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- 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