Overview

This documents reanalysis a dataset from an Experiment performed by Singmann and Klauer (2011) using the ANOVA functionality of afex followed by post-hoc tests using package emmeans (Lenth, 2017). After a brief description of the dataset and research question, the code and results are presented.

Description of Experiment and Data

Singmann and Klauer (2011) were interested in whether or not conditional reasoning can be explained by a single process or whether multiple processes are necessary to explain it. To provide evidence for multiple processes we aimed to establish a double dissociation of two variables: instruction type and problem type. Instruction type was manipulated between-subjects, one group of participants received deductive instructions (i.e., to treat the premises as given and only draw necessary conclusions) and a second group of participants received probabilistic instructions (i.e., to reason as in an everyday situation; we called this “inductive instruction” in the manuscript). Problem type consisted of two different orthogonally crossed variables that were manipulated within-subjects, validity of the problem (formally valid or formally invalid) and plausibility of the problem (inferences which were consisted with the background knowledge versus problems that were inconsistent with the background knowledge). The critical comparison across the two conditions was among problems which were valid and implausible with problems that were invalid and plausible. For example, the next problem was invalid and plausible:

If a person is wet, then the person fell into a swimming pool.
A person fell into a swimming pool.
How valid is the conclusion/How likely is it that the person is wet?

For those problems we predicted that under deductive instructions responses should be lower (as the conclusion does not necessarily follow from the premises) as under probabilistic instructions. For the valid but implausible problem, an example is presented next, we predicted the opposite pattern:

If a person is wet, then the person fell into a swimming pool.
A person is wet.
How valid is the conclusion/How likely is it that the person fell into a swimming pool?

Our study also included valid and plausible and invalid and implausible problems.

In contrast to the analysis reported in the manuscript, we initially do not separate the analysis into affirmation and denial problems, but first report an analysis on the full set of inferences, MP, MT, AC, and DA, where MP and MT are valid and AC and DA invalid. We report a reanalysis of our Experiment 1 only. Note that the factor plausibility is not present in the original manuscript, there it is a results of a combination of other factors.

Data and R Preperation

We begin by loading the packages we will be using throughout.

library("afex")     # needed for ANOVA functions.
library("emmeans")  # emmeans must now be loaded explicitly for follow-up tests.
library("multcomp") # for advanced control for multiple testing/Type 1 errors.
library("ggplot2")  # for customizing plots.
afex_options(emmeans_model = "multivariate") # use multivariate model for all follow-up tests.

Note that for ANOVAs involving repeated-measures factors, follow-up tests based on the multivariate model are generally preferrably to univariate follow-up tests. Consequently, we set this option globally. Future versions of afex will likely use the multivariate model as the default.

data(sk2011.1)
str(sk2011.1)

## 'data.frame':    640 obs. of  9 variables:
##  $ id          : Factor w/ 40 levels "8","9","10","12",..: 3 3 3 3 3 3 3 3 3 3 ...
##  $ instruction : Factor w/ 2 levels "deductive","probabilistic": 2 2 2 2 2 2 2 2 2 2 ...
##  $ plausibility: Factor w/ 2 levels "plausible","implausible": 1 2 2 1 2 1 1 2 1 2 ...
##  $ inference   : Factor w/ 4 levels "MP","MT","AC",..: 4 2 1 3 4 2 1 3 4 2 ...
##  $ validity    : Factor w/ 2 levels "valid","invalid": 2 1 1 2 2 1 1 2 2 1 ...
##  $ what        : Factor w/ 2 levels "affirmation",..: 2 2 1 1 2 2 1 1 2 2 ...
##  $ type        : Factor w/ 2 levels "original","reversed": 2 2 2 2 1 1 1 1 2 2 ...
##  $ response    : int  100 60 94 70 100 99 98 49 82 50 ...
##  $ content     : Factor w/ 4 levels "C1","C2","C3",..: 1 1 1 1 2 2 2 2 3 3 ...

