Multilevel models for family data

1
Multilevel models for family data
Tom OConnor Jon Rasbash
A presentation to the Research Methods
Festival Oxford, July 2004.
Work conducted for the ESRC research methods
programme project Methodologies for Studying
Families and Family Effects the systematic
assessment of research designs and data analytic
strategies
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The presentation looks at three applications of
multilevel modelling to family data

• Using multilevel models to explore the
  determinants of differential parental treatment
  of children.
• 2. Extending multilevel models to include genetic
  effects.
• 3. Applying multilevel models developed to handle
  social network data to family relationship data.

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Application 1 Understanding the sources of
differential parenting the role of child and
family level effects
Jenny Jenkins, Jon Rasbash and Tom
OConnor Developmental Psychology 2003(1) 99-113
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Mapping multilevel terminology to psychological
terminology

• Level 2 Family, shared environment
• Variables family ses, marital problems
• Level 1 Child, non-shared environment, child
  specific
• Variables age, sex, temperament

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Background

• Recent studies in developmental psychology and
  behavioural genetics emphasise non-shared
  environment is much more important in explaining
  childrens adjustment than shared environment has
  led to a focus on non-shared environment.(Plomin
  et al, 1994 TurkheimerWaldron, 2000)
• Has this meant that we have ignored the role of
  the shared family context both empirically and
  conceptually?

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Background

• 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?

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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
The Stress/Resources Hypothesis
Do family contexts(shared environment) increase
or decrease the extent to which children within
the same family are treated differently?
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How differential parental treatment has been
analysed
Previous analyses, in the literature exploring
the sources of differential parental treatment
ask mother to rate two siblings in terms of the
treatment(positive or negative) they give to each
child. The difference between these two treatment
scores is then analysed. This approach has
several major limitations
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The sibling pair difference difference model, for
exploring determinants of differential parenting

• Where y1i and y2i are parental ratings for
  siblings 1 and 2 in family I
• x1i is a family level variable for example family
  ses

• Problems
• One measurement per family makes it impossible
  to separate shared and non-shared random effects.
• All information about magnitude of response is
  lost (2,4) are the same as (22,24)
• It is not possible to introduce level
  1(non-shared) variables since the data has been
  aggregated to level 2.
• Family sizes larger than two can not be handled.

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With a multilevel model
Where yij is the jth mothers rating of her
treatment of her ith child x1ij are child
level(non-shared variables), x2j are child
level(shared variables) uj and eij are family and
child(shared and non-shared environment) random
effects. Note that the level 1 variance
is now a measure of differential parenting
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Advantages of the multilevel approach

• Can handle more than two kids per family
• Unconfounds family and child allowing estimation
  of family and child level fixed and random
  effects
• Can model parenting level and differential
  parenting in the same model.

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Overall Survey Design

• National Longitudinal Survey of Children and
  Youth (NLSCY)
• Statistics Canada Survey, representative sample
  of children across the provinces
• Nested design includes up to 4 children per
  family
• PMK respondent
• 4-11 year old children
• Criteria another sibling in the age range, be
  living with at least one biological parent, 4
  years of age or older
• 8, 474 children
• 3, 860 families
• 4 child 60, 3 child630, 2 child3157

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Measures of parental treatment of child

• Derived form factor analyses..
• PMK report of positive parenting frequency of
  praise of child, talk or play focusing on child,
  activities enjoyed together a.81
• PMK report of negative parenting frequency of
  disapproval, annoyance, anger, mood related
  punishment a.71
• Will talk today about positive parenting
• PMK is parent most known to the child.

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• Child specific factors
• Age
• Gender
• Child position in family
• Negative emotionality
• Biological relatedness to father and mother

• Family context factors
• Socioeconomic status
• Family size
• Single parent status
• Marital dissatisfaction

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Model 1 Null Model
The base line estimate of differential parenting
is 3.8. We can now add further shared and
non-shared explanatory variables and judge their
effect on differential parenting by the reduction
in the level 1 variance.
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Model 2 expanded model
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positive parenting

• Child level predictors
• Strongest predictor of positive parenting is age.
  Younger siblings get more attention. This
  relationship is moderated by family membership.
• Non-bio mother and Non_bio father reduce positive
  parenting
• Oldest sibling gt youngest sibling gt middle
  siblings
• Family level predictors
• Household SES increases positive parenting
• Marital dissatisfaction, increasing family size,
  mixed or all girl sib-ships all decrease positive
  parenting
• Lone parenthood has no effect.

