RESEARCH METHOD COHEN ok

REGRESSION ANALYSIS 537
we know or assume values of the other variable(s)’
(Cohen and Holliday 1996: 88). It
is a way of modelling the relationship between
variables. We concern ourselves here with
simple linear regression and multiple regression
(see http://www.routledge.com/textbooks/
9780415368780 – Chapter 24, file SPSS Manual
24.7).
Simple linear regression
In simple linear regression the model includes
one explanatory variable (the independent variable)
and one explained variable (the dependent
variable) (see http://www.routledge.com/
textbooks/9780415368780 – Chapter 24, file
24.15.ppt). For example, we may wish to see the
effect of hours of study on levels of achievement in
an examination, to be able to see how much improvement
will be made to an examination mark
by a given number of hours of study. Hours of study
is the independent variable and level of achievement
is the dependent variable. Conventionally,
as in the example in Box 24.29, one places the
independent variable in the vertical axis and the
dependent variable in the horizontal axis. In the
example in Box 24.29, we have taken 50 cases of
hours of study and student performance, and have
constructed a scatterplot to show the distributions
(SPSS performs this function at the click of two
or three keys). We have also constructed a line
of best fit (SPSS will do this easily) to indicate
the relationship between the two variables. The
line of best fit is the closest straight line that can
be constructed to take account of variance in the
scores, and strives to have the same number of
cases above it and below it and making each point
as close to the line as possible; for example, one
can see that some scores are very close to the
line and others are some distance away. There is a
formula for its calculation, but we do not explore
that here.
One can observe that the greater the number
of hours spent in studying, generally the greater
is the level of achievement. This is akin to
correlation. The line of best fit indicates not only
that there is a positive relationship, but also that
Box 24.29
Ascatterplotwiththeregressionline
Hours of study
6
5
4
3
2
1
0
20
30
40 50 60
Level of achievement
the relationship is strong (the slope of the line is
quite steep). However, where regression departs
from correlation is that regression provides an
exact prediction of the value – the amount – of
one variable when one knows the value of the
other. One could read off the level of achievement,
for example, if one were to study for two hours (43
marks out of 80) or for four hours (72 marks out
of 80), of course, taking no account of variance.
To help here scatterplots (e.g. in SPSS) can insert
grid lines, for example (Box 24.30).
It is dangerous to predict outside the limits of
the line; simple regression is to be used only to
calculate values within the limits of the actual line,
and not beyond it. One can observe, also, that
though it is possible to construct a straight line
of best fit (SPSS does this automatically), some
of the data points lie close to the line and some
lie a long way from the line; the distance of the
data points from the line is termed the residuals,
and this would have to be commented on in any
analysis (there is a statistical calculation to address
this but we do not go into it here).
Where the line strikes the vertical axis is named
the intercept. Wereturntothislater,butatthis
stage we note that the line does not go through
the origin but starts a little way up the vertical
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Chapter 24