An important feature in the data is that each participant provided two responses for each cell of the design (the content is different for each of those, each participant saw all four contents). These two data points will be aggregated automatically by afex.

with(sk2011.1, table(inference, id, plausibility))

## , , plausibility = plausible
## 
##          id
## inference 8 9 10 12 13 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
##        MP 2 2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2
##        MT 2 2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2
##        AC 2 2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2
##        DA 2 2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2
##          id
## inference 37 38 39 40 41 42 43 44 46 47 48 49 50
##        MP  2  2  2  2  2  2  2  2  2  2  2  2  2
##        MT  2  2  2  2  2  2  2  2  2  2  2  2  2
##        AC  2  2  2  2  2  2  2  2  2  2  2  2  2
##        DA  2  2  2  2  2  2  2  2  2  2  2  2  2
## 
## , , plausibility = implausible
## 
##          id
## inference 8 9 10 12 13 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
##        MP 2 2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2
##        MT 2 2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2
##        AC 2 2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2
##        DA 2 2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2
##          id
## inference 37 38 39 40 41 42 43 44 46 47 48 49 50
##        MP  2  2  2  2  2  2  2  2  2  2  2  2  2
##        MT  2  2  2  2  2  2  2  2  2  2  2  2  2
##        AC  2  2  2  2  2  2  2  2  2  2  2  2  2
##        DA  2  2  2  2  2  2  2  2  2  2  2  2  2

ANOVA

To get the full ANOVA table for the model, we simply pass it to aov_ez using the design as described above. We save the returned object for further analysis.

a1 <- aov_ez("id", "response", sk2011.1, between = "instruction", 
       within = c("inference", "plausibility"))

## 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: instruction

a1 # the default print method prints a data.frame produced by nice

## Anova Table (Type 3 tests)
## 
## Response: response
##                               Effect           df     MSE         F  ges p.value
## 1                        instruction        1, 38 2027.42      0.31 .003     .58
## 2                          inference 2.66, 101.12  959.12   5.81 **  .06    .002
## 3              instruction:inference 2.66, 101.12  959.12   6.00 **  .07    .001
## 4                       plausibility        1, 38  468.82 34.23 ***  .07  <.0001
## 5           instruction:plausibility        1, 38  468.82  10.67 **  .02    .002
## 6             inference:plausibility  2.29, 87.11  318.91    2.87 + .009     .06
## 7 instruction:inference:plausibility  2.29, 87.11  318.91    3.98 *  .01     .02
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1
## 
## Sphericity correction method: GG

The equivalent calls (i.e., producing exactly the same output) of the other two ANOVA functions aov_car or aov4 is shown below.

aov_car(response ~ instruction + Error(id/inference*plausibility), sk2011.1)
aov_4(response ~ instruction + (inference*plausibility|id), sk2011.1)

As mentioned before, the two responses per cell of the design and participants are aggregated for the analysis as indicated by the warning message. Furthermore, the degrees of freedom are Greenhouse-Geisser corrected per default for all effects involving inference, as inference is a within-subject factor with more than two levels (i.e., MP, MT, AC, & DA). In line with our expectations, the three-way interaction is significant.

The object printed per default for afex_aov objects (produced by nice) can also be printed nicely using knitr:

knitr::kable(nice(a1))

Effect	df	MSE	F	ges	p.value
instruction	1, 38	2027.42	0.31	.003	.58
inference	2.66, 101.12	959.12	5.81 **	.06	.002
instruction:inference	2.66, 101.12	959.12	6.00 **	.07	.001
plausibility	1, 38	468.82	34.23 ***	.07	<.0001
instruction:plausibility	1, 38	468.82	10.67 **	.02	.002
inference:plausibility	2.29, 87.11	318.91	2.87 +	.009	.06
instruction:inference:plausibility	2.29, 87.11	318.91	3.98 *	.01	.02

Alternatively, the anova method for afex_aov objects returns a data.frame of class anova that can be passed to, for example, xtable for nice formatting:

print(xtable::xtable(anova(a1), digits = c(rep(2, 5), 3, 4)), type = "html")

	num Df	den Df	MSE	F	ges	Pr(>F)
instruction	1.00	38.00	2027.42	0.31	0.003	0.5830
inference	2.66	101.12	959.12	5.81	0.063	0.0016
instruction:inference	2.66	101.12	959.12	6.00	0.065	0.0013
plausibility	1.00	38.00	468.82	34.23	0.068	0.0000
instruction:plausibility	1.00	38.00	468.82	10.67	0.022	0.0023
inference:plausibility	2.29	87.11	318.91	2.87	0.009	0.0551
instruction:inference:plausibility	2.29	87.11	318.91	3.98	0.013	0.0177

Post-Hoc Contrasts and Plotting

To further analyze the data we need to pass it to package emmeans, a package that offers great functionality for both plotting and contrasts of all kind. A lot of information on emmeans can be obtained in its vignettes and faq. emmeans can work with afex_aov objects directly as afex comes with the necessary methods for the generic functions defined in emmeans. When using the multivariate options as described above, emmeans uses the ANOVA model estimated via base R’s lm method (which in the case of a multivariate response is an object of class c("mlm", "lm")). In the default setting (i.e., emmeans_model = "univariate"), emmeans uses the object created by base R’s aov function, which for now is also part of an afex_aov object.

