mixed {afex} | R Documentation |
Estimates mixed models with lme4 and calculates p-values for all fixed effects. The default method "KR"
(= Kenward-Roger) as well as method="S"
(Satterthwaite) support LMMs and estimate the model with lmer
and then pass it to the lmerTest
anova
method (or Anova
). The other methods ("LRT"
= likelihood-ratio tests and "PB"
= parametric bootstrap) support both LMMs (estimated via lmer
) and GLMMs (i.e., with family
argument which invokes estimation via glmer
) and estimate a full model and restricted models in which the parameters corresponding to one effect (i.e., model term) are withhold (i.e., fixed to 0). Per default tests are based on Type 3 sums of squares. print
, nice
, anova
, and summary
methods for the returned object of class "mixed"
are available. summary
invokes the default lme4summary method and shows parameters instead of effects.
lmer_alt
is simply a wrapper for mixed that only returns the "lmerModLmerTest"
or "merMod"
object and correctly uses the ||
notation for removing correlations among factors. This function otherwise behaves like g/lmer
(as for mixed
, it calls glmer
as soon as a family
argument is present). Use afex_options
("lmer_function")
to set which function for estimation should be used. This option determines the class of the returned object (i.e., "lmerModLmerTest"
or "merMod"
).
mixed(formula, data, type = afex_options("type"), method = afex_options("method_mixed"), per_parameter = NULL, args_test = list(), test_intercept = FALSE, check_contrasts = afex_options("check_contrasts"), expand_re = FALSE, all_fit = FALSE, set_data_arg = afex_options("set_data_arg"), progress = TRUE, cl = NULL, return = "mixed", sig_symbols = afex_options("sig_symbols"), ...) lmer_alt(formula, data, check_contrasts = FALSE, ...)
formula |
a formula describing the full mixed-model to be fitted. As this formula is passed to |
data |
|
type |
type of test on which effects are based. Default is to use type 3 tests, taken from |
method |
character vector indicating which methods for obtaining p-values should be used: |
per_parameter |
|
args_test |
|
test_intercept |
logical. Whether or not the intercept should also be fitted and tested for significance. Default is |
check_contrasts |
|
expand_re |
logical. Should random effects terms be expanded (i.e., factors transformed into numerical variables) before fitting with |
all_fit |
logical. Should |
set_data_arg |
|
progress |
if |
cl |
A vector identifying a cluster; used for distributing the estimation of the different models using several cores (if seveal models are calculated). See examples. If |
return |
the default is to return an object of class |
sig_symbols |
Character. What should be the symbols designating significance? When entering an vector with |
... |
further arguments (such as |
For an introduction to mixed-modeling for experimental designs see our chapter (Singmann & Kellen, in press) or Barr, Levy, Scheepers, & Tily (2013). Arguments for using the Kenward-Roger approximation for obtaining p-values are given by Judd, Westfall, and Kenny (2012). Further introductions to mixed-modeling for experimental designs are given by Baayen and colleagues (Baayen, 2008; Baayen, Davidson & Bates, 2008; Baayen & Milin, 2010). Specific recommendations on which random effects structure to specify for confirmatory tests can be found in Barr and colleagues (2013) and Barr (2013), but also see Bates et al. (2015).
When method = "KR"
(the default, implemented via KRmodcomp
), the Kenward-Roger approximation for degrees-of-freedom is calculated using lmerTest
(if test_intercept=FALSE
) or Anova
(if test_intercept=TRUE
), which is only applicable to linear-mixed models (LMMs). The test statistic in the output is an F-value (F
). A similar method that requires less RAM is method = "S"
which calculates the Satterthwaite approximation for degrees-of-freedom via lmerTest
and is also only applicable to LMMs. method = "KR"
or method = "S"
provide the best control for Type 1 errors for LMMs (Luke, 2017).
method = "PB"
calculates p-values using parametric bootstrap using PBmodcomp
. This can be used for linear and also generalized linear mixed models (GLMMs) by specifying a family
argument to mixed
. Note that you should specify further arguments to PBmodcomp
via args_test
, especially nsim
(the number of simulations to form the reference distribution) or cl
(for using multiple cores). For other arguments see PBmodcomp
. Note that REML
(argument to [g]lmer
) will be set to FALSE
if method is PB
.
method = "LRT"
calculates p-values via likelihood ratio tests implemented in the anova
method for "merMod"
objects. This is the method recommended by Barr et al. (2013; which did not test the other methods implemented here). Using likelihood ratio tests is only recommended for models with many levels for the random effects (> 50), but can be pretty helpful in case the other methods fail (due to memory and/or time limitations). The lme4 faq also recommends the other methods over likelihood ratio tests.
