library(nlme); library(performance); library(emmeans)
library(dplyr); library(broom.mixed); library(multcompView)
library(multcomp); library(ggplot2)12 Marginal Means & Contrasts
12.1 Background
To start off with, we need to define estimated marginal means (EMM). Estimated marginal means are defined as marginal means of a variable across all levels of other variables in a model, essentially giving a “population-level” average.
The emmeans package is one of the most commonly used package in R in determine EMMs. This package provides methods for obtaining EMMs (also known as least-squares means) for factor combinations in a variety of models. The emmeans package is one of several alternatives to facilitate post hoc methods application and contrast analysis. It is a relatively recent replacement for the lsmeans package that some R users may be familiar with. It is intended for use with a wide variety of ANOVA models, including repeated measures and nested designs(mixed models). This is a flexible package that comes with a set of detailed vignettes and works with a lot of different model objects.
In this chapter, we will demonstrate the extended use of the emmeans package to calculate estimated marginal means and contrasts.
To demonstrate the use of the emmeans() package. We will pull the model from split plot lesson (Chapter 6), where we evaluated the effect of Nitrogen and Variety on Oat yield. This data contains 6 blocks, 3 main plots (Variety) and 4 subplots (Nitrogen). The primary outcome variable was oat yield. To read more about the experiment layout details please read RCBD split-plot section in Chapter 6.
Marginal means using lmer and nlme
For demonstration of the emmeans package, we are fitting model with nlme package. Please note that code below calculating marginal means works for both lmer and nlme models.
Let’s start the analysis by loading the required libraries for fitting linear mixed models using nlme package.
12.2 Analysis Examples
12.2.1 Import data
Let’s import oats data from the MASS package.
To read more about data and model fitting explanation please refer to Chapter 6.
data1 <- MASS::oats12.2.2 Model fitting
model1 <- lme(Y ~ V + N + V:N ,
random = ~1|B/V,
data = data1,
na.action = na.exclude)
tidy(model1)Warning in tidy.lme(model1): ran_pars not yet implemented for multiple levels
of nesting# A tibble: 12 × 7
effect term estimate std.error df statistic p.value
<chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl>
1 fixed (Intercept) 80 9.11 45 8.78 2.56e-11
2 fixed VMarvellous 6.67 9.72 10 0.686 5.08e- 1
3 fixed VVictory -8.50 9.72 10 -0.875 4.02e- 1
4 fixed N0.2cwt 18.5 7.68 45 2.41 2.02e- 2
5 fixed N0.4cwt 34.7 7.68 45 4.51 4.58e- 5
6 fixed N0.6cwt 44.8 7.68 45 5.84 5.48e- 7
7 fixed VMarvellous:N0.2cwt 3.33 10.9 45 0.307 7.60e- 1
8 fixed VVictory:N0.2cwt -0.333 10.9 45 -0.0307 9.76e- 1
9 fixed VMarvellous:N0.4cwt -4.17 10.9 45 -0.383 7.03e- 1
10 fixed VVictory:N0.4cwt 4.67 10.9 45 0.430 6.70e- 1
11 fixed VMarvellous:N0.6cwt -4.67 10.9 45 -0.430 6.70e- 1
12 fixed VVictory:N0.6cwt 2.17 10.9 45 0.199 8.43e- 112.2.3 Check Model Assumptions
check_model(model1, check = c('normality', 'linearity'))
12.2.4 Model Inference
anova(model1, type = "marginal") numDF denDF F-value p-value
(Intercept) 1 45 77.16729 <.0001
V 2 10 1.22454 0.3344
N 3 45 13.02273 <.0001
V:N 6 45 0.30282 0.932212.2.5 Estimated Marginal Means
Now that we have fitted a linear mixed model (model1) and it meets the model assumption. Let’s use the emmeans() function to obtain estimated marginal means for main (variety and nitrogen) and interaction (variety x nitrogen) effects.