Some First Contrasts

Main Effects Only

This object can now be passed to emmeans, for example to obtain the marginal means of the four inferences:

m1 <- emmeans(a1, ~ inference)
m1

##  inference   emmean       SE df lower.CL upper.CL
##  MP        87.51250 1.797265 38 83.87413 91.15087
##  MT        76.68125 4.064950 38 68.45219 84.91031
##  AC        69.41250 4.771297 38 59.75351 79.07149
##  DA        82.95625 3.837620 38 75.18740 90.72510
## 
## Results are averaged over the levels of: instruction, plausibility 
## Confidence level used: 0.95

This object can now also be used to compare whether or not there are differences between the levels of the factor:

pairs(m1)

##  contrast  estimate       SE df t.ratio p.value
##  MP - MT   10.83125 4.331479 38   2.501  0.0759
##  MP - AC   18.10000 5.017994 38   3.607  0.0047
##  MP - DA    4.55625 4.196484 38   1.086  0.7002
##  MT - AC    7.26875 3.983558 38   1.825  0.2778
##  MT - DA   -6.27500 4.702592 38  -1.334  0.5473
##  AC - DA  -13.54375 5.299024 38  -2.556  0.0672
## 
## Results are averaged over the levels of: instruction, plausibility 
## P value adjustment: tukey method for comparing a family of 4 estimates

To obtain more powerful p-value adjustments, we can furthermore pass it to multcomp (Bretz, Hothorn, & Westfall, 2011):

summary(as.glht(pairs(m1)), test=adjusted("free"))

## 
##   Simultaneous Tests for General Linear Hypotheses
## 
## Linear Hypotheses:
##              Estimate Std. Error t value Pr(>|t|)   
## MP - MT == 0   10.831      4.331   2.501  0.05907 . 
## MP - AC == 0   18.100      5.018   3.607  0.00457 **
## MP - DA == 0    4.556      4.196   1.086  0.31350   
## MT - AC == 0    7.269      3.984   1.825  0.19414   
## MT - DA == 0   -6.275      4.703  -1.334  0.31350   
## AC - DA == 0  -13.544      5.299  -2.556  0.05907 . 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## (Adjusted p values reported -- free method)

A Simple interaction

We could now also be interested in the marginal means of the inferences across the two instruction types. emmeans offers two ways to do so. The first splits the contrasts across levels of the factor using the by argument.

m2 <- emmeans(a1, "inference", by = "instruction")
## equal: emmeans(a1, ~ inference|instruction)
m2

## instruction = deductive:
##  inference  emmean       SE df lower.CL  upper.CL
##  MP        97.2875 2.541716 38 92.14206 102.43294
##  MT        70.4000 5.748708 38 58.76235  82.03765
##  AC        61.4875 6.747633 38 47.82763  75.14737
##  DA        81.8125 5.427214 38 70.82568  92.79932
## 
## instruction = probabilistic:
##  inference  emmean       SE df lower.CL  upper.CL
##  MP        77.7375 2.541716 38 72.59206  82.88294
##  MT        82.9625 5.748708 38 71.32485  94.60015
##  AC        77.3375 6.747633 38 63.67763  90.99737
##  DA        84.1000 5.427214 38 73.11318  95.08682
## 
## Results are averaged over the levels of: plausibility 
## Confidence level used: 0.95

Consequently, tests are also only performed within each level of the by factor:

pairs(m2)

## instruction = deductive:
##  contrast estimate       SE df t.ratio p.value
##  MP - MT   26.8875 6.125636 38   4.389  0.0005
##  MP - AC   35.8000 7.096515 38   5.045  0.0001
##  MP - DA   15.4750 5.934724 38   2.608  0.0599
##  MT - AC    8.9125 5.633601 38   1.582  0.4007
##  MT - DA  -11.4125 6.650469 38  -1.716  0.3297
##  AC - DA  -20.3250 7.493951 38  -2.712  0.0471
## 
## instruction = probabilistic:
##  contrast estimate       SE df t.ratio p.value
##  MP - MT   -5.2250 6.125636 38  -0.853  0.8287
##  MP - AC    0.4000 7.096515 38   0.056  0.9999
##  MP - DA   -6.3625 5.934724 38  -1.072  0.7084
##  MT - AC    5.6250 5.633601 38   0.998  0.7512
##  MT - DA   -1.1375 6.650469 38  -0.171  0.9982
##  AC - DA   -6.7625 7.493951 38  -0.902  0.8036
## 
## Results are averaged over the levels of: plausibility 
## P value adjustment: tukey method for comparing a family of 4 estimates

