12.7 - Maximum Likelihood Estimation Method

Assumption Section

Maximum Likelihood Estimation requires that the data are sampled from a multivariate normal distribution. This is a drawback of this method. Data is often collected on a Likert scale, especially in the social sciences. Because a Likert scale is discrete and bounded, these data cannot be normally distributed.

Using the Maximum Likelihood Estimation Method, we must assume that the data are independently sampled from a multivariate normal distribution with mean vector \(\mu\) and variance-covariance matrix of the form:

\(\boldsymbol{\Sigma} = \mathbf{LL' +\boldsymbol{\Psi}}\)

where \(\mathbf{L}\) is the matrix of factor loadings and \(\Psi\) is the diagonal matrix of specific variances.

We define additional notation: As usual, the data vectors for n subjects are represented as shown:

\(\mathbf{X_1},\mathbf{X_2}, \dots, \mathbf{X_n}\)

Maximum likelihood estimation involves estimating the mean, the matrix of factor loadings, and the specific variance.

The maximum likelihood estimator for the mean vector \(\mu\), the factor loadings \(\mathbf{L}\), and the specific variances \(\Psi\) are obtained by finding \(\hat{\mathbf{\mu}}\), \(\hat{\mathbf{L}}\), and \(\hat{\mathbf{\Psi}}\) that maximize the log-likelihood given by the following expression:

\(l(\mathbf{\mu, L, \Psi}) = - \dfrac{np}{2}\log{2\pi}- \dfrac{n}{2}\log{|\mathbf{LL' + \Psi}|} - \dfrac{1}{2}\sum_{i=1}^{n}\mathbf{(X_i-\mu)'(LL'+\Psi)^{-1}(X_i-\mu)}\)

The log of the joint probability distribution of the data is maximized. We want to find the values of the parameters, (\(\mu\), \(\mathbf{L}\), and \(\Psi\)), that are most compatible with what we see in the data. As was noted earlier the solutions for these factor models are not unique. Equivalent models can be obtained by rotation. If \(\mathbf{L'\Psi^{-1}L}\) is a diagonal matrix, then we may obtain a unique solution.

Computationally this process is complex. In general, there is no closed-form solution to this maximization problem so iterative methods are applied. Implementation of iterative methods can run into problems as we will see later.