This documents reanalysis response time data from an Experiment performed by Freeman, Heathcote, Chalmers, and Hockley (2010) using the mixed model functionality of afex implemented in function
mixed followed by post-hoc tests using package emmeans (Lenth, 2017). After a brief description of the data set and research question, the code and results are presented.
The data are lexical decision and word naming latencies for 300 words and 300 nonwords from 45 participants presented in Freeman et al. (2010). The 300 items in each
stimulus condition were selected to form a balanced \(2 \times 2\) design with factors neighborhood
density (low versus high) and
frequency (low versus high). The
task was a between subjects factor: 25 participants worked on the lexical decision task and 20 participants on the naming task. After excluding erroneous responses each participants responded to between 135 and 150 words and between 124 and 150 nonwords. We analyzed log RTs which showed an approximately normal picture.
We start with loading some packages we will need throughout this example. For data manipulation we will be using the
tidyr packages from the
tidyverse. A thorough introduction to these packages is beyond this example, but well worth it, and can be found in ‘R for Data Science’ by Wickham and Grolemund. For plotting we will be using
ggplot2, also part of the
After loading the packages, we will load the data (which comes with
afex), remove the errors, and take a look at the variables in the data.
library("afex") # needed for mixed() and attaches lme4 automatically. library("emmeans") # emmeans is needed for follow-up tests library("multcomp") # for advanced control for multiple testing/Type 1 errors. library("dplyr") # for working with data frames library("tidyr") # for transforming data frames from wide to long and the other way round. library("ggplot2") # for plots theme_set(theme_bw(base_size = 15) + theme(legend.position="bottom", panel.grid.major.x = element_blank())) data("fhch2010") # load fhch <- droplevels(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 ...
To make sure our expectations about the data match the data we use some
dplyr magic to confirm the number of participants per condition and items per participant.
##  TRUE
## # A tibble: 2 x 2 ## task n ## <fct> <int> ## 1 naming 20 ## 2 lexdec 25
## # A tibble: 2 x 4 ## stimulus min max mean ## <fct> <int> <int> <dbl> ## 1 word 135 150 145. ## 2 nonword 124 150 143.
Before running the analysis we should make sure that our dependent variable looks roughly normal. To compare
log_rt within the same figure we first need to transform the data from the wide format (where both rt types occupy one column each) into the long format (in which the two rt types are combined into a single column with an additional indicator column). To do so we use
tidyr::pivot_longer. Then we simply call
facet_wrap(vars(rt_type)) on the new
tibble. The plot shows that
log_rt looks clearly more normal than
rt, although not perfectly so. An interesting exercise could be to rerun the analysis below using a transformation that provides an even better ‘normalization’.
The main factors in the experiment were the between-subjects factor
lexdec), and the within-subjects factors
high). Before running an analysis it is a good idea to visually inspect the data to gather some expectations regarding the results. Should the statistical results dramatically disagree with the expectations this suggests some type of error along the way (e.g., model misspecification) or at least encourages a thorough check to make sure everything is correct. We first begin by plotting the data aggregated by-participant.
In each plot we plot the raw data in the background. To make the individual data points visible we use
alpha = 0.5 for semi-transparency. On top of this we add a (transparent) box plot as well as the mean and standard error.