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Differential parenting
Modelling age reduced the level 1 variance (our
measure of differential parenting) from 3.8 to
2.3, a reduction of 40. Other explanatory
variables both child specific and family(shared
environment) provide no significant reduction in
the level 1 variation. Does this mean that there
is no evidence to support the stress/resources
hypothesis.
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Testing the stress/resource hypothesis

• The mean and the variance are modelled
  simultaneously. So far we have modelled the mean
  in terms of shared environment but not the
  variance.
• We can elaborate model 2 by allowing the level 1
  variance to be a function of the family level
  variables household socioeconomic status, large
  family size, and marital conflict. That is

Reduction in the deviance with 7df is 78.
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Graphically
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Conclusion

• 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 as well
  as child specific factors.
• This challenges the current tendency in
  developmental psychology and behavioural genetics
  to focus on child specific factors.
• Multilevel models fitting complex level 1
  variation need to be employed to uncover these
  relationships.

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Application 2Including Genetic Effects in
Multilevel models
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Background
  Recent involvement in applying multilevel
models to family data, collaborating with
developmental psychologists.   They asked can we
include genetic effects in these models?   Long
tradition of quantitative genetics, arguably
begun with Fishers 1918 paper   The correlation
between relatives based on the supposition of
Mendalian inheritance   This work has been
developed by others and applied in Animal and
plant genetics, evolutionary biology, human
genetics and behavioural genetics.
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The basic multilevel model, for kids within
families
Given the standard independence assumptions of
multilevel models
The covariance of two children(i1 and i2 )in the
same family is
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Extending the basic model to include genetic
effects
Where gij is a genetic effect for the ith child
in the jth family.
For two individuals (i1,i2 )
BUT
The genetic covariance of two individuals in the
same family, is clearly not zero since there is a
non-zero probability that they share the same
genes.
What is F?   This where Fishers 1918 paper comes
in.
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A very little genetics background
  First remember, humans have 23 pairs of
chromosomes. A gene is a sequence of DNA at given
location(locus) on a chromosome.   In a
population there might be multiple different
versions of gene.   For example, with two
versions of a gene denoted by A and little a.
There are 3 possible genotypes   AA Aa
aa   (Note Aa is functional equivalent to
aA)   We can think of the genes conferring values
on an individual for a trait.  
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Given
a number of (strong) assumptions   1. A metric
trait is influenced by a large number of genes at
a large number of loci(effectively infinite)   2.
The effects of the genes add-up within and across
loci   3. The genes are transmitted independently
from parents to progeny.   4. The population
being studied is mating at random   5. The
population being studied is in evolutionary
equilibrium. That is gene frequencies are not
changing across generations.   6. There is no
correlation between genetic and environmental
effects.   Corrections to the theory exist for
all these assumptions, but I fear they are seldom
used(in BG), are often difficult to implement and
have not been thoroughly evaluated.
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Then..
With a lot of complicated argument and algebra,
Fisher shows that
Where r(i1,i2) is the relationship coefficient
between two individuals and equals
(0,0.125,0.25,0.5,1) for unrelated individuals,
cousins, half-sibs, full sibs and mz twins
respectively.
Thus the greater the relationship between
individuals the greater their genetic covariance
and therefore their phenotypic covariance. An
individuals gij is the sum of the effects of all
their genes. The variance of these gij is the
additive genetic variance(?g2). The size of the
additive genetic variance compared to other
environmental variances is often of interest.
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Data example
277 full sib pairs, 109 half sib pairs, 130
unrelated pairs, 93 DZ twins and 99 MZ twins aged
between 9 and 18 years.   Analysis of depression
scores
The total variance in the two models is
effectively the same 0.275 in model 1 and 0.285
in model 2
In model 2, which includes genetic effects, 70
of the family level variation and 60 of child
level variation are re-assigned to the genetic
variance
Like autocorrelation, time-series models except
covariance decays as a function of genetic
distance as opposed to distance in time between
measurements. Can use the same estimation
machinery.
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Adding covariates
From the fixed effects we see that depression
scores increase with child age, paternal and
maternal negativity girls and children in
stepfamilies also have higher depression scores.
The largest drop in the variance when these
explanatory variables are introduced occurs in
the genetic variance.
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Why the drop in the genetic variance?
The largest drop in the genetic variance occurs
when paternal and maternal negativity are added
to the model as covariates. Pike et al(1996)
analyse the same data using a series of
genetically calibrated bivariate structural
equations models. Two of the models they consider
are bivariate structural equations models for
maternal negativity and depression and paternal
negativity and depression. In each of these two
models they find 15 of the genetic variance in
depression is due to a shared genetic component
with parental negativity. When we add paternal
and maternal negativity to our model as fixed
effects we are sweeping out any common genetic
effects shared by parental negativity and
adolescent depression. We are also taking
account of any environmental correlations whereby
sibling pairs of greater relatedness experience
more similar parental treatment. Both these
factors will reduce the remaining additive
genetic variance in the model.
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Complex variation and gene environment
interactions
Currently our model for the variance partitions
the variance into three sources family, child and
genetic.   The model for the variance can be
further elaborated to allow each of the three
sources of variation to be modelled as functions
of explanatory variables, where the variables
may be measured at any level. That is
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Gene environment interaction with paternal
negativity
We now elaborate model 3 to allow all three
variances to be a function of paternal
negativity. That is
(4)
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Results from model 4
including the three extra parameters reduces the
deviance by 19.5. This reduction is almost
entirely driven by the gene environment
interaction term, ?1(g) . Removing the ?1(e) and
?1(u) terms from the model 4 results in a change
in only 1.5 in the deviance. The significant
coefficient constitutes a gene-environment
interaction because it implies the genetic
variance changes as a function of paternal
negativity.
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Graphing the gene environment interaction
One explanation of GXE interactions is in terms
of conditional gene expression.   Suppose we
have a gene A which gets switched on when an
individual is subject to persistent high levels
of cortisol. If some of the population have the A
gene and some dont then this genetic variation
only manifests in individuals under persistent
high levels of stress  
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Model Extensions
The multilevel model with genetic effects is
flexible and can be adapted to a variety of
situations where population structures have
further nested or crossed random classifications
in addition to the standard behavioural genetics
situation of children within families. For
example, Time repeated measures on kids within
families Institutions schools, hospitals Space
areas Multiple observers Complex example given
in next section.
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Application 3Applying social network models to
family relationship data-some preliminary work.
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Substantive focustrait-like versus context
specific behaviour
A question in personality theory is to what
extent particular emotions and behaviours are
trait-like in that they are constant across
different environments, to what extent are they
context sensitive in that behaviours expressed by
individuals are specific to particular
environments (Magnusson, 1990 Magnusson
Endler, 1977).
Clinically, trait-like behaviours are harder to
change since by definition altering the
environment will tend not to change the behaviour.
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Stability of behaviours over time
Studies in personality have shown that happiness
and positivity are very stable over time (Costa,
McCrae Zonderman, 1987) findings for the
stability of negativity vary as a function of
which aspect of negativity is being considered
with some aspects such as physical aggression
showing high stability (Broidy et al, 2003) and
other aspects such as whining showing low
stability (Capaldi et al., 1994).
Trait-like behaviours are often considered to be
driven by genetic factors (Plomin, 1994 Tellegen
et al., 1988)
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Stability of behaviours across family members
In this presentation we develop multilevel models
to explore the stability of an individuals
behaviour across their family members. 2 kids and
2 parents per family, 12 relationship scores per
family a relationship is made up of an actor and
a partner.
c1?c2 c1?m c1?f c2?c1 c2?m c2?f m?c1
m?c2 m?f f?c1 f?c2 f?m
We look at the traits of positivity and
negativity as responses.
Note that positivity and negativity are not
mirror images of the same underlying construct.
Clinically and statistically they show
independent patterns with evidence from
neuropsychology that they are controlled by
different brain systems (Caccioppo, Larsen, Smith
Berntson, 2004)
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The 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.