12.2.5.1 Main effects
m1 <- emmeans(model1, ~V, level = 0.95)NOTE: Results may be misleading due to involvement in interactionsm1 V emmean SE df lower.CL upper.CL
Golden.rain 104.5 7.8 5 84.5 125
Marvellous 109.8 7.8 5 89.7 130
Victory 97.6 7.8 5 77.6 118
Results are averaged over the levels of: N
Degrees-of-freedom method: containment
Confidence level used: 0.95 m2 <- emmeans(model1, ~N)NOTE: Results may be misleading due to involvement in interactionsm2 N emmean SE df lower.CL upper.CL
0.0cwt 79.4 7.17 5 60.9 97.8
0.2cwt 98.9 7.17 5 80.4 117.3
0.4cwt 114.2 7.17 5 95.8 132.7
0.6cwt 123.4 7.17 5 104.9 141.8
Results are averaged over the levels of: V
Degrees-of-freedom method: containment
Confidence level used: 0.95 Make sure to read and interpret EMMs carefully. Here, when we calculated EMMs for main effects of V and N, these were averaged over the levels of other factor in experiment. For example, estimated means for each variety were averaged over it’s N treatments, respectively.
12.2.5.2 Interaction effects
Now let’s evaluate the interaction effect EMMs for V and N. These can be calculated either using V*N or V|N.
m3 <- emmeans(model1, ~V*N)
m3 V N emmean SE df lower.CL upper.CL
Golden.rain 0.0cwt 80.0 9.11 5 56.6 103.4
Marvellous 0.0cwt 86.7 9.11 5 63.3 110.1
Victory 0.0cwt 71.5 9.11 5 48.1 94.9
Golden.rain 0.2cwt 98.5 9.11 5 75.1 121.9
Marvellous 0.2cwt 108.5 9.11 5 85.1 131.9
Victory 0.2cwt 89.7 9.11 5 66.3 113.1
Golden.rain 0.4cwt 114.7 9.11 5 91.3 138.1
Marvellous 0.4cwt 117.2 9.11 5 93.8 140.6
Victory 0.4cwt 110.8 9.11 5 87.4 134.2
Golden.rain 0.6cwt 124.8 9.11 5 101.4 148.2
Marvellous 0.6cwt 126.8 9.11 5 103.4 150.2
Victory 0.6cwt 118.5 9.11 5 95.1 141.9
Degrees-of-freedom method: containment
Confidence level used: 0.95 m4 <- emmeans(model1, ~V|N)
m4N = 0.0cwt:
V emmean SE df lower.CL upper.CL
Golden.rain 80.0 9.11 5 56.6 103.4
Marvellous 86.7 9.11 5 63.3 110.1
Victory 71.5 9.11 5 48.1 94.9
N = 0.2cwt:
V emmean SE df lower.CL upper.CL
Golden.rain 98.5 9.11 5 75.1 121.9
Marvellous 108.5 9.11 5 85.1 131.9
Victory 89.7 9.11 5 66.3 113.1
N = 0.4cwt:
V emmean SE df lower.CL upper.CL
Golden.rain 114.7 9.11 5 91.3 138.1
Marvellous 117.2 9.11 5 93.8 140.6
Victory 110.8 9.11 5 87.4 134.2
N = 0.6cwt:
V emmean SE df lower.CL upper.CL
Golden.rain 124.8 9.11 5 101.4 148.2
Marvellous 126.8 9.11 5 103.4 150.2
Victory 118.5 9.11 5 95.1 141.9
Degrees-of-freedom method: containment
Confidence level used: 0.95 12.3 Contrasts using emmeans
Firstly, the pairs() function from emmeans package can be used to evaluate the pairwise comparison among treatment objects. The emmean object (m1, m2) will be passed through pairs() function which will provide a p-value adjustment equivalent to the Tukey test.
