Introduction to mixed models

Session 10

MATH 80667A: Experimental Design and Statistical Methods
HEC Montréal

1 / 37

Outline

Blocking

Mixed effects

2 / 37

Blocking3 / 37

Terminology for nuisance

Block

Source of variation, but of no interest
known and controllable

Example timing
lab technician
machine

Noise factor

Under which setting is response least affected?

Example temperature
processing

4 / 37

Example 15.2.3 in Dean & Voss of Lipton soup mixture

A: number of mixer ports through which vegetable oil was added (two levels, 1 and 3); B: temperature of mixer jacket (two levels; ambient temperature, presence of cooling water); C: mixing time (two levels; 60 and 80 sec); D: batch weight (two levels; 1500 and 2000 lb); E: delay between mixing and packaging (two levels; 1 day and 7 days).

Why blocking?

Design experiment to reduce the effect of uncontrolled variations

In general, increases the power of the $F$ test for treatment effects.

Group units in sets as alike as possible.

(Often) compare only treatments, so interactions are not included.

5 / 37

Assignment to treatment

Divide subjects within each block

Randomly allocate to treatment within block

(stratified sampling)

6 / 37

Block-treatment design

Linear model (two-way ANOVA) without interaction,

$\underset{\text{response}\vphantom{b}}{Y_{ij}} = \underset{\text{global mean}}{\mu\vphantom{\beta_j}} + \underset{\text{treatment}\vphantom{b}}{\alpha_i\vphantom{\beta_j}} + \underset{\text{blocking}}{\beta_j}+ \underset{\text{error}\vphantom{b}}{\varepsilon_{ij}\vphantom{\beta_j}}$

Compromise between

reduced variability for residuals,
loss of degrees of freedom due to estimation of $\beta$ 's.

7 / 37

Example: Resting metabolic rate

From Dean, Voss and Draguljić (2017), Example 10.4.1 (p. 311)

experiment that was run to compare the effects of inpatient and outpatient protocols on the in-laboratory measurement of resting metabolic rate (RMR) in humans. A previous study had indicated measurements of RMR on elderly individuals to be 8% higher using an outpatient protocol than with an inpatient protocol. If the measurements depend on the protocol, then comparison of the results of studies conducted by different laboratories using different protocols would be difficult. The experimenters hoped to conclude that the effect on RMR of different protocols was negligible.

8 / 37

The experimental treatments consisted of three protocols: (1) an inpatient protocol in which meals were controlled—the patient was fed the evening meal and spent the night in the laboratory, then RMR was measured in the morning; (2) an outpatient protocol in which meals were controlled—the patient was fed the same evening meal at the laboratory but spent the night at home, then RMR was measured in the morning; and (3) an outpatient protocol in which meals were not strictly controlled—the patient was instructed to fast for 12 hours prior to measurement of RMR in the morning.

Interaction plot

9 / 37

ANOVA table (without blocking)
Analysis of variance table - without blocking
 
      
    Degrees of freedom 
    Sum of squares 
    Mean square 
    F statistic 
    p-value 
  


    protocol 
    2 
    0.04 
    0.02 
    0.02 
    0.982 
  

    Residuals 
    24 
    24.35 
    1.01 
     
     
  

10 / 37

Analysis of variance table - without blocking
	Degrees of freedom	Sum of squares	Mean square	F statistic	p-value
protocol	2	0.04	0.02	0.02	0.982
Residuals	24	24.35	1.01

ANOVA table (with blocking)
Analysis of variance table - with blocking
 
      
    Degrees of freedom 
    Sum of squares 
    Mean square 
    F statistic 
    p-value 
  


    subject 
    8 
    23.12 
    2.89 
    37.42 
    0.000 
  

    protocol 
    2 
    0.04 
    0.02 
    0.23 
    0.795 
  

    Residuals 
    16 
    1.24 
    0.08 
     
     
  

11 / 37

Analysis of variance table - with blocking
	Degrees of freedom	Sum of squares	Mean square	F statistic	p-value
subject	8	23.12	2.89	37.42	0.000
protocol	2	0.04	0.02	0.23	0.795
Residuals	16	1.24	0.08

Semipartial effect sizes

If there is a mix of experimental and blocking factors...

