Before you read any farther

Some constraints I’m placing on this post that may impact your desire to read it.

1. Yes ANOVA is a subset of the general linear model and specifically in R aov is just a wrapper around lm. I can’t tell you the number of times I read that and it’s true. But, in many disciplines and for me it is an important subset worthy of it’s own time. You won’t find any info here about how to do things with lm although you certainly could.
2. Yes contrasts can play an important role in understanding your results. This is especially true if you happen to have an unbalanced design. I’m only going to glance on the topic here since I’m going to run a balanced design and therefore contrasts are tangential. For a very good academic discussion of contrasts especially using R and especially at the introductory level I very strongly recommend Learning Statistics with R, (search and ye shall find) where Danielle has done a great job of covering the topic in several places. The text is free for the download and is invaluable although it doesn’t cover ANCOVA per se.
3. I’m going to use packages above and beyond base and stats as needed. Everything could be done with the basics but this is a practical approach not so much a theoretical approach. I’ll even put in a very subtle plug for a function I wrote and maintain on CRAN although it’s by no means required.

N.B. – I don’t expect this will reach the New York Times best-seller list but questions or comments if it useful or if I have missed something are most certainly welcome.

Background and problem statement

The Wikipedia definition of ANCOVA is actually quite good and I won’t bother to repeat it. Some other keys phrases you’ll hear are that ANCOVA allows you to “control for” or “partial out” the covariate which gives you the opportunity to estimate partial means or marginal means which at the end of the day is why one uses ANOVA/ANCOVA versus regression. They use the same math but ANOVA/ANCOVA typically reports in means and mean differences while regressions reports on the slopes of the regression terms. Yes you can move back and forth but many disciplines have a preference and it can be one less step.

I wanted to make sure I used a dataset that was easily accessible. I mean that both in terms of being available in a package that most users will all ready have as well on a topic that many will find intuitively understandable with no specialized knowledge. I selected the diamonds dataset from ggplot2. I’ll cut it down some and balance it but those are trivial steps that will hopefully make things clearer.

So imagine that you’re shopping for a diamond. You’d like to get the possible value for the money you spend but you have very little knowledge about what influences the price you pay versus the value you get. The sales people speak about various things that influence the price such as “cut” and “clarity” and “color” and “carats”. You don’t have a lot to spend so you’re going to limit yourself to something modest but you’d like to know you got good value for what you paid. Enter the diamond dataset from ggplot2 if you have library available you can get a terse description with ?diamonds.

Let’s say for the sake of argument you’d like to know more about how these factors of cut and color impact the price you’ll pay. Let’s go ahead and get things set up in R so we can proceed, load the right libraries etc.. You’ll see in the code I recommend grabbing the latest version of a package I maintain but it is totally optional and there’s nothing there you can’t do for yourself if you prefer. I just wrote it so I didn’t have to remember to repeat a bunch of steps in R to run a 2Way ANOVA. You can see the docs here. I’m going to assume you’re comfortable with a basic ANOVA although you’re welcome to review the vignette if that’s helpful. So let’s load the libraries (I’ve suppressed all the messages here), and check out the structure of the dataset.

require(car) # get the right sums of squares calculations
require(dplyr) # for manipulating our data
require(ggplot2) # for plotting and for our dataset
require(sjstats) # save us time computing key ANOVA stats beyond car
require(broom) # nice for making our results in neat tibbles
require(emmeans) # for marginal means calculations
# a shameless plug for a function I wrote called Plot2WayANOVA
# optional for you
devtools::install_github("ibecav/CGPfunctions")
library(CGPfunctions)
theme_set(theme_bw()) # set theme
str(diamonds)
## Classes 'tbl_df', 'tbl' and 'data.frame':    53940 obs. of  10 variables:
##  $ carat  : num  0.23 0.21 0.23 0.29 0.31 0.24 0.24 0.26 0.22 0.23 ...
##  $ cut    : Ord.factor w/ 5 levels "Fair"<"Good"<..: 5 4 2 4 2 3 3 3 1 3 ...
##  $ color  : Ord.factor w/ 7 levels "D"<"E"<"F"<"G"<..: 2 2 2 6 7 7 6 5 2 5 ...
##  $ clarity: Ord.factor w/ 8 levels "I1"<"SI2"<"SI1"<..: 2 3 5 4 2 6 7 3 4 5 ...
##  $ depth  : num  61.5 59.8 56.9 62.4 63.3 62.8 62.3 61.9 65.1 59.4 ...
##  $ table  : num  55 61 65 58 58 57 57 55 61 61 ...
##  $ price  : int  326 326 327 334 335 336 336 337 337 338 ...
##  $ x      : num  3.95 3.89 4.05 4.2 4.34 3.94 3.95 4.07 3.87 4 ...
##  $ y      : num  3.98 3.84 4.07 4.23 4.35 3.96 3.98 4.11 3.78 4.05 ...
##  $ z      : num  2.43 2.31 2.31 2.63 2.75 2.48 2.47 2.53 2.49 2.39 ...

