Statistics and Probability Commons^™

A Simple Index Of Smoking, Abhaya Indrayan Dr., Rajeev Kumar Mr., Shridhar Dwivedi Dr.

COBRA Preprint Series

Background: Cigarette smoking is implicated in a large number of diseases and other adverse health conditions. Among the dimensions of smoking are number of cigarettes smoked per day, duration of smoking, passive smoking, smoking of filter cigarettes, age at start, and duration elapsed since quitting by ex-smokers. The practice so far is to study most of these separately. We develop a simple index that integrates these dimensions of smoking into a single metric, and suggest that this index be developed further. Method: The index is developed under a series of natural assumptions. Broadly, these are (i) the burden of smoking …

The Strength Of Statistical Evidence For Composite Hypotheses With An Application To Multiple Comparisons, David R. Bickel

COBRA Preprint Series

The strength of the statistical evidence in a sample of data that favors one composite hypothesis over another may be quantified by the likelihood ratio using the parameter value consistent with each hypothesis that maximizes the likelihood function. Unlike the p-value and the Bayes factor, this measure of evidence is coherent in the sense that it cannot support a hypothesis over any hypothesis that it entails. Further, when comparing the hypothesis that the parameter lies outside a non-trivial interval to the hypotheses that it lies within the interval, the proposed measure of evidence almost always asymptotically favors the correct hypothesis …

A New Method For Constructing Exact Tests Without Making Any Assumptions, Karl H. Schlag

COBRA Preprint Series

We present a new method for constructing exact distribution-free tests (and con…fidence intervals) for variables that can generate more than two possible outcomes. This method separates the search for an exact test from the goal to create a non- randomized test. Randomization is used to extend any exact test relating to means of variables with fi…nitely many outcomes to variables with outcomes belonging to a given bounded set. Tests in terms of variance and covariance are reduced to tests relating to means. Randomness is then eliminated in a separate step. This method is used to create con…fidence intervals for the …

Joint Spatial Modeling Of Recurrent Infection And Growth With Processes Under Intermittent Observation, Farouk S. Nathoo

COBRA Preprint Series

In this article we present new statistical methodology for longitudinal studies in forestry where trees are subject to recurrent infection and the hazard of infection depends on tree growth over time. Understanding the nature of this dependence has important implications for reforestation and breeding programs. Challenges arise for statistical analysis in this setting with sampling schemes leading to panel data, exhibiting dynamic spatial variability, and incomplete covariate histories for hazard regression. In addition, data are collected at a large number of locations which poses computational difficulties for spatiotemporal modeling. A joint model for infection and growth is developed; wherein, a …

The Calculation Of The 97.5% Upper Confidence Bound: Application To Clustered Binary Data In A Binomial Non-Inferiority Two-Sample Trial., William F. Mccarthy

COBRA Preprint Series

This paper will discuss the analysis of a cluster randomized binomial non-inferiority two-sample trial. The determination of the intra-cluster correlation coefficient (ICC) and its use in the calculation of the 97.5% upper confidence bound for delta, the true difference in binomial proportions between the active control and the experimental treatment groups, will be outlined.

The Design And Sample Size Requirement For A Cluster Randomized Non-Inferiority Trial With Two Binary Co-Primary Outcomes., William F. Mccarthy

COBRA Preprint Series

This paper will discuss the design and sample size requirement for a cluster randomized non-inferiority trial with two binary co-primary outcomes. A hypothetical study (the EXAMPLE Trial) will be considered.

Lets assume the EXAMPLE Trial will consist of two separate binomial non-inferiority two-sample trials. Trial 1: the Coronary Artery Disease known population (co-primary 1) and Trial 2: the Coronary Artery Disease unknown population (co-primary 2). A physician-month cluster randomization scheme will be used. That is, for each trial (trial 1 and trial 2) every month for a 12-month period, each physician participating in the EXAMPLE Trial will be allocated a …

Bringing Game Theory To Hypothesis Testing: Establishing Finite Sample Bounds On Inference, Karl H. Schlag

COBRA Preprint Series

Small sample properties are of fundamental interest when only limited data is available. Exact inference is limited by constraints imposed by specific nonrandomized tests and of course also by lack of more data. These effects can be separated as we propose to evaluate a test by comparing its type II error to the minimal type II error among all tests for the given sample. Game theory is used to establish this minimal type II error, the associated randomized test is characterized as part of a Nash equilibrium of a fictitious game against nature. We use this method to investigate sequential …

Properties Of Monotonic Effects On Directed Acyclic Graphs, Tyler J. Vanderweele, James M. Robins

COBRA Preprint Series

Various relationships are shown hold between monotonic effects and weak monotonic effects and the monotonicity of certain conditional expectations. Counterexamples are provided to show that the results do not hold under less restrictive conditions. Monotonic effects are furthermore used to relate signed edges on a causal directed acyclic graph to qualitative effect modification. The theory is applied to an example concerning the direct effect of smoking on cardiovascular disease controlling for hypercholesterolemia. Monotonicity assumptions are used to construct a test for whether there is a variable that confounds the relationship between the mediator, hypercholesterolemia, and the outcome, cardiovascular disease.

