(mode of transport to work; walk, public transport, bicycle, own car); ordered category,
when frequency counts are arranged into ordered categories (social class group—I to V
where I refers to the most affluent group); or individual ranked when each subject or
object is assigned a numerical value designating its rank position relative to all other
objects (position of school in Local Education Authority school league tables). In this
chapter we examine the use of statistical inferences involving binary and nominal count
data.
A number of inferential statistical techniques have been developed for analyzing count
data. These nonparametric techniques do not require the data to be drawn from a
normally distributed population (the underlying distribution may be binomial,
multinomial, product multinomial or simply unknown) and therefore do not require
interpretation based on the normal distribution. However, some statistical tests used with
count data do approximate a normal distribution when sample sizes are large, such as
binomial and χ^2 distributions.
The examples used to illustrate the use and interpretation of inferential statistical tests,
in this and in subsequent chapters, are drawn from research questions and analyses that
have appeared in sections of research reports, journal articles and students’ theses. My
intention in this and the following chapters is to explain why and how statistical tests
should be used and to illustrate their application in a variety of research contexts. In
Chapter 5, a number of considerations important when choosing a statistical test were
outlined, namely research questions and design, data distribution, type of inference and
specific test assumptions. These considerations are used to organize in a logical way the
presentation of statistical tests in this and in subsequent chapters.
Each statistical procedure is introduced by beginning with a real research problem and
concentrating on the reasons why a particular test is appropriate. This helps to clarify the
relationship between statistical theory and statistical practice. Brevity of reporting of
statistical analyses in many research journals means that much statistical theory and many
assumptions are taken as understood. For the new researcher this can be frustrating. In the
examples of the use of statistical tests drawn from the literature, I have filled in details
about the inferential process used in a statistical analysis, that is the parameters estimated
and the hypotheses tested and how these relate to underlying statistical theory, necessary
test assumptions, and where appropriate I have commented on the original author(s)
interpretation.
To move the reader from understanding why a test is used to consideration of how a
statistical test works, simplified worked examples are presented. Formulae are
explanatory and kept to a minimum. These worked examples, intended to help
understanding, are for the most part based on sections of real data but necessarily
simplified. Interpretation of the analysis is related to both statistical theory, such as one-
sided or two-sided tests, alpha, statistical power, sample size and to the purpose behind
the study such as original research question(s). The examples address the usual kinds of
problems that a researcher will face when analyzing real data, for example, unequal
groups sizes, missing or out of range values, outlier observations and skewed
distributions. Emphasis is given to computer analysis of real data. Computer programmes
for statistical analysis are presented alongside related computer output and interpretation
of the analysis. Where relevant procedures for checking any specific test assumptions are
Inferences involving binomial and nominal count data 161