Lab Notebook

This article was motivated by my Work In Progress (WIP) seminar on 2/14/2018 for the Department of Genetics and Genomic Sciences at Mount Sinai. The figures are used in the corresponding presentation (2018-02-01-monoall-general.pdf).

Predict random regression coefficients of the mixed effect model M5 (the best fitting model). This uses the ranef function of the lme4 R package. As the documentation says: “[ranef is a] generic function to extract the conditional modes of the random effects from a fitted model object. For linear mixed models the conditional modes of the random effects are also the conditional means.”

Two characteristics of fit are checked for a set of models: (1) normality of residuals and (2) homogeneity of error variance. These are evaluated with diagnostic plots. Among the checked models the best fitting one is unlm.Q and wnlm.Q regardless of the terms in the linear predictor. However, the fit of wnlm.Q converges very slowly or fails to converge especially when more terms are present in the linear predictor.

The advantage of linear mixed models is demonstrated on a toy data (sleepstudy, bundled with R), in which study individuals act as grouping factors instead of genes. The mixed model is global in the sense that it may be fitted to the entire dataset covering all groups jointly. This is contrasted with local models fitted separately to each group, which is less powerful and tends to over fit the data. The mixed model is also contrasted with a fixed global model, in which the groups (individuals, genes) all have the same effect on the response or on the dependence of the response on predictors of potential interest.

For certain genes allelic bias was found to depend on biological predictors (Age, Dx, Ancestry) in our previous analysis. That analysis fitted various fixed effects multiple regression models (GLM family) using as response variable the read count ratio or its quasi-log transformed version . After checking model fit, we primarily based our results to a weighted, normal linear model using , which we called wnlm.Q (as opposed to its unweighted variation unlm.Q). Significant dependence of the allelic bias of gene on (some level of) the predictor was assessed in terms of the estimated regression coefficient .

For certain genes allelic bias was found to depend on biological predictors (Age, Dx, Ancestry) in our previous analysis. That analysis fitted various fixed effects multiple regression models (GLM family) using as response variable the read count ratio or its quasi-log transformed version . After checking model fit, we primarily based our results to a weighted, normal linear model using , which we called wnlm.Q (as opposed to its unweighted variation unlm.Q). Significant dependence of the allelic bias of gene on (some level of) the predictor was assessed in terms of the estimated regression coefficient .

This short article presents the testing of Gabriel Hoffman’s variancePartition package on our read count ratio data. In particular, a mixed effects model is fitted with Institution, Gender, Dx, and RNA_batch having random effects on the read count ratio and all other predictors having fixed effects. No interactions are considered. The large residual variance is probably due to the large variation among individuals.

This analysis presents plots of the conditional distribution of the read count ratio statistic or its quasi-log transformed version given each a gene a predictor of interest, which is either Gender or Dx. For the predictor Age see the previous post trellis display of data. In the present post a more detailed conditioning is done, using RIN and Institution for MEG3 and MEST. These are genes found in a previous analysis (see permutation test) to be significantly associated to Gender and Dx, respectively. The plots below show that the distributions greatly overlap between differing levels of gender or disease even for the most significantly associated genes, and that this holds in case of further conditioning on RIN and Institution.

This analysis compares some gene sets raised by our study to the set of differential expressed genes in schizophrenia (SCZ) presented by (Fromer et al 2016) in their article Gene expression elucidates functional impact of polygenic risk for schizophrenia. For discussion see the post about affected genes.

This analysis compares our findings on the biological regulation of parental expression bias to those published by Perez et al (eLife 2015;4:e07860), who presented and applied the BRAIM model to RNA-seq data from the cerebellum of P8 and P60 hybrid mice.

This analysis extends permuted observations to testing hypotheses for any gene. This is done by random permutation tests, which provide only approximate p-values in contrast with exact permutation tests (which are unpractical in the present case due to the large number of all permutations of cases). These approximations more or less agree with the “theoretical” p-values, which are based on the hypothesis that the Studentized follows a t-distribution on degrees of freedom (see ch8.3, p371 in A.C Davison: Statistical Models):

For a fixed set of terms in the linear predictor and for each gene, different data transformations, link functions, and error distributions are compared in terms of model fit. Fit is evaluated by diagnostic plots by inspecting the normality of residuals and the homogeneity of error (homoscedasticity). Although model fit varies across genes, taken all genes together the wnlm.Q and unlm.Q models emerge as the best fitting ones.

