Maximum Likelihood & Missing data Rachael Bedford Mplus: Longitudinal Analysis Workshop 23/06/2015

Overview Maximum Likelihood (ML) Maximum Likelihood: Robust Standard Errors (MLR) Missing data MCAR MAR NMAR Full Information Maximum Likelihood (FIML) Maximum likelihood vs. multiple imputation Summary

Maximum Likelihood (ML) Estimation ML is an estimator (like ordinary least squares) Why is estimation important? Influences quality and validity of estimates Most estimation techniques: Minimise something e.g. OLS Maximise something e.g. ML Simulation

Maximum Likelihood (ML) Estimation Asymptotic consistency – as sample size increases, estimator converges on true value Asymptotic normality – as sample size increases, the estimator distribution is normal Efficiency – small standard errors Wald test give se, mod indicies

Maximum Likelihood What does ML do? How does ML estimate parameters? ML identifies the population parameter values (e.g. population mean) that are most ‘likely’ or consistent with data (e.g. sample mean) It uses the observed data to find parameters with the highest likelihood (best fit) How does ML estimate parameters? By constraining search for parameters within a normal distribution Also good for Missing data

Maximum Likelihood Robust Robust Maximum Likelihood (MLR) still assumes data follow a multivariate normal distribution. BUT can deal with kurtosis “peakedness” of data MLR in Mplus uses a sandwich estimator to give robust standard errors E.g. Use for likert scale data Mahalanobis distance – tests for multivariate outliers

Full Information Maximum Likelihood Can apply ML to incomplete as well as complete data records i.e. where data is missing in response variables This is called Full Information Maximum Likelihood (FIML). If data are missing at random we can use FIML to estimate model parameters. Missing at random DOES NOT MEAN missing at random. Effect of attrition in LDA

Missing Data Missing completely at random (MCAR) missingness does not depend on the values of either observed or latent variables IF this holds can use listwise deletion Missing at random (MAR) (confusing name!) missingness is related to observed, but not latent variables or missing values Non-ignorable missing Missingness of data can be related to both observed but also to missing values and latent variables

FIML missing Craig Enders A case with an IQ 85 would likely have a performance rating of ~9. animations Based on this information, ML adjusts the job performance mean downward to account for the plausible (but missing) performance rating

Multiple imputation Another approach: Multiple Imputation Multiple copies of the data set are generated, each with different estimates of the missing values. Analyses performed on each of the imputed data sets, and the parameter estimates and standard errors are pooled into a single set of results. Maximum Likelihood Multiple imputation Estimate parameters does NOT fill in missing values Does fill in the missing data Usually best for continuous data Best for categorical or item level data In a large sample ML and MI give the same results

References http://vkc.library.uu.nl/vkc/ms/SiteCollectionDocuments/Mpl us- site/1%20nov%2011/Enders%20Pre%20Conference%20Worksh op.pdf http://jonathantemplin.com/files/sem/sem13psyc948/sem1 3psyc948_lecture02.pdf Tech 5 mplus

Considerations Recoverability Bias Power Is it possible to estimate what scores would have been if they were not missing? Bias Are statistics (e.g. means, variances, and covariances/correlations) the same as what they would have been had there not been any missing data? Power Do we have the same or similar rates of power (1 – Type II error rate) as we would without missing data? Item missing and parameter missing Power for specific parameter