For methods "KR"
and "S"
type 3 and 2 tests are implemented as in Anova
.
For all other methods, type 3 tests are obtained by comparing a model in which only the tested effect is excluded with the full model (containing all effects). For method "nested-KR"
(which was the default in previous versions) this corresponds to the (type 3) Wald tests given by car::Anova
for "lmerMod"
models. The submodels in which the tested effect is excluded are obtained by manually creating a model matrix which is then fitted in "lme4"
.
Type 2 tests are truly sequential. They are obtained by comparing a model in which the tested effect and all higher oder effect (e.g., all three-way interactions for testing a two-way interaction) are excluded with a model in which only effects up to the order of the tested effect are present and all higher order effects absent. In other words, there are multiple full models, one for each order of effects. Consequently, the results for lower order effects are identical of whether or not higher order effects are part of the model or not. This latter feature is not consistent with classical ANOVA type 2 tests but a consequence of the sequential tests (and I didn't find a better way of implementing the Type 2 tests). This does not correspond to the (type 2) Wald test reported by car::Anova
.
If check_contrasts = TRUE
, contrasts will be set to "contr.sum"
for all factors in the formula if default contrasts are not equal to "contr.sum"
or attrib(factor, "contrasts") != "contr.sum"
. Furthermore, the current contrasts (obtained via getOption("contrasts")
) will be set at the cluster nodes if cl
is not NULL
.
expand_re = TRUE
allows to expand the random effects structure before passing it to lmer
. This allows to disable estimation of correlation among random effects for random effects term containing factors using the ||
notation which may aid in achieving model convergence (see Bates et al., 2015). This is achieved by first creating a model matrix for each random effects term individually, rename and append the so created columns to the data that will be fitted, replace the actual random effects term with the so created variables (concatenated with +), and then fit the model. The variables are renamed by prepending all variables with rei (where i is the number of the random effects term) and replacing ":" with "_by_".
lmer_alt
is simply a wrapper for mixed
that is intended to behave like lmer
(or glmer
if a family
argument is present), but also allows the use of ||
with factors (by always using expand_re = TRUE
). This means that lmer_alt
per default does not enforce a specific contrast on factors and only returns the "lmerModLmerTest"
or "merMod"
object without calculating any additional models or p-values (this is achieved by setting return = "merMod"
). Note that it most likely differs from g/lmer
in how it handles missing values so it is recommended to only pass data without missing values to it!
One consequence of using expand_re = TRUE
is that the data that is fitted will not be the same as the passed data.frame which can lead to problems with e.g., the predict
method. However, the actual data used for fitting is also returned as part of the mixed
object so can be used from there. Note that the set_data_arg
can be used to change whether the data
argument in the call to g/lmer
is set to data
(the default) or the name of the data argument passed by the user.
An object of class "mixed"
(i.e., a list) with the following elements:
anova_table
a data.frame containing the statistics returned from KRmodcomp
. The stat
column in this data.frame gives the value of the test statistic, an F-value for method = "KR"
and a chi-square value for the other two methods.
full_model
the "lmerModLmerTest"
or "merMod"
object returned from estimating the full model. Use afex_options
("lmer_function")
for setting which function for estimation should be used. The possible options are "lmerTest"
(the default returning an object of class "lmerModLmerTest"
) and "lme4"
returning an object of class ("merMod"
). Note that in case a family
argument is present an object of class "glmerMod"
is always returned.
restricted_models
a list of "g/lmerMod"
(or "lmerModLmerTest"
) objects from estimating the restricted models (i.e., each model lacks the corresponding effect)
tests
a list of objects returned by the function for obtaining the p-values.
data
The data used for estimation (i.e., after excluding missing rows and applying expand_re if requested).
It also has the following attributes, "type"
and "method"
. And the attributes "all_fit_selected"
and "all_fit_logLik"
if all_fit=TRUE
.