agg_p <- fhch %>% group_by(id, task, stimulus, density, frequency) %>% summarise(mean = mean(log_rt)) %>% ungroup() ggplot(agg_p, aes(x = interaction(density,frequency), y = mean)) + ggbeeswarm::geom_quasirandom(alpha = 0.5) + geom_boxplot(fill = "transparent") + stat_summary(colour = "red") + facet_grid(cols = vars(task), rows = vars(stimulus))
Now we plot the same data but aggregated across items:
agg_i <- fhch %>% group_by(item, task, stimulus, density, frequency) %>% summarise(mean = mean(log_rt)) %>% ungroup() ggplot(agg_i, aes(x = interaction(density,frequency), y = mean)) + ggbeeswarm::geom_quasirandom(alpha = 0.3) + geom_boxplot(fill = "transparent") + stat_summary(colour = "red") + facet_grid(cols = vars(task), rows = vars(stimulus))
These two plots show a very similar pattern and suggest several things:
nonwordsappear slower than responses to
words, at least for the
lexdecresponses appear to be slower than
namingresponses, particularly in the
namingcondition we see a clear effect of
frequencywith slower responses to
frequencypattern appears to be in the opposite direction to the pattern described in the previous point: faster responses to
densityappears to have no effect, perhaps with the exception of the
To set up a mixed model it is important to identify which factors vary within which grouping factor generating random variability (i.e., grouping factors are sources of stochastic variability). The two grouping factors are participants (
id) and items (
item). The within-participant factors are
frequency. The within-item factor is
task. The ‘maximal model’ (Barr, Levy, Scheepers, and Tily, 2013) therefore is the model with by-participant random slopes for
frequency and their interactions and by-item random slopes for
It is rather common that a maximal model with a complicated random effect structure, such as the present one, does not converge successfully. The best indicator of this is a “singular fit” warning. A model with a singular fit warning should not be reported or used. Instead, one should make sure that qualitatively the same results are also observed with a model without singular fit warnings. If the maximal model that does not converge and a reduced model without a singular fit warning (i.e., the final model) diverge in their results, results should only be interpreted cautiously.
In case of a singular fit or another indicator of a convergence problem, the usual first step is removing the correlations among the random terms. In our example, there are two sets of correlations, one for each random effect grouping variable. Consequently, we can build four model that have the maximal structure in terms of random-slopes and only differ in which correlations among random terms are calculated:
The next decision to be made is which method to use for obtaining \(p\)-values. The default method is
KR (=Kenward-Roger) which provides the best control against anti-conservative results. However,
KR needs quite a lot of RAM, especially with complicated random effect structures and large data sets. As in this case we have both, relatively large data (i.e., many levels on each random effect, especially the item random effect) and a complicated random effect structure, it seems a reasonable decision to choose another method. The second ‘best’ method (in terms of controlling for Type I errors) is the ‘Satterthwaite’ approximation,
method='S'. It provides a similar control of Type I errors as the Kenward-Roger approximation and needs less RAM, however one downside is that it simply fails in some cases.
The following code fits the four models using the Satterthwaite method. To suppress random effects we use the
|| notation. Note that it is necessary to set
expand_re=TRUE when suppressing random effects among variables that are entered as factors and not as numerical variables (all independent variables in the present case are factors). Also note that
mixed automatically uses appropriate contrast coding if factors are included in interactions (
contr.sum) in contrast to the
R default (which is
contr.treatment). To make sure the estimation does not end prematurely we set the allowed number of function evaluations to a very high value (using
However, because fitting the models in R might take quite a while, you should also be able to load the fitted binary files them from this url and then load them in R with
## boundary (singular) fit: see ?isSingular
## boundary (singular) fit: see ?isSingular
## boundary (singular) fit: see ?isSingular
## boundary (singular) fit: see ?isSingular
We can see that fitting each of these models emits the “singular fit” message (it is technically a
message and not a
warning) indicating that the model is over-parameterized and did not converge successfully. In particular, this message indicates that the model is specified with more random effect parameters than can be estimated given the current data.
It is instructive to see that even with a comparatively large number of observations, 12960, a set of seven random slopes for the by-participant term and one random slope for the by-item term cannot be estimated successfully. And this holds even after removing all correlations. Thus, it should not be surprising that similar sized models regularly do not converge with smaller numbers of observations. Furthermore, we are here in the fortunate situation that each factor has only two levels. A factor with more levels corresponds to more parameters of the random effect terms.