The second version considers all factor levels together. Consequently, the number of pairwise comparisons is a lot larger:

m3 <- emmeans(a1, c("inference", "instruction"))
## equal: emmeans(a1, ~inference*instruction)
m3

##  inference instruction    emmean       SE df lower.CL  upper.CL
##  MP        deductive     97.2875 2.541716 38 92.14206 102.43294
##  MT        deductive     70.4000 5.748708 38 58.76235  82.03765
##  AC        deductive     61.4875 6.747633 38 47.82763  75.14737
##  DA        deductive     81.8125 5.427214 38 70.82568  92.79932
##  MP        probabilistic 77.7375 2.541716 38 72.59206  82.88294
##  MT        probabilistic 82.9625 5.748708 38 71.32485  94.60015
##  AC        probabilistic 77.3375 6.747633 38 63.67763  90.99737
##  DA        probabilistic 84.1000 5.427214 38 73.11318  95.08682
## 
## Results are averaged over the levels of: plausibility 
## Confidence level used: 0.95

pairs(m3)

##  contrast                            estimate       SE df t.ratio p.value
##  MP,deductive - MT,deductive          26.8875 6.125636 38   4.389  0.0020
##  MP,deductive - AC,deductive          35.8000 7.096515 38   5.045  0.0003
##  MP,deductive - DA,deductive          15.4750 5.934724 38   2.608  0.1848
##  MP,deductive - MP,probabilistic      19.5500 3.594529 38   5.439  0.0001
##  MP,deductive - MT,probabilistic      14.3250 6.285536 38   2.279  0.3310
##  MP,deductive - AC,probabilistic      19.9500 7.210470 38   2.767  0.1342
##  MP,deductive - DA,probabilistic      13.1875 5.992910 38   2.201  0.3741
##  MT,deductive - AC,deductive           8.9125 5.633601 38   1.582  0.7577
##  MT,deductive - DA,deductive         -11.4125 6.650469 38  -1.716  0.6772
##  MT,deductive - MP,probabilistic      -7.3375 6.285536 38  -1.167  0.9363
##  MT,deductive - MT,probabilistic     -12.5625 8.129901 38  -1.545  0.7783
##  MT,deductive - AC,probabilistic      -6.9375 8.864434 38  -0.783  0.9931
##  MT,deductive - DA,probabilistic     -13.7000 7.905839 38  -1.733  0.6666
##  AC,deductive - DA,deductive         -20.3250 7.493951 38  -2.712  0.1501
##  AC,deductive - MP,probabilistic     -16.2500 7.210470 38  -2.254  0.3446
##  AC,deductive - MT,probabilistic     -21.4750 8.864434 38  -2.423  0.2600
##  AC,deductive - AC,probabilistic     -15.8500 9.542594 38  -1.661  0.7111
##  AC,deductive - DA,probabilistic     -22.6125 8.659400 38  -2.611  0.1834
##  DA,deductive - MP,probabilistic       4.0750 5.992910 38   0.680  0.9971
##  DA,deductive - MT,probabilistic      -1.1500 7.905839 38  -0.145  1.0000
##  DA,deductive - AC,probabilistic       4.4750 8.659400 38   0.517  0.9995
##  DA,deductive - DA,probabilistic      -2.2875 7.675239 38  -0.298  1.0000
##  MP,probabilistic - MT,probabilistic  -5.2250 6.125636 38  -0.853  0.9885
##  MP,probabilistic - AC,probabilistic   0.4000 7.096515 38   0.056  1.0000
##  MP,probabilistic - DA,probabilistic  -6.3625 5.934724 38  -1.072  0.9588
##  MT,probabilistic - AC,probabilistic   5.6250 5.633601 38   0.998  0.9719
##  MT,probabilistic - DA,probabilistic  -1.1375 6.650469 38  -0.171  1.0000
##  AC,probabilistic - DA,probabilistic  -6.7625 7.493951 38  -0.902  0.9840
## 
## Results are averaged over the levels of: plausibility 
## P value adjustment: tukey method for comparing a family of 8 estimates

