afex: Analysis of Factorial EXperiments

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The main functionalities provided by afex are:

  1. Interfaces for estimating standard ANOVAs with any number or combination of within-subjects or between-subjects variables (the ANOVA functions are aov_car(), aov_ez(), and aov_4() which all fit the same model but differ in the way to specify the ANOVA model).
  2. Function mixed() provides an interface for mixed models analysis (estimated via lme4 lmer or glmer) that automatically obtains p-values for fixed effects model terms (i.e., main effects and interactions).
  3. afex_plot() visualizes results from factorial experiments combining estimated marginal means and uncertainties associated with the estimated means in the foreground with a depiction of the raw data in the background.
  4. All afex model objects (i.e., ANOVA and mixed models) can be passed to emmeans for follow-up/post-hoc/planned contrast analysis.

For afex support visit: afex.singmann.science

Installation

ANOVA functionality

To calculate an ANOVA, afex requires the data to be in the long format (i.e., one row per data point/observation). An ANOVA can then be calculated via one of three functions that only differ in how the model components are specified, but not in the output. Note that in contrast to base lm or aov, afex ANOVA functions always require the specification of a subject identifier column (the id-column), because in case there are multiple observations per participant and cell of the design, these multiple observations are aggregated (i.e., averaged) per default.

A further overview is provided by the vignette.

The following code provides a simple example for an ANOVA with both between- and within-subject factors. For this we use the lexical-decision and word naming latencies reported by Freeman, Heathcote, Chalmers, and Hockley (2010), see also ?fhch2010. As is commonly done, we use the natural logarithm of the response times, log_rt, as dependent variable. As independent variable we will consider the between-subjects factor task ("naming" or "lexdec") as well as the within-subjects-factors stimulus ("word" or "nonword") and length (with 3 levels, 3, 4, or 5 letters).

library("afex")
# examples data set with both within- and between-subjects factors (see ?fhch2010)
data("fhch2010", package = "afex")
fhch <- fhch2010[ fhch2010$correct,] # remove errors
str(fhch2010) # structure of the data
#> 'data.frame':    13222 obs. of  10 variables:
#>  $ id       : Factor w/ 45 levels "N1","N12","N13",..: 1 1 1 1 1 1 1 1 1 1 ...
#>  $ task     : Factor w/ 2 levels "naming","lexdec": 1 1 1 1 1 1 1 1 1 1 ...
#>  $ stimulus : Factor w/ 2 levels "word","nonword": 1 1 1 2 2 1 2 2 1 2 ...
#>  $ density  : Factor w/ 2 levels "low","high": 2 1 1 2 1 2 1 1 1 1 ...
#>  $ frequency: Factor w/ 2 levels "low","high": 1 2 2 2 2 2 1 2 1 2 ...
#>  $ length   : Factor w/ 3 levels "4","5","6": 3 3 2 2 1 1 3 2 1 3 ...
#>  $ item     : Factor w/ 600 levels "abide","acts",..: 363 121 202 525 580 135 42 368 227 141 ...
#>  $ rt       : num  1.091 0.876 0.71 1.21 0.843 ...
#>  $ log_rt   : num  0.0871 -0.1324 -0.3425 0.1906 -0.1708 ...
#>  $ correct  : logi  TRUE TRUE TRUE TRUE TRUE TRUE ...
# estimate mixed ANOVA on the full design:
aov_ez("id", "log_rt", fhch, between = "task", within = c("stimulus", "length"))
aov_car(log_rt ~ task * stimulus * length + Error(id/(stimulus * length)), 
        data = fhch)
## equivalent: aov_car(log_rt ~ task + Error(id/(stimulus * length)), data = fhch)

aov_4(log_rt ~ task * stimulus * length + (stimulus * length|id), data = fhch)
## equivalent: aov_4(log_rt ~ task  + (stimulus * length|id), data = fhch)

