GLMM Model Averaging with Predictor Multicollinearity

Question

I am running GLMM models to determine how environmental factors influence bird collisions. I've obtained a list of candidate models with delta AIC less than 2, and I want to perform model averaging.

I understand that using AIC-based average is a common approach. However, I have predictors that are highly correlated in different candidate models. For example, Model 1 includes "ntl500" (Mean nighttime light value within a 500m buffer), while Model 2 includes "ntl1000" (Mean nighttime light value within a 1000m buffer), and they are highly correlated.

In the paper Model averaging and muddled multimodel inference by BS Cade(2015), he mentioned that AIC-based model averaging for parameter coefficients is not reliable due to multicollinearity. For GLM models, he suggested standardizing predictors by partial standard deviation before fitting models. However, the calculation of partial standard deviation requires variance inflation factor (VIF) which is obtained from running models. Does this mean I need to run the models with unstandardized predictors first, then standardize them with partial standard deviation, then run the models again？

Or please let me know if there are any other ways of dealing highly correlated variables in candidate models when averaging models.

Thank you.

Answer 1

A common issue with this kind of analysis is the presence of multicollinearity among predictors. This problem becomes evident when different candidate models include highly correlated variables, such as in the resent question, ntl500 (Mean nighttime light value within a 500m buffer) and ntl1000 (Mean nighttime light value within a 1000m buffer).

In the paper "Model averaging and muddled multimodel inference" by B. S. Cade (2015), it is highlighted that AIC-based model averaging can be problematic due to multicollinearity. Cade suggests standardising predictors by partial standard deviation. This method requires some care and here we discusses how to approach this and explore alternative methods for dealing with highly correlated variables in candidate models when performing model averaging.

Standardising Predictors Using Partial Standard Deviation

1. Run Initial Models: Initially, run your GLMM models with unstandardised predictors to obtain the variance inflation factors (VIF) for each predictor. The VIF is calculated as $\text{VIF} = \frac{1}{1 - R^2}$, where $R^2$ is the coefficient of determination of the predictor regressed on all other predictors in the model.

2. Calculate Partial Standard Deviation: For each predictor, compute the partial standard deviation using the formula: $$ s^* = s \times \sqrt{\text{VIF} \times \left(\frac{1}{n-p}\right)} $$ where $ s $ is the sample standard deviation of the predictor, $ \text{VIF} $ is the variance inflation factor, ( n ) is the sample size, and $ p $ is the number of predictors in the model.

3. Standardise Predictors: Standardise each predictor by dividing it by its partial standard deviation: $$ X^* = \frac{X}{s^*} $$

4. Rerun Models: Fit the models again using these standardised predictors. This ensures that the scales of parameter estimates are comparable across models.

Standardising by partial standard deviations addresses the varying scales caused by multicollinearity, making parameter estimates more comparable across different models. This method effectively equates the scaling of predictors, allowing for more reliable model averaging .

Alternative Approaches

1. Variance Decomposition

An alternative method involves decomposing the variance of each predictor to understand its relative importance better. This method involves partitioning the variance to determine how much each predictor contributes to the model's explanatory power.

2. Hierarchical Partitioning

Hierarchical partitioning can help identify the independent contribution of each predictor by systematically partitioning the explained variance. This method accounts for multicollinearity by considering all possible combinations of predictors and their unique contributions.

3. Using Ratios of Standardised Estimates

Another practical approach is to use the ratios of standardised estimates. This involves taking the absolute values of standardised regression coefficients and comparing them. These ratios provide a measure of relative importance that incorporates both effect size and variance reduction .

Summing Up

Dealing with multicollinearity in model averaging requires careful consideration of predictor scaling. Standardising predictors by their partial standard deviations, as suggested by Cade (2015), can be an effective method for ensuring that parameter estimates are comparable across models. Alternative methods such as variance decomposition, hierarchical partitioning, and using ratios of standardised estimates can also be useful. Each approach provides different insights into the relative importance of predictors and helps mitigate the effects of multicollinearity.

References

• Cade, B. S. (2015). Model averaging and muddled multimodel inference. Ecology, 96(9), 2370-2382.