then is to look at the data using IDA and decide what model may best describe and
account for observed variation and in particular what model reduces the amount of error
variation.
A relatively simple family of statistical models which allows for systematic and error
or random variation is known as the general linear model. Put simply this states that
values of a response variable are given by the weighted sum of independent variables
specified in a model plus a term standing for error. In equation form this can be stated as,
yij=μi+εij, where yij is the value of the jth observation from the ith treatment, μi is a
constant representing the mean treatment score for the population and εij is a randomly
fluctuating error term. This model is sometimes called the ‘means model’ because it
accounts for the overall mean treatment effect and underlying variation only.
It is possible to formulate a linear model for the one-factor vocabulary teaching
methods experiment introduced in Chapter 1. Recall this was a fixed-effects model with
three conditions or levels of treatment. This is important in specifying the statistical
model. The statistical model for this design would be:
yij=μ+αi+εij, where i=1, 2 or 3 representing the effects of one of the treatments, so yij is
the observed value for an individual j in the ith treatment group, μ is a constant or mean
response in the population and εij is the deviation from the mean treatment response for
the jth individual in the ith treatment (the error term). This error term is assumed to have
a normal distribution. This model is sometimes called an ‘effects model’ because it
accounts for various treatment effects.
This statistical model implies that any score can be decomposed into three parts, one
component μ which is constant over all observations and treatments, a fixed treatment
effect αi making the same contribution to all vocabulary scores in a given treatment group
and an error component that will differ both between treatment groups and also within
treatment groups. If a random effects design was used then the fixed treatment effect αi
would be replaced by a random treatment effect ai representing the effect of the ith
treatment (teaching method) which would be sampled at random from a population of
teaching methods. This statistical model, sometimes referred to as a structural model,
states, for example, that an individual in the enhanced storytelling method group (say
method group 1) has a vocabulary score which is represented by the mean vocabulary
score of the population, μ, plus any difference between this population mean and the
mean score for all pupils exposed to method group 1, the treatment effect αi, plus any
difference between the individuals score and the contribution of the treatment effect.
SummaryThe foregoing chapter should have impressed upon you the importance of IDA. By now
you should also realize that the choice of descriptive statistics and data display methods is
not always as straightforward as it seems. You should by now be in a position to know
how to describe and summarize a data set and make a preliminary decision about whether
further inferential statistical analysis is justified.
Initial data analysis 79