bluuebunny/crossref_metadata_2025 huggingface dataset:
Discrimination and Clustering for Multivariate Time Series
Authors: Shumway Robert H.
Published: September 2014

This entry gives a summary of the time and frequency domain approaches to classification and clustering for biostatistical and medical data that have been collected as multivariate correlated time series.
D
urbin–
W
atson Test
Authors: Chatfield Chris
Published: September 2014

This text checks for independent errors when fitting a multiple regression model to time series data.
Fast
F
ourier Transform (
FFT
)
Authors: McCoy E. J.
Published: September 2014

Data can be described either in the “time domain” or in the “frequency domain”. The frequency domain representations present details about the same signal in a different, potentially more illuminating way. The fast Fourier transform (FFT) provides a fast computational algorithm to obtain the coefficients of the discrete‐frequency representation.
Instrumental Variables: Overview
Authors: Dunn Graham
Published: September 2014

We describe the application of instrumental variable methods to cope with model identification problems arising from either hidden selection effects or measurement errors. The problem of hidden selection effects is illustrated by reference to adjustment to non‐ or partial compliance with allocated treatment in a randomized controlled trial. Measurement error problems are illustrated by reference to two potential areas of application: (1) the estimation of the effect of environmental exposure on risk of disease when exposure is measured with error (i.e. measurement error models in epidemiology), and (2) comparative calibration of two measurement methods when both are subject to measurement errors (i.e. method comparison studies).
Intervention Analysis in Time Series
Authors: Helfenstein Ulrich
Published: September 2014

When data are collected in the form of dependent time series data, important questions are concerned with “changes” in the series. Changes may be “man made” or they may arise “naturally”: How efficient was a preventive program to decrease the monthly number of accidents? How did the pattern of morbidity change after an environmental accident? Notifications of diseases or accidents and so on, are often collected in regular intervals. ARIMA models allow the stochastic dependence of consecutive data to be modeled flexibly. Intervention analysis is an extension of ARIMA modeling allowing the study of the structure of changes of ARIMA processes. Since the series may be nonstationary, large changes of the series could occur even when no intervention takes place. Intervention analysis may allow an investigator to distinguish between what can be expected due to nonstationarity alone and what cannot. Analogous questions of “change” may arise when studying time series data recorded in an individual patient. This article reviews basic concepts of intervention analysis. The methods are illustrated by medical applications.
Non Fourier Waveforms
Authors: McCoy E. J.
Published: September 2014

The discrete Fourier transform (DFT), which gives us information about the “frequency” domain characteristics of data, is probably the most ubiquitous orthogonal data transform, and uses basis functions made up of sines and cosines at different frequencies. Often, we have to analyze signals whose frequency behavior changes over time. In such cases, we can utilize transforms with non‐Fourier basis functions, which can better represent nonstationary behavior.
Allometry
Authors: Sprent P.
Published: September 2014

Allometry is a study of shape changes associated with size. Most living organisms change shape as they grow in order to overcome biological or physical constraints. Simply, allometry occurs when two size measurements exhibit an approximately linear relationship between their logarithms. Suggestions made for multivariate extensions range from applications of principal component analysis to studies of size variables and shape vectors. The limitations imposed by Euclidean geometry and other disadvantages with these approaches have been removed by considering curvature and tangential directions at a series of points called landmarks and by coordinate transformations.
Chronomedicine
Authors: CornÉlissen Germaine , Halberg Franz
Published: September 2014

Chronobiology (fromchronos, time;bios, life; andlogos, science) investigates the mechanisms underlying variability in the otherwise unassessed physiological range, including rhythms found in us, resonating with cycles around us. Broad time structures (chronomes) consisting of deterministic chaos and trends organized by rhythms are found in organisms and in their environments. They are mapped by chronomics as the reference values for both an applied chronomedicine and a basic chronobiology. Chronomics quantify health, identifying new disease risks, diagnosing predisease and overt illness, enabling timely and timed treatment (Rx), and validating the short‐ and long‐term efficacy of a givenRxon an inferential statistical individualized (as well as population) basis.
Critical Care
Authors: Corwin Howard L.
Published: September 2014

The randomized controlled trial (RCT) has become the “gold” standard for clinical research. However, the complex nature of critical illness has made the design and conduct of RCTs difficult. Research in critically ill patients presents unique challenges. The disease processes are often not well defined. There are also multiple therapeutic interventions being administered and the disease processes themselves often have a variable clinical course. Finally, outcome selection is a challenge, mortality (short or long) versus nonmortality outcomes. The last few years have seen the publication of the results of five RCTs that have had a major impact on the practice of critical care.
Tree‐Structured Statistical Methods
Authors: Zhang Heping , Crowley John , Sox Harold C. , A. Olshen Richard
Published: September 2014

