This document discusses generalized linear mixed models (GLMMs) and their use in analyzing experimental data with more than one random source of variation, such as blocking or split plots. It provides an example of using a GLMM to analyze a split plot experiment with a proportional response variable. The GLMM allows modeling of non-normal data like counts or proportions while accounting for random effects. The document demonstrates how to set up the linear model for this type of split plot proportional data and analyze it using JMP's mixed modeling tools.

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Empowering Experimental Designs
(실험 설계의 강화)
: From Simplicity to Sophistication
(단순함에서 정교함으로)
Elizabeth A. Claassen, PhD
Senior Research Statistician Developer

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Why Mixed Models?
• Limitations can exist to our ability to perform experiments
– We may be restricted in the number of experimental units available
– Some factors may be easier or harder to change between runs
• After designing the experiment with these limitations, the analysis
should also account for the restrictions
• JMP makes this more complex (and sometimes daunting) analysis
easier
• While today’s focus is on designed experiments, mixed models are
often used in survey analysis and some observational studies

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Generalized Linear Mixed Models
Modeling non-Normal data with random effects
• Mixed Modeling has been the standard for analyzing data with more
than one source of random variation (blocking, split-plots, etc.).
• The Linear Mixed Model (LMM) assumes the response is continuous
with no bounds.
• What if your response is a discrete count? Or a binary response? Or a
proportion in terms of y/n?
• Enter the Generalized Linear Mixed Model (GLMM)

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▪Examples of GLMs
• Logistic regression
• Poisson regression
• Normal regression
• Analysis of variance models
Generalized Linear Model (GLM)

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Two Things a Model Must Do
5
▪ Plausibly describe the process that gives rise
to the observed data
– how explanatory variables affect response
– probability distributions involved
▪ Allow / facilitate addressing the objective that
motivated collecting the data
− test a hypothesis
− make a decision
− estimate a parameter

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Working Example
6
▪ Split plot experiment:
− 10 Blocks that are whole plots
− 2 x 3 Factorial
− Factor 1 is hard to change with Low/High levels
− Factor 2 is easier to change with Low/Medium/High
− Response: proportion, y/n
− Y is number of parts that passed QC
− N is the number of parts per “batch”, N=20 for all in
this case

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Linear Model for Split Plot Proportion Data
• ANOVA – linear model for split plot with categorical factors
• Model: proportion = intercept + whole plot factor + whole plot error +
split plot factor + interaction + residual error
–
• Implement in JMP with Standard Least Squares or JMP Pro with Mixed
Model
( )
ijk i j k ijk
ik
proportion w e
=  +  + + +  +

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8
Experiment Treatment Combined
Source d.f. Source d.f. Source d.f.
X1 1 X1 1
whole plot 9 Whole plot|X1 9-1=8
X2 2 X2 2
X1*X2 2 X1*X2 2
unit(wp)
10*(3-1)
=20
“parallels” 24
unit(wp) | X1 & X2
a.k.a. “residual”
a.k.a “wp x X2”
20-4=16
Total 29 Total 29 Total 29
Repurposed ANOVA Table

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JMP Demo

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Further Resources
JMP for Mixed Models (2021), Hummel, Claassen and Wolfinger
SAS for Mixed Models: Introduction and Basic Applications (2018), Stroup,
Milliken, Claassen and Wolfinger
Generalized Linear Mixed Models: Modern Concepts, Methods and
Applications (2012), Stroup
Statistically Speaking Webinar: The “What, Why, and How” of Generalized
Linear Mixed Models, Stroup and Claassen