Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM)...
Transcript of Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM)...
![Page 1: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/1.jpg)
Lesson 6, Part 1: Linear Mixed Effects Models
![Page 2: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/2.jpg)
Learn about linear mixed effects models (LMEM)
This Lesson’s Goals
Summarise results in an R Markdown document
Run some preliminary LMEMs in R
Make figures for data for LMEMs
![Page 3: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/3.jpg)
Learn about linear mixed effects models (LMEM)
This Lesson’s Goals
Summarise results in an R Markdown document
Run some preliminary LMEMs in R
Make figures for data for LMEMs
![Page 4: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/4.jpg)
But aren’t percentages reallyjust summarized count data?
But we had to drop a bunch of Union states, isn’t that a problem?
But Alabama was missing one Democrat data point, isn’t it not balanced?
linear mixed effects modelsgeneralized
End of Lesson 5 Questions
But what about the variance for ‘year’, shouldn’t we try and account for that too?
![Page 5: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/5.jpg)
Math (Part 1)
![Page 6: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/6.jpg)
yi = a + b1x1i + b2x2i + b3x1ix2i + ei
How do I add factorsfor random variance?
(i.e. things we’re not directly testing)
![Page 7: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/7.jpg)
yi = a + as + b1x1i + b2x2i + b3x1ix2i + ei
random effect
s = state
yi = a + b1x1i + b2x2i + b3x1ix2i + ei
![Page 8: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/8.jpg)
yi = a + as + b1x1i + b2x2i + b3x1ix2i + ei
random intercept
s = state
yi = a + b1x1i + b2x2i + b3x1ix2i + ei
![Page 9: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/9.jpg)
yi = a + as + ay + b1x1i + b2x2i + b3x1ix2i + ei
random intercept #1
s = state
random intercept #2
y = year
yi = a + b1x1i + b2x2i + b3x1ix2i + ei
![Page 10: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/10.jpg)
ANOVA language LMEM language
independent variable
error variable
In this paper we tested the effect of time on weight. A total of 50 baby chicks were included in the study.
dependent variable
fixed effect
random effect
dependent variable
time
weight
baby chick
mixed effects
![Page 11: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/11.jpg)
yi = a + as + ay + b1x1i + b2x2i +b3x1ix2i + ei
yi = specific y value
a = intercept
as = random intercept #1 for specific level
ay = random intercept #2 for specific level
b1 = slope of first variable
x1i = specific x value for first variable
b2 = slope of second variable
x2i = specific x value for second variable
b3 = slope of third variable (interaction)
ei = random variance or the residual
![Page 12: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/12.jpg)
yi = a + as + ay + b1x1i + b2x2i + b3x1ix2i + ei
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Percentage of Votes for Incumbentby Country in Civil War and Party of Incumbent
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Percentage of Votes for Incumbentby Country in Civil War and Party of Incumbent
yi = a + as + ay + b1x1i + b2x2i + b3x1ix2i + ei
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Percentage of Votes for Incumbentby Country in Civil War and Party of Incumbent
yi = a + as + ay + b1x1i + b2x2i + b3x1ix2i + ei
![Page 15: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/15.jpg)
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Percentage of Votes for Incumbentby Country in Civil War and Party of Incumbent
yi = a + as + ay + b1x1i + b2x2i + b3x1ix2i + ei
![Page 16: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/16.jpg)
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Rhode Island
Rhode Island
Rhode Island
Rhode Island
yi = a + as + ay + b1x1i + b2x2i + b3x1ix2i + ei
![Page 17: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/17.jpg)
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Kentucky
KentuckyKentucky
Kentucky
yi = a + as + ay + b1x1i + b2x2i + b3x1ix2i + ei
![Page 18: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/18.jpg)
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Percentage of Votes for Incumbentby Country in Civil War and Party of Incumbent
1964
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yi = a + as + ay + b1x1i + b2x2i + b3x1ix2i + ei
![Page 19: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/19.jpg)
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Percentage of Votes for Incumbentby Country in Civil War and Party of Incumbent
2012
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yi = a + as + ay + b1x1i + b2x2i + b3x1ix2i + ei
![Page 20: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/20.jpg)
But, at the end of the last lesson we said this was bad, because it was a
percentage of a count?
