A global goodness-of-fit statistic for Cox regression models

In this paper, a global goodness-of-fit test statistic for a Cox regression model, which has an approximate chi-squared distribution when the model has been correctly specified, is proposed. Our goodness-of-fit statistic is global and has power to detect if interactions or higher order powers of covariates in the model are needed. The proposed statistic is similar to the Hosmer and Lemeshow (1980, Communications in Statistics A10, 1043-1069) goodness-of-fit statistic for binary data as well as Schoenfeld's (1980, Biometrika 67, 145-153) statistic for the Cox model. The methods are illustrated using data from a Mayo Clinic trial in primary billiary cirrhosis of the liver (Fleming and Harrington, 1991, Counting Processes and Survival Analysis), in which the outcome is the time until liver transplantation or death. The are 17 possible covariates. Two Cox proportional hazards models are fit to the data, and the proposed goodness-of-fit statistic is applied to the fitted models.     

Assessing the goodness-of-fit of hidden Markov models

In this article, we propose a graphical technique for assessing the goodness-of-fit of a stationary hidden Markov model (HMM). We show that plots of the estimated distribution against the empirical distribution detect lack of fit with high probability for large sample sizes. By considering plots of the univariate and multidimensional distributions, we are able to examine the fit of both the assumed marginal distribution and the correlation structure of the observed data. We provide general conditions for the convergence of the empirical distribution to the true distribution, and demonstrate that these conditions hold for a wide variety of time-series models. Thus, our method allows us to compare not only the fit of different HMMs, but also that of other models as well. We illustrate our technique using a multiple sclerosis data set.     

An improved method for determining codon variability in a gene and its application to the rate of fixation of mutations in evolution

If one has the amino acid sequences of a set of homologous proteins as well as their phylogenetic relationships, one can easily determine the minimum number of mutations (nucleotide replacements) which must have been fixed in each codon since their common ancestor. It is found that for 29 species of cytochrome c the data fit the assumption that there is a group of approximately 32 invariant codons and that the remainder compose two Poisson-distributed groups of size 65 and 16 codons, the latter smaller group fixing mutations at about 3.2 times the rate of the larger. It is further found that the size of the invariant group increases as the range of species is narrowed. Extrapolation suggests that less than 10% of the codons in a given mammalian cytochrome c gene are capable of accepting a mutation. This is consistent with the view that at any one point in time only a very restricted number of positions can fix mutations but that as mutations are fixed the positions capable of accepting mutations also change so that examination of a wide range of species reveals a wide range of altered positions. We define this restricted group as the concomitantly variable codons. Given this restriction, the fixation rates for mutations in concomitantly variable codons in cytochrome c and fibrinopeptide A are not very different, a result which should be the case if most of these mutations are in fact selectively neutral as Kimura suggests.Paper number 1382 from the Laboratory of Genetics. Work performed in part at the University of Iowa, Department of Preventive Medicine and Environmental Health and Department of Statistics, Iowa City, Iowa. Computing supported by the Graduate College, University of Iowa.     

An information ratio-based goodness-of-fit test for copula models on censored data

Copula is a popular method for modeling the dependence among marginal distributions in multivariate censored data. As many copula models are available, it is essential to check if the chosen copula model fits the data well for analysis. Existing approaches to testing the fitness of copula models are mainly for complete or right-censored data. No formal goodness-of-fit (GOF) test exists for interval-censored or recurrent events data. We develop a general GOF test for copula-based survival models using the information ratio (IR) to address this research gap. It can be applied to any copula family with a parametric form, such as the frequently used Archimedean, Gaussian, and D-vine families. The test statistic is easy to calculate, and the test procedure is straightforward to implement. We establish the asymptotic properties of the test statistic. The simulation results show that the proposed test controls the type-I error well and achieves adequate power when the dependence strength is moderate to high. Finally, we apply our method to test various copula models in analyzing multiple real datasets. Our method consistently separates different copula models for all these datasets in terms of model fitness.     

