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Latent normal models for missing data Harvey Goldstein Centre for Multilevel Modelling University of Bristol The (multilevel) binary probit model Suppose that we have a variance components 2-level model for an underlying continuous variable written as y ij ( X ) ij u j e ij , u j ~ N ( 0 , u ), 2 e ij ~ N ( 0 ,1) and suppose a positive value is observed for a variable Z when y 0 . We then have ij ( X ) ij u Pr( z ij 1) Pr( y ij 0 ) Pr( e ij ( X ) ij u j ) ( t ) dt ( X ) ij u j which is in fact just the probit link function. j ( t ) dt Ordered and unordered latent normal models • Ordered categories, 1…p: Define k ( X ) ij u j Pr( z ij k ) ( t ) dt , 0 , 1 0 k 1 ( X ) ij u j Where we additionally need to estimate thresholds { k } • For unordered categories we can map to an underlying (p-1) variate normal distribution with covariance matrix I p Multivariate - mixed type - responses • Consider a bivariate (multilevel) response with a normal and binary response. • We can map onto a latent bivariate normal. • We can use ML for parameter estimation or MCMC • MCMC provides a chain for random draws from the latent normal distribution for the binary response. Each draw is conditioned on the observed (correlated) normal response. • Where a response is (randomly) missing we draw an impute from its (estimated) conditional distribution – this is easy for a MVN – and this is done for every MCMC iteration. For the binary response this can be mapped back onto (0,1). • Typically use uninformative priors. Multiple imputations • Every n-th iteration (say n~500 to ensure zero autocorrelation) we can choose a ‘completed’ bivariate dataset. This will then yield p imputed datasets to combine using ‘Rubin’ rules. • We can extend to include ordered, unordered, Poisson etc. responses all mapped onto a latent MVN with missing data mapped back to original scales. • We can also include responses defined at higher levels or classifications – correlated with higher level random effects for lower level responses. MI for multilevel GLMs • Every variable is treated as a response, possibly with fully observed variables as covariates in MVN model. For multilevel models this may include variables measured at higher levels. • Imputation carried out for the MVN model, mapped back to original scales, MOI fitted to multiple datasets and combined. • Note that for non-normal continuous variables we may be able to use e.g. a Box-Cox transform within the same model framework. • Note that for general discrete distributions we may be able to approximate by a set of ordered categories. A simulation ~ 30% records with randomly missing data Response is 16-year-old exam score Table 1. Simulation study model (PART). Parameter estimates and standard errors in brackets. One hundred simulated data sets. MCMC estimation used a burn in of 2000 with five imputed data sets at iterations 1, 500, 1000, 1500, 2000. Estimates are computed using restricted maximum likelihood. Parameter Complete data set Imputation Relative bias Imputation (%) standard error*** Intercept 0.260 (0.056) 0.263 1.2 0.0021 Reading test score 0.391 (0.017) 0.391 0.0 0.0007 Verbal reasoning -0.417 (0.032) -0.414 -0.7 0.0014 reasoning -0.765 (0.054) -0.768 0.4 0.0024 Level 2 variance 0.079 (0.016) 0.080 1.3 0.0005 Level 1 variance 0.536 (0.012) 0.536 0.0 0.0004 band 2* Verbal band 3* * Verbal reasoning band has 3 categories: category 1 (the reference category) is the top 25% of original verbal reasoning scores, category 2 is the middle 50% of verbal reasoning scores and category 3 is the bottom 25% of verbal reasoning scores. ***The imputation standard error is the standard error for each parameter over the 100 simulations. See Goldstein, Carpenter, Kenward and Levin. (2009). Statist. Modelling, 9, 173-197