# Is maximum likelihood a form of data substitution? Or not?

I’m using maximum likelihood with missing data.

In this case of missing data, is maximum likelihood a form of data substitution?

I’m significantly more familiar with multiple imputation which I would consider a form of data substitution. However, I’m trying to work out if ML would be considered the same or not. Is there a reason why it would not be considered data substitution?

• Could you be a little more specific about how you are proposing to apply Maximum Likelihood?
– whuber
Commented May 16, 2015 at 19:55

I'm not exactly sure what you mean by substitution because, even in the context of MI, I find this notion highly misleading. The point of MI is not to substitute the missing data, it is to obtain unbiased parameter estimates.

However, I think you're asking: Does ML also create "replacements" for the missing data in order to obtain parameter estimates?

The answer is no. Maximum likelihood estimation (also called FIML in that context) simply maximizes the likelihood from the observed data alone. You can find a very readable introduction to ML estimation with missing data in the book by Enders (2010).

• Maximum likelihood simply chooses the estimates for the parameters that maximize the likelihood. Obtaining unbiased estimates doesn't even come into consideration. There is no mathematical law that states one must or should use maximum likelihood but it is found to have (many times but not always) good sampling properties and can be shown to have good properties as the sample size gets large.
– JimB
Commented Jun 28, 2015 at 18:44
• I agree. I see I have a typo in my answer that might've been confusing. Corrected as per your suggestion. Commented Jun 28, 2015 at 18:45

Multiple imputation is similar to maximum likelihood in that they both provide unbiased and efficient parameter estimates even with missing at random (MAR: or missingness with likelihood depending on observed factors) data. The reason for this is that imputation models and reduced likelihoods both plausibly account for what a reasonable range of missing values would have been had they been observed.

I would argue that neither really "substitutes data". However, with multiple imputation, you can inspect the multiply imputed datasets and explain to non-analysts what values were actually imputed. Just the same, you can use your maximum likelihood estimates to create a parametric bootstrap for the missing values after performing EM and "sample from the posterior" in that fashion. MI has been described as an approximate Bayesian bootstrap for that reason.