A smooth test of goodness-of-fit for growth curves and monotonic nonlinear regression models

We propose a goodness-of-fit test for growth curves based on an adaptation of the data-driven smooth test paradigm. It is simple to apply and can assess the fit of a model to a set of growth experiences. A simulation study shows that for small samples, the test holds its level. Moreover, its power is found to be generally greater than existing tests. The article concludes by revisiting the long-standing problem of validating a model for the growth of human stature.     

Statistical alignment: computational properties, homology testing and goodness-of-fit

The model of insertions and deletions in biological sequences, first formulated by Thorne, Kishino, and Felsenstein in 1991 (the TKF91 model), provides a basis for performing alignment within a statistical framework. Here we investigate this model.Firstly, we show how to accelerate the statistical alignment algorithms several orders of magnitude. The main innovations are to confine likelihood calculations to a band close to the similarity based alignment, to get good initial guesses of the evolutionary parameters and to apply an efficient numerical optimisation algorithm for finding the maximum likelihood estimate. In addition, the recursions originally presented by Thorne, Kishino and Felsenstein can be simplified. Two proteins, about 1500 amino acids long, can be analysed with this method in less than five seconds on a fast desktop computer, which makes this method practical for actual data analysis.Secondly, we propose a new homology test based on this model, where homology means that an ancestor to a sequence pair can be found finitely far back in time. This test has statistical advantages relative to the traditional shuffle test for proteins.Finally, we describe a goodness-of-fit test, that allows testing the proposed insertion-deletion (indel) process inherent to this model and find that real sequences (here globins) probably experience indels longer than one, contrary to what is assumed by the model.     

Bayesian comparisons of codon substitution models

In 1994, Muse and Gaut (MG) and Goldman and Yang (GY) proposed evolutionary models that recognize the coding structure of the nucleotide sequences under study, by defining a Markovian substitution process with a state space consisting of the 61 sense codons (assuming the universal genetic code). Several variations and extensions to their models have since been proposed, but no general and flexible framework for contrasting the relative performance of alternative approaches has yet been applied. Here, we compute Bayes factors to evaluate the relative merit of several MG and GY styles of codon substitution models, including recent extensions acknowledging heterogeneous nonsynonymous rates across sites, as well as selective effects inducing uneven amino acid or codon preferences. Our results on three real data sets support a logical model construction following the MG formulation, allowing for a flexible account of global amino acid or codon preferences, while maintaining distinct parameters governing overall nucleotide propensities. Through posterior predictive checks, we highlight the importance of such a parameterization. Altogether, the framework presented here suggests a broad modeling project in the MG style, stressing the importance of combining and contrasting available model formulations and grounding developments in a sound probabilistic paradigm.     

A common goodness-of-fit framework for neural population models using marked point process time-rescaling

The noisy threshold regime, where even a small set of presynaptic neurons can significantly affect postsynaptic spike-timing, is suggested as a key requisite for computation in neurons with high variability. It also has been proposed that signals under the noisy conditions are successfully transferred by a few strong synapses and/or by an assembly of nearly synchronous synaptic activities. We analytically investigate the impact of a transient signaling input on a leaky integrate-and-fire postsynaptic neuron that receives background noise near the threshold regime. The signaling input models a single strong synapse or a set of synchronous synapses, while the background noise represents a lot of weak synapses. We find an analytic solution that explains how the first-passage time (ISI) density is changed by transient signaling input. The analysis allows us to connect properties of the signaling input like spike timing and amplitude with postsynaptic first-passage time density in a noisy environment. Based on the analytic solution, we calculate the Fisher information with respect to the signaling input’s amplitude. For a wide range of amplitudes, we observe a non-monotonic behavior for the Fisher information as a function of background noise. Moreover, Fisher information non-trivially depends on the signaling input’s amplitude; changing the amplitude, we observe one maximum in the high level of the background noise. The single maximum splits into two maximums in the low noise regime. This finding demonstrates the benefit of the analytic solution in investigating signal transfer by neurons.     