Estimation Of Dose-Response Functions For Longitudinal Data, Erica E M Moodie, David A. Stephens

COBRA Preprint Series

In a longitudinal study of dose-response, the presence of confounding or non-compliance compromises the estimation of the true effect of a treatment. Standard regression methods cannot remove the bias introduced by patient-selected treatment level, that is, they do not permit the estimation of the causal effect of dose. Using an approach based on the Generalized Propensity Score (GPS), a generalization of the classical, binary treatment propensity score, it is possible to construct a balancing score that provides a more meaningful estimation procedure for the true (unconfounded) effect of dose. Previously, the GPS has been applied only in a single interval …

Bootstrap Confidence Regions For Optimal Operating Conditions In Response Surface Methodology, Roger D. Gibb, I-Li Lu, Walter H. Carter Jr

COBRA Preprint Series

This article concerns the application of bootstrap methodology to construct a likelihood-based confidence region for operating conditions associated with the maximum of a response surface constrained to a specified region. Unlike classical methods based on the stationary point, proper interpretation of this confidence region does not depend on unknown model parameters. In addition, the methodology does not require the assumption of normally distributed errors. The approach is demonstrated for concave-down and saddle system cases in two dimensions. Simulation studies were performed to assess the coverage probability of these regions.

AMS 2000 subj Classification: 62F25, 62F40, 62F30, 62J05.

Key words: Stationary …

An Example Of How To Write The Statistical Section Of A Bioequivalence Study Protocol For Fda Review, William F. Mccarthy

COBRA Preprint Series

This paper provides a detailed example of how one should write the statistical section of a bioequivalence study protocol for FDA review. Three forms of bioequivalence are covered: average bioequivalence (ABE), population bioequivalence (PBE) and individual bioequivalence (IBE). The method of analysis is based on Jones and Kenward (2003) and a modification of their SAS Macro is provided.

Adjustment To The Mcnemar’S Test For The Analysis Of Clustered Matched-Pair Data, William F. Mccarthy

COBRA Preprint Series

This paper presents how one can adjust the McNemar’s test for the analysis of clustered matched-pair data. A McNemar’s-like table for K clusters of matched-pair data is used.

Assessment Of Sample Size And Power For The Analysis Of Clustered Matched-Pair Data, William F. Mccarthy

COBRA Preprint Series

This paper outlines how one can determined the sample size or power of a study design that is based on clustered matched-pair data. Detailed examples are provided.

Lachenbruch’S Method For Determining The Sample Size Required For Testing Interactions: How It Compares To Nquery Advisor And O’Brien’S Sas Unifypow., William F. Mccarthy

COBRA Preprint Series

Lachenbruch (1988) proposed a simple method based on the use of orthogonal contrasts to determine the sample size or power for testing main effects and interactions, and uses the normal distribution instead of the non-central F distribution. This method can be used for factorial designs of various size. The example illustrated in this paper considers a 2 x 2 factorial design. This paper will determine both sample size and power of a particular study design with anticipated (assumed) means for each cell of the 2 x 2 factorial design. Lachenbruch’s method will be compared to nQuery Advisor 6.0 (2005) and …

The Existence Of Maximum Likelihood Estimates For The Binary Response Logistic Regression Model, William F. Mccarthy

COBRA Preprint Series

The existence of maximum likelihood estimates for the binary response logistic regression model depends on the configuration of the data points in your data set. There are three mutually exclusive and exhaustive categories for the configuration of data points in a data set: Complete Separation, Quasi-Complete Separation, and Overlap. For this paper, a binary response logistic regression model is considered. A 2 x 2 tabular presentation of the data set to be modeled is provided for each of the three categories mentioned above. In addition, the paper will present an example of a data set whose data points have a …

The Assessment Of The Degree Of Concordance Between The Observed Values And The Predicted Values Of A Mixed-Effect Model Using “Method Of Comparison” Techniques, William F. Mccarthy, Nan Guo

COBRA Preprint Series

In this paper, we present a methodology for determining the degree of concordance between observed and model-based predicted values of a mixed-effect model. In particular, we will compare the degree to which observed and model-based predicted values agree by using ‘method of comparison’ techniques. We will also present the results of the concordance correlation coefficient (CCC).