# the three calls return the same ANOVA table:
#> Warning: More than one observation per cell, aggregating the data using mean
#> (i.e, fun_aggregate = mean)!
#> Contrasts set to contr.sum for the following variables: task
#> Anova Table (Type 3 tests)
#> 
#> Response: log_rt
#>                 Effect          df  MSE          F   ges p.value
#> 1                 task       1, 43 0.23  13.38 ***  .221   <.001
#> 2             stimulus       1, 43 0.01 173.25 ***  .173   <.001
#> 3        task:stimulus       1, 43 0.01  87.56 ***  .096   <.001
#> 4               length 1.83, 78.64 0.00  18.55 ***  .008   <.001
#> 5          task:length 1.83, 78.64 0.00       1.02 <.001    .358
#> 6      stimulus:length 1.70, 72.97 0.00       1.91 <.001    .162
#> 7 task:stimulus:length 1.70, 72.97 0.00       1.21 <.001    .298
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1
#> 
#> Sphericity correction method: GG

Plotting with afex_plot

ANOVA models can be used for plotting via afex_plot:

a <- aov_ez("id", "log_rt", fhch, between = "task", within = c("stimulus", "length"))
#> Warning: More than one observation per cell, aggregating the data using mean
#> (i.e, fun_aggregate = mean)!
afex_plot(a, "task", "stimulus", "length")
#> Warning: Panel(s) show a mixed within-between-design.
#> Error bars do not allow comparisons across all means.
#> Suppress error bars with: error = "none"

afex_plot returns a ggplot2 plot object which allows simple customization:

library("ggplot2")
afex_plot(a, "task", "stimulus", "length") + 
  theme_bw()
#> Warning: Panel(s) show a mixed within-between-design.
#> Error bars do not allow comparisons across all means.
#> Suppress error bars with: error = "none"

Follow-up Tests with emmeans

Follow-up tests with emmeans need to be specified in two steps.

  1. Decide which factors of model should be involved in tests. Use these factors to set-up reference grid of marginal means using emmeans().
  2. Specify set of tests on reference grid from step 1. Either custom contrasts as a list and using contrast() or a convenience function such as pairs().
library("emmeans")
## set up reference grid using only length
em1 <- emmeans(a, "length")
em1
#>  length  emmean     SE df lower.CL upper.CL
#>  X4     -0.1087 0.0299 43   -0.169 -0.04834
#>  X5     -0.0929 0.0296 43   -0.153 -0.03310
#>  X6     -0.0653 0.0290 43   -0.124 -0.00679
#> 
#> Results are averaged over the levels of: task, stimulus 
#> Confidence level used: 0.95

## test all pairwise comparisons on reference grid:
pairs(em1)
#>  contrast estimate      SE df t.ratio p.value
#>  X4 - X5   -0.0159 0.00768 43 -2.065  0.1092 
#>  X4 - X6   -0.0434 0.00782 43 -5.555  <.0001 
#>  X5 - X6   -0.0276 0.00602 43 -4.583  0.0001 
#> 
#> Results are averaged over the levels of: task, stimulus 
#> P value adjustment: tukey method for comparing a family of 3 estimates

## only test specified tests
con <- list(
  "4vs5" = c(-1, 1, 0),
  "5vs6" = c(0, -1, 1)
)
contrast(em1, con, adjust = "holm")
#>  contrast estimate      SE df t.ratio p.value
#>  4vs5       0.0159 0.00768 43 2.065   0.0449 
#>  5vs6       0.0276 0.00602 43 4.583   0.0001 
#> 
#> Results are averaged over the levels of: task, stimulus 
#> P value adjustment: holm method for 2 tests

Mixed Models

Function mixed() fits a mixed model with lme4::lmer (or lme4::glmer if a family argument is passed) and then calculates p-values for fixed effects model terms using a variety of methods. The formula to mixed needs to be the same as in a call to lme4::lmer. The default method for calculation of p-values is 'S' (Satterthwaite) which only works for linear mixed models (i.e., no family argument). A similar method that provides a somewhat better control of Type I errors for small data sets is 'KR' (Kenward-Roger), but it can require considerable RAM and time. Other methods are , similar to 'KR' but requires less RAM), 'PB' (parametric bootstrap), and 'LRT' (likelihood-ratio test).

