Why use multilevel modelling?
Clio Day
Jon Rasbash
GSOE
November 2006
A simple question
How much of the variability in pupil attainment is attributable to
schools level factors and how much to pupil level factors?
First of all we have to define what we mean by “variability”
A toy example – first we calculate the mean
Two schools each with two pupils.
attainment
3
2
-1
-4
School 1
School 2
Overall mean= (3+2+(-1)+(-4))/4=0
Overall mean(0)
Calculating the “variance”
attainment
3
2
-1
Overall mean(0)
-4
School 1
School 2
The total variance is the sum of the squares of the departures of the
observations around mean divided by the sample size(4) =
(9+4+1+16)/4=7.5
The variance of the school means around the overall mean
attainment
3
2
2.5
Overall mean(0)
-1
-2.5
-4
School 1
School 2
The variance of the school means around the overall mean=
(2.52+(-2.5)2)/2=6.25
Total variance =7.5
The variance of the pupils scores around their school’s mean
attainment
3
2
2.5
-1
-2.5
-4
School 1
School 2
The variance of the pupils scores around their school’s mean=
((3-2.5)2 + (2-2.5)2 + (-1-(-2.5))2 + (-4-(-2.5))2 )/4 =1.25
The variance of the school means around overall mean = (2.52+(2.5)2)/2=6.25
Total variance =7.5=6.25+1.25
Returning to our question
How much of the variability in pupil attainment is attributable to
schools level factors and how much to pupil level factors?
In terms of our toy example we can now say
6.25/7.5= 82% of the total variation of pupils attainment is
attributable to school level factors
1.25/7.5= 18% of the total variation of pupils attainment is
attributable to pupil level factors
Now lets do the same thing on real data(65 schools+4000
pupils)
Overall mean=0
(attainment scaled
to have 0 mean
overall)
Total variation = 1
Variance of school
means around
overall mean=0.15
85% of variability due to pupil level factors, 15% due to
school level factors
Variance of
pupil’s attainment
scores around
school mean=0.85
Estimating parameters of distributions
The multilevel model assumes the school means and the pupils
departures around their school means are Normally distributed.
The Normal distribution has two
parameters: the mean and the
variance. Our model estimates
N(0,0.85) for pupil within school
effects and N(0,0.15) for school
effects
We gain a great deal of
modelling power and
flexibility by making these
Normality assumption
Can we explain the variation at the school and pupil levels?
With educational data we typically want to take account of pupil intake ability
when they enter school. In our data set the plot looks like this:
Can we explain the variation at the school and pupil levels?
What might happen to the between school and between pupil within school
variances when we correct for prior ability?
Prior ability
Prior ability
Both the between school and between pupil within school
variation will be reduced when we model mean attainment as a
function of prior ability
And on the real world data…
Recall that the between school and between pupil variances before
taking account of prior ability were 0.15 and 0.85 respectively.
After taking account of prior ability between school and
between pupil variances are reduced to 0.092 an 0.566.
So accounting for prior ability explains 39% and 35% of between school
and between pupil variation.
The effect of taking account of prior attainment is important but not
entirely surprising. School variability is partly determined by school
intake profile and pupil prior attainment is a good predictor of
subsequent attainment.
Obvious next questions
Are there other school level variables that can explain why schools
differ?
Does school gender(mixed school, boy school, girl school) explain
some of the between school variation?
If we (additionally to prior
ability) allow the mean to
be a function of school
gender the between school
variance is reduced by
13%.
Focussing on school gender differences
mixed school
boy school
girl school
Problems with traditional techniques
How much between school variation is there? What school level predictor
variables can explain some of this variation?
This line of enquiry is powerful. Traditional statistical analysis techniques can
not pursue this exploratory avenue.
Estimate the between school variability OR
Estimate school level predictors such as school gender
But not both
It gets worse
If we fit a single level
model we get incorrect
uncertainty intervals
leading to incorrect
inferences
This is because the
single level model
ignores the clustering
effect of schools
mixed school
boy school
girl school
The distributional assumptions made by the multilevel model allows the
estimation of between school variance and school level predictors.
Variance is our business
We have modelled mean attainment as a function of prior ability
and school gender
We have simultaneously modelled the total variation in attainment as
function of school and pupil levels.