Okay just crossing cut and color would give us 35 cells in a table of means ( 5 levels times 7 levels). I’ve also admitted I’m a cheapskate and don’t want a huge diamond so let’s pare our data down a bit to something more manageable. So let’s use dplyr to create a subset of the data where we focus on “fair” and “good” cuts, and colors “E”, “F”, and “G” with a carat weight of less than 1.75. This is also a good time to admit I cheated and peeked and saw that the data were likely to be very unbalanced table(diamonds$color, diamonds$cut), so at this point I’m also going to force our data into a balanced design by ensuring that we randomly sample the same number of data points into each cell. I’ve used set.seed so you should be able to reproduce the same dataset if you choose.

set.seed(1234)
diamonds2 <- filter(diamonds, 
                    cut %in% c("Fair", "Good") & 
                    color %in% c("E", "F", "G") & 
                    carat < 1.75) 
diamonds2 <- droplevels(diamonds2)
one <- diamonds2 %>% filter(cut == "Fair" & color == "E") %>% sample_n(218)
two <- diamonds2 %>% filter(cut == "Fair" & color == "F") %>% sample_n(218)
three <- diamonds2 %>% filter(cut == "Fair" & color == "G") %>% sample_n(218)
four <- diamonds2 %>% filter(cut == "Good" & color == "E") %>% sample_n(218)
five <- diamonds2 %>% filter(cut == "Good" & color == "F") %>% sample_n(218)
six <- diamonds2 %>% filter(cut == "Good" & color == "G") %>% sample_n(218)
diamonds2 <- bind_rows(one, two, three, four, five, six)
str(diamonds2)
## Classes 'tbl_df', 'tbl' and 'data.frame':    1308 obs. of  10 variables:
##  $ carat  : num  0.92 1.01 0.5 1.32 1 0.5 0.31 0.51 0.34 0.4 ...
##  $ cut    : Ord.factor w/ 2 levels "Fair"<"Good": 1 1 1 1 1 1 1 1 1 1 ...
##  $ color  : Ord.factor w/ 3 levels "E"<"F"<"G": 1 1 1 1 1 1 1 1 1 1 ...
##  $ clarity: Ord.factor w/ 8 levels "I1"<"SI2"<"SI1"<..: 3 3 3 2 6 2 4 3 7 4 ...
##  $ depth  : num  65.6 65.3 66.5 65.2 65.4 58 54.2 55.8 54 64.7 ...
##  $ table  : num  57 59 58 58 56 67 63 61 56 58 ...
##  $ price  : int  3407 4844 1098 6221 7918 851 814 1340 1012 813 ...
##  $ x      : num  6.1 6.13 4.87 6.86 6.28 5.26 4.61 5.43 4.8 4.63 ...
##  $ y      : num  6.07 6.19 4.9 6.79 6.2 5.17 4.51 5.35 4.76 4.67 ...
##  $ z      : num  3.99 4.02 3.25 4.45 4.08 3.02 2.47 3.01 2.58 3.01 ...

A note on balanced designs

As I noted earlier I’m really not interested in digressing to talk about why the concept of a balanced design is important to your work. Please do consult Learning Statistics with R for more details. At this point I’m simply going to encourage you to always use “Type II” sums of squares from the car package if there is any chance your design is unbalanced. If your design is balanced they give the same answer. The contrast you choose is also if you are unbalanced and are using Type III.

Are unbalanced designs completely wrong and to be avoided at all costs? Not exactly… here are three things they impact in order of likelihood.

1. They always impact your power, you ability to detect significant differences. Your power is limited by the size of your smallest cell.
2. They usually impact your ability to divide the sums of squares cleanly to 100%. You may wind up with unexplained variance that is due to an effect but you won’t which effect. This is different than unexplained (residual variance).
3. The least likely but most worrisome is that it will mask an important relationship in your data.

