Biostatistics

8.1 INTRODUCTION 305
8.1 INTRODUCTION
In the preceding chapters the basic concepts of statistics have been examined, and they
provide a foundation for this and the next several chapters. In this chapter and the three that
follow, we provide an overview of two of the most commonly employed analytical tools
used by applied statisticians, analysis of variance and linear regression. The conceptual
foundations of these analytical tools are statistical models that provide useful representations
of the relationships among several variables simultaneously.
Linear Models A statistical model is a mathematical representation of the relationships
among variables. More specifically for the purposes of this book, a statistical model is
most often used to describe how random variables are related to one another in a context in
which the value of one outcome variable, often referred to with the letter “y,” can be
modeled as a function of one or more explanatory variables, often referred to with the letter
“x.” In this way, we are interested in determining how much variability in outcomes can be
explained by random variables that were measured or controlled as part of an experiment.
The linear model can be expanded easily to the more generalized form, in which we include
multiple outcome variables simultaneously. These models are referred to as General Linear
Models, and can be found in more advanced statistics books.
DEFINITION
An outcome variable is represented by the set of measured values that
result from an experiment or some other statistical process. An
explanatory variable, on the other hand, is a variable that is useful for
predicting the value of the outcome variable.
A linear model is any model that is linear in the parameters that define the model. We
can represent such models generically in the form:
Y j ¼ b 0 þ b 1 X 1j þ b 2 X 2j þ ...þ b k X kj þ e j (8.1.1)
In this equation, b j represents the coefficients in the model and e j represents random error.
Therefore, any model that can be represented in this form, where the coefficients are
constants and the algebraic order of the model is one, is considered a linear model. Though
at first glance this equation may seem daunting, it actually is generally easy to find values
for the parameters using basic algebra or calculus, as we shall see as the chapter progresses.
We will see many representations of linear models in this and other forms in the next
several chapters. In particular, we will focus on the use of linear models for analyzing data
using the analysis of variance for testing differences among means, regression for making
predictions, and correlation for understanding associations among variables. In the context
of analysis of variance, the predictor variables are classification variables used to define
factors of interest (e.g., differentiating between a control group and a treatment group), and
in the context of correlation and linear regression the predictor variables are most often
continuous variables, or at least variables at a higher level than nominal classes. Though the
underlying purposes of these tasks may seem quite different, studying these techniques and