Traditional modelling techniques are unable to partition the variation in
this way and just estimate a single term and refer to it as “error”.
We think this variation is not error, it contains a lot of interesting
structure. Multilevel modelling is a great tool for exploring the structure
in the “error” term.
Is there a family effect?
Recent studies in developmental psychology and behavioural
genetics(BG) emphasise non-shared environment and genetic
influences are much more important in explaining children’s
adjustment than shared environment has led to a focus on nonshared environment.(Plomin et al, 1994; Turkheimer&Waldron,
2000)
Multilevel modelling can replicate the BG analysis. It can also
extend them to more reasonably represent the complexities of
family structures and processes. When this is done persistent
family effects are found.
10 schools two scenarios
Is there a family effect?
Recent studies in developmental psychology and behavioural
genetics emphasise non-shared environment and genetic
influences are much more important in explaining children’s
adjustment than shared environment has led to a focus on nonshared environment.(Plomin et al, 1994; Turkheimer&Waldron,
2000)
My collaborators from psychology Jenny Jenkins(Toronto University)
and Tom O’Connor(Rochester University) were concerned that perhaps
the analytic techniques being used might have some simplifying
assumptions that made it difficult to pick up the shared family context.
They were interested to see if applying multilevel models, with the
recognised strengths in exploring contextual effects might turn up some
different findings.
Two analyses
1. Understanding the sources of differential parenting: the role of child
and family level effects. Jenny Jenkins, Jon Rasbash and Tom O’Connor
Developmental Psychology 2003(1) 99-113
2. Applying social network models to within family processes.
Currently being written up for publication.
Differential parental treatment
•One key aspect of the non-shared environment that has been
investigated is differential parental treatment of siblings.
•Differential treatment predicts differences in sibling
adjustment
•What are the sources of differential treatment?
•Child specific/non-shared: age, temperament, biological
relatedness
•Can family level shared environmental factors influence
differential treatment?
The Stress/Resources Hypothesis
Do family contexts(shared environment) increase or decrease the extent
to which children within the same family are treated differently?
“Parents have a finite amount of resources in terms of time, attention,
patience and support to give their children. In families in which most of
these resources are devoted to coping with economic stress, depression
and/or marital conflict, parents may become less consciously or
intentionally equitable and more driven by preferences or child
characteristics in their childrearing efforts”. Henderson et al 1996.
This is the hypothesis we wish to test. We operationalised the
stress/resources hypothesis using four contextual variables:
socioeconomic status, single parenthood, large family size, and marital
conflict
positive parenting
Modelling the mean and variance simultaneously
We show a possible pattern of how the mean, within family variance and
between family variance might behave as functions of HSES in the schematic
diagram below.
Here are 5 families of increasing
HSES(in the actual data set there
are 3900 families.
We can fit a linear function of SES
to the mean.
The family means now vary around
the dashed trend line. This is now
the between family variation;
which is pretty constant wrt HSES
HSES
However, the within family variation(measure of differential
parenting) decreases with HSES – this supports the SR hypothesis.
Conclusion on differential parental treatment
• We have found strong support for the stress/resources hypothesis. That
is although differential parenting is a child specific factor that drives
differential adjustment, differential parenting itself is influenced by
family factors such as HSES.
• This challenges the current tendency in developmental psychology and
behavioural genetics to focus on child specific factors.
• Multilevel models which model both the mean and the variability
simultaneously are needed to uncover these relationships.
Deconstructing relationships: what determines how people
get on within a family?
family Culture
the individual
the dyad(the two people relating)
genes
Applying models from social network theory to family data
Non-Shared Environment Adolescent Development(NEAD) data set, Reiss et al(1994).
•2 wave longitudinal family study, designed for testing hypothesis about genetic
and environmental effects
•277 full-sib pairs, 109 half-sib pairs, 130 unrelated pairs, 93 DZ twins and 99 MZ
twins, aged between 9 and 18 years
•Wave 2 followed 3 years after wave 1 and any families where the older sib was
older than 18 were not followed up.
•A wide range of self-report, parental-report and observer variables were collected.
•All families had 2 parents and 2 kids of the same sex.
•We focus here on data on relationship quality collected by observers.