Include the variance of all blocking factors and interactions (only with the effect!) in denominator.

e.g., if $A$ is effect of interest, $B$ is a blocking factor and $C$ is another experimental factor, use $\eta_{\langle A \rangle}^2 = \frac{\sigma^2_A}{\sigma^2_A + \sigma^2_B + \sigma^2_{AB} + \sigma^2_{\text{resid}}}.$

In R, use effectsize::omega_squared(model, partial = TRUE, generalized = "blocking") where blocking gets replaced with a vector containing the name of the blocking factors.

12 / 37

Random effects and mixed models13 / 37

Fixed effects

All experiments so far treated factors as fixed effects.

We estimate a mean parameter for each factor (including blocking factors in repeated measures).

Change of scenery

14 / 37

Change of scenery

Assume that the levels of a factor form a random sample from a large population.

We are interested in making inference about the variability of the factor.

measures of performance of employees
results from different labs in an experiment
subjects in repeated measures

We treat these factors as random effects.

15 / 37

Fixed vs random effects

There is no consensual definition, but Gelman (2005) lists a handful, of which:

When a sample exhausts the population, the corresponding variable is fixed; when the sample is a small (i.e., negligible) part of the population the corresponding variable is random [Green and Tukey (1960)].

Effects are fixed if they are interesting in themselves or random if there is interest in the underlying population (e.g., Searle, Casella and McCulloch [(1992), Section 1.4])

16 / 37

Random effect model

Consider a one-way model

$\underset{\text{response}}{Y_{ij}} = \underset{\text{global mean}}{\mu} + \underset{\text{random effect}}{\alpha_j} + \underset{\text{error term}}{\varepsilon_{ij}}.$

where

$\alpha_j \sim \mathsf{Normal}(0, \sigma^2_\alpha)$ is normal with mean zero and variance $\sigma^2_\alpha$ .
$\varepsilon_{ij}$ are independent $\mathsf{Normal}(0, \sigma^2_\varepsilon)$

17 / 37

Fictional example

Consider the weekly number of hours spent by staff members at HEC since September.

We collect a random sample of 40 employees and ask them to measure the number of hours they work from school (as opposed to remotely) for eight consecutive weeks.

18 / 37

Fitting mixed models in R

We use the lme4 package in R to fit mixed models.

The lmerTest package provides additional functionalities for testing.

lmer function fits linear mixed effect regression

Random effects are specified using the notation (1 | factor).

19 / 37

Model fit

library(lmerTest) # also loads lme4
rmod <- lmer(time ~ (1 | id), data = hecedsm::workhours)
summary_rmod <- summary(rmod)

Random effects:
 Groups   Name        Variance Std.Dev.
 id       (Intercept) 38.63    6.215   
 Residual              5.68    2.383   
Number of obs: 320, groups:  id, 40
Fixed effects:
            Estimate Std. Error      df t value Pr(>|t|)    
(Intercept)  23.3016     0.9917 39.0000    23.5   <2e-16 ***

Note that std. dev is square root of variance

20 / 37

Intra-class correlation

We are interested in the variance of the random effect, $\sigma^2_\alpha$ .

Measurements from the same individuals are correlated. The intra-class correlation between measurements $Y_{ij}$ and $Y_{ik}$ from subject $i$ at times $j\neq k$ is

$\rho = \frac{\sigma^2_\alpha}{\sigma^2_\alpha + \sigma^2_\varepsilon}.$

In the example, $\widehat{\sigma}^2_\alpha=38.63$ , $\widehat{\sigma}^2_\varepsilon = 5.68$ and $\widehat{\rho} = 0.87$ .

The mean number of working hours on the premises is $\widehat{\mu}=23.3$ hours.

21 / 37

Confidence intervals

We can use confidence intervals for the parameters.

