2.2 Model Estimation and Inference

Section 2.2 of Exam MAS-II

MLE: If theta is known/unknown, Beta hat is the ____

Best Linear Unbiased Estimator (BLUE)

Empirical Best Linear Unbiased Estimator (EBLUE)

Restricted MLE

Adjusts for the loss of degrees of freedom from estimating the fixed effects to produce an unbiased estimate for theta

Biased?

Computational Algorithms: Expectation Maximization: Pro and Con

Pros: good at finding starting values for other algorithms

Cons: converges slowly and produces "optimistic" estimators

Computational Algorithms: Newton-Raphson: Pro and Con

Pros: converges in a small number of iterations and can be used to obtain an asymptotic covariance matrix for the covariance parameters in theta

Cons: each iteration takes a while

Computational Algorithms: Fisher Scoring: Pro and Con

Pros: less computationally intensive and more likely to converge

Cons: difficult to obtain the expected Hessian matrix, which is needed in order for the estimates to be accurate

Computational Algorithms: General Advice

In general, start with a few iterations of EM to generate starting values, finish with N-R, and avoid Fisher scoring.

Handling Computational Problems (If D does not converge)

1. Choose different starting values for the covariance parameters.

2. Rescale the covariates.

3. Remove potentially unnecessary random effects.

4. Fit the implied marginal model instead.

5. Fit the marginal model with an unstructured covariance matrix.

The ICC for level j of a two-level variance components model

For a subject i in cluster j nested within group of k clusters, the Level 2 ICC is:

the level 3 ICC is:

General intuition for ICC

In general, the ICC for a given level of clustering can be thought of as the proportion of the total observed variation due to the random effects at that level and higher levels. It must be positive.

Marginal ICC: For the implied marginal model arising from a variance components model

The marginal correlation between two different observations within the same group

EBLUPs: Predict ____

For a variance components model, we can use what to compute the shrinkage mean?

Buhlmann Credibility

Define the components of this equation

Likelihood Ratio Test Equation

Alternative Tests: t-tests: What it tests and shortcoming

Alternative Tests: F-tests: What it tests, df, Type I and III, Kenward-Roger importance

Alternative Tests: Omnibus Wald Test: What it tests, shortcoming, distribution

Alternative Tests: Wald Test: What it tests, shortcoming

Information Criterion: AIC and BIC equations

Top Down Strategy

Start with a complex model and reduce it to a simpler model.

1. Build a model with a "loaded" mean structure

2. Select a structure for the random effects

3. Select a residual error covariance structure

4. Reduce the model by removing non-significant fixed effects

Step-Up Strategy

More commonly utilized for constructing models in hierarchical form. Start with a simple model and gradually add terms.

1. Build a "means-only" model

2. Check the random intercepts

3. Add the Level 1 covariates and related Level 2 random coefficients

4. Add the Level 2 covariates and related Level 3 random coefficients

5. Repeat Step 4 if necessary for any Level 3 covariates