I need to identify outliers and high leverage points, and perform model diagnostics, in an
lme4 model. For outliers and high leverage points, simply making a plot to visually inspect would be nice, but is insufficient. I have 10,800 data points, and need to mark each point via some analytic or computational test as either outlier/high-leverage or not outlier/high-leverage. After identification of outliers/high-leverage points, I will go through a separate process to decide whether or not to exclude the points from the data set.
Exclusion of points will take into consideration prior detailed analysis of each observation's raw data source (an audio recording), in addition to the automated identification mentioned above. Here, I refer to this process as "selective deletion".
I also need to understand if my outliers should be based on "marginal" or "conditional" residuals, and if my leverage should be based on fixed or fixed plus random effects. For defintions of "marginal" and "conditional", as well as potential defintions of leverage, I am following Diagnostic and Treatment for Linear Mixed Models, Singer et al, 2013.
I.e., with a mixed model of the form... $$ y= X\beta + Zb + e $$ $$ e \thicksim N(0,\sigma^2I) $$ $$ b \thicksim N(0,G) $$ Where $G$ is a symmetric, positive definite matrix. By marginal, I mean residuals of the form: $$ \zeta = y-E[y]=y - X\beta $$ By conditional, I means residuals of the form: $$ e = y - X\beta - Zb $$ My questions are:
- How to identify outliers via an automated procedure based on an
- Whether marginal or conditional residuals should be used to identify candidates for selective deletion.
- What kind of residuals should be used for assessment of normality, linearity, homoscedasticity, etc.
- How to identify high leverage points for the purposes of selective deletion, and whether to use leverage from fixed or all effects (see Singer et al, above).
- How to test that $b$ is distributed as $N(0,G)$, i.e., general multi-variate normal? Is this done simply by looking at QQ plots of the random effects? What if $G$ has covariances, i.e., non-zero off-diagonal terms? Is looking at 1-dimensional QQ plots for each random effect still adequate to evaluate this type of normality? Or is some sort of transformation required?