Back to the diamonds

Before we look at ANCOVA lets run an ANOVA. We have two ordinal factors for independent (predictor) variables cut and color and one dependent (outcome) variable the price. A classic two-way ANOVA. We could simply run `aov(price ~ cut * color, diamonds2) and then a bunch of other commands to get the information we need. I found that a bit tedious and annoying plus I wanted to be able to plot the results to look at any possible interactions. So I wrote a function. Everything in it you can do by hand but I think it does a pretty good job wrapping the process in one function. So …

Plot2WayANOVA(price ~ color * cut, 
              diamonds2,
              mean.label = TRUE)
## 
## You have a balanced design.
##        term        sumsq     meansq   df statistic p.value etasq
## 1     color 4.807090e+06 2403545.03    2     0.325   0.722 0.000
## 2       cut 1.235500e+02     123.55    1     0.000   0.997 0.000
## 3 color:cut 7.664167e+06 3832083.61    2     0.519   0.595 0.001
## 4 Residuals 9.614813e+09 7384649.03 1302        NA      NA    NA
##   partial.etasq omegasq partial.omegasq cohens.f power
## 1         0.000  -0.001          -0.001    0.022 0.102
## 2         0.000  -0.001          -0.001    0.000 0.050
## 3         0.001  -0.001          -0.001    0.028 0.136
## 4            NA      NA              NA       NA    NA
## 
## Measures of overall model fit
## # A tibble: 1 x 11
##   r.squared adj.r.squared sigma statistic p.value    df  logLik    AIC
##       <dbl>         <dbl> <dbl>     <dbl>   <dbl> <int>   <dbl>  <dbl>
## 1   0.00130      -0.00254 2717.     0.338   0.890     6 -12196. 24406.
## # … with 3 more variables: BIC <dbl>, deviance <dbl>, df.residual <int>
## 
## Table of group means
## # A tibble: 6 x 9
## # Groups:   color [3]
##   color cut   TheMean TheSD TheSEM CIMuliplier LowerBound UpperBound     N
##   <ord> <ord>   <dbl> <dbl>  <dbl>       <dbl>      <dbl>      <dbl> <int>
## 1 E     Fair    3406. 2485.   168.        1.97      3075.      3738.   218
## 2 E     Good    3425. 3260.   221.        1.97      2989.      3860.   218
## 3 F     Fair    3392. 2484.   168.        1.97      3061.      3724.   218
## 4 F     Good    3196. 2698.   183.        1.97      2836.      3556.   218
## 5 G     Fair    3340. 2327.   158.        1.97      3030.      3651.   218
## 6 G     Good    3517. 2940.   199.        1.97      3125.      3910.   218
## 
## Post hoc tests for all effects that were significant
## [1] "No signfiicant effects"
## 
## Testing Homogeneity of Variance with Brown-Forsythe
##    *** Possible violation of the assumption ***
## Levene's Test for Homogeneity of Variance (center = median)
##         Df F value  Pr(>F)   
## group    5  3.2086 0.00695 **
##       1302                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Testing Normality Assumption with Shapiro-Wilk
##    *** Possible violation of the assumption.  You may 
##             want to plot the residuals to see how they vary from normal ***
## 
##  Shapiro-Wilk normality test
## 
## data:  MyAOV_residuals
## W = 0.84893, p-value < 2.2e-16
## 
## Interaction graph plotted...

It does it’s job of plotting the results and providing you with nice summaries of not just the ANOVA table but the table of means, post hoc tests if needed and even testing classic assumptions. it also tests for whether your design is balanced and always uses type II sums of squares.

Well this is unfortunate! Looks like I won’t be publishing the results in JODR - The Journal of Obscure Diamond Results since not a single one of my ANOVA terms is significant at the p<.05 level. The model seems to be a terrible fit whether I look at R Squared or AIC or BIC, neither cut nor color seem to matter although the interaction term is “marginally significant”.

Should we conclude cut and color don’t matter? Are they just immaterial with no discernible impact on pricing?

ANCOVA helps our understanding

As you have probably already guessed that’s not where we’re heading. Remember that Wikipedia article? A telling quote from Tabachnick, B. G. and Fidell, L. S. (2007) is in there…

ANCOVA can be used to increase statistical power (the probability a significant difference is found between groups when one exists) by reducing the within-group error variance.

