# Missing data and maximum likelihood

I've heard it said that maximum likelihood estimation is an alternative to imputation methods for missing data. Does that mean any model fitted using maximum likelihood such as logistic regression, Poission regression, generalised linear model etc.? If that is so, then is a complete case analysis (i.e. retaining only rows with no non missing values for all variables in the model) followed by any model fitted using maximum liklihood robust to data missing at random? This is how most software would run the analysis anyway. It doesn't seem that simple to me as then, why do all these fancy imputation methods exist?!

• I think there might be a slight misunderstanding: maximum likelihood is a means to deal with missing data if the likelihood function includes a model of the missingness mechanism (e.g., Heckman selection). So generally what you've heard does not refer to "MLE of a logistic regression", but "MLE of a logistic regression with a missingness model." Commented Jan 19 at 17:57
• MLE of what? A joint model? Conditional modeling? Commented Jan 20 at 11:47
• MLE of a joint model in this case. Commented Jan 23 at 0:14

The likelihood to be maximized when you have missing data is different from the likelihood for complete data. So, just because a computer program uses maximum likelihood, does not mean that it will do the correct thing when you have missing data.

Sometimes the correct likelihood with missing data (missing at random) is much more complicated than the complete data likelihood and that is why techniques like multiple imputation and the E-M algorithm are used. They can use existing tools instead of having to create a tool that uses the more complicated likelihood. But if you have the full/correct likelihood, then yes the maximum likelihood estimation from that likelihood is a good (probably better) alternative to imputation.

Yes, full information maximum likelihood (FIML) estimation in many situations offers a very straightforward way to address missing data under the "missing at random" (MAR) mechanism. FIML is equivalent to multiple imputation (MI) in terms of the underlying mathematical theory. MI has certain advantages in some situations such as when you are dealing with categorical data. Otherwise, FIML is often a lot more straightforward to apply. Many software programs for structural equation modeling and related techniques (e.g., Mplus) use FIML as the default estimation method for many different statistical procedures. You can even automatically include auxiliary variables (variables related to missingness) in a statistical model with FIML in Mplus to better approximate the MAR condition.

A complete case analysis (listwise deletion) is usually not recommended because it rests on the stronger assumption of "missing completely at random" (MCAR) data and because listwise deletion leads to reduced statistical power even when MCAR holds (because you are loosing cases/information). See:

Enders, C. K. (2022). Applied missing data analysis (2nd ed.). Guilford Press.

Graham, J. W. (2003). Adding missing-data-relevant variables to FIML-based structural equation models. Structural Equation Modeling, 10(1), 80-100.

Schafer, J. L., & Graham, J. W. (2002). Missing data: Our view of the state of the art. Psychological Methods, 7(2), 147–177. https://doi.org/10.1037/1082-989X.7.2.147

• Thanks Christian. Mixed models e.g. lmer in R uses complete case by default so why are these superior to e.g. glm with cluster sandwich estimator? Both use maximum likelihood on complete cases. So why do people say mixed models are better for handling missing data? Commented Jan 26 at 16:59
• Multilevel models automatically use FIML (at least in some cases) without this being obvious to the analyst. For example, when you apply multilevel models to longitudinal data (e.g., growth curve models), time points are nested within individuals. Individuals with incomplete data (missing time points) are automatically included in the analysis. Commented Jan 26 at 18:22
• That is extremely helpful Christian. Though I will need to do some extra reading to attempt to understand this. Commented Jan 27 at 16:53