In this article, we describe the binary tree‐structured methods and illustrate the methods with applications in biomedical research. The methods include decision trees, regression trees, survival trees, and trees for a vector response. We also provide information with regard to the software.
Medico–Legal Cases and Statistics
Authors: Gastwirth Joseph L.
Published: September 2014

Medico‐legal statistics concerns the use of data from animal studies, case–control and cohort studies and clinical trials in the context of the legal and regulatory system. Statistical findings are submitted to courts and governmental agencies to demonstrate that a new drug is safe and efficacious; exposure to a chemical in the workplace is harmful or not as well as in product liability cases. The validity of clinical findings that are used in advertisements claiming a product is superior to a competitor's or in support of a patent are subject to legal challenge. Recently, the adequacy of the information provided to patients who participate in clinical trials and similar studies has become an issue. This aspect of trials is especially important when the main investigators or the University have a financial interest in the outcome either through stock ownership or options or a royalty agreement.
Sampling With Probability Proportional to Size
Authors: Czaja Ronald
Published: September 2014

Sampling with probability to proportionate to size (PPS), as the name implies, involves sampling elements that are typically arranged into natural groupings of different sizes. Sampling PPS is an efficient design when a list of the population is not available; the elements are in clusters or natural groups that differ greatly in size; it is important to control the sample size; costs and equalizing interviewer workloads are important; the use of natural divisions is possible and supplementary information is available by the natural divisions; and when the sample must be drawn in multiple stages. This entry presents a detailed example of a PPS design, a multistage area probability design, and discusses a random digit telephone sample design that uses PPS sampling.
Systematic Sampling Methods
Authors: Bellhouse D. R.
Published: September 2014

Systematic sampling is a sampling technique that is used for its simplicity and convenience. At its simplest, a systematic sample is obtained by selecting a random start near the beginning of the population list and then taking every unit equally spaced thereafter. The technique can be generalized to include systematic sampling schemes with probability proportional to an auxiliary variable. In many situations, systematic sampling is statistically efficient when compared to other sampling schemes. This is especially true in populations with linear and quadratic trends and autocorrelated populations. For populations in random order systematic sampling, it is equivalent to simple random sampling. A major drawback to systematic sampling is that it does not admit an unbiased estimator of variance with respect to the sampling design. Variance estimation must rely on an assumed underlying structure in the population.
Forward Search
Authors: Atkinson A. C. , Riani Marco
Published: September 2014

This is a powerful robust statistical method for exploring the relationship between data and fitted models. It aids discovery of clusters of observations and previously unidentified important subsets of the data as well as revealing groups of outliers. Examples are given with regard to the use of the forward search for regression and generalized linear models, and for multivariate data. The method is to search through the data, fitting subsets of increasing numbers of observations, with outliers being included towards the end.
Akaike's Criterion: Details
Authors: Stone M.
Published: September 2014

This entry covers Akaike's Final Prediction Error and information criteria for model selection, their links with Mallows'Cpand cross‐validation, and user's expectations of such criteria.
Dummy Variables
Authors: Darlington G. A.
Published: September 2014

Categorical explanatory variables can be included in models via the use of dummy variables. In a regression model with an intercept,
K
 − 1 linearly independent dummy variables must be defined to represent a
K
level categorical variable. Indicator variables are commonly used to define a set of dummy variables. In a regression model with an intercept, indicators are derived for
K
 − 1 of the
K
possible levels and the remaining level is the referent or baseline category. Overall tests regarding the categorical variable will not depend on the choice of dummy variable definitions. However, the interpretation of the coefficient for each dummy variable will depend on the coding used.
Interaction
Authors: Farewell Vern T.
Published: September 2014

A basic introduction to interaction terms in regression models is given.
Measurement Scale
Authors: Hand David J.
Published: September 2014

The measured variables analyzed in statistics are reflections of aspects of the system being studied. As such, the relationships between values of the variables must reflect properties of the attributes being studied. Different types of measurement scale, amongst which the most familiar are nominal, ordinal, interval, and ratio, reflect different properties of the attributes. Incorrect conclusions can be drawn if a statistical analysis fails to take appropriate account of this relationship between the objects of study and their numerical representations.
Model, Choice of
Authors: S. Matthews John N.
Published: September 2014

The choice of which statistical model to use in a given application is a complex matter, which draws not only on the technical expertise of the statistician but also on the nature of the application and the purpose of the analysis. Different purposes may require quite different models to be applied to the same dataset. The role of modeling in clinical trials, epidemiology, and general biostatistics is discussed. The relative contribution of expertise both in the area of application and in the relevant statistical issues is explored. The use of sensitivity analyses to ensure that substantive conclusions are not heavily dependent on model assumptions is stressed. A strong theme throughout the article is that the judgment of the biostatistician plays an inescapable role in an activity, which nonstatisticians too often see as wholly objective.
Multiple Linear Regression
Authors: Matthews David E.
Published: September 2014