![Page 21: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/21.jpg)
log[p/(1-p)]i = a + as + ay + b1x1i + b2x2i + b3x1ix2i + ei
logistic regression —> generalized linear mixed effects model
yi = a + as + ay + b1x1i + b2x2i +b3x1ix2i + ei
![Page 22: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/22.jpg)
R Code (Part 1)
![Page 23: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/23.jpg)
lmer(perc_votes_incumbent ~incumbent_party * civil_war
+ (1|state)+ (1|year))
lme4
yi = a + as + ay + b1x1i + b2x2i + b3x1ix2i + ei
![Page 24: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/24.jpg)
lmer(perc_votes_incumbent ~incumbent_party * civil_war
+ (1|state)+ (1|year))
lme4
yi = a + as + ay + b1x1i + b2x2i + b3x1ix2i + ei
![Page 25: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/25.jpg)
But, in the ANOVA we accounted for the fact that a variable could be
within- or between-subject?
![Page 26: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/26.jpg)
Math (Part 2)
![Page 27: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/27.jpg)
yi = a + as + ay + (bs1+b1)x1i + (by1+b2)x2i +
b3x1ix2i + eirandom slope
s = state
random slope
y = year
yi = a + as + ay + b1x1i + b2x2i +b3x1ix2i + ei
![Page 28: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/28.jpg)
yi = a + as + ay + (bs1+b1)x1i + (by1+b2)x2i +
b3x1ix2i + eiyi = specific y value
a = intercept
as = random intercept #1
ay = random intercept #2
bs1 = slope of r.e. #1
b1 = slope of variable #1
x1i = x value for variable #1
by1 = slope of r.e. #2
b2 = slope of variable #2
x2i = x value for variable #2
b3 = slope of variable #3
ei = random variance
![Page 29: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/29.jpg)
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Union ConfederacyCivil War country
Perc
enta
ge o
f vot
e fo
r inc
umbe
nt● ●Democrat Republican
Percentage of Votes for Incumbentby Country in Civil War and Party of Incumbent
Rhode Island
Rhode Island
yi = a + as + ay + (bs1+b1)x1i + (by1+b2)x2i + b3x1ix2i + ei
Rhode Island
Rhode Island
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0
25
50
75
100
Union ConfederacyCivil War country
Perc
enta
ge o
f vot
e fo
r inc
umbe
nt● ●Democrat Republican
Percentage of Votes for Incumbentby Country in Civil War and Party of Incumbent
Rhode Island
Rhode Island
Rhode Island
Rhode Island
Rhode Island
Rhode Island
Rhode Island
Rhode Island
yi = a + as + ay + (bs1+b1)x1i + (by1+b2)x2i + b3x1ix2i + ei
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R Code (Part 2)
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lmer(perc_votes_incumbent ~incumbent_party * civil_war + (1+incumbent_party|state)
+ (1+civil_war|year))
lme4
yi = a + as + ay + (bs1+b1)x1i + (by1+b2)x2i + b3x1ix2i + ei
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lmer(perc_votes_incumbent ~incumbent_party * civil_war + (1+incumbent_party|state)
+ (1+civil_war|year))
lme4
yi = a + as + ay + (bs1+b1)x1i + (by1+b2)x2i + b3x1ix2i + ei
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LMEM with only random intercepts
LMEM with only random intercepts and slopes
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Lab
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Data set: Stroop Task
source: real students!
