1. Modelling the Diluting Effect of Social Mobility
on Health Inequality
Heather Turner1,2 and David Firth1
1
University of Warwick, UK
2 Independent statistical/R consultant
5 September 2012
H. Turner and D. Firth (Warwick, UK) Social Mobility & Health Inequality RSS 2012 1 / 19

2. Setting
Given intergenerational data on
socio-economic position (origin, destination)
health outcome
covariates
how can we analyse the effect of social mobility on health?
H. Turner and D. Firth (Warwick, UK) Social Mobility & Health Inequality RSS 2012 2 / 19

3. Trajectory Analysis
A common approach is to analyse the effect of social mobility by
considering all possible moves from origin class to destination class.
These trajectories are then used to produce
descriptive statistics
statistical models
Often the social classes are merged into two categories, so that the
trajectories are simplified to up/stable high/stable low/down.
H. Turner and D. Firth (Warwick, UK) Social Mobility & Health Inequality RSS 2012 3 / 19

4. Bartley & Plewis Models
Bartley & Plewis (JRSSA, 2007) used a more sophisticated approach,
in which social mobility effects were combined first with the effect of
origin class and second with the effect of destination class.
For example the origin + mobility model includes the term

αi + δ0 i > j

θij = αi + δ1 i = j

αi + δ2 i < j

where αi is the effect for origin i and i > j denotes moving to a more
favourable destination class.
H. Turner and D. Firth (Warwick, UK) Social Mobility & Health Inequality RSS 2012 4 / 19

5. Case Study: Long-term Limiting Illness
This application presented by Bartley & Plewis concerns data from
the ONS Longitudinal Survey, which links census and vital event data
for 1% of the population of England and Wales.
The outcome of interest is long-term limiting illness (LLTI)
a long-standing illness, health problem or handicap that
limits a person’s activities or the work they can do
H. Turner and D. Firth (Warwick, UK) Social Mobility & Health Inequality RSS 2012 5 / 19

6. Model Scope
The probability of long-term limiting illness in 2001 was modelled via
logistic regression.
The social mobility effects θij were based on the National Statistics
socio-economic classification (7 classes, high man/prof to routine) in
1991 and 2001.
Age in 1991 was included as a covariate and men and women were
modelled separately.
H. Turner and D. Firth (Warwick, UK) Social Mobility & Health Inequality RSS 2012 6 / 19

7. Social Mobility Effects
The exponential of the mobility parameters gives the odds ratios of
LLTI, shown here for men
Origin + Mobility Destination + Mobility
To more favourable 1.00 1.00
Stable 1.21 0.71
To less favourable 1.45 0.52
given origin, odds of LLTI increased by downward mobility
given destination, odds of LLTI decreased by downward mobility
H. Turner and D. Firth (Warwick, UK) Social Mobility & Health Inequality RSS 2012 7 / 19

8. Weighted Residuals
The working residuals from the IWLS iterations can be averaged over
each origin-destination combination to provide an indicator of fit for
each trajectory
rijs wijs
i,j,s wijs
where rijs is the s th residual for origin i and destination j, and wijs
is the corresponding working weight.
These average residuals can be standardized to be approximately
N(0, 1) assuming the model is correct.
H. Turner and D. Firth (Warwick, UK) Social Mobility & Health Inequality RSS 2012 8 / 19

9. > mosaic(model1men, ~mnssec9 + mnssec0, set_varnames = list(mnssec9 = "Origin class",
> mosaic(model2men, ~mnssec9 + mnssec0, set_varnames =
+ mnssec0 = "Destination class"), set_labels = list(mnssec9 = 1:7,
+ mnssec0 = "Destination class"), set_labels = lis
+ mnssec0 = 1:7), rot_labels = c(0, 0), margins = c(1, 1, 0,
+ mnssec0 = 1:7), rot_labels = c(0, 0), margins =
+ 4))
+ 4))
Origin + Mobility Destination + Mobility
Destination class Destination class
1 2 3 4 5 67 1 2 3 4 5 67
Mean Mean
1 residual residual
1
3.99
4.00
2 2
2.00
2.00
3
Origin class
3
Origin class
4 0.00
4
0.00
5 5
−2.00
6 −2.00
6
−4.17 −3.45
7 7
p−value = p−value =
6.1194e−16 < 2.22e−16
Despite being two sides of the same coin, the two models capture
> mosaic(model2men, ~mnssec9 + mnssec0, set_varnames = list(mnssec9 = "Origin class",
different features of the data. = list(mnssec9 = 1:7,
+ mnssec0 = "Destination class"), set_labels
17
+ mnssec0 = 1:7), rot_labels = c(0, 0), margins = c(1, 1, 0,
+ 4))
Destination class
1 2 3 4 5 67
H. Turner and D. Firth (Warwick, UK) Social Mean
Mobility & Health Inequality RSS 2012 9 / 19

10. Diagonal Reference Model
Sobel (Amer. Soc. Rev, 1991) proposed the diagonal reference
model, which combines origin, destination and social mobility effects
w1 γi + (1 − w1 )γj
The effect of moving from class i to class j is a weighted sum of the
diagonal effects γi , where γi is the effect for stable individuals in that
class.
H. Turner and D. Firth (Warwick, UK) Social Mobility & Health Inequality RSS 2012 10 / 19