Multiple linear regression involves finding the best‐fitting surface of a suitable functional form that relates the values of explanatory variables,X1, …,Xk, and the mean value of a response variable,Y, given values ofX1, …,Xk. The objectives of regression modeling are to determine whetherYand one or more of the explanatory variables are associated in some systematic way, and to estimate or predict the value ofY, or its mean, corresponding to known values of a selected subset ofX1, …,Xk. We describe methods of estimation, variable selection, statistical inference, and diagnostic checking of the assumed model and any associated unknown parameters, giving due importance to their statistical and scientific interpretations. An example concerning the relationship between oxygen uptake and age, weight, sex, the time required to run 1.5 miles, and various pulse rates for participants in a physical fitness workshop illustrates these concepts concretely. Finally, we outline the close connection between ordinary and weighted linear regression.
Regression
Authors: Farewell Vern T.
Published: September 2014

Statistical usage of the term regression is discussed.
Separate Families of Hypotheses
Authors: Braganca Pereira Basilio
Published: September 2014

In the last forty years since statistical procedures were first developed to test separate (nonnested) families of hypotheses, a huge amount of literature has developed for testing a null model against one or more separate alternatives. In the first two decades, mainly statisticians were interested in this problem. During the eighties the econometricians took a great interest and in the nineties, Bayesian statisticians developed some new results on the subject. The purpose of this paper is to given an introductory overview of the subject and an updated bibliography.
Univariate Response
Authors: Farewell Vern T.
Published: September 2014

A univariate response variable is contrasted with a multivariate response variable. The terms univariate analysis and multivariate analysis are also discussed.
Variance Components
Authors: Solomon P. J.
Published: September 2014

Random variation may have a complex structure, with several identifiable sources and corresponding components of variance. In a hierarchical or multilevel model, parameters may change, for example, between individuals, their variation being measured by variance components. This article reviews the history of the study of variance components and describes methods of estimation. Topics include synthesis of variance, empirical Bayes methods, nonnormal models, and model assessment and prediction.
Cox Regression Model
Authors: Sasieni Peter
Published: September 2014

In 1972, D.R. Cox proposed the semiparametric modelfor the hazard at timetfor an individual with covariatesz. Here, the hazard is the instantaneous conditional probability of failure attgiven that failure has not happened beforet.

The Cox model has had enormous influence in the theory and practice of biostatistics, and this article surveys its definition, history, basic use, estimators and algorithms, asymptotic properties, use of time‐dependent covariates, model checking, and alternatives and extensions.
Structural Time Series Models
Authors: Harvey A. , Koopman S. J.
Published: September 2014

Structural time series models are set up in terms of components, such as trends and cycles, which have a direct interpretation. Their statistical treatment is based on the state space form and the Kalman filter. Signal extraction is carried out by smoothing algorithms. Models may be constructed for multivariate series, and explanatory and intervention variables may be included. Nonlinear and non‐Gaussian models are handled using simulation techniques.
Canonical Correlation
Authors: Das Shubhabrata , Sen Pranab K.
Published: September 2014

The article starts with the genesis of canonical correlation from the product moment correlation and defines canonical correlation from the various angles including algebraic and geometric approaches. The different issues related to interpreting canonical correlation analysis, for example, canonical coefficients, canonical loadings, and redundancy coefficients are highlighted. Relationship with other multivariate methods like principal component analysis, factor analysis, and MANOVA have also been outlined. Then the article dwells on the sampling distribution of the sample canonical correlation and the related inference problems, while touching upon the relevant resampling methods in this context. Various generalizations and extensions are taken up next, with emphasis on the treatment of dependence between more than two sets of variables as well as constrained dependence under problem specific restrictions on the coefficients. The article concludes with a narration of issues related to applications, specially in the field of Biostatistics and a reference to key articles in this domain.
Biostatistics, an overview
Authors: Byron Brown W.
Published: September 2014

Biostatistics is concerned with the development and use of statistical methods to solve problems and to answer questions that arise in human biology and medicine. Its development is traced through early work in vital and health statistics, and in heredity, with the important influence of R.A. Fisher on experimental design and statistical inference. Applications in clinical trials, epidemiology, psychometrics, and health services research are described. Current challenges include the impact of computers, Bayesian methods, the use of data banks, longitudinal studies, and research on the human genome. The organization of biostatistics as a professional discipline is discussed.
B
ayes' Theorem
Authors: Lindley Dennis V.
Published: September 2014

