parameters is based on a generalized estimating equations (GEE) approach (Liang and Zeger, 1986). The proposed method yields estimates of the model parameters that are valid when nonresponse is nonignorable under a variety of assumptions concerning the dropout process. Furthermore, the proposed method can be implemented using widely available statistical software. Finally, an example using data from a clinical trial of contracepting women is used to illustrate the methodology.     

On power and sample size calculations for likelihood ratio tests in generalized linear models

A direct extension of the approach described in Self, Mauritsen, and Ohara (1992, Biometrics 48, 31-39) for power and sample size calculations in generalized linear models is presented. The major feature of the proposed approach is that the modification accommodates both a finite and an infinite number of covariate configurations. Furthermore, for the approximation of the noncentrality of the noncentral chi-square distribution for the likelihood ratio statistic, a simplification is provided that not only reduces substantial computation but also maintains the accuracy. Simulation studies are conducted to assess the accuracy for various model configurations and covariate distributions.