The Analysis Of Pixel Intensity (Myocardial Signal Density) Data: The Quantification Of Myocardial Perfusion By Imaging Methods., William F. Mccarthy, Douglas R. Thompson

COBRA Preprint Series

This paper described a number of important issues in the analysis of pixel intensity data, as well as approaches for dealing with these. We particularly emphasized the issue of clustering, which may be ubiquitous in studies of pixel intensity data. Clustering can take many forms, e.g., measurements of different sections of a heart or repeated measurements of the same research participant. Clustering typically has the effect of increasing variance estimates. When one fails to account for clustering, variance estimates may be unrealistically small, resulting in spurious significance. We illustrated several possible approaches to account for clustering, including adjusting standard errors …

Coronary Evaluation Using Multi-Detector Spiral Computed Tomography Angiography: Statistical Design And Analysis, William F. Mccarthy, Douglas R. Thompson, Bruce A. Barton

COBRA Preprint Series

Contrast-enhanced multi-detector row spiral computed tomography (MDCT) has been introduced as a method for non-invasive visualization of coronary artery stenosis. To determine the diagnostic accuracy of MDCT coronary angiography, as compared to the “gold standard” invasive coronary angiography, sensitivity and specificity are estimated (95% Confidence Intervals). Three separate levels of estimation are computed: at the patient level, at the coronary artery level, and at the coronary artery segment level. We review the methodology for the estimation of sensitivity and specificity of non-clustered binary data (patient level analysis) and present a methodology for the estimation of sensitivity and specificity that considers …

Review Of The Maximum Likelihood Functions For Right Censored Data. A New Elementary Derivation., Stefano Patti, Elia Biganzoli, Patrizia Boracchi

COBRA Preprint Series

Censoring is a well known feature recurrent in the analysis of lifetime data, occurring in the model when exact lifetimes can be collected for only a representative portion of the surveyed individuals. If lifetimes are known only to exceed some given values, it is referred to as right censoring. In this paper we propose a systematization and a new derivation of the likelihood function for right censored sampling schemes; calculations are reported and assumptions are carefully stated. The sampling schemes considered (Type I, II and Random Censoring) give rise to the same ML function. Only the knowledge of elementary probability …

A Flexible Semi-Parametric Approach To Estimating A Dose-Response Relationship: The Treatment Of Childhood Amblyopia. , David A. Stephens, Erica E M Moodie

COBRA Preprint Series

In a study of a dose-response relationship, flexibility in modelling is essential to capturing the treatment effect when the mean effect of other covariates is not fully understood, so that observed treatment effect is not due to the imposition of a rigid model for the relationship between response, treatment, and other variables. A semiparametric additive linear mixed (SPALM) model (Ruppert et al. 2003) provides a tractable and flexible approach to modelling the influence of potentially confounding variables. In this paper, we present pure likelihood and Bayesian versions of the SPALM model. Both methods of inference are readily implementable, but the …

A Bayesian Hierarchical Model For Spot Fluorescence In Microarrays, Federico Mattia Stefanini

COBRA Preprint Series

Microarray experiments are characterized by the presence of many sources of experimental bias and a remarkably large technical variability. The assessment of differential expression for genes transcribed into a small number of mRNA copies heavily depends on the proper quantification of background fluorescence within spot. The rough model `observed = hybridization plus background' fluorescence is at first reformulated at spot level, then it is embedded into a Bayesian hierarchical model suited for fitting control spots. The novelties of the approach include the background correction performed on the latent mean of replicated spots, and an explicit model for outlying observations at …

False Discovery Rate Analysis Of Brain Diffusion Direction Maps, Armin Schwartzman, Robert F. Dougherty, Jonathan E. Taylor

COBRA Preprint Series

Diffusion tensor imaging (DTI) is a novel modality of magnetic resonance imaging that allows non-invasive mapping of the brain’s white matter. A particular map derived from DTI measurements is a map of water principal diffusion directions, which are proxies for neural fiber directions. We consider an experiment in which diffusion direction maps were acquired for two groups of subjects. The objective of the analysis is to find regions of the brain in which the corresponding diffusion directions differ between the groups. This is attained by first computing a test statistic for the difference in direction at every brain location using …

Properties Of Monotonic Effects, Tyler J. Vanderweele, James M. Robins

COBRA Preprint Series

Various relationships are shown hold between monotonic effects and weak monotonic effects and the monotonicity of certain conditional expectations. This relationship is considered for both binary and non-binary variables. Counterexamples are provide to show that the results do not hold under less restrictive conditions. The ideas of monotonic effects are furthermore used to relate signed edges on a directed acyclic graph to qualitative effect modification.