Two similar methods exist for objects of class "mixed"
: print
and anova
. They print a nice version of the anova_table
element of the returned object (which is also invisibly returned). This methods omit some columns and nicely round the other columns. The following columns are always printed:
Effect
name of effect
p.value
estimated p-value for the effect
For LMMs with method="KR"
or method="S"
the following further columns are returned (note: the Kenward-Roger correction does two separate things: (1) it computes an effective number for the denominator df; (2) it scales the statistic by a calculated amount, see also http://stackoverflow.com/a/25612960/289572):
F
computed F statistic
ndf
numerator degrees of freedom (number of parameters used for the effect)
ddf
denominator degrees of freedom (effective residual degrees of freedom for testing the effect), computed from the Kenward-Roger correction using pbkrtest::KRmodcomp
F.scaling
scaling of F-statistic computing from Kenward-Roger approximation (only printed if method="nested-KR"
)
For models with method="LRT"
the following further columns are returned:
df.large
degrees of freedom (i.e., estimated paramaters) for full model (i.e., model containing the corresponding effect)
df.small
degrees of freedom (i.e., estimated paramaters) for restricted model (i.e., model without the corresponding effect)
chisq
2 times the difference in likelihood (obtained with logLik
) between full and restricted model
df
difference in degrees of freedom between full and restricted model (p-value is based on these df).
For models with method="PB"
the following further column is returned:
stat
2 times the difference in likelihood (obtained with logLik
) between full and restricted model (i.e., a chi-square value).
Note that anova
can also be called with additional mixed and/or merMod
objects. In this casethe full models are passed on to anova.merMod
(with refit=FALSE
, which differs from the default of anova.merMod
) which produces the known LRT tables.
The summary
method for objects of class mixed
simply calls summary.merMod
on the full model.
If return = "merMod"
, an object of class "merMod"
, as returned from g/lmer
, is returned.
When method = "KR"
, obtaining p-values is known to crash due too insufficient memory or other computational limitations (especially with complex random effects structures). In these cases, the other methods should be used. The RAM demand is a problem especially on 32 bit Windows which only supports up to 2 or 3GB RAM (see R Windows FAQ). Then it is probably a good idea to use methods "S", "LRT", or "PB".
"mixed"
will throw a message if numerical variables are not centered on 0, as main effects (of other variables then the numeric one) can be hard to interpret if numerical variables appear in interactions. See Dalal & Zickar (2012).
Per default mixed
uses lmer
, this can be changed to lmer
by calling: afex_options(lmer_function = "lme4")
Formulas longer than 500 characters will most likely fail due to the use of deparse
.
Please report bugs or unexpected behavior by opening a guthub issue: https://github.com/singmann/afex/issues
Henrik Singmann with contributions from Ben Bolker and Joshua Wiley.
Baayen, R. H. (2008). Analyzing linguistic data: a practical introduction to statistics using R. Cambridge, UK; New York: Cambridge University Press.
Baayen, R. H., Davidson, D. J., & Bates, D. M. (2008). Mixed-effects modeling with crossed random effects for subjects and items. Journal of Memory and Language, 59(4), 390-412. doi:10.1016/j.jml.2007.12.005
Baayen, R. H., & Milin, P. (2010). Analyzing Reaction Times. International Journal of Psychological Research, 3(2), 12-28.
Barr, D. J. (2013). Random effects structure for testing interactions in linear mixed-effects models. Frontiers in Quantitative Psychology and Measurement, 328. doi:10.3389/fpsyg.2013.00328
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. doi:10.1016/j.jml.2012.11.001
Bates, D., Kliegl, R., Vasishth, S., & Baayen, H. (2015). Parsimonious Mixed Models. arXiv:1506.04967 [stat]. Retrieved from http://arxiv.org/abs/1506.04967
Dalal, D. K., & Zickar, M. J. (2012). Some Common Myths About Centering Predictor Variables in Moderated Multiple Regression and Polynomial Regression. Organizational Research Methods, 15(3), 339-362. doi:10.1177/1094428111430540
Judd, C. M., Westfall, J., & Kenny, D. A. (2012). Treating stimuli as a random factor in social psychology: A new and comprehensive solution to a pervasive but largely ignored problem. Journal of Personality and Social Psychology, 103(1), 54-69. doi:10.1037/a0028347
Luke, S. (2017). Evaluating significance in linear mixed-effects models in R. Behavior Research Methods. https://doi.org/10.3758/s13428-016-0809-y
Maxwell, S. E., & Delaney, H. D. (2004). Designing experiments and analyzing data: a model-comparisons perspective. Mahwah, N.J.: Lawrence Erlbaum Associates.
aov_ez
and aov_car
for convenience functions to analyze experimental deisgns with classical ANOVA or ANCOVA wrapping Anova
.
see the following for the data sets from Maxwell and Delaney (2004) used and more examples: md_15.1
, md_16.1
, and md_16.4
.