More examples are provided in the vignette, here we use the same example data as above, the lexical decision and word naming latencies collected by Freeman et al. (2010). To avoid long computation times we only consider the two factors task and length (omitting stimulus is probably not a fully sensible model). Because mixed models easily allow it, we will consider crossed-random effects for participants (id) and items (tem).

library("afex")
# examples data set with both within- and between-subjects factors (see ?fhch2010)
data("fhch2010", package = "afex")
fhch <- fhch2010[ fhch2010$correct,] # remove errors
str(fhch2010) # structure of the data
#> 'data.frame':    13222 obs. of  10 variables:
#>  $ id       : Factor w/ 45 levels "N1","N12","N13",..: 1 1 1 1 1 1 1 1 1 1 ...
#>  $ task     : Factor w/ 2 levels "naming","lexdec": 1 1 1 1 1 1 1 1 1 1 ...
#>  $ stimulus : Factor w/ 2 levels "word","nonword": 1 1 1 2 2 1 2 2 1 2 ...
#>  $ density  : Factor w/ 2 levels "low","high": 2 1 1 2 1 2 1 1 1 1 ...
#>  $ frequency: Factor w/ 2 levels "low","high": 1 2 2 2 2 2 1 2 1 2 ...
#>  $ length   : Factor w/ 3 levels "4","5","6": 3 3 2 2 1 1 3 2 1 3 ...
#>  $ item     : Factor w/ 600 levels "abide","acts",..: 363 121 202 525 580 135 42 368 227 141 ...
#>  $ rt       : num  1.091 0.876 0.71 1.21 0.843 ...
#>  $ log_rt   : num  0.0871 -0.1324 -0.3425 0.1906 -0.1708 ...
#>  $ correct  : logi  TRUE TRUE TRUE TRUE TRUE TRUE ...

For the random-effects grouping factors we begin with the maximal random effect structure justified by the design (see Barr, Levy, Scheepers, & Tily, 2013). In this case this is by-subject random intercepts and by-subjects random slopes for stimulus and by-item random intercepts and by-item random slopes for task.

m1 <- mixed(log_rt ~ task * length + (length | id) + (task | item), 
            fhch)
#> Contrasts set to contr.sum for the following variables: task, length, id, item
#> Fitting one lmer() model.
#> boundary (singular) fit: see ?isSingular
#> [DONE]
#> Calculating p-values. [DONE]

Fitting this model produces a critical convergence warning, that the fit is singular. This warning usually indicates that the data does not provide enough information for the request random effect parameters. In a real analysis it would therefore be a good idea to iteratively reduce the random effect structure until the warning disappears. A good first step would be to remove the correlations among random effect terms as shown below.

This warning is also shown if we simply print the model object, but not if we call the nice() method.

m1
#> Warning: lme4 reported (at least) the following warnings for 'full':
#>   * boundary (singular) fit: see ?isSingular
#> Mixed Model Anova Table (Type 3 tests, S-method)
#> 
#> Model: log_rt ~ task * length + (length | id) + (task | item)
#> Data: fhch
#>        Effect        df         F p.value
#> 1        task  1, 44.79 13.47 ***   <.001
#> 2      length 2, 325.75   6.03 **    .003
#> 3 task:length 2, 303.20      0.33    .722
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1
nice(m1)
#> Mixed Model Anova Table (Type 3 tests, S-method)
#> 
#> Model: log_rt ~ task * length + (length | id) + (task | item)
#> Data: fhch
#>        Effect        df         F p.value
#> 1        task  1, 44.79 13.47 ***   <.001
#> 2      length 2, 325.75   6.03 **    .003
#> 3 task:length 2, 303.20      0.33    .722
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1

If we call the anova() method a slightly different output is shown in which the p-values are not rounded in the same way and the warning is shown again.