pairs(m1, adjust = "tukey") contrast estimate SE df t.ratio p.value
Golden.rain - Marvellous -5.29 7.08 10 -0.748 0.7419
Golden.rain - Victory 6.88 7.08 10 0.971 0.6104
Marvellous - Victory 12.17 7.08 10 1.719 0.2458
Results are averaged over the levels of: N
Degrees-of-freedom method: containment
P value adjustment: tukey method for comparing a family of 3 estimates pairs(m2) contrast estimate SE df t.ratio p.value
0.0cwt - 0.2cwt -19.50 4.44 45 -4.396 0.0004
0.0cwt - 0.4cwt -34.83 4.44 45 -7.853 <.0001
0.0cwt - 0.6cwt -44.00 4.44 45 -9.919 <.0001
0.2cwt - 0.4cwt -15.33 4.44 45 -3.457 0.0064
0.2cwt - 0.6cwt -24.50 4.44 45 -5.523 <.0001
0.4cwt - 0.6cwt -9.17 4.44 45 -2.067 0.1797
Results are averaged over the levels of: V
Degrees-of-freedom method: containment
P value adjustment: tukey method for comparing a family of 4 estimates Here if we look at the results from code chunk above, it’s easy to interpret results from pairs() function in case of varietey comparison becuase there were only 3 groups. It’s bit confusing in case of Nitrogen treatments where we had 4 groups. We can further simplify it by using custom contrasts.
pairs()
Remember!! The pairs() function can be used to calculate pairwise comparison when treatment groups are less than equal to 3.
12.3.1 Custom contrasts
Firstly, let’s run emmean object ‘m2’ for nitrogen treatment comparison.
m2 N emmean SE df lower.CL upper.CL
0.0cwt 79.4 7.17 5 60.9 97.8
0.2cwt 98.9 7.17 5 80.4 117.3
0.4cwt 114.2 7.17 5 95.8 132.7
0.6cwt 123.4 7.17 5 104.9 141.8
Results are averaged over the levels of: V
Degrees-of-freedom method: containment
Confidence level used: 0.95 Now, lets a create a vector for each nitrogen treatment in the same order as presented in output from m2.
A1 = c(1, 0, 0, 0)
A2 = c(0, 1, 0, 0)
A3 = c(0, 0, 1, 0)
A4 = c(0, 0, 0, 1)These vectors (A1, A2, A3, A4) represent each Nitrogen treatment in an order as presented in m2 emmeans object. A1, A2, and A3, A4 vectors represents 0.0cwt, 0.2cwt, 0.4cwt, and 0.6cwt treatments, respectively.
Let’s create custom contrasts for comparing ‘0.0cwt’ (A1) treatment to ‘0.2cwt’ (A2), ‘0.4cwt’ (A3), and ‘0.6cwt’ (A4) treatments. This can be evaluated as shown below: Here the output shows the difference in mean yield between these two varieties
contrast(m2, method = list(A1 - A2) ) contrast estimate SE df t.ratio p.value
c(1, -1, 0, 0) -19.5 4.44 45 -4.396 0.0001
Results are averaged over the levels of: V
Degrees-of-freedom method: containment contrast(m2, method = list(A1 - A3) ) contrast estimate SE df t.ratio p.value
c(1, 0, -1, 0) -34.8 4.44 45 -7.853 <.0001
Results are averaged over the levels of: V
Degrees-of-freedom method: containment contrast(m2, method = list(A1 - A4) ) contrast estimate SE df t.ratio p.value
c(1, 0, 0, -1) -44 4.44 45 -9.919 <.0001
Results are averaged over the levels of: V
Degrees-of-freedom method: containment Using custom contrasts is strongly recommended instead of pairs() when you are comparing multiple treatment groups (>5).
12.4 Compact letter displays
Compact letter displays (CLDs) are a popular way to display multiple comparisons when there are more than few group means to compare. However, they are problematic as they are more prone to misinterpretation. The R package multcompView (Graves et al., 2019) provides an implementation of CLDs creating a display where any two means associated with same symbol are not statistically different.