Those are based on profile likelihood methods (asymmetric).

(conf <- confint(rmod, oldNames = FALSE))

##                       2.5 %    97.5 %
## sd_(Intercept)|id  4.978127  7.799018
## sigma              2.198813  2.595272
## (Intercept)       21.335343 25.267782

The variability of the measurements and the week-to-week correlation of employee measures are different from zero.

22 / 37

Shrinkage

Predictions of random effects are shrunk towards global mean, more so for larger values and when there are fewer measurements.

23 / 37

Even with old school, we can get confidence intervals but only in simple designs like this one-way.

Mixed models

Mixed models include both fixed effects and random effects.

Fixed effects for experimental manipulations
Random effects for subject, lab factors

Mixed models make it easier to

handle correlations between measurements and
account for more complex designs.

24 / 37

Repeated measures ANOVA using mixed model

Data need to be in long format, i.e., one response per line with an id column.

Illustration by Garrick Adden-Buie

25 / 37

Example: two-way ANOVA

We consider a repeated measure ANOVA (2 by 2 design, within-between) from Hatano et al. (2022).

data(HOSM22_E3, package = "hecedsm")
fmod <- afex::aov_ez(
  id = "id", 
  dv = "imscore", 
  between = "waiting",
  within = "ratingtype",
  data = HOSM22_E3)
anova(fmod)

26 / 37

## Anova Table (Type 3 tests)
## 
## Response: imscore
##                    num Df den Df     MSE       F      ges    Pr(>F)    
## waiting                 1     61 2.48926 11.2551 0.126246   0.00137 ** 
## ratingtype              1     61 0.68953 38.4330 0.120236 5.388e-08 ***
## waiting:ratingtype      1     61 0.68953  0.0819 0.000291   0.77575    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

27 / 37

Repeated measures with linear mixed models

Results are the same as for repeated measures ANOVA if the correlation estimate between observations of the same id are nonnegative.

mixmod <- lmerTest::lmer(
  imscore ~ waiting*ratingtype +
    (1 | id), # random intercept per id
  data = HOSM22_E3)
anova(mixmod, type = 3)

## Type III Analysis of Variance Table with Satterthwaite's method
##                     Sum Sq Mean Sq NumDF DenDF F value    Pr(>F)    
## waiting             7.7608  7.7608     1    61 11.2551   0.00137 ** 
## ratingtype         26.5009 26.5009     1    61 38.4330 5.388e-08 ***
## waiting:ratingtype  0.0565  0.0565     1    61  0.0819   0.77575    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

28 / 37

Theory

Full coverage of linear mixed models and general designs is beyond the scope of the course, but note

Estimation is performed via restricted maximum likelihood (REML)
Testing results may differ from repeated measure ANOVA
Different approximations for F degrees of freedom, either
- Kenward–Roger (costly) or
- Satterthwaite's approximation

29 / 37

Structure of the design

It is important to understand how data were gathered.

Oelhert (2010) guidelines

Identify sources of variation
Identify whether factors are crossed or nested
Determine whether factors should be fixed or random
Figure out which interactions can exist and whether they can be fitted.

30 / 37

Crossed vs nested effects

Nested effects if a factor appears only within a particular level of another factor.

Crossed is for everything else (typically combinations of factors are possible).

Russian dolls

Example of nested random effects: class nested within schools

class 1 is not the same in school 1 than in school 2

31 / 37

Matroschka from Wikimedia Commons CC-BY-SA 3.0 https://en.wikipedia.org/wiki/Matryoshka_doll#/media/File:Matryoshka_transparent.png

Specifying interactions

Consider factors $A$ , $B$ and $C$ .

If factor $A$ is treated as random, interactions with $A$ must be random too.
There must be repeated measurements to estimate variability of those interactions.
Testing relies on the variance components.

32 / 37

Mixed models vs repeated measures

In repeated measure ANOVA, repetitions over each sub-condition are averaged across subject.
Treat subject identifier as a blocking factor (fixed effect).
Impose sum-to-zero for coefficients.

