Two-way Repeated Measures ANOVA

A repeated measures ANOVA can also be analyzed under a mixed effects framework using the lme4 package. Mixed effects models contain a mixture of fixed and random effects and are also sometimes referred to as multi-level models or hierarchical linear models. Fixed effects can be thought of as factors in which all possible levels that a researcher is interested in are represented in the data. A random effect on the other hand is a factor for which the levels in the experimental data represent a sample of a larger set. Mixed effects models may be advantageous because they do not require the assumption of independence in ANOVA, the assumption of homogeneity of regression slopes in ANCOVA, and may be better suited for handling unbalanced designs or situations with missing data. While mixed effect models are estimated using slightly different procedures than more traditional models, there are cases in which the results will overlap. The balanced repeated measures ANOVA with two within-subjects factors is one of those cases.

The lmer() function from the lme4 package is designed to fit linear mixed-effects regression models via REML or maximum likelihood. Just like the base R lm() function, lmer() takes a formula specifying the dependent variable predicted by (~) a combination of fixed- and random- effect variables. Predictor variables encased in parentheses specify random effects, while variables without parentheses are specified as fixed effects.

The first part of the formula in the example below specifies to predict Latency from the interaction between Condition and Angle as fixed effects. Including just the interaction term is a shortcut for “Latency ~ Condition + Angle + Condition * Angle.” The term “(1|id)” specifies a random intercept for each subject. This allows each subject to have their own mean (starting point) modeled. The remaining terms with the colons indicate random intercepts for the interactions between subject and Condition (1|Condition:id) and subject and Angle (1|Angle:id). The following results will not only mirror those displayed under the rstatix approach, but will also mirror the results generated by SAS.

library(lme4)
library(lmerTest)
library(broom.mixed)

rm.mod <- lmerTest::lmer(Latency ~ Condition + Angle + Condition:Angle + 
                 (1|id) +
                 (1|Condition:id) + 
                 (1|Angle:id), 
               data=data)    

anova(rm.mod)

## Type III Analysis of Variance Table with Satterthwaite's method
##                 Sum Sq Mean Sq NumDF DenDF F value    Pr(>F)    
## Condition        39169   39169     1     9  33.766  0.000256 ***
## Angle            94468   47234     2    18  40.719 2.087e-07 ***
## Condition:Angle 105120   52560     2    18  45.310 9.424e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

summary(rm.mod)

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: Latency ~ Condition + Angle + Condition:Angle + (1 | id) + (1 |  
##     Condition:id) + (1 | Angle:id)
##    Data: data
## 
## REML criterion at convergence: 616.1
## 
## Scaled residuals: 
##      Min       1Q   Median       3Q      Max 
## -1.49491 -0.46023  0.02919  0.50472  1.41413 
## 
## Random effects:
##  Groups       Name        Variance Std.Dev.
##  Angle:id     (Intercept) 1200     34.64   
##  Condition:id (Intercept) 2433     49.33   
##  id           (Intercept) 3600     60.00   
##  Residual                 1160     34.06   
## Number of obs: 60, groups:  Angle:id, 30; Condition:id, 20; id, 10
## 
## Fixed effects:
##                         Estimate Std. Error     df t value Pr(>|t|)    
## (Intercept)               462.00      28.97  19.79  15.947 9.28e-13 ***
## ConditionPresent           30.00      26.81  14.08   1.119  0.28183    
## Angle4                     48.00      21.73  28.60   2.209  0.03532 *  
## Angle8                     66.00      21.73  28.60   3.038  0.00505 ** 
## ConditionPresent:Angle4   120.00      21.54  18.00   5.571 2.75e-05 ***
## ConditionPresent:Angle8   204.00      21.54  18.00   9.470 2.05e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##             (Intr) CndtnP Angle4 Angle8 CnP:A4
## CondtnPrsnt -0.463                            
## Angle4      -0.375  0.199                     
## Angle8      -0.375  0.199  0.500              
## CndtnPrs:A4  0.186 -0.402 -0.496 -0.248       
## CndtnPrs:A8  0.186 -0.402 -0.248 -0.496  0.500

tidy(rm.mod)

## # A tibble: 10 × 8
##    effect   group        term            estim…¹ std.e…² stati…³    df   p.value
##    <chr>    <chr>        <chr>             <dbl>   <dbl>   <dbl> <dbl>     <dbl>
##  1 fixed    <NA>         (Intercept)       462.     29.0   15.9   19.8  9.28e-13
##  2 fixed    <NA>         ConditionPrese…    30.0    26.8    1.12  14.1  2.82e- 1
##  3 fixed    <NA>         Angle4             48.0    21.7    2.21  28.6  3.53e- 2
##  4 fixed    <NA>         Angle8             66.0    21.7    3.04  28.6  5.05e- 3
##  5 fixed    <NA>         ConditionPrese…   120.     21.5    5.57  18.0  2.75e- 5
##  6 fixed    <NA>         ConditionPrese…   204.     21.5    9.47  18.0  2.05e- 8
##  7 ran_pars Angle:id     sd__(Intercept)    34.6    NA     NA     NA   NA       
##  8 ran_pars Condition:id sd__(Intercept)    49.3    NA     NA     NA   NA       
##  9 ran_pars id           sd__(Intercept)    60.0    NA     NA     NA   NA       
## 10 ran_pars Residual     sd__Observation    34.1    NA     NA     NA   NA       
## # … with abbreviated variable names ¹​estimate, ²​std.error, ³​statistic

For these data, the lmer() function produces the exact same results as the anova_test() function. The