Mutation-selection models of codon substitution and their use to estimate selective strengths on codon usage

Current models of codon substitution are formulated at the levels of nucleotide substitution and do not explicitly consider the separate effects of mutation and selection. They are thus incapable of inferring whether mutation or selection is responsible for evolution at silent sites. Here we implement a few population genetics models of codon substitution that explicitly consider mutation bias and natural selection at the DNA level. Selection on codon usage is modeled by introducing codon-fitness parameters, which together with mutation-bias parameters, predict optimal codon frequencies for the gene. The selective pressure may be for translational efficiency and accuracy or for fine-tuning translational kinetics to produce correct protein folding. We apply the models to compare mitochondrial and nuclear genes from several mammalian species. Model assumptions concerning codon usage are found to affect the estimation of sequence distances (such as the synonymous rate d(S), the nonsynonymous rate d(N), and the rate at the 4-fold degenerate sites d(4)), as found in previous studies, but the new models produced very similar estimates to some old ones. We also develop a likelihood ratio test to examine the null hypothesis that codon usage is due to mutation bias alone, not influenced by natural selection. Application of the test to the mammalian data led to rejection of the null hypothesis in most genes, suggesting that natural selection may be a driving force in the evolution of synonymous codon usage in mammals. Estimates of selection coefficients nevertheless suggest that selection on codon usage is weak and most mutations are nearly neutral. The sensitivity of the analysis on the assumed mutation model is discussed.     

Analysis of codon usage in genes for nitrogen fixation from phylogenetically diverse diazotrophs

Summary A cluster analysis based on codon usage in genes for biological nitrogen fixation (nif genes) grouped diazotrophs into three distinct classes: anaerobes, cyanobacteria, and aerobes. In thenif genes ofKlebsiella pneumoniae there was no evidence for selection pressure in favor of highly translatable codons. However, in the nitrogen regulatory operonglnAntrBntrC of enteric bacteria the stoichiometrically high level of glutamine synthetase may be facilitated by the presence of efficiently translatable codons inglnA. Thenif genes of the cyanobacteriumAnabaena showed codon selection in favor of translational efficiency. Computation of codon adaptation indices for expression in heterologous systems indicated that the reading frames most suitable for expression ofnif genes inEscherichia coli, Bacillus subtilis, andSaccharomyces cerevisiae were present in azotobacters, clostridia, and cyanobacteria, respectively. In codon-usage-based cluster analysis, type 3 nitrogenase genes ofAzotobacter vinelandii grouped along with type 1 and type 2 genes. This is in contrast to the nucleotide sequence-based multiple alignment in which type 3 nitrogenase genes ofA. vinelandii have been reported to cluster with entirely unrelated diazotrophs such as methanogens and clostridia. This may be indicative of lateral transfer ofnif genes among widely divergent taxons. The chromosomal- and plasmid-locatednif genes of rhizobia also cluster separately in nucleotide sequence-based analysis but showed similar codon usage. These analyses suggested that the phylogeny ofnif genes drawn on the basis of nucleotide sequence homology was not masked by the taxon-specific pressure on codon usage.     

Estimation and testing of parent-of-origin effects for quantitative traits

Recent progress in developing family-based association methods has extended their use to the analysis of quantitative traits in the offspring and to the estimation, for dichotomous traits, of the relative contribution of genetic and environmental mechanisms for parent-of-origin effects. However, many traits of interest are not naturally measured on a binary scale yet are suspected or known to be influenced by imprinted genes, and there is consequent interest in seeking evidence for parent-of-origin effects at these loci. Here we show how simple linear models can be used to estimate these parent-of-origin effects for a broad class of phenotypes; in particular, normally distributed quantitative traits are easily dealt with.     

Applications of codon and rate variation models in molecular phylogeny

The current article illustrates the practical advantages of some new models and statistical algorithms for codon substitution and spatial rate variation in molecular phylogeny. Our companion paper in this issue discusses at length the mathematical properties of these models for nucleotide and codon substitution, for site-to-site and branch-to-branch heterogeneity in rates of evolution, and for spatial correlation in the assignment of rates. In this study we summarize the theoretical background and apply the models and algorithms to data on beta-globin, the complete HIV genome, and the mitochondrial genome. Our complex but realistic models enhance biological interpretation of sequence data and show substantial improvements in model fit over existing models. All the new statistical algorithms applied are incorporated in our phylogeny software LINNAEUS, which is tuned for performance and modeling flexibility.