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Within family structure
We start with 12 relationship scores in each
family. These can be classified
partner
and dyad
actor
Family 1
Dyad d1 d2
d3 d4 d5
d6
Relationship c1?c2 c1?m c1?f c2?c1 c2?m
c2?f m?c1 m?c2 m?f f?c1 f?c2 f?m
Partner c1
c2 m
f
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Diagrams to represent the 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
diagrammatically with
A unit diagram one node per unit
A classification diagram with one node per
classification
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The multilevel social relations model-Snijders
and Kenny(1999)
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Interpretation of variance components
Familythe 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-gtfred is similar of that from fred-gtjoe.
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.
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Results of SRM more detail
Table shows variance partition coefficients
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.
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Adding fixed effects for 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.
We use child as the reference category for
actor_role and partner_role variables.
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Including actor and partner roles-positivity
Modelling actor and partner role drops likelihood
by over 1000 units with 4df.
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.
These actor_role role variables explain over 50
of the actor level variance.
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.
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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
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Including actor and partner roles-negativity
Now an interaction is required between actor_role
and partner_role. Note the interaction categories
a_mothp_moth and a_fathp_fath structurally do
not exist.
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.
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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.
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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.
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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
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Stability results of two bivariate SRM
The basic patterns of the vpcs 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 (?120.8).
Family effects are a bit stronger for negativity.
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.
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A comment on family effects
Developmental psychology and behavioural
genetics, .(Plomin et al, 1994
TurkheimerWaldron, 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.
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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.
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In conclusion
The multilevel social relations model offers a
powerful framework for exploring within family
dynamics and processes.