Within family structure
We start with 12 relationship scores in each family. These can be classified :
actor partner dyad
Family 1…
Dyad
d1
Actor:
d2
c1
Relationship: c1c2 c1m c1f
Partner:
d3
c1
d4
d5
c2
c2c1 c2m c2f
c2
d6
m
mc1 mc2 mf
m
f
fc1 fc2 fm
f
This model is the multilevel social relations model-Snijders+Kenny(1999)
Useful diagrams for thinking about multilevel structure
The relationship scores are contained within a cross classification of actor, dyad and
partner and all of this structure is nested within families. This can structure can be
shown diagramatically with:
A unit diagram – one node per unit
A classification diagram with one node per
classification
family
Family 1…
Dyad
d1
Actor:
d2
c1
Relationship: c1c2 c1m c1f
Partner:
d3
c1
d4
c2
d6
m
c2c1 c2m c2f
c2
d5
m
mc1 mc2 mf
f
actor
partner
fc1 fc2 fm
f
Relationship score
dyad
Interpretation of variance components
Family:the extent to which family level factors effect all the relationships in a
family.
Actor: the extent to which individuals act similarly across relationships with
other family members(actor stability, trait-like behaviour)
Partner: We actually have two traits operating, in addition to the trait of
common acting to other family members we also have the trait of elicitation
from other family members. The greater the partner variance component the
greater the evidence for such a trait operating.
Dyad: The extent to which relationship quality is specific to the dyad. A high
dyad random effect means that the relationship score from joe->fred is similar of
that from fred->joe. In social network theory this is known as reciprocity.
Reciprocity is a context specific effect(non trait-like)
Relationship: residual variation across relationships in relationship quality.
Results of SRM more detail
Table shows variance partition coefficients
Pos
SRM
Neg
SRM
Family
0.12
0.19
Actor
0.44
0.12
Partner
0.01
0.03
Dyad
0.18
0.41
Relat.
0.25
0.24
-2loglike 10225.7
17800.9
For positivity 44% of the
variablity is attributable to actors
indicating that individuals act in a
consistent way across relationships
with other family members. There
is a strong actor trait component to
positivity.
For negativity 0.41 of the
variability is attributable to dyad.
Indicating the dyad is an
important structure in
determining negativity in
relationships. There is a strong
context specific component to
negativity.
There is little evidence of an elicitation or partner trait for either response.
At the family level there are stronger effects for negativity than positivity.
Modelling the mean relationship quality in terms of role
The basic unit, a relationship, has an actor and a partner. Actors and
partners are classified into the roles of children, mothers and fathers by the
two categorical variables actor_role and partner_role.
relation
ship
child
Actor_role
Partner_role
mother
father
child
mother
father
c1c2
1
0
0
1
0
0
c1m
1
0
0
0
1
0
c1f
1
0
0
0
0
1
c2c1
1
0
0
1
0
0
c2m
1
0
0
1
1
0
c2f
1
0
0
0
0
1
mc1
0
1
0
1
0
0
mc2
0
1
0
1
0
0
mf
0
1
0
0
0
1
fc1
0
0
1
1
0
0
fc2
0
0
1
1
0
0
fm
0
0
1
0
1
0
We use child as the
reference category for
actor_role and
partner_role
variables.
Including actor and partner roles-positivity
Modelling actor and partner role
drops likelihood by over 1000
units with 4df.
param(se)
param(se)
intercept
2.834(0.011)
2.263(0.014)
a_mother
-
0.502(0.016)
a_father
-
0.351(0.016)
p_mother
-
0.021(0.011)
p_father
-
-0.032(0.011)
These actor_role role variables
explain over 50% of the actor level
variance.
family
0.034(0.004)
0.050(0.004)
actor
0.124(0.005)
0.061(0.004)
partner
0.003(0.002)
0.001(0.002)
dyad
0.050(0.003)
0.051(0.003)
relationship
0.073(0.002)
0.073(0.002)
-2loglike
10225.7
9092.64
Adding interactions between
actor_role and partner-role does not
improve the model.
Since we have explained actor level
variance this means actor role
explains the some of the trait
component of relationship positivity.
fixed
random
The effect is dominated by the actor
role categories. With mothers and
then fathers being much more
positive as actors than the reference
category child.
