Quality & Quantity 38: 425–433, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Scaling for Residual Variance Components of
Ordered Category Responses in Generalised Linear
Mixed Multilevel Models
Department of Economics, University of Birmingham, B15 2TT, UK
Abstract. A method is proposed that enables changes in variance components to be computed from
the results of ﬁtting ordered response generalised models with multilevel and random effects. This
deals with the rescaling of the response that occurs as we add new features to a developing model.
Key words: multilevel, ordered responses, variance components
1. Introduction and Motivation
In linear mixed models for random effects and multilevel models in particular, it
is of interest to see how the random effects residual variance component estimates
change as additional effects and features are added. It is also of interest to see
how the estimates of the coefﬁcients of ﬁxed parameters change. Table I below
shows the results of two-level models ﬁtted to the achievements in Key Stage 1
Mathematics Standard Assessment Task taken at or around the age seven by 1444
children in 114 Birmingham Primary Schools. The response is based on equal
interval scoring of six grades and is standardised to have mean zero and variance
unity over the sample data. Model A is base two level variance components model.
It estimates that school (level 2) variance is 19% of total. Model B uses as controls
initial ability of children on reception around two years previously. These take the
form of standard baseline achievement scores in six tested areas. On this adjustment
residual pupil variance is reduced by 34% to 0.55. The school variance is reduced
very slightly. The intra-school correlation increases to 0.24. It might be noted that
it is possible for the level 2 school variance to increase with the addition of extra
ﬁxed effects (see Fielding, 1999a, or Snijders and Bosker, 1999). The relative sizes
of the components in each model or changes as the model develops is motivated by
real substantive concerns.
The model in Table I has used arbitrary points scores for what are in effect rather
few categories of an ordered response. This enables standard multilevel models for
continuous responses to be ﬁtted. However, it must be emphasised that this is for
illustrative purposes only. There is accumulating evidence that ordered responses