High-throughput sequencing assays such as RNA-Seq, ChIP-Seq or barcode counting provide quantitative readouts in the form of count data. To infer differential signal in such data correctly and with good statistical power, estimation of data variability throughout the dynamic range and a suitable error model are required. We propose a method based on the negative binomial distribution, with variance and mean linked by local regression and present an implementation, DESeq, as an R/Bioconductor package.

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Differential expression analysis for sequence count data.

Summary 1. While teaching statistics to ecologists, the lead authors of this paper have noticed common statistical problems. If a random sample of their work (including scientific papers) produced before doing these courses were selected, half would probably contain violations of the underlying assumptions of the statistical techniques employed. 2. Some violations have little impact on the results or ecological conclusions; yet others increase type I or type II errors, potentially resulting in wrong ecological conclusions. Most of these violations can be avoided by applying better data exploration. These problems are especially troublesome in applied ecology, where management and policy decisions are often at stake. 3. Here, we provide a protocol for data exploration; discuss current tools to detect outliers, heterogeneity of variance, collinearity, dependence of observations, problems with interactions, double zeros in multivariate analysis, zero inflation in generalized linear modelling, and the correct type of relationships between dependent and independent variables; and provide advice on how to address these problems when they arise. We also address misconceptions about normality, and provide advice on data transformations. 4. Data exploration avoids type I and type II errors, among other problems, thereby reducing the chance of making wrong ecological conclusions and poor recommendations. It is therefore essential for good quality management and policy based on statistical analyses.

https://besjournals.onlinelibrary.wiley.com/doi/epdf/10.1111/j.2041-210X.2009.00001.x

A protocol for data exploration to avoid common statistical problems

Although economists have long been aware of Jensen's inequality, many econometric applications have neglected an important implication of it: the standard practice of interpreting the parameters of log-linearized models estimated by ordinary least squares as elasticities can be highly misleading in the presence of heteroskedasticity. This paper explains why this problem arises and proposes an appropriate estimator. Our criticism to conventional practices and the solution we propose extends to a broad range of economic applications where the equation under study is log-linearized. We develop the argument using one particular illustration, the gravity equation for trade, and apply the proposed technique to provide new estimates of this equation. We find significant differences between estimates obtained with the proposed estimator and those obtained with the traditional method. These discrepancies persist even when the gravity equation takes into account multilateral resistance terms or fixed effects

The Log of Gravity

We propose a framework for analyzing and comparing distributions, which we use to construct statistical tests to determine if two samples are drawn from different distributions. Our test statistic is the largest difference in expectations over functions in the unit ball of a reproducing kernel Hilbert space (RKHS), and is called the maximum mean discrepancy (MMD).We present two distribution free tests based on large deviation bounds for the MMD, and a third test based on the asymptotic distribution of this statistic. The MMD can be computed in quadratic time, although efficient linear time approximations are available. Our statistic is an instance of an integral probability metric, and various classical metrics on distributions are obtained when alternative function classes are used in place of an RKHS. We apply our two-sample tests to a variety of problems, including attribute matching for databases using the Hungarian marriage method, where they perform strongly. Excellent performance is also obtained when comparing distributions over graphs, for which these are the first such tests.

/pdf/a-kernel-two-sample-test-2mr8hwo4ml.pdf

A kernel two-sample test

Preface 1. Introduction 2. The concept of risk 3. Overview of count response models 4. Methods of estimation and assessment 5. Assessment of count models 6. Poisson regression 7. Overdispersion 8. Negative binomial regression 9. Negative binomial regression: modeling 10. Alternative variance parameterizations 11. Problems with zero counts 12. Censored and truncated count models 13. Handling endogeneity and latent class models 14. Count panel models 15. Bayesian negative binomial models Appendix A. Constructing and interpreting interactions Appendix B. Data sets and Stata files References Index.

/pdf/negative-binomial-regression-29q4k5nu4u.pdf

Negative Binomial Regression

Three nonlinear count models, Poisson R.egression (PR), Negative Binomial Regression (NBR), and Generalized Poisson Regression (GPR) are used for assessing the effects of risk factors on agricultural injuries from farm injury data. A sample of 1,322 respondents who participated in the farm safety/injury baseline survey in nine rural counties in Alabama and Mississippi, aged 18 years and older are considered for analysis. The dispersion parameter estimates and their standard errors for GPR models were consistently smaller than that of NBR models. Estimated dispersion parameters in the NBR and GPR models were positive and significantly different from zero. Estimated goodness-of-fit measures showed that GPR models outperformed the NBR and PR models.

Regression analysis of count data

This paper introduces a general class of distributions generated from the logit of the beta random variable. A special case of this family is the beta-normal distribution. The shape properties of the beta-normal distribution are discussed. Estimation of parameters of the beta-normal distribution by the maximum likelihood method is also discussed. The beta-normal distribution provides great flexibility in modeling not only symmetric heavy-tailed distributions, but also skewed and bimodal distributions. The flexibility of this distribution is illustrated by applying it to two empirical data sets and comparing the results to previously used methods.

Beta-normal distribution and its applications

In this paper, a new method is proposed for generating families of continuous distributions. A random variable $$X$$
 , “the transformer”, is used to transform another random variable $$T$$
 , “the transformed”. The resulting family, the $$T$$
 -
 $$X$$
 family of distributions, has a connection with the hazard functions and each generated distribution is considered as a weighted hazard function of the random variable $$X$$
 . Many new distributions, which are members of the family, are presented. Several known continuous distributions are found to be special cases of the new distributions.

/pdf/a-new-method-for-generating-families-of-continuous-2e75jsuwp2.pdf

A new method for generating families of continuous distributions

(1995). Continuous Univariate Distributions, Volume 1. Technometrics: Vol. 37, No. 4, pp. 466-466.

Continuous Univariate Distributions, Volume 1

Some properties of a four-parameter beta-Weibull distribution are discussed. The beta-Weibull distribution is shown to have bathtub, unimodal, increasing, and decreasing hazard functions. The distribution is applied to censored data sets on bus-motor failures and a censored data set on head-andneck-cancer clinical trial. A simulation is conducted to compare the beta-Weibull distribution with the exponentiated Weibull distribution.

/pdf/beta-weibull-distribution-some-properties-and-applications-1gd4fjldjo.pdf

Felix Famoye

Papers

Regression analysis of count data

Beta-normal distribution and its applications

A new method for generating families of continuous distributions

Continuous Univariate Distributions, Volume 1

Beta-Weibull Distribution: Some Properties and Applications to Censored Data