Two--Way Contingency Tables. Three--Way Contingency Tables. Generalized Linear Models. Logistic Regression. Loglinear Models for Contingency Tables. Building and Applying Logit and Loglinear Models. Multicategory Logit Models. Models for Matched Pairs. A Twentieth--Century Tour of Categorical Data Analysis. Appendix. Table of Chi--Squared Distribution Values for Various Right--Tail Probabilities. Bibliography. Indexes.

An introduction to categorical data analysis

The algorithm described in this paper discovers one or more motifs in a collection of DNA or protein sequences by using the technique of expectation maximization to fit a two-component finite mixture model to the set of sequences Multiple motifs are found by fitting a mixture model to the data, probabilistically erasing the occurrences of the motif thus found, and repeating the process to find successive motifs The algorithm requires only a set of unaligned sequences and a number specifying the width of the motifs as input It returns a model of each motif and a threshold which together can be used as a Bayes-optimal classifier for searching for occurrences of the motif in other databases The algorithm estimates how many times each motif occurs in each sequence in the dataset and outputs an alignment of the occurrences of the motif The algorithm is capable of discovering several different motifs with differing numbers of occurrences in a single dataset

/pdf/fitting-a-mixture-model-by-expectation-maximization-to-1ci3599stf.pdf

Fitting a mixture model by expectation maximization to discover motifs in biopolymers.

For interval estimation of a proportion, coverage probabilities tend to be too large for “exact” confidence intervals based on inverting the binomial test and too small for the interval based on inverting the Wald large-sample normal test (i.e., sample proportion ± z-score × estimated standard error). Wilson's suggestion of inverting the related score test with null rather than estimated standard error yields coverage probabilities close to nominal confidence levels, even for very small sample sizes. The 95% score interval has similar behavior as the adjusted Wald interval obtained after adding two “successes” and two “failures” to the sample. In elementary courses, with the score and adjusted Wald methods it is unnecessary to provide students with awkward sample size guidelines.

https://www.jstor.org/stable/pdfplus/2685469.pdf

Approximate is Better than “Exact” for Interval Estimation of Binomial Proportions

/pdf/modern-applied-statistics-with-s-fourth-edition-fw5louzblu.pdf

Modern Applied Statistics with S Fourth edition

We revisit the problem of interval estimation of a binomial proportion. The erratic behavior of the coverage probability of the stan- d ardWaldconfid ence interval has previously been remarkedon in the literature (Blyth andStill, Agresti andCoull, Santner andothers). We begin by showing that the chaotic coverage properties of the Waldinter- val are far more persistent than is appreciated. Furthermore, common textbook prescriptions regarding its safety are misleading and defective in several respects andcannot be trusted . This leads us to consideration of alternative intervals. A number of natural alternatives are presented, each with its motivation and con- text. Each interval is examinedfor its coverage probability andits length. Basedon this analysis, we recommendthe Wilson interval or the equal- tailedJeffreys prior interval for small n andthe interval suggestedin Agresti andCoull for larger n. We also provide an additional frequentist justification for use of the Jeffreys interval.

/pdf/interval-estimation-for-a-binomial-proportion-3t1vha8pn0.pdf

Interval Estimation for a Binomial Proportion

1 Introduction.- 2 Univariate Discrete Responses.- 3 Loglinear Models.- 4 Cross-Classified Data.- 5 Univariate Discrete Data with Covariates.- Appendix 1. Some Results from Linear Algebra.- Appendix 2. Maximization of Concave Functions.- Appendix 3. Proof of Proposition 3.3.1 (ii) and (iii).- Appendix 4. Elements of Large Sample Theory.- Problems.- References.- List of Notation.- Index to Data Sets.- Author Index.

The Statistical Analysis of Discrete Data

This paper develops alternatives to maximum likelihood estimators (MLE) for logistic regression models and compares the mean squared error (MSE) of the estimators. The MLE for the vector of underlying success probabilities has low MSE only when the true probabilities are extreme (i.e., near 0 or 1). Extreme probabilities correspond to logistic regression parameter vectors which are large in norm. A competing “restricted” MLE and an empirical version of it are suggested as estimators with better performance than the MLE for central probabilities. An approximate EM-algorithm for estimating the restriction is described. As in the case of normal theory ridge estimators, the proposed estimators are shown to be formally derivable by Bayes and empirical Bayes arguments. The small sample operating characteristics of the proposed estimators are compared to the MLE via a simulation study; both the estimation of individual probabilities and of logistic parameters are considered.

On the small sample properties of norm-restricted maximum likelihood estimators for logistic regression models

The Statistical Analysis of Discrete Data.

This chapter describes the use of classical likelihood methods and loglinear models to analyze cross-classified data. Cross-classified data arise when a random sample W 1, W 2,…, W m , say, is drawn from a discrete d-variate distribution where each trial W k =(W 1 k ,...,W d k )’ has common joint probability mass function: 

$${p_i}: = P[W_1^k = {i_1}, \ldots ,W_d^k = {i_d}],$$

(4.1.1)

. Here the support of W j k is taken to be $\{ 1, \ldots ,{L_j}\} $ without loss of generality. The symbol W, without a superscript, will be used to denote a generic classification variable with probability mass function (4.1.1). By sufficiency, the data can be summarized as the counts $\{ {Y_i}:i \in x\} $ in a d-dimensional contingency table where Y i is the number of vectors W which equal i. Thus the counts $\{ {Y_i}:i \in x\} $ have the M t ( m, p ) multinomial distribution where $p = \{ {p_i}:i \in X\} ,\sum\nolimits_{i \in x} {{p_i}} = 1,t = \Pi _{j = 1}^d{L_j},$, and $m = \sum\nolimits_{i \in x} {{Y_{i\cdot }}} $.

Cross-Classified Data

Polychotomous response regression data has the form (Y i , m i , x i ),1≤i≤T, where, without loss of generality, each response takes one of the values{0,...,g}.Each vector Y i =(Y i0,…, Y ig )’,giving the umber of times the (g +1) outcomes occur at the ith design point, is assumed to follow an independent multinomial distribution with m i trials $\left( {{{\sum\nolimits_{j = 0}^g Y }_{ij}} = {m_i}} \right)$. The vector x i =( x i1,..., x ik )’contains covariate values affecting the cell probabilities of Y i . Example 1.2.7 on the severity of nausea in cancer patients undergoing chemotherapy is typical of such data. There are six possible outcomes (g = 5), two design points (T = 2), a scalar covariate (k = 1) which is 1 or 0 according as the chemotherapy includes cisplatinum or not, and the numbers of patients are m 1 = 58 and m 2 = 161 for the cisplatinum and no-cisplatinum groups, respectively.

Diane E. Duffy

Papers

The Statistical Analysis of Discrete Data

On the small sample properties of norm-restricted maximum likelihood estimators for logistic regression models

The Statistical Analysis of Discrete Data.

Cross-Classified Data

Univariate Discrete Data with Covariates