Example: happy fakes

Jittered scatterplot of measurements per participant and stimulus type.

33 / 37

Interaction with random and fixed effect

Add student id as random effect, stimulus as fixed effect and their interaction as random effect (since one parent is random)

data(AA21, package = "hecedsm")
anova(ddf = "Kenward-Roger", # other option is "Satterthwaite"
  lmerTest::lmer(
    data = AA21 |> dplyr::filter(latency > -40), 
    latency ~ stimulus + (1 | id) + (1 | id:stimulus)))

## Type III Analysis of Variance Table with Kenward-Roger's method
##          Sum Sq Mean Sq NumDF  DenDF F value Pr(>F)
## stimulus 65.573  32.786     2 21.878  0.8465 0.4425

# Approximately 22 degrees of freedom (as for repeated measures)

34 / 37

Data structure

35 / 37

Example: chocolate rating

Example from L. Meier, adapted from Oehlert (2010)

A group of 10 rural and 10 urban raters rated 4 different chocolate types. Every rater got to eat two samples from the same chocolate type in random order.

36 / 37

Example: Curley et al. (2022)

Two variables were manipulated within participants: (a) evidence anchor (strong-first versus weak-first); (b) verdict system (two- versus three-verdict systems). Total pre-trial bias score was used as a covariate in the analysis (this score is based on the PJAQ and is explained further in the Materials section). Participants were also given two vignettes (Vignette 1 and Vignette 2); thus, the vignette variable was included in the data analysis [...]

The dependent variable was the final belief of guilt score, which was measured on an accumulated scale from 0–14, with 0 representing no belief of guilt and 14 representing a total belief that the person is guilty

This study is an instance of incomplete design.

37 / 37

Introduction to mixed models

Session 10

MATH 80667A: Experimental Design and Statistical Methods
HEC Montréal

1 / 37

Outline

Blocking

Mixed effects

2 / 37

Blocking3 / 37

Terminology for nuisance

Block

Source of variation, but of no interest
known and controllable

Example timing
lab technician
machine

Noise factor

Under which setting is response least affected?

Example temperature
processing

4 / 37

Example 15.2.3 in Dean & Voss of Lipton soup mixture

Why blocking?

Design experiment to reduce the effect of uncontrolled variations

In general, increases the power of the $F$ test for treatment effects.

Group units in sets as alike as possible.

(Often) compare only treatments, so interactions are not included.

5 / 37

Assignment to treatment

Divide subjects within each block

Randomly allocate to treatment within block

(stratified sampling)

6 / 37

Block-treatment design

Linear model (two-way ANOVA) without interaction,

Compromise between

reduced variability for residuals,
loss of degrees of freedom due to estimation of $\beta$ 's.

7 / 37

Example: Resting metabolic rate

From Dean, Voss and Draguljić (2017), Example 10.4.1 (p. 311)

experiment that was run to compare the effects of inpatient and outpatient protocols on the in-laboratory measurement of resting metabolic rate (RMR) in humans. A previous study had indicated measurements of RMR on elderly individuals to be 8% higher using an outpatient protocol than with an inpatient protocol. If the measurements depend on the protocol, then comparison of the results of studies conducted by different laboratories using different protocols would be difficult. The experimenters hoped to conclude that the effect on RMR of different protocols was negligible.

8 / 37

Interaction plot

9 / 37

ANOVA table (without blocking)
Analysis of variance table - without blocking
 
      
    Degrees of freedom 
    Sum of squares 
    Mean square 
    F statistic 
    p-value 
  


    protocol 
    2 
    0.04 
    0.02 
    0.02 
    0.982 
  

    Residuals 
    24 
    24.35 
    1.01 
     
     
  

10 / 37

Analysis of variance table - without blocking
	Degrees of freedom	Sum of squares	Mean square	F statistic	p-value
protocol	2	0.04	0.02	0.02	0.982
Residuals	24	24.35	1.01