anova(m1)
#> Warning: lme4 reported (at least) the following warnings for 'full':
#>   * boundary (singular) fit: see ?isSingular
#> Mixed Model Anova Table (Type 3 tests, S-method)
#> 
#> Model: log_rt ~ task * length + (length | id) + (task | item)
#> Data: fhch
#>             num Df  den Df       F    Pr(>F)    
#> task             1  44.789 13.4676 0.0006431 ***
#> length           2 325.751  6.0252 0.0026948 ** 
#> task:length      2 303.203  0.3263 0.7218281    
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

We can also get the default lme4 output if we call the summary method. However, note that in contrast to the previous methods, results are shown for factor-levels and not model-terms which is usually not interpretable for factors with more than two levels. This is the case for length here. The problem is that factors with k levels are mapped to k − 1 parameters and at the same time the intercept represent the (unweighted) grand mean. This means that factor-levels cannot be mapped in a 1-to-1 manner to the parameters and thus cannot be uniquely interpreted.

summary(m1)
#> Linear mixed model fit by REML. t-tests use Satterthwaite's method [
#> lmerModLmerTest]
#> Formula: log_rt ~ task * length + (length | id) + (task | item)
#>    Data: data
#> 
#> REML criterion at convergence: 7624.2
#> 
#> Scaled residuals: 
#>     Min      1Q  Median      3Q     Max 
#> -5.9267 -0.5900 -0.1018  0.4789  5.2673 
#> 
#> Random effects:
#>  Groups   Name        Variance  Std.Dev. Corr       
#>  item     (Intercept) 0.0115697 0.10756             
#>           task1       0.0104578 0.10226  0.47       
#>  id       (Intercept) 0.0374095 0.19342             
#>           length1     0.0003298 0.01816   0.16      
#>           length2     0.0001008 0.01004   0.11 -0.96
#>  Residual             0.0925503 0.30422             
#> Number of obs: 12960, groups:  item, 600; id, 45
#> 
#> Fixed effects:
#>                 Estimate Std. Error         df t value Pr(>|t|)    
#> (Intercept)    -0.089098   0.029470  44.980778  -3.023 0.004119 ** 
#> task1          -0.108035   0.029439  44.788920  -3.670 0.000643 ***
#> length1        -0.020756   0.007810 226.871782  -2.658 0.008425 ** 
#> length2        -0.003746   0.007466 380.252651  -0.502 0.616203    
#> task1:length1   0.005719   0.007568 206.598906   0.756 0.450727    
#> task1:length2  -0.004627   0.007214 353.248363  -0.641 0.521640    
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Correlation of Fixed Effects:
#>             (Intr) task1  lngth1 lngth2 tsk1:1
#> task1        0.118                            
#> length1      0.056  0.007                     
#> length2      0.021  0.002 -0.526              
#> tsk1:lngth1  0.007  0.058  0.329 -0.173       
#> tsk1:lngth2  0.003  0.022 -0.174  0.349 -0.528
#> optimizer (nloptwrap) convergence code: 0 (OK)
#> boundary (singular) fit: see ?isSingular

Reducing the Random Effect Structure

Because of the singular fit warning, we reduce the random effect structure. Usually a good starting point is removing the correlations among the random effects parameters. This can be done in afex::mixed even for factors by combining the double bar notation || with expand_re = TRUE. We do so for both random effects terms.

m2 <- mixed(log_rt ~ task * length + (length || id) + (task || item), 
            fhch, expand_re = TRUE)
#> Contrasts set to contr.sum for the following variables: task, length, id, item
#> Fitting one lmer() model.
#> boundary (singular) fit: see ?isSingular
#> [DONE]
#> Calculating p-values. [DONE]

However, the singular fit warning remains. We therefore inspect the random effect estimates to see which random effect parameter is estimated to be near to zero.

summary(m2)$varcor
#>  Groups   Name        Std.Dev.
#>  item     re2.task1   0.101197
#>  item.1   (Intercept) 0.106844
#>  id       re1.length2 0.000000
#>  id.1     re1.length1 0.012294
#>  id.2     (Intercept) 0.193419
#>  Residual             0.304373

As shown above, one parameter of the by-participant random slope for length is estimated to be almost zero, re1.length2. We therefore remove the by-participant random slope for length in the next model which does not show any convergence warnings.

