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The purpose of acquiring data is ultimately to help us answer a research question we have proposed. Statistics is the toolkit we use to do so. As we said in the opening chapter, there are three basic goals of statistics: summarization, inference, and prediction. This chapter focuses on summarization. Summarization is the process of taking a potentially large volume of data and reducing it to a few quantities that adequately represent the data so that the data can be easily understood. For example, suppose I collected data on heights and weights of a sample of 500 people. I could simply report all of this information, but it would not be clear exactly what height and weight look like in the sample, let alone the general population, nor would it be clear what the relationship is between them. I could, however, summarize both height and weight in terms of their means and standard deviations, and I could compute the correlation between them to show how they are related. We will begin this chapter by discussing methods for summarizing univariate data, that is, data on one variable. Later in the chapter we will discuss basic methods for summarizing bivariate data, that is, data on two variables. Subsequent chapters will discuss methods for examining relationships between variables.

The problem of testing equality of survival distributions on the basis of paired censored survival data has received considerable attention in literature. Some of the important statistics used for such purposes can be expressed as linear combinations of two statistics, one based on uncensored pairs and the other based on the censored pairs. Raychaudhuri and Rao (Nonparametric Statistics, 1996, 6, 1–11) investigated properties of two classes of such statistics and derived expressions for the optimal coefficients (weights) for the linear combination that will maximize efficacy within each class. As the optimal weights depend upon the form of the underlying survival and censoring distributions, statistics with optimal weights can only be used with estimated weights. This article presents a method of estimating optimal weights on the basis of an assumed model that specifies the distribution of the difference between the observed survival times conditional on the censoring pattern. The model, in addition to dispensing with the usual assumption that the survival and censoring variables are independent, also permits a graphical check of its lack of fit on the basis of observed data. The performance of statistics with the estimated weights is evaluated by using two simulation studies – one with data generated under the assumed model and the other assuming independence of the survival and censoring times. Simulation results show that the optimal statistics with estimated weights have good power properties in all cases considered, and that they compare well with other commonly used tests for paired censored survival data. An advantage of the tests with optimal weights is that, unlike their competitors, these tests have demonstrated performance characteristics in some cases where the assumption of independent censoring may not be justified.

A measure of dependence indicates in some particular manner how closely the variables X and Y are related; one extreme will include a case of complete linear dependence, and the other extreme will be complete mutual independence. Although it is customary in bivariate data analysis to compute a correlation measure of some sort, one number (or index) alone can never fully reveal the nature of dependence; hence a variety of measures are needed.

In this chapter we discuss bivariate discrete distributions. Bivariate means that there are two factors (categorical variables) defining cells. The response values are frequencies, that is, counts or instances of observations, at each cell.

The shared frailty model described in Chapter 7 is very useful for bivariate data with common risk dependence, but in many cases, we do need extensions. In particular, for truly multivariate data, that is, when there are three or more observations, we need more models with varying degrees of dependence. This general frailty approach can be used to create a random treatment by group interaction, or other models with several sources of variation. Secondly, combining subgroups with different degrees of dependence in a single model, for example, monozygotic and dizygotic twins, is difficult in a shared frailty model. Furthermore, this extension can be an improvement for the consideration of effects of covariates in frailty models.

We study the joint and the marginal distributions of the maximum and minimum of a pair of random variables (X 1, X 2) having the bivariate exponential distribution due to Freund, Marshall and Olkin, and Raftery. We explore their reliability properties and copula functions and compare these properties with the corresponding ones of the order statistics when X 1 and X 2 are assumed to be independent exponential random variables.

Exploring and identifying structure is even more important for multivariate data than univariate data, given the difficulties in graphically presenting multivariate data and the comparative lack of parametric models to represent it. Unfortunately, such exploration is also inherently more difficult.

In this chapter we study relationships between two columns of the data matrix which contain observations on quantitative random variables. The first section of the chapter will be devoted to a review of theory for bivariate distributions for both discrete and continuous random variables. This section will also include an introduction to the theory of correlation and regression and an introduction to the bivariate normal distribution. The second and third sections of the chapter will be devoted to the techniques of inference for correlation and regression respectively. The last section of the chapter will discuss bivariate inference in the presence of other variables including the topics of partial correlation and lurking variables. In each of the four sections a review of techniques normally presented in an introductory course will be combined with additional material.

The paired observation approach to the two-sample problem discussed in Chapter 12 assumes that either the data become available in natural pairs or that it is possible to arrange test subjects in pairs in such way that the two test subjects in a pair can be expected to perform very similarly. When pairing test subjects, a practical question arises: how successful has the matching process been? Have we succeeded in matching high performers with high performers and low performers with low performers?

This chapter deals with a number of diverse representations which are all aimed at providing meaningful, simple and interesting presentations of univariate and bivariate data. All these representations may be regarded as types of exploratory techniques that are used without any preconceived ideas of possible further analyses.

In this chapter the different measures of angular association are compared numerically. In section 8.1 the case of independent and identically distributed bivariate data is considerd, while in section 8.2 the different measures are studied in the context of a wrapped autoregressive process. It is found that amongst all the coefficients considered the proposal σQ shows the best performance.