Graphing actor and partner role effects for positivity
The graph shows actor_role having a
big effect on relationship quality and
partner role having a marginal effect.
actor child
actor m
actor f
Including actor and partner roles-negativity
param(se)
param(se)
Now an interaction is required
between actor_role and
partner_role. Note the interaction
categories a_moth*p_moth and
a_fath*p_fath structurally do not
exist.
fixed
intercept
0.348(0.018) 0.729(0.027)
a_mother
-
-0.375(0.030)
a_father
-
-0.516(0.031)
p_mother
-
-0.319(0.028)
p_father
-
-0.625(0.028)
a_moth*p_fath
-
0.359(0.040)
a_fath*p_moth
-
0.563(0.040)
random
family
0.137(0.012) 0.144(0.012)
actor
0.082(0.006) 0.087(0.006)
partner
0.022(0.005) 0.018(0.004)
dyad
0.282(0.010) 0.239(0.009)
relationship
0.165(0.005) 0.162(0.005)
-2loglike
17800.9
17305.18
Modelling actor and partner role
and the interaction drops the
loglike by 500 units with 6df.
}
Note the main drop in the variance
occurs at the dyad level which
reduces by 15%. This means
modelling actor and partner roles
has explained context specific
variation in relationship quality for
negativity.
Graphing actor and partner role effects for negativity
With respect to actor and partner roles
the main context specific effects for
relationship quality occur in
relationships where the child is an
actor..
Whether the partner is another child, a
mother or a father greatly effects the
negativity of the predicted relationship
quality
actor child
actor m
actor f
A possible psychological explanation for this pattern is that negativity is “ high
stakes” behaviour. The amount of negativity a child feels “safe” to express is
determined by the power/authority of the partner.
Note that parents are trait-like wrt actor negativity effects.
Genetic effects
Individuals exhibit some trait-like behaviour for both relationship positivity and
negativity. With individuals exhibiting stronger trait-like behaviour for relationship
positivity.
Such trait-like behaviour may have a genetic component.
The standard behavioural genetics model for children within families estimates
shared environment(family), non-shared environment(individual) and genetic
components of variation.
Our structure is more complex in that the lowest level is not the individual but a
relationship between two individuals. Also we have a dyad component of variation
and the individual component of variation is split into actor and partner components.
However, we can extend the basic BG model (which incorporates some questionable
assumptions) to our structure. The extended model gives heritabilities (genetic
variance)/(total variance) of 0.42 and 0.16 for positivity and negativity respectively.
The actor and partner variance components were reduced with the inclusion of genetic
effects but the family variance component was undiminished.
Stability of effects over time
The data has two waves where the same relationships were measured three years later.
This allows us to explore the stability of family, actor, partner, dyad and relationship
effects over time.
We can operationalise the longitudinal structure by fitting a multivariate response social
relations model where the first response is the time 1 relationship score and the second the
time 2 relationship score.
We simultaneously estimate all variance components for each response
and the following correlations
time 1 relationship score
time 2 relationship score
family
family
actor
partner
dyad
Relationship score
actor
partner
dyad
Relationship score
Stability – results of two bivariate SRM
Positivity
Negativity
w1 vpc w2 vpc
12
w1 vpc
w2 vpc
12
family
0.11
0.12
0.77
0.20
0.17
0.8
actor
0.44
0.46
0.87
0.11
0.11
0.67
partner
0.01
0.01
1.5??
0.03
0.04
0.88
dyad
0.17
0.12
0.15
0.42
0.41
0.34
relat.
0.26
0.29
0.11
0.25
0.27
0.16
Actor effects are stronger for positivity than negativity but
stability across time is high for both actor behaviours(0.87
and 0.67)
Dyad effects are much stronger for negativity than positivity. But
the stability of dyad effects for both behaviours is lower than
actor, partner and family effect stabilities. Dyads are more stable
for negativity than positivity.
The basic patterns of the
vpc’s found in wave 1 are
repeated in wave 2 for
both positivity and
negativity.
Family effects are
very stable over time
for both positivity (12
= 0.77) and negativity
(12=0.8). Family
effects are a bit
stronger for negativity.
A comment on family effects
Developmental psychology and behavioural genetics, .(Plomin et al, 1994;
Turkheimer&Waldron, 2000). Have suggested that after taking account of
genetic and individual level factors there is scant evidence for family level
effects. Our work shows strong family level effects, that persist over time, even
when genetic, actor, partner, dyad and relationship level variance components
are included in the model.