m3 <- mixed(log_rt ~ task * length + (1 | id) + (task || item), 
            fhch, expand_re = TRUE)
#> Contrasts set to contr.sum for the following variables: task, length, id, item
#> Fitting one lmer() model. [DONE]
#> Calculating p-values. [DONE]
m3
#> Mixed Model Anova Table (Type 3 tests, S-method)
#> 
#> Model: log_rt ~ task * length + (1 | id) + (task || item)
#> Data: fhch
#>        Effect        df         F p.value
#> 1        task  1, 44.74 13.52 ***   <.001
#> 2      length 2, 597.20   6.67 **    .001
#> 3 task:length 2, 592.82      0.40    .668
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1

Plotting with afex_plot

Objects returned by mixed can be used for plotting with afex_plot. However, two things need to be considered.

library("ggplot2")
## all data points shown
afex_plot(m3, "task", "length") + 
  theme_bw()

## data points show IDs
afex_plot(m3, "task", "length", id = "id") + 
  theme_bw()

## data points show items
afex_plot(m3, "task", "length", id = "item") + 
  theme_bw()

Follow-up Tests with emmeans

Follow-up tests with emmeans need to be specified in two steps.

  1. Decide which factors of model should be involved in tests. Use these factors to set-up reference grid of marginal means using emmeans().
  2. Specify set of tests on reference grid from step 1. Either custom contrasts as a list and using contrast() or a convenience function such as pairs().

For mixed models, emmeans attempts to estimate the degrees of freedom. The method can be set via emm_options(lmer.df = ...). Here we use "asymptotic" which does not estimate the degrees of freedom, but sets them to infinity.

library("emmeans")
emm_options(lmer.df = "asymptotic")
## set up reference grid using only length
em2 <- emmeans(m3, "length")
#> NOTE: Results may be misleading due to involvement in interactions
em2
#>  length  emmean     SE  df asymp.LCL asymp.UCL
#>  4      -0.1099 0.0304 Inf    -0.169  -0.05040
#>  5      -0.0924 0.0304 Inf    -0.152  -0.03296
#>  6      -0.0642 0.0304 Inf    -0.124  -0.00469
#> 
#> Results are averaged over the levels of: task 
#> Degrees-of-freedom method: asymptotic 
#> Confidence level used: 0.95

## test all pairwise comparisons on reference grid:
pairs(em2)
#>  contrast estimate     SE  df z.ratio p.value
#>  4 - 5     -0.0175 0.0126 Inf -1.384  0.3495 
#>  4 - 6     -0.0457 0.0126 Inf -3.618  0.0009 
#>  5 - 6     -0.0282 0.0126 Inf -2.238  0.0649 
#> 
#> Results are averaged over the levels of: task 
#> Degrees-of-freedom method: asymptotic 
#> P value adjustment: tukey method for comparing a family of 3 estimates

## only test specified tests
con <- list(
  "4vs5" = c(-1, 1, 0),
  "5vs6" = c(0, -1, 1)
)
contrast(em2, con, adjust = "holm")
#>  contrast estimate     SE  df z.ratio p.value
#>  4vs5       0.0175 0.0126 Inf 1.384   0.1665 
#>  5vs6       0.0282 0.0126 Inf 2.238   0.0504 
#> 
#> Results are averaged over the levels of: task 
#> Degrees-of-freedom method: asymptotic 
#> P value adjustment: holm method for 2 tests

References

Barr, D. J., Levy, R., Scheepers, C., & Tily, H. J. (2013). Random effects structure for confirmatory hypothesis testing: Keep it maximal. Journal of Memory and Language, 68(3), 255-278. https://doi.org/10.1016/j.jml.2012.11.001

Freeman, E., Heathcote, A., Chalmers, K., & Hockley, W. (2010). Item effects in recognition memory for words. Journal of Memory and Language, 62(1), 1-18. https://doi.org/10.1016/j.jml.2009.09.004

Code of Conduct

Please note that afex is released with a Contributor Code of Conduct. By contributing to this project, you agree to abide by its terms.