Before deciding what to do next, we take a look at the estimated random effect estimates. We do so for the model without any correlations. Note that for
afex models we usually do not want to use the
summary method as it prints the results on the level of the model coefficients and not model terms. But for the random effects we have to do so. However, we are only interested in the random effect terms so we only print those using
## Groups Name Std.Dev. ## item re2.task1 0.0568715 ## item.1 (Intercept) 0.0537587 ## id re1.stimulus1_by_density1_by_frequency1 0.0077923 ## id.1 re1.density1_by_frequency1 0.0000000 ## id.2 re1.stimulus1_by_frequency1 0.0000000 ## id.3 re1.stimulus1_by_density1 0.0000000 ## id.4 re1.frequency1 0.0179993 ## id.5 re1.density1 0.0000000 ## id.6 re1.stimulus1 0.0445333 ## id.7 (Intercept) 0.1936292 ## Residual 0.3001905
The output shows that the estimated SDs of the random slopes of the two-way interactions are all zero. However, because we cannot remove the random slopes for the two way interaction while retaining the three-way interaction, we start by removing the three-way interaction first.
## boundary (singular) fit: see ?isSingular
## Groups Name Std.Dev. ## item re2.task1 5.6826e-02 ## item.1 (Intercept) 5.3736e-02 ## id re1.density1_by_frequency1 0.0000e+00 ## id.1 re1.stimulus1_by_frequency1 0.0000e+00 ## id.2 re1.stimulus1_by_density1 0.0000e+00 ## id.3 re1.frequency1 1.7966e-02 ## id.4 re1.density1 1.4422e-05 ## id.5 re1.stimulus1 4.4539e-02 ## id.6 (Intercept) 1.9374e-01 ## Residual 3.0030e-01
Not too surprisingly, this model also produces a singular fit. Inspection of the estimates shows that the two-way interaction of the slopes are still estimated to be zero. So we remove those in the next step.
## boundary (singular) fit: see ?isSingular
This model still shows a singular fit warning. The random effect estimates below show a potential culprit.
## Groups Name Std.Dev. ## item re2.task1 0.056831 ## item.1 (Intercept) 0.053737 ## id re1.frequency1 0.017972 ## id.1 re1.density1 0.000000 ## id.2 re1.stimulus1 0.044524 ## id.3 (Intercept) 0.193659 ## Residual 0.300297
m4s above, the random effect SD for the density term is estimated to be zero. Thus, we remove this as well in the next step.
This model finally does not emit a singular fit warning. Is this our final model? Before deciding on this, we see whether we can add the correlation terms again without running into any problems. We begin by adding the correlation to the by-participant term.
## Warning: Model failed to converge with max|grad| = 0.00347544 (tol = 0.002, component 1)
This model does not show a singular fit message but emits another warning. Specifically, a warning that the absolute maximal gradient at the final solution is too high. This warning is not necessarily critical (i.e., it can be a false positive), but can also indicate serious problems. Consequently, we try adding the correlation between the by-item random terms instead:
This model also does not show any warnings. Thus, we have arrived at the end of the model selection process.
We now have the following two relevant models.
m1s: The maximal random effect structure justified by the design (i.e., the maximal model)
m9s: The final model
Robust results are those that hold regardless across maximal and final (i.e., reduced) model. Therefore, let us compare the pattern of significant and non-significant effects.
