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One can find information on the covariance matrix on the Wikipedia website where there is much information about the mathematics involved. One can also find information on Mathworks.

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2013-08-07 02:05:26
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Q: Where can one find information on the covariance matrix?
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What is full information maximum likelihood?

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. 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. 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.

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