Bayes' Theorem, which plays a central role in Bayesian inference, relates the probability of an uncertain event after additional knowledge has been obtained, to the probability before the acquisition of that knowledge, by multiplying by the likelihood (i.e. the probability of the additional information given the occurrence of the event). An alternative formulation is that the posterior odds on the event is obtained from the prior odds by multiplying by the likelihood ratio.
Density Estimation
Authors: Scott D. W.
Published: September 2014

The “density estimation” article highlights the dual roles of the graphical exploration of data as well as the nonparametric estimation of probabilities. Estimators described in detail include the histogram, frequency polygon, averaged shifted histogram, and kernel method. The basic theory of estimation is outlined. Data‐based calibration by cross‐validation is described. Several concrete examples are given.
Empirical Bayes, with Application to Genomics
Authors: Morris Carl N. , Christiansen Cindy L.
Published: September 2014

When certain estimation situations are encountered many times, the data from all these situations can be used to construct estimates for each individual case. This overview shows how empirical Bayes has evolved and expanded since its 1955 origins in order to useful practical procedures, which are good not only asymptotically but also in moderately sized samples. Various distributional models are described, as is an application to gene expression data.
The Theory of Estimating Functions
Authors: Desmond A. F. , Godambe V. P.
Published: September 2014

The aim of this article is twofold. First, we demonstrate that the methodology of estimating functions produces a unification and extensions of approaches to statistical modeling and inference, both parametric and semiparametric. Secondly, we illustrate via selected examples, how this methodology applies in biostatistics. We have limited our discussion, here, to case–control studies, prospective and retrospective sampling, and overdispersion.
Expectation
Authors: Proschan Michael A.
Published: September 2014

The expectation of a random variable is its mean, and may be thought of as the long‐run average of a random series of observations. Attention is paid here to the definition of conditional expectation, and to the properties of marginal and conditional expectations.
Fiducial probabiltiy including confidence intervals, nuisance factors and prediction
Authors: Seidenfeld Teddy
Published: September 2014

Fiducial probability was introduced by R. A. Fisher in 1930, as a method by which probability statements could (apparently) be made about an unknown parameter value without recourse to the Bayesian tools of prior and posterior probability. In a simple univariate one‐parameter case, this involves inversion of a probability statement from the distribution function. In other cases, it may involve a pivotal quantity whose distribution is independent of the parameter. The resulting statements are often similar to Bayesian posteriors with uniform priors, and fiducial intervals are often numerically equivalent to confidence intervals, although the logical structure is quite different. In some cases, notably the ratio of two normal means (the Creasy–Fieller problem) and the difference between two normal means with different variances (the Behrens–Fisher problem) fiducial and confidence intervals differ. The relationship between fiducial and Bayesian inference is discussed here in some detail.
James‐Stein Estimator: Introduction
Authors: Casella George
Published: September 2014

For a multivariate normal distribution with independent components, the observed vector is an inadmissible estimator of the mean vector. The James–Stein estimator defines a statistic with lower squared error loss than the observed vector. It has led to research into families of minimax estimators, some of which is summarized.
Law of Large Numbers: Theory
Authors: Janssen Paul
Published: September 2014

It is well known that the sample mean, based on a sequence of independent random variables with common distribution, is a weakly consistent estimator as well as a strongly consistent estimator for the population mean. The first property is the famous weak law of large numbers (Khinchin), the second property is the strong law of large numbers (Kolmogorov). Both laws are key results of modern probability theory. Similar properties hold for more complicated sequences of random variables, for example, for sequences satisfying the stationarity assumption or for martingale sequences. These extensions are of crucial importance for statistical applications in ergodic theory and survival analysis.
Likelihood Ratio
Authors: Birkes D.
Published: September 2014

In a statistical model for a set of data, the likelihood of a particular value of the parameter is the density function for that parameter evaluated at the observed data. For any two values of the parameter, the ratio of their likelihoods indicates how likely the first parameter value is relative to the second.
Limit Theorems
Authors: Zabell S. L.
Published: September 2014

These concern the asymptotic behavior of distributions of random variables in large samples. Classical results include the weak and strong laws of large numbers, the central limit theorem, and the law of the iterated logarithm. Other results discussed briefly are the ergodic theorem, the martingale convergence theorem, Skorokhod's theorem, and the use of coupling methods.
L
indley's Paradox
Authors: Shafer Glenn
Published: September 2014

A test of significance can disagree strongly with a Bayesian analysis when the Bayesian analysis assigns a substantial probability to the point hypothesis and distributes the remaining probability diffusely. The disagreement is sometimes called
Lindley's paradox
. If the diffuse distribution under the alternative hypothesis is based on evidence, the Bayesian analysis is convincing, but if the distribution is diffuse only because of ignorance, the test of significance may be more convincing.
Locally Most Powerful Tests
Authors: Peña Edsel A. , Rohatgi Vijay K.
Published: September 2014