## Mixed Model Anova Table (Type 3 tests, S-method) ## ## Model: log_rt ~ task * stimulus * density * frequency + (stimulus * ## Model: density * frequency | id) + (task | item) ## Data: fhch ## Effect df_full F_full p.value_full df_final F_final p.value_final ## 1 task 1, 43.58 13.71 *** <.001 1, 43.51 13.68 *** <.001 ## 2 stimulus 1, 50.50 151.03 *** <.001 1, 50.57 151.38 *** <.001 ## 3 density 1, 175.53 0.33 .569 1, 584.58 0.36 .547 ## 4 frequency 1, 71.17 0.55 .459 1, 70.27 0.56 .456 ## 5 task:stimulus 1, 51.45 70.91 *** <.001 1, 51.50 71.32 *** <.001 ## 6 task:density 1, 184.47 16.22 *** <.001 1, 578.72 17.89 *** <.001 ## 7 stimulus:density 1, 247.85 1.13 .289 1, 584.60 1.19 .275 ## 8 task:frequency 1, 75.03 81.10 *** <.001 1, 74.09 82.77 *** <.001 ## 9 stimulus:frequency 1, 165.95 55.85 *** <.001 1, 584.78 63.29 *** <.001 ## 10 density:frequency 1, 215.83 0.11 .739 1, 584.62 0.11 .742 ## 11 task:stimulus:density 1, 259.93 14.52 *** <.001 1, 578.74 14.87 *** <.001 ## 12 task:stimulus:frequency 1, 178.33 110.95 *** <.001 1, 578.91 124.16 *** <.001 ## 13 task:density:frequency 1, 228.12 5.53 * .020 1, 578.75 5.93 * .015 ## 14 stimulus:density:frequency 1, 101.11 3.98 * .049 1, 584.64 4.62 * .032 ## 15 task:stimulus:density:frequency 1, 108.10 10.19 ** .002 1, 578.77 11.72 *** <.001 ## --- ## Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘+’ 0.1 ‘ ’ 1
What this shows is that the pattern of significant and non-significant effect is the same for both models. The only significant effect for which the evidence is not that strong is the 3-way interaction between
stimulus:density:frequency. It is only just below .05 for the full model and has a somewhat lower value for the final model.
We can also see that one of the most noticeable differences between the maximal and the final model is the number of denominator degrees of freedom. This is highly influenced by the random effect structure and thus considerable larger in the final (i.e., reduced) model. The difference in the other statistics is lower.
It is instructive to compare those results with results obtained using the comparatively ‘worst’ method for obtaining \(p\)-values implemented in
afex::mixed, likelihood ratio tests. Likelihood ratio-tests should in principle deliver reasonable results for large data sets. A common rule of thumb is that the number of levels for each random effect grouping factor needs to be large, say above 50. Here, we have a very large number of items (600), but not that many participants (45). Thus, qualitative results should be the very similar, but it still is interesting to see exactly what happens. We therefore fit the final model using
## Mixed Model Anova Table (Type 3 tests, LRT-method) ## ## Model: log_rt ~ task * stimulus * density * frequency + (stimulus + ## Model: frequency || id) + (task | item) ## Data: fhch ## Df full model: 23 ## Effect df Chisq p.value ## 1 task 1 12.44 *** <.001 ## 2 stimulus 1 70.04 *** <.001 ## 3 density 1 0.37 .545 ## 4 frequency 1 0.57 .449 ## 5 task:stimulus 1 45.52 *** <.001 ## 6 task:density 1 17.81 *** <.001 ## 7 stimulus:density 1 1.21 .272 ## 8 task:frequency 1 53.81 *** <.001 ## 9 stimulus:frequency 1 60.64 *** <.001 ## 10 density:frequency 1 0.11 .741 ## 11 task:stimulus:density 1 14.85 *** <.001 ## 12 task:stimulus:frequency 1 114.21 *** <.001 ## 13 task:density:frequency 1 5.96 * .015 ## 14 stimulus:density:frequency 1 4.65 * .031 ## 15 task:stimulus:density:frequency 1 11.71 *** <.001 ## --- ## Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘+’ 0.1 ‘ ’ 1
The results in this case match the results of the Satterthwaite method. With lower numbers of levels of the grouping factor (e.g., less participants) this would not necessarily be expected.
Fortunately, the results from all models converged on the same pattern of significant and non-significant effects providing a high degree of confidence in the results. This might not be too surprising given the comparatively large number of total data points and the fact that each random effect grouping factor has a considerable number of levels (above 30 for both participants and items). In the following we focus on the final model using the Satterthwaite method,
In terms of the significant findings, there are many that seem to be in line with the descriptive results described above. For example, the highly significant effect of
task:stimulus:frequency with \(F(1, 578.91) = 124.16\), \(p < .001\), appears to be in line with the observation that the frequency effect appears to change its sign depending on the
task:stimulus cell (with
naming showing the opposite patterns than the other three conditions). Consequently, we start by investigating this interaction further below.