################################## ## Simple Examples (from MEMSS) ## ################################## data("Machines", package = "MEMSS") # simple model with random-slopes for repeated-measures factor m1 <- mixed(score ~ Machine + (Machine|Worker), data=Machines) m1 # suppress correlation among random effect parameters with expand_re = TRUE m2 <- mixed(score ~ Machine + (Machine||Worker), data=Machines, expand_re = TRUE) m2 ## compare: summary(m1)$varcor summary(m2)$varcor # for wrong solution see: # summary(lmer(score ~ Machine + (Machine||Worker), data=Machines))$varcor # follow-up tests (emm1 <- emmeans(m1, "Machine")) pairs(emm1, adjust = "holm") # all pairwise comparisons con1 <- list( c1 = c(1, -0.5, -0.5), # 1 versus other 2 c2 = c(0.5, -1, 0.5) # 1 and 3 versus 2 ) contrast(emm1, con1, adjust = "holm") # plotting emmip(m1, ~Machine, CIs = TRUE) emmip(m2, ~Machine, CIs = TRUE) ## Not run: ####################### ### Further Options ### ####################### ## Multicore: require(parallel) (nc <- detectCores()) # number of cores cl <- makeCluster(rep("localhost", nc)) # make cluster # to keep track of what the function is doindg redirect output to outfile: # cl <- makeCluster(rep("localhost", nc), outfile = "cl.log.txt") data("Machines", package = "MEMSS") ## There are two ways to use multicore: # 1. Obtain fits with multicore: mixed(score ~ Machine + (Machine|Worker), data=Machines, cl = cl) # 2. Obtain PB samples via multicore: mixed(score ~ Machine + (Machine|Worker), data=Machines, method = "PB", args_test = list(nsim = 50, cl = cl)) # better use 500 or 1000 ## Both ways can be combined: # 2. Obtain PB samples via multicore: mixed(score ~ Machine + (Machine|Worker), data=Machines, cl = cl, method = "PB", args_test = list(nsim = 50, cl = cl)) #### use all_fit = TRUE and expand_re = TRUE: data("sk2011.2") # data described in more detail below sk2_aff <- droplevels(sk2011.2[sk2011.2$what == "affirmation",]) require(optimx) # uses two more algorithms sk2_aff_b <- mixed(response ~ instruction*type+(inference*type||id), sk2_aff, expand_re = TRUE, all_fit = TRUE) attr(sk2_aff_b, "all_fit_selected") attr(sk2_aff_b, "all_fit_logLik") # considerably faster with multicore: clusterEvalQ(cl, library(optimx)) # need to load optimx in cluster sk2_aff_b2 <- mixed(response ~ instruction*type+(inference*type||id), sk2_aff, expand_re = TRUE, all_fit = TRUE, cl=cl) attr(sk2_aff_b2, "all_fit_selected") attr(sk2_aff_b2, "all_fit_logLik") stopCluster(cl) ## End(Not run) ################################################### ## Replicating Maxwell & Delaney (2004) Examples ## ################################################### ## Not run: ### replicate results from Table 15.4 (Maxwell & Delaney, 2004, p. 789) data(md_15.1) # random intercept plus random slope (t15.4a <- mixed(iq ~ timecat + (1+time|id),data=md_15.1)) # to also replicate exact parameters use treatment.contrasts and the last level as base level: contrasts(md_15.1$timecat) <- contr.treatment(4, base = 4) (t15.4b <- mixed(iq ~ timecat + (1+time|id),data=md_15.1, check_contrasts=FALSE)) summary(t15.4a) # gives "wrong" parameters extimates summary(t15.4b) # identical parameters estimates # for more examples from chapter 15 see ?md_15.1 ### replicate results from Table 16.3 (Maxwell & Delaney, 2004, p. 837) data(md_16.1) # original results need treatment contrasts: (mixed1_orig <- mixed(severity ~ sex + (1|id), md_16.1, check_contrasts=FALSE)) summary(mixed1_orig$full_model) # p-value stays the same with afex default contrasts (contr.sum), # but estimates and t-values for the fixed effects parameters change. (mixed1 <- mixed(severity ~ sex + (1|id), md_16.1)) summary(mixed1$full_model) # data for