Locally most powerful tests, that is, tests that have maximum power against local alternatives are discussed.
Loss Function
Authors: Tritchler D.
Published: September 2014

In decision theory, the loss function expresses the consequences of a specific decision when a particular state of nature holds. Negative loss is called
gain or utility
. The aim in Bayesian theory is to minimize the expected loss, integrating over the prior distribution. In frequentist theory, various restrictive conditions may be imposed, or a minimax solution may be sought. Statistical procedures such as hypothesis testing and estimation may be interpreted in terms of loss functions. Practical applications include the design of clinical trials and screening procedures.
Minimax Theory
Authors: Brown L. D.
Published: September 2014

In statistical decision theory and the theory of games, a minimax strategy or decision is one that minimizes (over decisions) the maximum risk (over possible distributions). The fundamental minimax theorem asserts that, under certain conditions, there is a minimax procedure, which is a Bayes procedure for a corresponding prior distribution.
Minimum Variance Unbiased (
MVU
) Estimator
Authors: Bethel J.
Published: September 2014

Determining the minimum variance of an unbiased estimator constitutes a fundamental topic in mathematical statistics. Two primary results are the Cramér–Rao inequality, which gives a lower bound on the variance of an unbiased estimator, and the Rao–Blackwell theorem, which supplies a method for improving the variance of unbiased estimators that are not functions of sufficient statistics. The strongest results are available for sufficient statistics from complete exponential families. However, any functions of sufficient statistics will attain the Cramér–Rao lower bound for sufficiently large sample sizes.
Mode
Authors: Everitt Brian S.
Published: September 2014

The mode of a distribution is the value of the variable with the highest frequency (for discrete data) or density (for continuous data). An example is given of a bimodal (two‐mode) distribution.
Paired

t

Test
Authors: Hsu Henry , Lachenbruch Peter A.
Published: September 2014

This is used to compare mean differences between treatments when the observations have been obtained in pairs. The difference between the paired values is assumed to be normally distributed, and the null hypothesis that the expectation is zero is tested by Student's
t
test. The robustness properties are discussed, as is the asymptotic relative efficiency of nonparametric alternatives.
Pitman Efficiency
Authors: Zacks S.
Published: September 2014

The Pitman efficiency is an index for comparing test procedures or estimators. It is especially important for comparing procedures in large samples. If procedure P1 requires n1 observations to attain a certain power of a test, or a specified mean squared error, and procedure P2 requires n2 observations to achieve the same precision, the Pitman relative efficiency of P1 against P2 is n2/n1. Accordingly, if the mean squared error of an estimator E1 is v1/n + o(1/n), and if that of another estimator, E2, is v2/n + o(1/n), then the asymptotic (large sample) relative efficiency of E1 against E2 is v2/v1. Under smoothness regularity conditions, the variance of an unbiased estimator, in the one‐parameter case, is bounded below by the Cramer–Rao (CR) lower bound. Accordingly, the ratio of the CR‐lower bound to the variance of the estimator is defined as the efficiency index of the unbiased estimator. This is the Pitman relative efficiency of the estimator against the best possible one. Moreover, the CR‐lower bound contains the Fisher information function as a factor, and the inverse of the Fisher information function is the asymptotic variance of the maximum likelihood estimator (bf MLE). Therefore, the efficiency of an unbiased estimator is measured relative to the asymptotic variance of the maximum likelihood estimator.

The Pitman relative efficiency of consistent and asymptotically normal test statistics is defined as the ratio of their corresponding
efficacy ratios
. The asymptotic efficacy relates the rate of change of the mean function of the statistic to its asymptotic standard deviation. For further information, see the article.
Signed‐rank Statistics
Authors: Woolson R. F.
Published: September 2014

In a single sample, or with differences between paired observations in a matched‐pair study, the observations may be ranked in order of their absolute values. The sign of the rank is then restored, and an appropriate statistic is calculated from the signed ranks. The most usual example is the Wilcoxon signed‐rank test.
Simultaneous Confidence Intervals
Authors: Chen Tuhao , Hoppe Fred M.
Published: September 2014

These provide a confidence region for a multivariate parameter, comprising individual intervals for the separate components of the parameter. There is a specified confidence level of the simultaneous correctness of all the statements. The approach is useful for multiple comparisons of treatment means. The methods of Bonferroni, Tukey, Scheffé and Duncan are described, with examples.
Statistical Dependence and Independence
Authors: Wermuth Nanny , Cox D. R.
Published: September 2014

Statistical dependence is a type of relation between different characteristics measured on the same units. At one extreme is deterministic dependence; at the other is statistical independence, where the distribution of one variable is the same for all levels of the other. With more than two variables, an important distinction is between marginal and conditional dependence. In many contexts, the degree of dependence may be summarized by a suitable measure of association, perhaps as part of a general model. Reference is made to graphical models.
Sufficiency
Authors: Strawderman William E.
Published: September 2014

