RESEARCH METHOD COHEN ok

MULTILEVEL MODELLING 583
Degrees of freedom is given by:
df = rcl − (cl − 1) − (r − 1) − 1
= rcl − cl − r + 1 = 8 − 4 − 2 + 1 = 3
Whiteley (1983) observes:
Note that we are assuming c and l are interrelated
so that once, say, p +11 is calculated, then P +12 , P +21
and P +22 are determined, so we have only 1 degree
of freedom; that is to say, we lose (cl − 1) degrees of
freedom in calculating that relationship.
(Whiteley 1983)
From chi-square tables we see that the critical
value of χ 2 with three degrees of freedom is 7.81 at
p = 0.05. Our obtained value is less than this. We
therefore accept the null hypothesis and conclude
that there is no relationship between sex on the one
hand and voting preference and social class on the
other.
Suppose now that instead of casting our
data into a three-way classification as shown in
Box 25.17, we had simply used a 2 × 2contingency
table and that we had sought to test the null
hypothesis that there is no relationship between
sex and voting preference. The data are shown in
Box 25.21.
When we compute chi-square from the above
data our obtained value is χ 2 = 4.48. Degrees
of freedom are given by (r − 1)(c − 1) = (2 − 1)
(2 − 1) = 1.
From chi-square tables we see that the critical
value of χ 2 with 1 degree of freedom is 3.84 at
p = 0.05. Our obtained value exceeds this. We
reject the null hypothesis and conclude that sex is
significantly associated with voting preference.
But how can we explain the differing
conclusions that we have arrived at in respect
of the data in Boxes 25.17 and 25.21 These
examples illustrate an important and general
point, Whiteley (1983) observes. In the bivariate
analysis (Box 25.21) we concluded that there
was a significant relationship between sex and
voting preference. In the multivariate analysis
(Box 25.17) that relationship was found to be nonsignificant
when we controlled for social class. The
Box 25.21
Sex and voting preference: a two-way classification
table
Conservative Labour
Men 120 160
Women 140 130
Source:adaptedfromWhiteley1983
lesson is plain: use a multivariate approach to the
analysis of contingency tables wherever the data
allow.
Multilevel modelling
Multilevel modelling (also known as multilevel
regression) is a statistical method that recognizes
that it is uncommon to be able to assign students
in schools randomly to control and experimental
groups, or indeed to conduct an experiment that
requires an intervention with one group while
maintaining a control group (Keeves and Sellin
1997: 394).
Typically in most schools, students are brought
together in particular groupings for specified
purposes and each group of students has its own
different characteristics which renders it different
from other groups. Multilevel modelling addresses
the fact that, unless it can be shown that different
groups of students are, in fact, alike, it is generally
inappropriate to aggregate groups of students or
data for the purposes of analysis. Multilevel models
avoid the pitfalls of aggregation and the ecological
fallacy (Plewis 1997: 35), i.e. making inferences
about individual students and behaviour from
aggregated data.
Data and variables exist at individual and
group levels, indeed Keeves and Sellin (1997)
break down analysis further into three main
levels: between students over all groups, between
groups, and between students within groups. One
could extend the notion of levels, of course, to
include individual, group, class, school, local, regional,
national and international levels (Paterson
Chapter 25