Exploration Of Distributional Models For A Novel Intensity-Dependent Normalization , Nicola Lama, Patrizia Boracchi, Elia Mario Biganzoli

COBRA Preprint Series

Currently used gene intensity-dependent normalization methods, based on regression smoothing techniques, usually approach the two problems of location bias detrending and data re-scaling without taking into account the censoring characteristic of certain gene expressions produced by experiment measurement constraints or by previous normalization steps. Moreover, the bias vs variance balance control of normalization procedures is not often discussed but left to the user's experience. Here an approximate maximum likelihood procedure to fit a model smoothing the dependences of log-fold gene expression differences on average gene intensities is presented. Central tendency and scaling factor were modeled by means of B-splines smoothing …

Crude Cumulative Incidence In The Form Of A Horvitz-Thompson Like And Kaplan-Meier Like Estimator, Laura Antolini, Elia Mario Biganzoli, Patrizia Boracchi

COBRA Preprint Series

The link between the nonparametric estimator of the crude cumulative incidence of a competing risk and the Kaplan-Meier estimator is exploited. The equivalence of the nonparametric crude cumulative incidence to an inverse-probability-of-censoring weighted average of the sub-distribution function is proved. The link between the estimation of crude cumulative incidence curves and Gray's family of nonparametric tests is considered. The crude cumulative incidence is proved to be a Kaplan-Meier like estimator based on the sub-distribution hazard, i.e. the quantity on which Gray's family of tests is based. A standard probabilistic formalism is adopted to have a note accessible to applied statisticians.

A Flexible Statistical Method For Detecting Genomic Copy-Number Changes Using Hidden Markov Models With Reversible Jump Mcmc , Oscar M. Rueda, Ramon Diaz-Uriarte

COBRA Preprint Series

We have developed a statistical method for the analysis of array based CGH data to detect genomic DNA copy number changes. Our method allows us to answer the biologically relevant questions (what is the probability that a given gene or region has increased or decreased copy number changes) in a clear and simple way, within a rigorous statistical framework. We use a non-homogeneous Hidden Markov Model that incorporates distance between genes, a crucial requirement to analyze data from platforms where distances between probes is highly variable. As the true number of hidden states (states of copy number changes) is not …

Survival Analysis Of Longitudinal Microarrays, Natasa Rajicic, Dianne M. Finkelstein, David A. Schoenfeld

COBRA Preprint Series

Motivation: The development of methods for linking gene expressions to various clinical and phenotypic characteristics is an active area of genomic research. Scientists hope that such analysis may, for example, describe relationships between gene function and clinical events such as death or recovery. Methods are available for relating gene expression to measurements that are categorized or continuous, but there is less work in relating expressions to an observed event time such as time to death, response, or relapse. When gene expressions are measured over time, there are methods for differentiating temporal patterns. However, no methods have yet been proposed for …

Causal Comparisons In Randomized Trials Of Two Active Treatments: The Effect Of Supervised Exercise To Promote Smoking Cessation, Jason Roy, Joseph W. Hogan

COBRA Preprint Series

In behavioral medicine trials, such as smoking cessation trials, two or more active treatments are often compared. Noncompliance by some subjects with their assigned treatment poses a challenge to the data analyst. Causal parameters of interest might include those defined by subpopulations based on their potential compliance status under each assignment, using the principal stratification framework (e.g., causal effect of new therapy compared to standard therapy among subjects that would comply with either intervention). Even if subjects in one arm do not have access to the other treatment(s), the causal effect of each treatment typically can only be identified from …

New Spiked-In Probe Sets For The Affymetrix Hgu-133a Latin Square Experiment, Monnie Mcgee, Zhongxue Chen

COBRA Preprint Series

The Affymetrix HGU-133A spike in data set has been used for determining the sensitivity and specificity of various methods for the analysis of microarray data. We show that there are 22 additional probe sets that detect spike in RNAs that should be considered as spike in probe sets. We assign each proposed spiked-in probe set to a concentration group within the Latin Square design, and examine the effects of the additional spiked-in probe sets on assessing the accuracy of analysis methods currently in use. We show that several popular preprocessing methods are more sensitive and specific when the new spike-ins …

Sample Size And Power Calculations For Body Weight In Beef Cattle, Claudia Cristina Paro Paz, Alfredo Ribeiro De Freitas, Irineu Umberto Packer, Daniela Tambasco-Talhari, Luciana Correa De Almeida Regitano, Mauricio Mello Alencar

COBRA Preprint Series

Estimates of minimum sample sizes are calculated in order to test differences in rates of changes over time for longitudinal designs. In this study, body weight of crossbred beef cattle, considering 14 measurements on individuals, taken at birth, weaning (7 months of age) and monthly from 8 to 19 months of age, were analyzed by an usual mixed model for repeated measures. The number of individuals n required to detect significant differences (delta) between any two consecutive measurements on the individual, was obtained by a SAS program considering a t-variate normal distribution (t = 14), sample variance–covariance matrix among the …