Part of the previous failure to find family effects may be the analytical strategy
of breaking down families into series of overlapping dyads and analyising each
dyad separately. This strategy is probably in part determined by the
methodology available to the researchers.
A comment on dyad effects for relationship negativity
For relationship negativity we saw large dyad effects and relatively low stability
over time.
This means that at wave 1 there is a large within family variability in dyad
negativity and likewise at wave 2. However the dyads which are most and least
negative within the family are to an extent switching around. The next step is to see
if we can find some systematic pattern to these dyadic dynamics for relationship
negativity.
Alspac data – an example of highly complex multilevel
structure
All the children born in the Avon area in 1990 followed up
longitudinally
Many measurements made including educational
attainment measures
Children span 3 school year cohorts(say 1994,1995,1996)
Suppose we wish to model development of numeracy over
the schooling period. We may have the following attainment
measures on a child :
m1 m2
m3 m4
primary school
m5
m6 m7 m8
secondary school
Structure for primary schools
Primary school
Area
P School Cohort
Pupil
P. Teacher
M. Occasion
•Measurement occasions within pupils
•At each occasion there may be a different teacher
•Pupils are nested within primary school cohorts
•All this structure is nested within primary school
• Pupils are nested within residential areas
A mixture of nested and crossed relationships
Primary school
P School Cohort
Area
Pupil
P. Teacher
M. occasions
Nodes directly connected by a single arrow are nested, otherwise nodes are crossclassified. For example, measurement occasions are nested within pupils. However,
cohort are cross-classified with primary teachers, that is teachers teach more than one
cohort and a cohort is taught by more than one teacher.
T1
T2
T3
Cohort 1
95
96
97
Cohort 2
96
97
98
Cohort 3
98
99
00
Multiple membership
It is reasonable to suppose the attainment of a child in a particualr year is
influenced not only by the current teacher, but also by teachers in previous
years. That is measurements occasions are “multiple members” of teachers.
m1
t1
m2
t2
m3
t3
m4
t4
Primary school
We represent this in
the classification
diagram by using a
double arrow.
Area
P School Cohort
Pupil
M. occasions
P. Teacher
What happens if pupils move area?
Primary school
Area
P School Cohort
P. Teacher
Classification diagram
without pupils moving
residential areas
Pupil
M. occasions
If pupils move area, then pupils are no longer nested within areas. Pupils and areas are cross-classified.
Also it is reasonable to suppose that pupils measured attainments are effected by the areas they have
previously lived in. So measurement occasions are multiple members of areas
Primary school
P School Cohort
Area
P. Teacher
Classification diagram
where pupils move between
residential areas
Pupil
M. occasions
BUT…
If pupils move area they will also move schools
Primary school
P School Cohort
Area
P. Teacher
Classification diagram
where pupils move between
areas but not schools
Pupil
M. occasions
If pupils move schools they are no longer nested within primary school or primary school
cohort. Also we can expect, for the mobile pupils, both their previous and current cohort
and school to effect measured attainments
Primary school
Area
P School Cohort
Pupil
M. occasions
P. Teacher
Classification diagram
where pupils move
between schools and
areas
If pupils move area they will also move schools cnt’d
And secondary schools…
Primary school
Area
P School Cohort
Pupil
P. Teacher
M. occasions
We could also extend the above model to take account of Secondary school,
secondary school cohort and secondary school teachers.
So why use multilevel models?
It gives the correct answers for the standard errors of regression
coefficients(in the presence of clustering). Thereby protecting against
incorrect inferences(school gender example).
Modelling the variance(in addition to the mean) gives a framework that
allows a greater range of questions. For example, how does variability in
parental treatment of sibs partition between and within families? Does the
within family variance change as a function of social class?(As in the
differential parenting example)
Multilevel models extend to handle situations where there are multiple
classifications arranged in nested, crossed and multiple membership
relations. For example in the social relations model with relationship score,
actor, partner, dyad, family and genetic effects.
Other predictor variables
Remember we are partitioning the variability in attainment over time
between primary school, residential area, pupil, p. school cohort,
teacher and occasion. We also have predictor variables for these
classifications, eg pupil social class, teacher training, school budget and
so on. We can introduce these predictor variables to see to what extent
they explain the partitioned variability.