According to the principle of sufficiency, introduced by R.A. Fisher in 1922, a “sufficient statistic” is equivalent, for all purposes of estimation, to the original data. Connections with the likelihood principle are noted, as is the case for using robust methods, which do not demand full confidence in the proposed statistical model.
Randomized complete blocks designs
Authors: Murray L.
Published: September 2014

This article discusses the general need for blocking when experimental units are not homogeneous. Complete, generalized, and incomplete block designs are discussed, together with issues of randomization, missing cells, and the usual assumption of “no interaction”. Two specific cases of the Randomized Complete Block Design are considered: the fixed case (treatments and blocks are both fixed) and the mixed case (treatments are fixed, blocks are random). For each case, the scalar linear models are defined in full‐rank and non‐full‐rank versions, together with necessary assumptions. Point estimates and standard errors are given for main‐effect means and cell means, and for variance components. The analysis of variance table is given, with expected mean squares, for testing hypotheses.
Coarsening at Random
Authors: Nielsen Søren Feodor
Published: September 2014

Statistical analysis is often complicated by the presence of incomplete observations. Unfortunately, incomplete observations are unavoidable in many studies. However, if the incompleteness is ignorable, some simplifications to the statistical analysis occur. The concept of coarsening at random is a formalization of what is meant by ignorability of incomplete data.
Grouped Survival Times
Authors: Zhang Mei‐Jie
Published: September 2014

The grouped survival data consists of occurrence and exposure data over given time intervals and possible covariate strata. Grouping data from such large sample sizes into tabular presentations (life tables) often provides a convenient format for presenting and summarizing life information. Grouped survival data can involve censored data (right censoring, left censoring, or double censoring) and/or truncated data. Moreover, the exact censoring or truncation times may be unknown for grouped data. In this article, we provide the literature on analyzing grouped survival data, and discuss the estimating procedure for the hazard function and survival function under nonparametric or parametric models, test statistics for comparing the survival probabilities among different population groups, and large sample properties for these estimators and test statistics.
Hazard Ratio Estimator
Authors: Andersen Per Kragh
Published: September 2014

The two‐sample proportional hazards model is studied and a series of semiparametric estimators for the (constant) hazard ratio are presented. These include both Cox's maximum partial likelihood estimator and explicit rank‐type estimators.
Influence Function in Survival Analysis
Authors: Reid N.
Published: September 2014

The influence function of a statistic is a type of derivative that indicates how perturbations in the data affect the statistic. If the influence function is bounded, then the effect of outliers on the statistic is limited. Influence functions can be used in deriving the limiting distribution of a statistic, and can also be used to develop diagnostics for regression models. This latter application has been useful in proportional hazards regression.
Joint modelling of longitudinal and event time data
Authors: Henderson Robin
Published: September 2014

The relationship between the development over time of a quantitative response variable and a parallel process recording events in time can be investigated through a joint model for the two outcome types. Interest in this topic arose from two separate backgrounds: the possibility and consequences of informative dropout in longitudinal trials, and measurement error problems for survival studies with time‐varying covariates. Modeling strategies include combining a marginal distribution for one response type with a conditional distribution for the other, or adopting a correlated random effects model with otherwise conditional independence. Naive methods can lead to significant bias, yet more sophisticated modeling approaches are often sensitive to assumptions, some of which cannot be checked from observable data. When simple summaries such as treatment effect estimators are of interest, sensitivity analyses can bring some reassurance as to whether important conclusions are maintained as assumptions are allowed to change. Overall, the topic is relatively new and research is continuing in many directions.
Measurement Error in Survival Analysis
Authors: Prentice Ross L.
Published: September 2014

This entry describes various approaches to estimating the regression coefficient in the Cox failure time regression model when some of the regression variables are poorly measured. Regression parameter estimation may be considered if there is a validation subsample within which the “true” regression variable is available, or if there is a reliability subsample within which multiple estimates of the regression variable having independent measurement errors are available. The former context is closely connected to corresponding missing data estimation problems. Simplead hocestimators are reviewed, along with effective but inconsistent regression calibration estimators, and more complex nonparametric maximum likelihood and augmented inverse probability weighted estimators. With only a reliability sample available, estimation procedures include regression calibration and an extended risk set regression calibration approach, along with corrected score estimation procedures. Some comments on procedures for measurement error accommodation in an additive hazards model are also provided. This rapidly developing measurement error research area has implications for the analysis of follow‐up studies in a number of important application areas.
Repeated Events
Authors: Lawless J. F.
Published: September 2014

