Full Information Maximum Likelihood method(Estimation method to assign missing values)

I would like to see "Full Information Maximum Likelihood (FIML)" added as a missing value assignment method in SPSS Statistics.

FIML is a powerful method for handling missing data in statistical modeling. By using all available information in the data, it provides more efficient and accurate parameter estimates compared to traditional methods that discard missing data. 

Full Information Maximum Likelihood (FIML) is an estimation method used in statistical modeling, particularly in structural equation modeling (SEM) and other forms of latent variable analysis. FIML is designed to handle missing data effectively by using all available information in the data, rather than relying on traditional techniques like listwise or pairwise deletion.

Key Concepts of FIML

Maximum Likelihood Estimation (MLE):

• MLE is a method for estimating the parameters of a statistical model. It finds parameter values that maximize the likelihood function, which measures how likely it is to observe the given data under different parameter values.

Handling Missing Data:

• FIML handles missing data by using all available data points without imputing missing values. It does this by maximizing the likelihood function for each observation based on the observed data, rather than excluding cases with missing values.

Model Specification:

• FIML requires a correctly specified model to produce unbiased parameter estimates. This involves defining the relationships among observed and latent variables in the model.

Advantages:

• Efficiency: FIML uses all available data, leading to more efficient and unbiased estimates compared to methods that discard missing data.
• Flexibility: It can be applied to a wide range of models, including SEM, growth models, and factor analysis.
• Accuracy: FIML often provides more accurate parameter estimates in the presence of missing data compared to traditional methods.

Implementation:

• FIML is implemented in various statistical software packages such as Mplus, LISREL, Amos, and lavaan (in R). Each software may have specific syntax and options for performing FIML.

Mathematical Formulation

In FIML, the likelihood function is constructed for each observation based on the observed data. If yi\mathbf{y}_iyi​ represents the vector of observed data for the iii-th individual, and θ\mathbf{\theta}θ represents the vector of model parameters, the likelihood function for yi\mathbf{y}_iyi​ given θ\mathbf{\theta}θ is:

L(θ∣yi)=∏j=1mf(yij∣θ)L(\mathbf{\theta} | \mathbf{y}_i) = \prod_{j=1}^m f(y_{ij} | \mathbf{\theta})L(θ∣yi​)=∏j=1m​f(yij​∣θ)

where f(yij∣θ)f(y_{ij} | \mathbf{\theta})f(yij​∣θ) is the probability density function for the jjj-th variable given the parameters θ\mathbf{\theta}θ, and the product is over all observed variables for the iii-th individual. The overall likelihood function is the product of the individual likelihoods across all observations.

FIML maximizes this likelihood function to estimate the parameters θ\mathbf{\theta}θ.