GLMM Models

Often times our data conspires against us violating essential assumptions of the standard linear model.  One critical assumption is the errors term of a model must be uncorrelated. There can be many causes of such correlation, repeated data sources (as with customers over time) or missing variables or data which is skewed over time.  If those variations are linear you can model them by adding linear random effects, (i.e. allow the intercept to vary by the source of the variation). One such solution is Generalized Linear Mixed Models (GLMM). It is called a mixed model because it has both random effects and fixed effects. Remember fixed effects are assumed have no measurement error and the same generalization (no group effects). Random effects can have measurement bias or group effects.

Example one: suppose the probability of a customer returning increases after each visit. If this increase in probability is linear, (e.g. first visit 10% more likely to return, second visit 20% likely to return, third visit 30% likely to return,…), then by adding a linear random effects that varies by customer you can correctly model this. Often times if you do not correct for this relationship between variables such as customer age (because the older you are the more likely you have returned more than once) may proxy for this leading to incorrect conclusion.

Example two:  suppose you are collecting data at the county level.  Each county may have different fixed effects.  Imagine crime rates across counties.  The base rate of crime will vary while the influence of say poverty rate will be the same.  If you did not allow the intercept to vary the relation between poverty and crime may be obscured by the base rate variation across counties.

GLMM do not correct for non-linear variations across sampling units.  If you believe customer retention is a non-linear relationship visit or the influence of poverty varies by county than GLMM alone cannot correct for this.  In many circumstances however, such as influence of poverty on crime, the variation across counties is due to missing or latent variables such as the support programs available for individuals at or below the poverty line.  Likewise the fixed effect can also be caused by omitted or latent variables.  The base crime rate across counties is a function of general underline conditions that oftentimes cannot be measured.  Such latent variables are often impossible to uncover so using GLMM is an acceptable solution to correct for this missing variable bias. Further Reading

http://support.sas.com/rnd/app/papers/glimmix.pdf
http://www.wiley.com/legacy/wileychi/eosbs/pdfs/bsa251.pdf
http://www.stat.umu.se/forskning/reports/glmmML.pdf
http://web.maths.unsw.edu.au/~wand/kowpap.pdf
http://arxiv.org/PS_cache/math/pdf/0606/0606491v1.pdf
http://staff.pubhealth.ku.dk/~pd/mixed-jan.2006/glmm.pdf
http://www.stat.umn.edu/geyer/bernor/library/bernor/doc/examples.pdf