So with ANCOVA we’re going to add one or more continuous variables known as covariates which are “hiding” the relationship between our factors of interest cut and color, if we can control for or partial the covariate out then we’ll hopefully be likely to “see” the impact of cut and color on price. It should be something we know is linearly related to price but distinct from cut and color.

So looking at our data above it hopefully is becoming obvious to you. On surface to meet the criteria of being a continuous numeric variable we have "carat", "depth", "table", "x", "y" and "z". Since clarity is ordered we could force it to an integer and use it but let’s not. Speaking for me, I’m pretty sure size as measured in weight carat is going to be related to price of the diamond. I don’t know much about diamonds but I’ve heard size adds to the cost…

So what we want to do now is confirm our hunch it is linearly related as well as hope that it is not strongly correlated with our current predictors cut and color we’re looking for a variable that reduces the variance but is not entangled with our current IVs. So first a scatter plot with the regression and loess lines followed by a glance at the linear models price ~ carat, carat ~ cut, and carat ~ color.

ggplot(diamonds2, aes(x = carat, y= price)) + 
  geom_point(alpha = 1/10) + 
  geom_smooth(method = "loess", color = "red") + 
  geom_smooth(method = "lm", color = "blue")

broom::glance(lm(price ~ carat, diamonds2))
## # A tibble: 1 x 11
##   r.squared adj.r.squared sigma statistic p.value    df  logLik    AIC
##       <dbl>         <dbl> <dbl>     <dbl>   <dbl> <int>   <dbl>  <dbl>
## 1     0.756         0.756 1340.     4057.       0     2 -11273. 22552.
## # … with 3 more variables: BIC <dbl>, deviance <dbl>, df.residual <int>
broom::glance(lm(carat ~ cut, diamonds2))
## # A tibble: 1 x 11
##   r.squared adj.r.squared sigma statistic p.value    df logLik   AIC   BIC
##       <dbl>         <dbl> <dbl>     <dbl>   <dbl> <int>  <dbl> <dbl> <dbl>
## 1    0.0190        0.0182 0.332      25.2 5.75e-7     2  -411.  828.  843.
## # … with 2 more variables: deviance <dbl>, df.residual <int>
broom::glance(lm(carat ~ color, diamonds2))
## # A tibble: 1 x 11
##   r.squared adj.r.squared sigma statistic p.value    df logLik   AIC   BIC
##       <dbl>         <dbl> <dbl>     <dbl>   <dbl> <int>  <dbl> <dbl> <dbl>
## 1   0.00292       0.00139 0.334      1.91   0.148     3  -421.  851.  872.
## # … with 2 more variables: deviance <dbl>, df.residual <int>
caratonly <- lm(price ~ carat, diamonds2)

Excellent news! Our potential covariate carat is highly correlated with price (r = 0.87) while having near zero correlations with cut (r = 0.14) and color (r = 0.05).

Comparing models

We’re going to compare two different models in a step by step fashion using the same tools in R for each step, aov to create the model, car::Anova to display the ANOVA table and ensure we’re using type 2 sums of squares. broom::glance to get information about overall fir like R squared and AIC, and finally sjstats::anova_stats to give us a nice clean display including effect sizes for the terms.

So our original model without a covariate looked like this (you can scroll back and look but I assure you this is it).

noCOVmodel <- aov(price ~ cut * color, diamonds2)
car::Anova(noCOVmodel, type = 2)
## Anova Table (Type II tests)
## 
## Response: price
##               Sum Sq   Df F value Pr(>F)
## cut              124    1  0.0000 0.9967
## color        4807090    2  0.3255 0.7222
## cut:color    7664167    2  0.5189 0.5953
## Residuals 9614813032 1302
broom::glance(noCOVmodel)
## # A tibble: 1 x 11
##   r.squared adj.r.squared sigma statistic p.value    df  logLik    AIC
##       <dbl>         <dbl> <dbl>     <dbl>   <dbl> <int>   <dbl>  <dbl>
## 1   0.00130      -0.00254 2717.     0.338   0.890     6 -12196. 24406.
## # … with 3 more variables: BIC <dbl>, deviance <dbl>, df.residual <int>
sjstats::anova_stats(car::Anova(noCOVmodel, type = 2)) %>% select(1:7)
##        term        sumsq     meansq   df statistic p.value etasq
## 1       cut 1.235500e+02     123.55    1     0.000   0.997 0.000
## 2     color 4.807090e+06 2403545.03    2     0.325   0.722 0.000
## 3 cut:color 7.664167e+06 3832083.61    2     0.519   0.595 0.001
## 4 Residuals 9.614813e+09 7384649.03 1302        NA      NA    NA

Adding carat to the model

Let’s add another term to the model to include carat, we’re not going to let it interact with the other factors so we’ll use a plus sign. There’s nothing especially tricky about this, we’re just adding another predictor to our model, it’s not exactly traditional ANOVA because it’s a continuous numeric variable rather than a factor, but it’s simple to imagine.