11. Model Estimation
The diagonal reference model has predominantly been used to model
political and social attitudes, where a nonlinear least squares model is
appropriate.
Here we have a binary outcome and wish to model the log odds,
producing a logistic model with nonlinear terms.
Most statistical software packages do not have the facilities to
estimate such a model “out-of-the-box”. However this is a particular
example of a generalized nonlinear model which may be fitted using
the gnm package for R (Turner and Firth, R News, 2007).
H. Turner and D. Firth (Warwick, UK) Social Mobility & Health Inequality RSS 2012 11 / 19

12. Generalized Nonlinear Models
Given a response variable Y , a generalized nonlinear model maps the
mean response E(Y ) = µ to a parameteric model or predictor via a
link function g:
g(µ) = η(x, β)
The model is completed by a variance function V (µ) describing how
Var(Y ) depends on µ.
For our logistic model g is the logit function and V (µ) is determined
by assuming a binomial distribution for the response.
Following the previous analysis we fit the diagonal reference model
with age as a covariate and fit models for men and women separately.
H. Turner and D. Firth (Warwick, UK) Social Mobility & Health Inequality RSS 2012 12 / 19

13. Diagonal Effects
The model for the stable individuals is an ordinary logistic regression
logit(piik ) = β0 + β1 agek + γi
Therefore the diagonal effects are log odds ratios of LLTI in class i
against the reference class for a given age
piik /(1 − piik )
log
p11k /(1 − p11k )
=(β0 + β1 agek + γi ) − (β0 + β1 agek )
=γi
H. Turner and D. Firth (Warwick, UK) Social Mobility & Health Inequality RSS 2012 13 / 19

14. Health Inequality
q men
q women
q
4
q
q
Odds Ratio
3
q q
q
q
q
2
q q q q
q q
1
high prof low prof intermed self empl low sup semi−routine routine
Socio−economic Position
H. Turner and D. Firth (Warwick, UK) Social Mobility & Health Inequality RSS 2012 14 / 19

15. Diluting Effect of Social Mobility
The ratio of origin weight to destination weight quantifies the
diluting effect of social mobility on health inequality
1:0 social mobility has no effect on individual
0:1 social mobility has no effect on inequality
otherwise social mobility increases P(LLTI) in the upper classes
and decreases P(LLTI) in the lower classes.
The larger the origin weight, the greater the diluting effect.
H. Turner and D. Firth (Warwick, UK) Social Mobility & Health Inequality RSS 2012 15 / 19

16. Diagonal Weights
The diagonal weights for the LLTI models are
Men Women
Origin 0.62 (0.03) 0.41 (0.03)
Destination 0.38 (0.03) 0.59 (0.03)
Since their destination class is given more weight, social mobility has
a greater impact for women.
H. Turner and D. Firth (Warwick, UK) Social Mobility & Health Inequality RSS 2012 16 / 19

17. Model Comparison
The models can be compared by the difference in deviance from the
null model:
Men Women
Deviance Df Deviance Df
Origin + mobility 4050 9 3194 9
Destination + mobility 4026 9 3273 9
Diagonal reference 4121 8 3312 8
The diagonal reference model reduces the deviance the most despite
requiring fewer degrees of freedom.
H. Turner and D. Firth (Warwick, UK) Social Mobility & Health Inequality RSS 2012 17 / 19

18. > mosaic(drefmen, ~mnssec9 + mnssec0, set_varnames = list(mnssec9 = "Origin class",+ mnssec0, set_varnames = list
> mosaic(drefwomen, ~mnssec9
+ mnssec0 = "Destination class"), set_labels = list(mnssec9 = 1:7,"Destination class"), set_labels = list(mns
+ mnssec0 =
+ mnssec0 = 1:7), rot_labels = c(0, 0), margins = +
c(1, 1, 0,
mnssec0 = 1:7), rot_labels = c(0, 0), margins = c(1,
+ 4)) + 4))
Men Women
Destination class Destination class
1 2 3 4 5 67 1 2 3 45 67
Mean Mean
1 residual 1
residual
3.38 2.71
2
2 2.00
2.00
3
Origin class
3
Origin class
4 0.00 0.00
4
5 5
−2.00 6
6 −2.00
−3.28 −2.91
7
p−value = 7 p−value =
4.6821e−05 0.00028829
The presence of large residuals on the diagonal in the model for
women suggests that the covariate adjustment is inadequate.
> mosaic(model1men, ~mnssec9 + mnssec0, type = "expected", set_varnames = list(mnssec9 mnssec0, type = "expected"
+
> mosaic(model1women, ~mnssec9 + = "Origin class",
mnssec0 = "Destination class"), set_labels = list(mnssec9 = 1:7,"Destination class"), set_labels = list(mns
+ mnssec0 =
+ mnssec0 = 1:7), rot_labels = c(0, 0), margins = +
c(1, 1, 0,
mnssec0 = 1:7), rot_labels = c(0, 0), margins = c(1,
+ 4)) + 4))
Destination class Destination class
1 2 3 4 5 67 1 2 3 45 67
H. Turner and D. Firth (Warwick, UK)
1 SocialMean
Mobility & Health Inequality
1 RSS 2012
Mean 18 / 19

19. Summary
Diagonal reference models provide a parsimonious and interpretable
model for inequality between classes and the effects of social mobility
on this inequality.
gnm (www.cran.r-project.org/package=gnm) enables these
models to be easily applied to binary as well as continuous responses.
Further examples are provided in the package vignette including
allowing the diagonal weight to depend on covariates, e.g. to fit
separate weights for upwardly/downwardly mobile.
H. Turner and D. Firth (Warwick, UK) Social Mobility & Health Inequality RSS 2012 19 / 19