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Say the color of the ink not the written word.
blue
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Say the color of the ink not the written word.
blue
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Say the color of the ink not the written word.
blueword = ink color congruent trial
blueword ≠ ink color incongruent trial
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Data set: Stroop Task
Congruency: Are responses to incongruent trials less accurate and slower than to congruent trials?
Experiment half: Are responses more accurate and faster in the second half of the experiment than the first half of the experiment?
Congruency x Experiment half: Is there an interaction between these variables?
accuracy
subject
source: real students!
logit pi x1 x2 r1 r2
= = = = =
congruencyexperiment half
item
reaction times
subject
yi x1 x2 r1 r2
= = = = =
congruencyexperiment half
item
accuracy (logistic) reaction times (linear)
![Page 41: Lesson 6, Part 1: Linear Mixed Effects Models · Learn about linear mixed effects models (LMEM) This Lesson’s Goals Summarise results in an R Markdown document Run some preliminary](https://reader030.fdocuments.net/reader030/viewer/2022040707/5e088796e75aa446441de883/html5/thumbnails/41.jpg)
dplyr
data_clean = data_results
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dplyr
data_clean = data_results %>% rename(trial_number = SimpleRTBLock.TrialNr.)
change variable name
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dplyr
data_clean = data_results %>% rename(trial_number = SimpleRTBLock.TrialNr.) %>% rename(congruency = Congruency) %>% rename(correct_response = StroopItem.CRESP.) %>% rename(given_response = StroopItem.RESP.) %>% rename(accuracy = StroopItem.ACC.) %>% rename(rt = StroopItem.RT.) %>%
change variable name
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dplyr
data_clean = data_results %>% rename(trial_number = SimpleRTBLock.TrialNr.) %>% rename(congruency = Congruency) %>% rename(correct_response = StroopItem.CRESP.) %>% rename(given_response = StroopItem.RESP.) %>% rename(accuracy = StroopItem.ACC.) %>% rename(rt = StroopItem.RT.) %>% select(subject_id, block, item, trial_number, congruency, correct_response, given_response, accuracy, rt)
change variable name
choose subset of variables
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RColorBrewer
cols = brewer.pal(5, “PuOr")
call to make palette
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RColorBrewer
cols = brewer.pal(5, “PuOr")
call to make palette
number of colors
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RColorBrewer
cols = brewer.pal(5, “PuOr")
call to make palette
number of colors
palette name
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RColorBrewer
cols = brewer.pal(5, “PuOr")
call to make palette
number of colors
palette name
> cols[1] "#E66101" "#FDB863" "#F7F7F7" "#B2ABD2" "#5E3C99"
0.0
2.5
5.0
7.5
60 90 120 1500.0
2.5
5.0
7.5
60 90 120 1500.0
2.5
5.0
7.5
60 90 120 1500.0
2.5
5.0
7.5
60 90 120 1500.0
2.5
5.0
7.5
60 90 120 150
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RColorBrewer
cols = brewer.pal(5, “PuOr")col_con = cols[1] call to
make palette
number of colors
palette name
> cols[1] "#E66101" "#FDB863" "#F7F7F7" "#B2ABD2" "#5E3C99"
0.0
2.5
5.0
7.5
60 90 120 1500.0
2.5
5.0
7.5
60 90 120 1500.0
2.5
5.0
7.5
60 90 120 1500.0
2.5
5.0
7.5
60 90 120 1500.0
2.5
5.0
7.5
60 90 120 150
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RColorBrewer
cols = brewer.pal(5, “PuOr")col_con = cols[1]col_incon = cols[5]
call to make palette
number of colors
palette name
> cols[1] "#E66101" "#FDB863" "#F7F7F7" "#B2ABD2" "#5E3C99"
0.0
2.5
5.0
7.5
60 90 120 1500.0
2.5
5.0
7.5
60 90 120 1500.0
2.5
5.0
7.5
60 90 120 1500.0
2.5
5.0
7.5
60 90 120 1500.0
2.5
5.0
7.5
60 90 120 150