Running Custom Contrasts

Objects returned from emmeans can also be used to test specific contrasts. For this, we can simply create a list, where each element corresponds to one contrasts. A contrast is defined as a vector of constants on the reference grid (i.e., the object returned from emmeans, here m3). For example, we might be interested in whether there is a difference between the valid and invalid inferences in each of the two conditions.

c1 <- list(
  v_i.ded = c(0.5, 0.5, -0.5, -0.5, 0, 0, 0, 0),
  v_i.prob = c(0, 0, 0, 0, 0.5, 0.5, -0.5, -0.5)
  )

contrast(m3, c1, adjust = "holm")

##  contrast estimate      SE df t.ratio p.value
##  v_i.ded  12.19375 4.11901 38    2.96  0.0105
##  v_i.prob -0.36875 4.11901 38   -0.09  0.9291
## 
## Results are averaged over the levels of: plausibility 
## P value adjustment: holm method for 2 tests

summary(as.glht(contrast(m3, c1)), test = adjusted("free"))

## 
##   Simultaneous Tests for General Linear Hypotheses
## 
## Linear Hypotheses:
##               Estimate Std. Error t value Pr(>|t|)  
## v_i.ded == 0   12.1937     4.1190    2.96   0.0105 *
## v_i.prob == 0  -0.3687     4.1190   -0.09   0.9291  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## (Adjusted p values reported -- free method)

The results can be interpreted as in line with expectations. Responses are larger for valid than invalid problems in the deductive, but not the probabilistic condition.

Plotting

Since version 0.22, afex comes with its own plotting function based on ggplot2, afex_plot, which works directly with afex_aov objects.

As said initially, we are interested in the three-way interaction of instruction with inference, plausibility, and instruction. As we saw above, this interaction was significant. Consequently, we are interested in plotting this interaction.

Basic Plots

For afex_plot, we need to specify the x-factor(s), which determine which factor-levels or combinations of factor-levels are plotted on the x-axis. We can also define trace factor(s), which determine which factor levels are connected by lines. Finally, we can also define panel factor(s), which determine if the plot is split into subplots. afex_plot then plots the estimated marginal means obtained from emmeans, confidence intervals, and the raw data in the background. Note that the raw data in the background is per default drawn using an alpha blending of .5 (i.e., 50% semi-transparency). Thus, in case of several points lying directly on top of each other, this point appears noticeably darker.

afex_plot(a1, x = "inference", trace = "instruction", panel = "plausibility")

## Warning: Panel(s) show a mixed within-between-design.
## Error bars do not allow comparisons across all means.
## Suppress error bars with: error = "none"

In the default settings, the error bars show 95%-confidence intervals based on the standard error of the underlying model (i.e., the lm model in the present case). In the present case, in which each subplot (defined by x- and trace-factor) shows a combination of a within-subjects factor (i.e., inference) and a between-subjects (i.e., instruction) factor, this is not optimal. The error bars only allow to assess differences regarding the between-subjects factor (i.e., across the lines), but not inferences regarding the within-subjects factor (i.e., within one line). This is also indicated by a warning.

An alternative would be within-subject confidence intervals:

afex_plot(a1, x = "inference", trace = "instruction", panel = "plausibility", 
          error = "within")

## Warning: Panel(s) show a mixed within-between-design.
## Error bars do not allow comparisons across all means.
## Suppress error bars with: error = "none"

However, those only allow inferences regarding the within-subject factors and not regarding the between-subjecta factor. So the same warning is emitted again.

A further alternative is to suppress the error bars altogether. This is the approach used in our original paper and probably a good idea in general when figures show both between- and within-subjects factors within the same panel. The presence of the raw data in the background still provides a visual depiction of the variability of the data.

afex_plot(a1, x = "inference", trace = "instruction", panel = "plausibility", 
          error = "none")

Customizing Plots

afex_plot allows to customize the plot in a number of different ways. For example, we can easily change the aesthetic mapping associated with the trace factor. So instead of using lineytpe and shape of the symbols, we can use color. Furthermore, we can change the graphical element used for plotting the data points in the background. For example, instead of plotting the raw data, we can replace this with a boxplot. Finally, we can also make both the points showing the means and the lines connecting the means larger.

p1 <- afex_plot(a1, x = "inference", trace = "instruction", 
                panel = "plausibility", error = "none", 
                mapping = c("color", "fill"), 
                data_geom = geom_boxplot, data_arg = list(width = 0.4), 
                point_arg = list(size = 1.5), line_arg = list(size = 1))
p1

ANOVA and Post-Hoc Contrasts: Reanalysis of Singmann and Klauer (2011)

Henrik Singmann

2018-09-23