between X and Y in the population can be described by two parameters: beta(0), (β 0 ) the
intercept (value of Y when X=0) and beta(1) (β 1 ) the regression coefficient, a measure of
the slope of the regression line (change in Y per unit change in X). The parameters are
estimated by the regression equation and hypothesis tests can be constructed for beta(0)
and beta(1). Some introductory statistical texts state that the response variable in a linear
regression has to be normally distributed; this is not strictly correct. The important
assumption is that the residuals are normally distributed and independent (see Chapter 8).
The response variable does not have to be normally distributed or even a continuous
measure.
The idea of simple linear regression can be extended to find the best fitting statistical
model that describes the relationship between a response variable and more than one
predictor (explanatory independent) variable. This is called multiple regression. When
the outcome variable is binary, then an appropriate underlying probability model is the
logistic regression model. Not surprisingly the regression approach is called a logistic
regression, predictor variables may be continuous or binary.
Differences between two samplesWhen a research study is designed to assess treatment effectiveness, probably the first
statistic to come to mind is the t-test. This ubiquitous statistic is generally appropriate for
two sample comparison designs (sometimes called two group comparison designs). Being
a parametric statistical procedure, several assumptions have to be met before the t-test
can be properly used (see Chapter 8). Another important consideration is whether the
two-sample comparison is between independent or related samples. If a response variable
such as height is measured for two independent samples of individuals, for example, boys
and girls, then to test whether there was any statistically significant difference between
the mean height for boys and the mean height for girls (a difference between two
independent samples) an independent t-test could be considered. However, if a group of
boys were weighed on two occasions, for example, before and after dieting, two measures
are taken for the one group of subjects, then a related t-test (paired, repeated measures)
should be considered, because the measures on the two samples of weights are related or
correlated. An alternative design would be to match pairs of subjects on certain variables.
For a matched subjects design the paired t-test should be considered. One advantage of
the related t-test over the independent t-test is that statistical significance is attained, at a
specified p-level, with a smaller difference between the two means (assuming other
important attributes are equal). The null hypothesis for the independent t-test is that the
two means are equal, μ 1 =μ 2 , and for the related t-test the mean difference is zero,
μ 1 −μ 2 =0.
Should the independent t-test be considered inappropriate then an alternative
nonparametric procedure is the Wilcoxon Mann-Whitney test. The null hypothesis is
that the two samples have the same population distribution. An alternative nonparametric
procedure to the repeated measures t-test is the Wilcoxon Signed Ranks test.
A nonparametric repeated measures test for change, when any change is indicated
simply as + or −, is the Sign test. This test makes use of medians only, has only one
distributional assumption, the response variable, which theoretically has a continuous
distribution. The sign test provides an indication of only the direction of any difference,
Choosing a statistical test 123