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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. References
us- site/1%20nov%2011/Enders%20Pre%20Conference%20Worksh op.pdf
3psyc948_lecture02.pdf
Tech 5 mplus

12. 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