next examples (Maxwell & Delaney, Table 16.4) data(md_16.4) str(md_16.4) ### replicate results from Table 16.6 (Maxwell & Delaney, 2004, p. 845) # Note that (1|room:cond) is needed because room is nested within cond. # p-value (almost) holds. (mixed2 <- mixed(induct ~ cond + (1|room:cond), md_16.4)) # (differences are dut to the use of Kenward-Roger approximation here, # whereas M&W's p-values are based on uncorrected df.) # again, to obtain identical parameter and t-values, use treatment contrasts: summary(mixed2) # not identical # prepare new data.frame with contrasts: md_16.4b <- within(md_16.4, cond <- C(cond, contr.treatment, base = 2)) str(md_16.4b) # p-value stays identical: (mixed2_orig <- mixed(induct ~ cond + (1|room:cond), md_16.4b, check_contrasts=FALSE)) summary(mixed2_orig$full_model) # replicates parameters ### replicate results from Table 16.7 (Maxwell & Delaney, 2004, p. 851) # F-values (almost) hold, p-values (especially for skill) are off (mixed3 <- mixed(induct ~ cond + skill + (1|room:cond), md_16.4)) # however, parameters are perfectly recovered when using the original contrasts: mixed3_orig <- mixed(induct ~ cond + skill + (1|room:cond), md_16.4b, check_contrasts=FALSE) summary(mixed3_orig) ### replicate results from Table 16.10 (Maxwell & Delaney, 2004, p. 862) # for this we need to center cog: md_16.4b$cog <- scale(md_16.4b$cog, scale=FALSE) # F-values and p-values are relatively off: (mixed4 <- mixed(induct ~ cond*cog + (cog|room:cond), md_16.4b)) # contrast has a relatively important influence on cog (mixed4_orig <- mixed(induct ~ cond*cog + (cog|room:cond), md_16.4b, check_contrasts=FALSE)) # parameters are again almost perfectly recovered: summary(mixed4_orig) ## End(Not run) ########################### ## Full Analysis Example ## ########################### ## Not run: ### split-plot experiment (Singmann & Klauer, 2011, Exp. 2) ## between-factor: instruction ## within-factor: inference & type ## hypothesis: three-way interaction data("sk2011.2") # use only affirmation problems (S&K also splitted the data like this) sk2_aff <- droplevels(sk2011.2[sk2011.2$what == "affirmation",]) # set up model with maximal by-participant random slopes sk_m1 <- mixed(response ~ instruction*inference*type+(inference*type|id), sk2_aff) sk_m1 # prints ANOVA table with nicely rounded numbers (i.e., as characters) nice(sk_m1) # returns the same but without printing potential warnings anova(sk_m1) # returns and prints numeric ANOVA table (i.e., not-rounded) summary(sk_m1) # lmer summary of full model # same model but using Satterthwaite approximation of df # very similar results but faster sk_m1b <- mixed(response ~ instruction*inference*type+(inference*type|id), sk2_aff, method="S") nice(sk_m1b) # identical results as: anova(sk_m1$full_model) # suppressing correlation among random slopes: # very similar results, but significantly faster and often less convergence warnings. sk_m2 <- mixed(response ~ instruction*inference*type+(inference*type||id), sk2_aff, expand_re = TRUE) sk_m2 ## mixed objects can be passed to emmeans directly: # recreates basically Figure 4 (S&K, 2011, upper panel) # only the 4th and 6th x-axis position are flipped emmip(sk_m1, instruction~type+inference) # use lattice instead of ggplot2: emm_options(graphics.engine = "lattice") emmip(sk_m1, instruction~type+inference) emm_options(graphics.engine = "ggplot") # reset options # set up reference grid for custom contrasts: # this can be made faster via: emm_options(lmer.df = "Kenward-Roger") # set df for emmeans to KR # emm_options(lmer.df = "Satterthwaite") # the default # emm_options(lmer.df = "asymptotic") # the fastest, no df (rg1 <- emmeans(sk_m1, c("instruction", "type", "inference"))) # set