The cld() function from the multcomp package is used to implement CLDs in the form of symbols or letters. The emmeans package provides a emmGrid objects for cld() method.
Let’s start evaluating CLDs for main effects. We will use emmean objects m1 and m2 for this. In the output below, groups sharing a letter in the .group are not statistically different from each other.
cld(m1, alpha=0.05, Letters=letters) V emmean SE df lower.CL upper.CL .group
Victory 97.6 7.8 5 77.6 118 a
Golden.rain 104.5 7.8 5 84.5 125 a
Marvellous 109.8 7.8 5 89.7 130 a
Results are averaged over the levels of: N
Degrees-of-freedom method: containment
Confidence level used: 0.95
P value adjustment: tukey method for comparing a family of 3 estimates
significance level used: alpha = 0.05
NOTE: If two or more means share the same grouping symbol,
then we cannot show them to be different.
But we also did not show them to be the same. cld(m2, alpha=0.05, Letters=letters) N emmean SE df lower.CL upper.CL .group
0.0cwt 79.4 7.17 5 60.9 97.8 a
0.2cwt 98.9 7.17 5 80.4 117.3 b
0.4cwt 114.2 7.17 5 95.8 132.7 c
0.6cwt 123.4 7.17 5 104.9 141.8 c
Results are averaged over the levels of: V
Degrees-of-freedom method: containment
Confidence level used: 0.95
P value adjustment: tukey method for comparing a family of 4 estimates
significance level used: alpha = 0.05
NOTE: If two or more means share the same grouping symbol,
then we cannot show them to be different.
But we also did not show them to be the same. Let’s have a look at the CLDs for the interaction effect:
cld3 <- cld(m3, alpha=0.05, Letters=letters)
cld3 V N emmean SE df lower.CL upper.CL .group
Victory 0.0cwt 71.5 9.11 5 48.1 94.9 a
Golden.rain 0.0cwt 80.0 9.11 5 56.6 103.4 abcde
Marvellous 0.0cwt 86.7 9.11 5 63.3 110.1 abc fg
Victory 0.2cwt 89.7 9.11 5 66.3 113.1 ab d f h
Golden.rain 0.2cwt 98.5 9.11 5 75.1 121.9 abcdefghi
Marvellous 0.2cwt 108.5 9.11 5 85.1 131.9 abcdefghi
Victory 0.4cwt 110.8 9.11 5 87.4 134.2 bcdefghi
Golden.rain 0.4cwt 114.7 9.11 5 91.3 138.1 fghi
Marvellous 0.4cwt 117.2 9.11 5 93.8 140.6 de hi
Victory 0.6cwt 118.5 9.11 5 95.1 141.9 c e g i
Golden.rain 0.6cwt 124.8 9.11 5 101.4 148.2 fghi
Marvellous 0.6cwt 126.8 9.11 5 103.4 150.2 hi
Degrees-of-freedom method: containment
Confidence level used: 0.95
P value adjustment: tukey method for comparing a family of 12 estimates
significance level used: alpha = 0.05
NOTE: If two or more means share the same grouping symbol,
then we cannot show them to be different.
But we also did not show them to be the same. Interpretation of these letters is: Here we have a significant difference in grain yield with varieties “victory”, with N treatments of 0.0cwt, 0.2cwt, 0.4cwt, and 0.6wt. Grain yield for Golden.rain variety was significantly lower with 0.0cwt N treatment compared to the 0.2cwt, 0.4cwt, and 0.6wt treatments.
In the data set we used for demonstration here, we had equal number of observations in each group. However, this might not be a case every time as it is common to have missing values in the data set. In such cases, readers usually struggle to interpret significant differences among groups. For example, estimated means of two groups are substantially different but they are no statistically different. This normally happens when SE of one group is large due to its small sample size, so it’s hard for it to be statistically different from other groups. In such cases, we can use alternatives to CLDs as shown below.