Subjects in longitudinal studies often experience repeated events. Data on the timing of such events allow us to study patterns of event occurrence, to characterize within‐ and between‐subject variability, and to relate event occurrence to other processes. This article describes statistical models for recurrent events and associated methods of analysis.
Smoothing Hazard Rates
Authors: Wang Jane‐Ling
Published: September 2014

The nonparametric approach to estimate hazard rates for lifetime data is flexible, model‐free, and data‐driven. No shape assumption is imposed other than that the hazard function is a smooth function. Such an approach typically involves smoothing of an initial hazard estimate, with arbitrary choice of smoother. We describe methods for grouped lifetime data observed at certain time intervals and for continuously observed lifetime data. There are some intrinsic differences between the smoothing approaches for these two types of data. More specifically, smoothing an initial hazard estimate based on the life table is adopted for grouped lifetime data; while for continuous data, smoothing is employed to increments of the Nelson–Aalen cumulative hazard estimate aiming at the derivative of the cumulative hazard function. A few nonparametric hazard regression methods are also discussed.
Tied Survival Times
Authors: Zhang Mei‐Jie
Published: September 2014

Tied survival times occur in survival studies frequently. We discuss adjustment for ties for some common statistical procedures.
Time Origin, Choice of
Authors: Keiding Niels
Published: September 2014

Subject‐matter issues as well as technical issues regarding choice of suitable time origin in survival and event history analysis are discussed. In rare cases, alternative time
scales
may be relevant.
Truncated Survival Times
Authors: Gruttola Victor , Liao Q.
Published: September 2014

Truncation of survival data arises when observation can only occur if the failure time (survival time) lies within a certain interval (l,r) wherelorrmay be infinite. When the failure time is outside the truncation interval, no information about the subject is available, so the subject is “sampled from a conditional distribution”. This feature distinguishes truncation from censoring. Left truncation (delayed entry) is relatively easily handled by minor modification of hazard‐based survival analysis methods. Right truncation has been extensively studied in connection with latency times from HIV infection to onset of clinical AIDS.
Bayesian Approaches to Cure Rate Models
Authors: Ibrahim Joseph G. , Chen Ming‐Hui , Sinha Debajyoti
Published: September 2014

Bayesian approaches to cure rate modeling have gained great popularity in the analysis of survival data. In this chapter, we review several Bayesian approaches, discussing various alternative formulations of the cure rate model, their interpretations, and their properties.
Standardization Methods
Authors: Inskip Hazel
Published: September 2014

Standardization methods are used to adjust for the effects of age and sex, and possibly other factors, in the comparison of mortality or disease rates between two or more populations. Such methods have a long history, particularly for the comparison of mortality rates. The two methods that are most widely used are known as direct and indirect standardization and the mortality ratios that are derived from these are known respectively as the Comparative Mortality Figure and the Standardized Mortality Ratio. When the populations at risk are hard to define, methods involving the proportions of deaths from specific causes can be used. More recently, different approaches involving modeling have been developed to adjust rates for the effects of age and other confounding factors.
Age–Period–Cohort Analysis
Authors: Holford Theodore R.
Published: September 2014

Age–period–cohort analysis refers to the interpretation of temporal trends in disease incidence or mortality rates in terms of the three time scales, age, calendar date (period), and year of birth (cohort). We present graphical and analytic methods for understanding trends in disease rates. A fundamental issue of identifiability arises in some models because of a linear constraint on these time scales: year of birth plus age equals the current calendar year. We review parametric and nonparametric approaches to estimate identifiable aspects of the data, such as the curvature of temporal trends.
Subjective Probability and Human Judgement
Authors: Ayton Peter
Published: September 2014

Psychological research on Subjective Probability has used a variety of strategies to investigate how reliable people are at judging probabilities. The issue is of considerable practical importance as all sorts of critical decisions depend on assessment of the likelihood of events for which there is neither a method for mathematically computing its probability nor actuarial data for measuring it empirically. Since the development of Decision theory, a plethora of management technologies have emerged that can take advantage of likelihood information for decision support. While the concept of subjective probability remains a theoretically contentious notion for some, people can and do express and request judgments of the likelihood of uncertain events. Here, research investigating the validity of subjective probabilities – judged likelihood – is reviewed.
Configural Frequency Analysis
Authors: Eye Alexander
Published: September 2014

Configural Frequency Analysis (CFA) is a method for the analysis of cross‐classifications of categorical variables. Using CFA, one examines individual cells or groups of cells of cross‐classifications with the goal of determining whether they contain more or fewer cases than expected based on some chance model. The specification of the chance model determines the interpretation of results. Depending on chance model, cells with more or fewer cases than expected can be interpreted as indicative of local associations, but also of predictor‐criterion relationships; they allow one to distinguish between groups, or they reflect typical and atypical patterns of change over time.
Multigraph Modeling
Authors: Khamis Harry J.
Published: September 2014

