## 17.5 Estimation Methods

• Similar to logistic regression, estimates from multi-level models typically aren’t estimated directly using maximum likelihood (ML) methods.
• Iterative methods like Restricted (residual) Maximum Likelihood (REML) are used to get approximations.
• REML is typically the default estimation method for most packages.

Details of REML are beyond the scope of this class, but knowing the estimation method is important for two reasons

1. Some type of testing procedures that use the likelihood ratio may not be valid.
• Comparing models with different fixed effects using a likelihood ratio test is not valid. (Must use Wald Test instead)
• Can still use AIC/BIC as guidance (not as formal tests)
2. Iterative procedures are procedures that perform estimation steps over and over until the change in estimates from one step to the next is smaller than some tolerance.
• Sometimes this convergence to an answer never happens.
• You will get some error message about the algorithm not converging.
• The more complex the model, the higher chance this can happen
• scaling, centering, and avoiding collinearity can alleviate these problems with convergence.

You can change the fitting algorithm to use the Log Likelihood anyhow, it may be slightly slower but for simple models the estimates are going to be very close to the REML estimate. Below is a table showing the estimates for the random intercepts,

REML MLE
AITKIN 1.1107726 1.1143652
ANOKA 0.9427046 0.9438525
BECKER 1.2688325 1.2700351
BELTRAMI 1.2694025 1.2702493
BENTON 1.3243796 1.3245917
BIGSTONE 1.4081134 1.4068867

and the same estimates for the variance terms.

VarCorr(fit_partpool)
##  Groups   Name        Std.Dev.
##  county   (Intercept) 0.29767
##  Residual             0.76607
VarCorr(fit_partpool_MLE)
##  Groups   Name        Std.Dev.
##  county   (Intercept) 0.29390
##  Residual             0.76607

So does it matter? Yes and no. In general you want to fit the models using REML, but if you really want to use a Likelihood Ratio test to compare models then you need to fit the models using ML.