12.5 Alternatives to CLD
- Equivalence test
Let’s assume based on subject matter considerations, if mean yield of two groups differ by less than 30 can be considered equivalent. Let’s try equivalence test on clds of nitrogen treatment emmeans (m2)
cld(m2, delta = 30, adjust = "none") N emmean SE df lower.CL upper.CL .equiv.set
0.0cwt 79.4 7.17 5 60.9 97.8 1
0.2cwt 98.9 7.17 5 80.4 117.3 12
0.4cwt 114.2 7.17 5 95.8 132.7 23
0.6cwt 123.4 7.17 5 104.9 141.8 3
Results are averaged over the levels of: V
Degrees-of-freedom method: containment
Confidence level used: 0.95
Statistics are tests of equivalence with a threshold of 30
P values are left-tailed
significance level used: alpha = 0.05
Estimates sharing the same symbol test as equivalent Here, two treatment groups ‘0.0cwt’ and ‘0.2cwt’, ‘0.4cwt’ and ‘0.6cwt’ can be considered equivalent.
- Significance Sets
Another alternative is to simply reverse all the boolean flags we used in constructing CLDs for m3 first time.
cld(m2, signif = TRUE) N emmean SE df lower.CL upper.CL .signif.set
0.0cwt 79.4 7.17 5 60.9 97.8 12
0.2cwt 98.9 7.17 5 80.4 117.3 12
0.4cwt 114.2 7.17 5 95.8 132.7 1
0.6cwt 123.4 7.17 5 104.9 141.8 2
Results are averaged over the levels of: V
Degrees-of-freedom method: containment
Confidence level used: 0.95
P value adjustment: tukey method for comparing a family of 4 estimates
significance level used: alpha = 0.05
Estimates sharing the same symbol are significantly different
Cautionary Note about CLD
It’s important to note that we cannot conclude that treatment levels with the same letter are the same. We can only conclude that they are not different.
There is a separate branch of statistics, “equivalence testing” that is for ascertaining if things are sufficiently similar to conclude they are equivalent.
See Section 2.0.4 for additional warnings about problems with using compact letter display.
12.6 Export emmeans to excel sheet
The outputs from emmeans() or cld() objects can exported by firstly converting outputs to a data frame and then using writexlsx() function from the ‘writexl’ package to export the outputs.
result_n <- as.data.frame(summary(m1))writexl::write_xlsx(result_n)12.7 Graphical display of emmeans
The results of emmeans() object can be plotted in two different ways. Firstly, we can use base plot() function in R.
plot(m1)
plot(m4)
Or we can use ‘ggplot2’ library. We can plot cld3 object in ggplot, with Variety on x-axis and estimated means of yield on y-axis. Different N treatments are presented in groups of different colors.
ggplot(cld3) +
aes(x = V, y = emmean, color = N) +
geom_point(position = position_dodge(width = 0.9)) +
geom_errorbar(mapping = aes(ymin = lower.CL, ymax = upper.CL),
position = position_dodge(width = 1),
width = 0.1) +
geom_text(mapping = aes(label = .group, y = upper.CL * 1.05),
position = position_dodge(width = 0.8),
show.legend = F)+
theme_bw()+
theme(axis.text= element_text(color = "black",
size =12))
Recall: groups that do not differ significantly from each other share the same letter.
we can also use emmip() built in emmeans package to look at the trend in interaction of variety and nitrogen factors.
emmip(model1, N ~ V)
More details on emmeans
If you want to read more about emmeans, please refer to vignettes on this CRAN page.
12.8 Conclusion
Be cautious with the terms “significant” and “nonsignificant”, and don’t ever interpret a “nonsignificant” result as saying that there is no effect. Follow good statistical practices such as getting the model right first, and using adjusted P values for appropriately chosen families of comparisons or contrasts.
P values, “significance”, and recommendations
P values are often misinterpreted, and the term “statistical significance” can be misleading. Please refer to this link to read more about basic principles outlined by the American Statistical Association when considering p-values.
If you want to read more about emmeans, please refer to vignettes on this CRAN page.