The present analysis investigates more thoroughly the significance of effect on read count ratio of various predictors for various genes. This is done by random permutations of the observed values of any given predictor and fitting the same models to both observed (unpermuted) and permuted data. This approach is thought to provide more accurate p-values and confidence intervals (CIs) than deriving those from only the observed data using the standard way based on normal distribution theory (see ch8.3, p371 in A.C Davison: Statistical Models), which is very sensitive to the quality of model fit. Indeed, a general finding from this analysis is that some genes show more unexpected distribution of CI than others and that variability is explained by diagnostics of model fit for the corresponding genes. See model checking for details.

Some shortcomings of the previous design matrix are inspected and corrected here. These include highly collinear terms as well as awkward allocation of control and treatment factors.

This article illustrates some of the properties of the various regression models (logi.S, logi2.S, wnlm.S, wnlm.R, unlm.S, unlm.R) used to fit the Common Mind data as well as the properties of the data themselves

In the present observational study the effects of predictors are random. Associations among predictors has a negative impact on maximum likelihood estimation (MLE) of the vector of regression coefficients . This is because the association (in the probabilistic sense, i.e. dependence) among predictors induces an association among the coefficients themselves in the sense of a specific directionality in the gradients of the likelihood surface; that directionality is what makes MLE less reliable. Confidence regions of all coefficients are quasi-ellipsoids on the -dimensional parameter space on which the likelihood function is defined.

Imprinted genes form clusters of one or more genes. Previous work by Ifat established the imprinting status of each gene as either known to be imprinted, not known to be imprinted but near an “known” gene, or else neither. I refer to these three categories as “known”, “nearby candidate” and “candidate”, respectively.

It is crucial for publication to ensure that the genes selected for regression analysis were chosen on objective criteria. This requires the reproduction of the set of genes that were called monoallelic in the manuscript drafted by Ifat. I’ll refer to that in this document as the previous manuscript. If perfect reproduction fails then minimum discordance is desired, and the regression analysis will potentially need to be repeated accordingly with the new set of genes called monoallelic.

Various plots present the distribution of read count ratio across individuals and genes jointly. Genes are also ranked using a summary statistic derived from the empirical distribution of read count ration across individuals for each given gene.

The present article extends previous inference of regression parameters (see extending ANOVA) with a conditional analysis. Operationally, conditioning takes a subset of all observations (i.e. individuals/RNA preparations) defined by a condition. In the present, rather special, case a condition is defined by a single level of a single factor-type (i.e. categorical) predictor/explanatory variable. However, more general conditioning setups could also be explored. In that case multiple predictors might be selected, of which some might be continuous (i.e. a covariate) while others categorical, and for each predictor a possibly compound event (e.g. multiple levels of a factor) might be taken.

Improved graphical overview of data and results pertaining to the recently extended regression analysis and ANOVA

The analysis presented here follows up on the previous ANOVA and extends it in two ways:

The analysis presented here follows up on earlier fitting of regression models on statistics (like ) based on read counts; somewhat less relevant the subsequent comparison of regression models. Data processing steps as well as several notations and computational objects were introduced in the article on data import.

This post follows up on a previous regression analysis, which used fitted several regression model types (nlm, logi, log2) to the data concerning monoallelic expression, and the effect of age and other explanatory variables. The goal here is to compare these models to select the one(s) with the most desirable properties.

Previous analysis by Ifat analyzed the relationship between age and allelic imbalance by fitting a normal linear model to rank-transformed values averaged over a set of genes. My follow-up analysis extended this to a set of genes. Here alternative models are fitted directly to the (untransformed) values. Furthermore, all models are fitted to not only aggregated gene sets but also single genes. Models are compared based on the Akaike information criterion (AIC).

This is an explanatory, qualitative, analysis on the distribution of the age of death among the studied individuals from the CommonMind Consortium. The conclusion is that biological variables such as gender and psychiatric state do seem to affect age of death but such qualitative analysis is insufficient to confidently disentangle these biological variables from psychological, social, institutional, and other biomedical variables.

This document supplements the article Binomial Models.