ANOVA table (with blocking)
Analysis of variance table - with blocking
 
      
    Degrees of freedom 
    Sum of squares 
    Mean square 
    F statistic 
    p-value 
  


    subject 
    8 
    23.12 
    2.89 
    37.42 
    0.000 
  

    protocol 
    2 
    0.04 
    0.02 
    0.23 
    0.795 
  

    Residuals 
    16 
    1.24 
    0.08 
     
     
  

11 / 37

Analysis of variance table - with blocking
	Degrees of freedom	Sum of squares	Mean square	F statistic	p-value
subject	8	23.12	2.89	37.42	0.000
protocol	2	0.04	0.02	0.23	0.795
Residuals	16	1.24	0.08

Semipartial effect sizes

If there is a mix of experimental and blocking factors...

Include the variance of all blocking factors and interactions (only with the effect!) in denominator.

e.g., if $A$ is effect of interest, $B$ is a blocking factor and $C$ is another experimental factor, use $\eta_{\langle A \rangle}^2 = \frac{\sigma^2_A}{\sigma^2_A + \sigma^2_B + \sigma^2_{AB} + \sigma^2_{\text{resid}}}.$

In R, use effectsize::omega_squared(model, partial = TRUE, generalized = "blocking") where blocking gets replaced with a vector containing the name of the blocking factors.

12 / 37

Random effects and mixed models13 / 37

Fixed effects

All experiments so far treated factors as fixed effects.

We estimate a mean parameter for each factor (including blocking factors in repeated measures).

Change of scenery

14 / 37

Change of scenery

Assume that the levels of a factor form a random sample from a large population.

We are interested in making inference about the variability of the factor.

measures of performance of employees
results from different labs in an experiment
subjects in repeated measures

We treat these factors as random effects.

15 / 37

Fixed vs random effects

There is no consensual definition, but Gelman (2005) lists a handful, of which:

When a sample exhausts the population, the corresponding variable is fixed; when the sample is a small (i.e., negligible) part of the population the corresponding variable is random [Green and Tukey (1960)].

Effects are fixed if they are interesting in themselves or random if there is interest in the underlying population (e.g., Searle, Casella and McCulloch [(1992), Section 1.4])

16 / 37

Random effect model

Consider a one-way model

$\underset{\text{response}}{Y_{ij}} = \underset{\text{global mean}}{\mu} + \underset{\text{random effect}}{\alpha_j} + \underset{\text{error term}}{\varepsilon_{ij}}.$

where

$\alpha_j \sim \mathsf{Normal}(0, \sigma^2_\alpha)$ is normal with mean zero and variance $\sigma^2_\alpha$ .
$\varepsilon_{ij}$ are independent $\mathsf{Normal}(0, \sigma^2_\varepsilon)$

17 / 37

Fictional example

Consider the weekly number of hours spent by staff members at HEC since September.

We collect a random sample of 40 employees and ask them to measure the number of hours they work from school (as opposed to remotely) for eight consecutive weeks.

18 / 37

Fitting mixed models in R

We use the lme4 package in R to fit mixed models.

The lmerTest package provides additional functionalities for testing.

lmer function fits linear mixed effect regression

Random effects are specified using the notation (1 | factor).

19 / 37

Model fit

library(lmerTest) # also loads lme4
rmod <- lmer(time ~ (1 | id), data = hecedsm::workhours)
summary_rmod <- summary(rmod)

Random effects:
 Groups   Name        Variance Std.Dev.
 id       (Intercept) 38.63    6.215   
 Residual              5.68    2.383   
Number of obs: 320, groups:  id, 40
Fixed effects:
            Estimate Std. Error      df t value Pr(>|t|)    
(Intercept)  23.3016     0.9917 39.0000    23.5   <2e-16 ***

Note that std. dev is square root of variance

20 / 37

Intra-class correlation

We are interested in the variance of the random effect, $\sigma^2_\alpha$ .

