What is full information maximum likelihood?

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Full information maximum likelihood is almost universally

abbreviated FIML, and it is often pronounced like "fimmle" if

"fimmle" was an English Word. FIML is often the ideal tool to use

when your data contains missing values because FIML uses the raw

data as input and hence can use all the available information in

the data. This is opposed to other methods which use the observed

covariance matrix which necessarily contains less information than

the raw data. An observed covariance matrix contains less

information than the raw data because one data set will always

produce the same observed covariance matrix, but one covariance

matrix could be generated by many different raw data sets.

Mathematically, the mapping from a data set to a covariance matrix

is not one-to-one (i.e. the function is non-injective), but rather

many-to-one.

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Although there is a loss of information between a raw data set and

an observed covariance matrix, in structural equation modeling we

are often only modeling the observed covariance matrix and the

observed means. We want to adjust the model parameters to make the

observed covariance and means matrices as close as possible to the

model-implied covariance and means matrices. Therefore, we are

usually not concerned with the loss of information from raw data to

observed covariance matrix. However, when some raw data is missing,

the standard maximum likelihood method for determining how close

the observed covariance and means matrices are to the

model-expected covariance and means matrices fails to use all of

the information available in the raw data. This failure of maximum

likelihood (ML) estimation, as opposed to FIML, is due to ML

exploiting for the sake of computational efficiency some

mathematical properties of matrices that do not hold true in the

presence of missing data. The ML estimates are not wrong per se and

will converge to the FIML estimates, rather the ML estimates do not

use all the information available in the raw data to fit the

model.

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The intelligent handling of missing data is a primary reason to use

FIML over other estimation techniques. The method by which FIML

handles missing data involves filtering out missing values when

they are present, and using only the data that are not missing in a

given row.