up contrasts on reference grid: contr_sk2 <- list( ded_validity_effect = c(rep(0, 4), 1, rep(0, 5), -1, 0), ind_validity_effect = c(rep(0, 5), 1, rep(0, 5), -1), counter_MP = c(rep(0, 4), 1, -1, rep(0, 6)), counter_AC = c(rep(0, 10), 1, -1) ) # test the main double dissociation (see S&K, p. 268) contrast(rg1, contr_sk2, adjust = "holm") # only plausibility effect is not significant here. ## End(Not run) #################### ## Other Examples ## #################### ## Not run: # use the obk.long data (not reasonable, no random slopes) data(obk.long) mixed(value ~ treatment * phase + (1|id), obk.long) # Examples for using the per.parameter argument # note, require method = "nested-KR", "LRT", or "PB" # also we use custom contrasts data(obk.long, package = "afex") obk.long$hour <- ordered(obk.long$hour) contrasts(obk.long$phase) <- "contr.sum" contrasts(obk.long$treatment) <- "contr.sum" # tests only the main effect parameters of hour individually per parameter. mixed(value ~ treatment*phase*hour +(1|id), per_parameter = "^hour$", data = obk.long, method = "nested-KR", check_contrasts = FALSE) # tests all parameters including hour individually mixed(value ~ treatment*phase*hour +(1|id), per_parameter = "hour", data = obk.long, method = "nested-KR", check_contrasts = FALSE) # tests all parameters individually mixed(value ~ treatment*phase*hour +(1|id), per_parameter = ".", data = obk.long, method = "nested-KR", check_contrasts = FALSE) # example data from package languageR: # Lexical decision latencies elicited from 21 subjects for 79 English concrete nouns, # with variables linked to subject or word. data(lexdec, package = "languageR") # using the simplest model m1 <- mixed(RT ~ Correct + Trial + PrevType * meanWeight + Frequency + NativeLanguage * Length + (1|Subject) + (1|Word), data = lexdec) m1 # Mixed Model Anova Table (Type 3 tests, KR-method) # # Model: RT ~ Correct + Trial + PrevType * meanWeight + Frequency + NativeLanguage * # Model: Length + (1 | Subject) + (1 | Word) # Data: lexdec # Effect df F p.value # 1 Correct 1, 1627.73 8.15 ** .004 # 2 Trial 1, 1592.43 7.57 ** .006 # 3 PrevType 1, 1605.39 0.17 .68 # 4 meanWeight 1, 75.39 14.85 *** .0002 # 5 Frequency 1, 76.08 56.53 *** <.0001 # 6 NativeLanguage 1, 27.11 0.70 .41 # 7 Length 1, 75.83 8.70 ** .004 # 8 PrevType:meanWeight 1, 1601.18 6.18 * .01 # 9 NativeLanguage:Length 1, 1555.49 14.24 *** .0002 # --- # Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘+’ 0.1 ‘ ’ 1 # Fitting a GLMM using parametric bootstrap: require("mlmRev") # for the data, see ?Contraception gm1 <- mixed(use ~ age + I(age^2) + urban + livch + (1 | district), method = "PB", family = binomial, data = Contraception, args_test = list(nsim = 10)) ## note that nsim = 10 is way too low for all real examples! ## End(Not run) ## Not run: ##################################### ## Interplay with effects packages ## ##################################### data("Machines", package = "MEMSS") # simple model with random-slopes for repeated-measures factor m1 <- mixed(score ~ Machine + (Machine|Worker), data=Machines, set_data_arg = TRUE) ## necessary for it to work! library("effects") Effect("Machine", m1$full_model) # not correct: # Machine effect # Machine # A B C # 59.65000 52.35556 60.32222 # compare: emmeans(m1, "Machine") # Machine emmean SE df asymp.LCL asymp.UCL # A 52.35556 1.680711 Inf 49.06142 55.64969 # B 60.32222 3.528546 Inf 53.40640 67.23804 # C 66.27222 1.806273 Inf 62.73199 69.81245 ## necessary to set contr.sum globally: set_sum_contrasts() Effect("Machine", m1$full_model) # Machine effect # Machine # A B C # 52.35556 60.32222 66.27222 plot(Effect("Machine", m1$full_model)) ## End(Not run)