Measurements from the same individuals are correlated. The intra-class correlation between measurements $Y_{ij}$ and $Y_{ik}$ from subject $i$ at times $j\neq k$ is

$\rho = \frac{\sigma^2_\alpha}{\sigma^2_\alpha + \sigma^2_\varepsilon}.$

In the example, $\widehat{\sigma}^2_\alpha=38.63$ , $\widehat{\sigma}^2_\varepsilon = 5.68$ and $\widehat{\rho} = 0.87$ .

The mean number of working hours on the premises is $\widehat{\mu}=23.3$ hours.

21 / 37

Confidence intervals

We can use confidence intervals for the parameters.

Those are based on profile likelihood methods (asymmetric).

(conf <- confint(rmod, oldNames = FALSE))

##                       2.5 %    97.5 %
## sd_(Intercept)|id  4.978127  7.799018
## sigma              2.198813  2.595272
## (Intercept)       21.335343 25.267782

The variability of the measurements and the week-to-week correlation of employee measures are different from zero.

22 / 37

Shrinkage

Predictions of random effects are shrunk towards global mean, more so for larger values and when there are fewer measurements.

23 / 37

Even with old school, we can get confidence intervals but only in simple designs like this one-way.

Mixed models

Mixed models include both fixed effects and random effects.

Fixed effects for experimental manipulations
Random effects for subject, lab factors

Mixed models make it easier to

handle correlations between measurements and
account for more complex designs.

24 / 37

Repeated measures ANOVA using mixed model

Data need to be in long format, i.e., one response per line with an id column.

Illustration by Garrick Adden-Buie

25 / 37

Example: two-way ANOVA

We consider a repeated measure ANOVA (2 by 2 design, within-between) from Hatano et al. (2022).

data(HOSM22_E3, package = "hecedsm")
fmod <- afex::aov_ez(
  id = "id", 
  dv = "imscore", 
  between = "waiting",
  within = "ratingtype",
  data = HOSM22_E3)
anova(fmod)

26 / 37

## Anova Table (Type 3 tests)
## 
## Response: imscore
##                    num Df den Df     MSE       F      ges    Pr(>F)    
## waiting                 1     61 2.48926 11.2551 0.126246   0.00137 ** 
## ratingtype              1     61 0.68953 38.4330 0.120236 5.388e-08 ***
## waiting:ratingtype      1     61 0.68953  0.0819 0.000291   0.77575    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

27 / 37

Repeated measures with linear mixed models

Results are the same as for repeated measures ANOVA if the correlation estimate between observations of the same id are nonnegative.

mixmod <- lmerTest::lmer(
  imscore ~ waiting*ratingtype +
    (1 | id), # random intercept per id
  data = HOSM22_E3)
anova(mixmod, type = 3)

## Type III Analysis of Variance Table with Satterthwaite's method
##                     Sum Sq Mean Sq NumDF DenDF F value    Pr(>F)    
## waiting             7.7608  7.7608     1    61 11.2551   0.00137 ** 
## ratingtype         26.5009 26.5009     1    61 38.4330 5.388e-08 ***
## waiting:ratingtype  0.0565  0.0565     1    61  0.0819   0.77575    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

28 / 37

Theory

Full coverage of linear mixed models and general designs is beyond the scope of the course, but note

Estimation is performed via restricted maximum likelihood (REML)
Testing results may differ from repeated measure ANOVA
Different approximations for F degrees of freedom, either
- Kenward–Roger (costly) or
- Satterthwaite's approximation

29 / 37

Structure of the design

It is important to understand how data were gathered.

Oelhert (2010) guidelines

Identify sources of variation
Identify whether factors are crossed or nested
Determine whether factors should be fixed or random
Figure out which interactions can exist and whether they can be fitted.

30 / 37

Crossed vs nested effects

Nested effects if a factor appears only within a particular level of another factor.

Crossed is for everything else (typically combinations of factors are possible).