Before investigating the significant interaction in detail it is a good idea to remind oneself what a significant interaction represents on a conceptual level; that one or multiple of the variables in the interaction moderate (i.e., affect) the effect of the other variable or variables. Consequently, there are several ways to investigate a significant interaction. Each of the involved variables can be seen as the moderating variables and each of the variables can be seen as the effect of interest. Which one of those possible interpretations is of interest in a given situation highly depends on the actual data and research question and multiple views can be ‘correct’ in a given situation.
In addition to this conceptual issue, there are also multiple technical ways to investigate a significant interaction. One approach not followed here is to split the data into subsets according to the moderating variables and compute the statistical model again for the subsets with the effect variable(s) as remaining fixed effect. This approach, also called simple effects analysis, is, for example, recommended by Maxwell and Delaney (2004) as it does not assume variance homogeneity and is faithful to the data at each level. The approach taken here is to simply perform the test on the estimated final model. This approach assumes variance homogeneity (i.e., that the variances in all groups are homogeneous) and has the added benefit that it is computationally relatively simple. In addition, it can all be achieved using the framework provided by
emmeans (Lenth, 2017).
Our interest in the beginning is on the effect of
task:stimulus combination. So let us first look at the estimated marginal means of this effect. In
emmeans parlance these estimated means are called ‘least-square means’ because of historical reasons, but because of the lack of least-square estimation in mixed models we prefer the term estimated marginal means, or EMMs for short. Those can be obtained in the following way. To prevent
emmeans from calculating the df for the EMMs (which can be quite costly), we use asymptotic dfs (i.e., \(z\) values and tests).
emmeans requires to first specify the variable(s) one wants to treat as the effect variable(s) (here
frequency) and then allows to specify condition variables.
## NOTE: Results may be misleading due to involvement in interactions
## stimulus = word, task = naming: ## frequency emmean SE df asymp.LCL asymp.UCL ## low -0.32333 0.0455 Inf -0.4125 -0.2342 ## high -0.38210 0.0455 Inf -0.4713 -0.2929 ## ## stimulus = nonword, task = naming: ## frequency emmean SE df asymp.LCL asymp.UCL ## low -0.14294 0.0455 Inf -0.2321 -0.0538 ## high 0.06405 0.0455 Inf -0.0252 0.1533 ## ## stimulus = word, task = lexdec: ## frequency emmean SE df asymp.LCL asymp.UCL ## low 0.02337 0.0413 Inf -0.0576 0.1044 ## high -0.04017 0.0413 Inf -0.1211 0.0408 ## ## stimulus = nonword, task = lexdec: ## frequency emmean SE df asymp.LCL asymp.UCL ## low 0.10455 0.0413 Inf 0.0235 0.1856 ## high -0.00632 0.0413 Inf -0.0873 0.0746 ## ## Results are averaged over the levels of: density ## Degrees-of-freedom method: asymptotic ## Confidence level used: 0.95
The returned values are in line with our observation that the
naming condition diverges from the other three. But is there actual evidence that the effect flips? We can test this using additional
emmeans functionality. Specifically, we first use the
pairs function which provides us with a pairwise test of the effect of
frequency in each
task:stimulus combination. Then we need to combine the four tests within one object to obtain a family-wise error rate correction which we do via
update(..., by = NULL) (i.e., we revert the effect of the
by statement from the earlier
emmeans call) and finally we select the
holm method for controlling for family wise error rate (the Holm method is uniformly more powerful than the Bonferroni).