Multigraph modeling makes use of mathematical graph theory to analyze and interpret hierarchical loglinear models. For a given such model, a multigraph is constructed, and from the multigraph, decomposability is determined from a simple combinatorial identity; factorization of the joint distribution is based on a maximum spanning tree; and the conditional independence interpretation comes from the edge cutsets.
Breslow–Day Statistic
Authors: Wang Molin , Berger Vance W.
Published: September 2014

When analyzing a series of stratified 2 × 2 tables, the analysis can be simplified if each table has a common odds ratio. The Breslow–Day‐statistic is a test statistic for testing this homogeneity against the global alternative of heterogeneity.
Simulation Methods for Categorical Variables
Authors: Weber Stefan , Eye Alexander
Published: September 2014

The results of a simulation study on the performance of measures for the analysis of 2 × 2 tables are reported. The simulations included 11 measures for 2 × 2 tables: Pearson's?, the standard normal z, the log‐odds ratio, the log‐linear interaction, Goodman's (1991) weighted log‐linear interaction, Vogel's z, the binomial test, Lehmacher's (1981) asymptotic hypergeometric test, Perli, Hommel, and Lehmacher's (1985) asymptotic test, Lindner's (1984) exact hypergeometric test, and Lausch and von Weber's (2002) adaptation of Dunkl and von Eye's (1990) test. The factors varied in the simulations were (1) type of sampling distribution, (2) sample size, (3) strength of association in the 2 × 2 table, (4) the symmetry of the distribution in the 2 × 2 table, and (5) the nominal a. Results suggest that the distribution of these 11 tests is very near the normal under all conditions. Of the five factors, only the type of the sampling distribution has no strong effects on the ß‐curves of the 11 measures. Finally, it was found that the 11 measures respond differently to the five factors such that the rank order of performance varies with the simulated conditions.
Optimization Methods
Authors: Lewis Robert Michael , Trosset Michael W.
Published: September 2014

Optimization is concerned with identifying the best element in a set of competing possibilities. Except in special circumstances, one must use numerical methods to search for solutions iteratively. If one's problem is well‐behaved, for example, the objective to be minimized is smooth and the feasible possibilities are smoothly defined by finite numbers of equality and inequality constraints, then one can use a number of sophisticated methods to identify local solutions, that is, possibilities that are as good or better than any other nearby possibilities. However, finding global solutions is much harder.
Linear Models: Permutation Methods
Authors: Cade Brian S.
Published: September 2014

Permutation tests of hypotheses about parameters in linear models derive their validity from exchangeability of the errors in the null model. These errors are unobservable but differ from the observable residuals by a constant for any hypothesis that posits a null, reduced parameter model with just an intercept term. For these hypotheses, exact permutation inferences are possible for iid error distributions of any form. Other hypotheses on subsets of parameters in multiple regression models are not exact because the residuals are not exchangeable with equal probability. Several options to improve exchangeability for these hypotheses have been developed and are presented for permutation tests on quantile regression estimates of the linear model.
Randomized Response Technique: Definition
Authors: Everitt Brian S.
Published: September 2014

A technique that can be used in surveys to collect honest answers to sensitive questions.
Balanced Incomplete Block Designs
Authors: Cotton John W.
Published: September 2014

Balanced incomplete block designs (BIBD) are experimental designs that can be described by a labeled, fully filled rectangular array or by a larger equivalent rectangular display with some empty cells. A cell in the rectangle identifies one treatment in a given block. The number of different treatments for a given block (individual) is less than the total number of treatments; this precludes having each treatment appear once in every block. Hence, the design is anincompleteblock design. Without some constraints on the number of occurrences of certain treatments in different rows, the average performance in different parts of the design can be biased. This bias yields different averages for two treatments, for example, because of differences in the cells in which those treatments were employed. To eliminate the bias, a special form of balance is employed in which every possible pair of treatments appears in an equal number of blocks. Accordingly, there also is balance in the total number of times each treatment occurs. Analysis methods are discussed for general linear fixed, mixed, and Bayesian models. I discuss a special case of a BIBD design, Youden squares, that permits but does not require the assessment of one more independent variable than other BIBDs. In the behavioral sciences, the rows of Youden squares can be different blocks, typically different people or animals, and the columns can be different observation occasions that permit the assessment of so‐called period effects.
Field Experiment
Authors: Busk Patricia L.
Published: September 2014

An experiment conducted in a natural setting to show a causal relationship. These studies attempt to control the influence of environmental constraints on the relationship between the independent and dependent variables. They do not rely exclusively on natural conditions of the environment in that the independent variables are manipulated by the experimenter. Field experiments typically have less rigorous control features than laboratory experiments.