Russian dolls

Example of nested random effects: class nested within schools

class 1 is not the same in school 1 than in school 2

31 / 37

Matroschka from Wikimedia Commons CC-BY-SA 3.0 https://en.wikipedia.org/wiki/Matryoshka_doll#/media/File:Matryoshka_transparent.png

Specifying interactions

Consider factors $A$ , $B$ and $C$ .

If factor $A$ is treated as random, interactions with $A$ must be random too.
There must be repeated measurements to estimate variability of those interactions.
Testing relies on the variance components.

32 / 37

Mixed models vs repeated measures

In repeated measure ANOVA, repetitions over each sub-condition are averaged across subject.
Treat subject identifier as a blocking factor (fixed effect).
Impose sum-to-zero for coefficients.

Example: happy fakes

Jittered scatterplot of measurements per participant and stimulus type.

33 / 37

Interaction with random and fixed effect

Add student id as random effect, stimulus as fixed effect and their interaction as random effect (since one parent is random)

data(AA21, package = "hecedsm")
anova(ddf = "Kenward-Roger", # other option is "Satterthwaite"
  lmerTest::lmer(
    data = AA21 |> dplyr::filter(latency > -40), 
    latency ~ stimulus + (1 | id) + (1 | id:stimulus)))

## Type III Analysis of Variance Table with Kenward-Roger's method
##          Sum Sq Mean Sq NumDF  DenDF F value Pr(>F)
## stimulus 65.573  32.786     2 21.878  0.8465 0.4425

# Approximately 22 degrees of freedom (as for repeated measures)

34 / 37

Data structure

35 / 37

Example: chocolate rating

Example from L. Meier, adapted from Oehlert (2010)

A group of 10 rural and 10 urban raters rated 4 different chocolate types. Every rater got to eat two samples from the same chocolate type in random order.

36 / 37

Example: Curley et al. (2022)

Two variables were manipulated within participants: (a) evidence anchor (strong-first versus weak-first); (b) verdict system (two- versus three-verdict systems). Total pre-trial bias score was used as a covariate in the analysis (this score is based on the PJAQ and is explained further in the Materials section). Participants were also given two vignettes (Vignette 1 and Vignette 2); thus, the vignette variable was included in the data analysis [...]

The dependent variable was the final belief of guilt score, which was measured on an accumulated scale from 0–14, with 0 representing no belief of guilt and 14 representing a total belief that the person is guilty

This study is an instance of incomplete design.

37 / 37

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Introduction to mixed models

Outline

Blocking

Terminology for nuisance

Why blocking?

Assignment to treatment

Block-treatment design

Example: Resting metabolic rate

Interaction plot

ANOVA table (without blocking)

ANOVA table (with blocking)

Semipartial effect sizes

Random effects and mixed models

Fixed effects

Change of scenery

Fixed vs random effects

Random effect model

Fictional example

Fitting mixed models in R

Model fit

Intra-class correlation

Confidence intervals

Shrinkage

Mixed models

Repeated measures ANOVA using mixed model

Example: two-way ANOVA

Repeated measures with linear mixed models

Theory

Structure of the design

Crossed vs nested effects

Specifying interactions

Example: happy fakes

Interaction with random and fixed effect

Data structure

Example: chocolate rating

Example: Curley et al. (2022)

Outline

Help

Introduction to mixed models

Introduction to mixed models

Outline

Blocking

Terminology for nuisance

Why blocking?

Assignment to treatment

Block-treatment design

Example: Resting metabolic rate

Interaction plot

ANOVA table (without blocking)

ANOVA table (with blocking)

Semipartial effect sizes

Random effects and mixed models

Fixed effects

Change of scenery

Fixed vs random effects

Random effect model

Fictional example

Fitting mixed models in R

Model fit

Intra-class correlation

Confidence intervals

Shrinkage

Mixed models

Repeated measures ANOVA using mixed model

Example: two-way ANOVA

Repeated measures with linear mixed models

Theory

Structure of the design

Crossed vs nested effects

Specifying interactions

Example: happy fakes

Interaction with random and fixed effect

Data structure

Example: chocolate rating

Example: Curley et al. (2022)