## NOTE: Results may be misleading due to involvement in interactions
## contrast stimulus task estimate SE df z.ratio p.value ## low - high word naming 0.0588 0.0149 Inf 3.941 0.0002 ## low - high nonword naming -0.2070 0.0150 Inf -13.823 <.0001 ## low - high word lexdec 0.0635 0.0167 Inf 3.807 0.0002 ## low - high nonword lexdec 0.1109 0.0167 Inf 6.619 <.0001 ## ## Results are averaged over the levels of: density ## Degrees-of-freedom method: asymptotic ## P value adjustment: holm method for 4 tests
We could also use a slightly more powerful method than the Holm method, method
free from package
multcomp. This method takes the correlation of the model parameters into account. However, results do not differ much here:
## ## Simultaneous Tests for General Linear Hypotheses ## ## Linear Hypotheses: ## Estimate Std. Error z value Pr(>|z|) ## low - high, word, naming == 0 0.05877 0.01491 3.941 0.000162 *** ## low - high, nonword, naming == 0 -0.20699 0.01497 -13.823 < 2e-16 *** ## low - high, word, lexdec == 0 0.06353 0.01669 3.807 0.000162 *** ## low - high, nonword, lexdec == 0 0.11086 0.01675 6.619 1.11e-10 *** ## --- ## Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 ## (Adjusted p values reported -- free method)
We see that the results are exactly as expected. In the
naming condition we have a clear negative effect of frequency while in the other three conditions it is clearly positive.
We could now also use
emmeans and re-transform the estimates back onto the original RT response scale. For this, we can again
emmeans object by using
tran = "log" to specify the transformation and then indicating we want the means on the response scale with
type = "response". These values might be used for plotting.
## frequency stimulus task response SE df asymp.LCL asymp.UCL ## low word naming 0.724 0.0329 Inf 0.662 0.791 ## high word naming 0.682 0.0310 Inf 0.624 0.746 ## low nonword naming 0.867 0.0394 Inf 0.793 0.948 ## high nonword naming 1.066 0.0485 Inf 0.975 1.166 ## low word lexdec 1.024 0.0423 Inf 0.944 1.110 ## high word lexdec 0.961 0.0397 Inf 0.886 1.042 ## low nonword lexdec 1.110 0.0459 Inf 1.024 1.204 ## high nonword lexdec 0.994 0.0410 Inf 0.916 1.077 ## ## Results are averaged over the levels of: density ## Degrees-of-freedom method: asymptotic ## Confidence level used: 0.95 ## Intervals are back-transformed from the log scale
A more direct approach for plotting the interaction is via
afex_plot. For a plot that is not too busy it makes sense to specify across which grouping factor the individual level data should be aggregated. We use the participant variable
"id" here. We also use
ggbeeswarm::geom_quasirandom as the geom for the data in the background following the example in the
As the last example, let us take a look at the significant four-way interaction of
task:stimulus:density:frequency, \(F(1, 578.77) = 11.72\), \(p < .001\). Here we might be interested in a slightly more difficult question namely whether the
density:frequency interaction varies across
task:stimulus conditions. If we again look at the figures above, it appears that there is a difference between
high:low in the
lexdec condition, but not in the other conditions.
Looking at the 2-way interaction of
density:frequency by the
task:stimulus interaction can be done using
emmeans using the
joint_test function. We simply need to specify the appropriate
by variables and get conditional tests this way.
## stimulus = word, task = naming: ## model term df1 df2 F.ratio p.value ## density 1 Inf 1.292 0.2556 ## frequency 1 Inf 15.530 0.0001 ## density:frequency 1 Inf 0.196 0.6578 ## ## stimulus = nonword, task = naming: ## model term df1 df2 F.ratio p.value ## density 1 Inf 17.926 <.0001 ## frequency 1 Inf 191.069 <.0001 ## density:frequency 1 Inf 3.656 0.0559 ## ## stimulus = word, task = lexdec: ## model term df1 df2 F.ratio p.value ## density 1 Inf 0.359 0.5491 ## frequency 1 Inf 14.494 0.0001 ## density:frequency 1 Inf 1.669 0.1964 ## ## stimulus = nonword, task = lexdec: ## model term df1 df2 F.ratio p.value ## density 1 Inf 15.837 0.0001 ## frequency 1 Inf 43.811 <.0001 ## density:frequency 1 Inf 14.806 0.0001
This test indeed shows that the
density:frequency interaction is only significant in the
lexdec condition. Next, let’s see if we can unpack this interaction in a meaningful manner. For this we compare
high:low in each of the four groups. And just for the sake of making the example more complex, we also compare
To do so, we first need to setup a new set of EMMs. Specifically, we get the EMMs of the two variables of interest, density and frequency, using the same
by specification as the
joint_test call. We can then setup custom contrasts that tests our hypotheses.
