general linear model
DESCRIPTION
General Linear Model. Generalized Linear Model. Generalized Linear Mixed Model. General Linear Model. Generalized Linear Model. Generalized Linear Mixed Model. GLMM. LMM. Generalized Linear Mixed Model. LMEM. HLM. Multilevel Model. Tagliamonte & Baayen (2012 : 7 of preprint). - PowerPoint PPT PresentationTRANSCRIPT
GeneralLinear Model
GeneralizedLinear Model
GeneralizedLinearMixed Model
GeneralLinear Model
GeneralizedLinear Model
GeneralizedLinearMixed Model
GLMM
LMM
LMEM
HLM
GeneralizedLinearMixed Model
MultilevelModel
Tagliamonte & Baayen (2012: 7 of preprint)
Tagliamonte, S. A., & Baayen, R. H. (2012). Models, forests, and trees of York English: Was/were variation as a case study for statistical
practice. Language Variation and Change, 24(02), 135-178.
The Beauty of Mixed Models
• Account for clusters without averaging• Different distributions (generalized LMM)• Interpretation at the trial-level• Everything in one model• Excellent for individual differences studies
(cf. Drager & Hay, 2012; Dan Mirman’s work)
More Power!! (see e.g.,
Barr et al., 2013)
Problems of Mixed Models
• Issues surrounding p-values• People misuse them … in a way that doesn’t
improve Type I error rate(Schielzeth & Forstmeier, 2009; Barr et al., 2013)
• Sometimes take A LOT of time• Some models don’t converge
response ~ intercept + slope * fixed effect + error
The Linear Model
structural partsystematic
partdeterministic
part
probabilistic part
stochastic partrandom part
response ~ intercept + slope * fixed effect + error
structural partsystematic
partdeterministic
part
probabilistic part
stochastic part
random part
The Linear Mixed Effects Model
Important terminology
- repeatable - non-repeatable
- systematic influence - random influence
- exhaust the population - sample the population
- generally of interest - often not of interest
- can be continuous - have to be categorical
or categorical
Fixed effectRandom effect
“Fixed-effects factors are those in which the populations to which we wish to generalize are precisely the levels represented in our
analysis.”
assumed to be constantacross experimentsStructural
PartStochastic
Part
Crawley (2013: 681)
Subjects as a fixed effect?
NO… why:
not repeatable not systematic often, not of interest small subset of population
Repetitions as a fixed effect?
Yes… why:
repeatable systematic[ often, not of interest] “exhausts the population”
Rep 1
Rep 2
Rep 3
Item #1
Subject
Common experimental data
Item...
Item...
German
French
English
Spanish Italian
Swedish
NorwegianFinnish
Hungarian
Turkish
Romanian
library(lme4)lmer(y ~ x + (1|subject), mydata)
In R:
Random interceptsversus
Random slopes
RT (m
s)
Subjects
Random intercepts
Random slopes
Experiment time
RT (m
s)
Randomintercepts
Experiment time
RT (m
s)
Randominterceptsand slopes
Random intercept vs. slope models
Random intercept model= the fixed effect is evaluated against an error term that captures subject- or item-specific variability in the response
Random slope model= the fixed effect is evaluated against an error term that captures subject- or item-specific variability in how the fixed effect affects the response
In R: (1|subject)
In R: (1+pred|subject)
http://anythingbutrbitrary.blogspot.com/2012/06/random-regression-coefficients-using.html
Random intercept examples
• Some people are fast responders, some people are slow responders (their “intercepts” for response time are different)
• Some people are very sensitive / accurate listeners, some are less sensitive (their “intercepts” for accuracy are different)
• Some people have high or low voices with respect to their gender (their “intercepts” for pitch are different)
Random slope examples
• Some people speed up during a long experiment, some slow down
• Some people become more accurate during a long experiment, some less
• Some people raise their pitch more for focus than others
An example
RT ~
An example
RT ~ Condition+ (1|Subject)
An example
RT ~ Condition ++ (1+Condition|Subject)
An example
RT ~ Condition ++ (1+Condition|Subject)+ (1|Item)
An example
RT ~ Condition ++ (1+Condition|Subject)+ (1+Condition|Item)
An example
RT ~ Condition + TrialOrder ++ (1+Condition|Subject)+ (1+Condition|Item)
An example
RT ~ Condition + TrialOrder ++ (1+Condition+
TrialOrder|Subject)+ (1+Condition|Item)
Model specificationfor random effects
(1|subject)random intercept
(0+fixedeffect|subject) random slope
(1+fixedeffect|subject) … with correlation
term
Assumptions
Absence ofCollinearity
Normality of Errors
Homoskedasticity of Errors
No influentialdata points
Independence