## stimulus = word, task = naming: ## density frequency emmean SE df asymp.LCL asymp.UCL ## low low -0.31384 0.0448 Inf -0.4016 -0.22613 ## high low -0.33268 0.0408 Inf -0.4126 -0.25276 ## low high -0.37741 0.0466 Inf -0.4687 -0.28611 ## high high -0.38644 0.0472 Inf -0.4789 -0.29399 ## ## stimulus = nonword, task = naming: ## density frequency emmean SE df asymp.LCL asymp.UCL ## low low -0.10399 0.0499 Inf -0.2019 -0.00611 ## high low -0.18230 0.0441 Inf -0.2688 -0.09580 ## low high 0.07823 0.0519 Inf -0.0236 0.18004 ## high high 0.04902 0.0494 Inf -0.0478 0.14588 ## ## stimulus = word, task = lexdec: ## density frequency emmean SE df asymp.LCL asymp.UCL ## low low 0.03713 0.0403 Inf -0.0419 0.11617 ## high low 0.00933 0.0368 Inf -0.0627 0.08138 ## low high -0.04512 0.0419 Inf -0.1272 0.03696 ## high high -0.03479 0.0424 Inf -0.1179 0.04828 ## ## stimulus = nonword, task = lexdec: ## density frequency emmean SE df asymp.LCL asymp.UCL ## low low 0.04480 0.0449 Inf -0.0432 0.13278 ## high low 0.16331 0.0398 Inf 0.0852 0.24140 ## low high -0.00729 0.0466 Inf -0.0987 0.08411 ## high high -0.00564 0.0444 Inf -0.0927 0.08140 ## ## Degrees-of-freedom method: asymptotic ## Confidence level used: 0.95
These contrasts can be specified via a list of custom contrasts on the EMMs (or reference grid in
emmeans parlance) which can be passed to the
contrast function. The contrasts are a
list where each element should sum to one (i.e., be a proper contrast) and be of length equal to the number of EMMs (although we have 16 EMMs in total, we only need to specify it for a length of four due to conditioning by
To control for the family wise error rate across all tests, we again use
update(..., by = NULL) on the result to revert the effect of the conditioning.
## contrast stimulus task estimate SE df z.ratio p.value ## ll_hl word naming 0.01884 0.0210 Inf 0.895 1.0000 ## lh_hh word naming 0.00904 0.0212 Inf 0.426 1.0000 ## ll_hl nonword naming 0.07831 0.0222 Inf 3.525 0.0030 ## lh_hh nonword naming 0.02921 0.0210 Inf 1.391 0.9847 ## ll_hl word lexdec 0.02780 0.0200 Inf 1.391 0.9847 ## lh_hh word lexdec -0.01033 0.0199 Inf -0.520 1.0000 ## ll_hl nonword lexdec -0.11850 0.0211 Inf -5.628 <.0001 ## lh_hh nonword lexdec -0.00165 0.0197 Inf -0.084 1.0000 ## ## Degrees-of-freedom method: asymptotic ## P value adjustment: holm method for 8 tests
In contrast to our expectation, the results show two significant effects and not only one. In line with our expectations, in the
lexdec condition the EMM of
low:low is smaller than the EMM for
high:low, \(z = -5.63\), \(p < .0001\). However, in the
naming condition we found the opposite pattern; the EMM of
low:low is larger than the EMM for
high:low, \(z = 3.53\), \(p = .003\). For all other effects \(|z| < 1.4\), \(p > .98\). In addition, there is no difference between
high:high in any condition.