COVmodel <- aov(price ~ cut * color + carat, diamonds2)
car::Anova(COVmodel, type = 2)
## Anova Table (Type II tests)
## 
## Response: price
##               Sum Sq   Df   F value    Pr(>F)    
## cut        141145906    1   84.2441 < 2.2e-16 ***
## color       17038543    2    5.0848  0.006314 ** 
## carat     7435066381    1 4437.6815 < 2.2e-16 ***
## cut:color    7214868    2    2.1531  0.116534    
## Residuals 2179746650 1301                        
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
broomExtra::glance(COVmodel)
## # A tibble: 1 x 11
##   r.squared adj.r.squared sigma statistic p.value    df  logLik    AIC
##       <dbl>         <dbl> <dbl>     <dbl>   <dbl> <int>   <dbl>  <dbl>
## 1     0.774         0.773 1294.      741.       0     7 -11225. 22467.
## # … with 3 more variables: BIC <dbl>, deviance <dbl>, df.residual <int>
sjstats::anova_stats(car::Anova(COVmodel, type = 2)) %>% select(1:7)
##        term      sumsq     meansq   df statistic p.value etasq
## 1       cut  141145906  141145906    1    84.244   0.000 0.014
## 2     color   17038543    8519272    2     5.085   0.006 0.002
## 3     carat 7435066381 7435066381    1  4437.681   0.000 0.760
## 4 cut:color    7214868    3607434    2     2.153   0.117 0.001
## 5 Residuals 2179746650    1675439 1301        NA      NA    NA

Wow that sure changed our results didn’t it? Suddenly cut and color matter! Make no mistake they don’t have nearly the impact that carat does but at least we can reliably see their impact on price by measures such as eta squared (etasq).

That’s because we originally had 9,553,303,035 in residual sums of squares but by adding carat we’ve reduced that number to 2,308,863,799 which makes our factors (cut and color) (the numerators) much more potent.

A quick peek into what’s changing may help. Let’s build a little tibble that shows us what’s going on. The first three columns are straight from the diamonds2 dataset, our price, cut, and color. The next column shows the “prediction” made using our initial model. Every row with the same condition e.g. “Fair” & “E” gets the same entry “3406” the mean for the cell. We can see that when we use just carat as the predictor we get very different predictions (although of course the same size gets the same prediction). Our COVmodel predictions yield a third set of answers in the final column that makes use of all the information available.

diamonds3 <- diamonds2 %>% 
   mutate(OriginalPred = predict(noCOVmodel), 
          WithCaratPred = predict(COVmodel), 
          CaratOnlyPred = predict(caratonly)) %>% 
   select(price, cut, color, OriginalPred, carat, CaratOnlyPred, WithCaratPred)
diamonds3
## # A tibble: 1,308 x 7
##    price cut   color OriginalPred carat CaratOnlyPred WithCaratPred
##    <int> <ord> <ord>        <dbl> <dbl>         <dbl>         <dbl>
##  1  3407 Fair  E            3406.  0.92         4193.        4094. 
##  2  4844 Fair  E            3406.  1.01         4828.        4743. 
##  3  1098 Fair  E            3406.  0.5          1230.        1065. 
##  4  6221 Fair  E            3406.  1.32         7015.        6978. 
##  5  7918 Fair  E            3406.  1            4757.        4671. 
##  6   851 Fair  E            3406.  0.5          1230.        1065. 
##  7   814 Fair  E            3406.  0.31         -111.        -304. 
##  8  1340 Fair  E            3406.  0.51         1300.        1138. 
##  9  1012 Fair  E            3406.  0.34          101.         -88.2
## 10   813 Fair  E            3406.  0.4           524.         344. 
## # … with 1,298 more rows

More progress with emmeans

Okay, we’re making progress here but this isn’t all we can or should do. We have a good sense that adding carat as a covariate makes for a much more accurate model. But, we’re not interested in carat per se, it’s not that it’s unimportant (clearly it matters) we’re just interested in what happens to cut and color when we control for carat. That’s where the emmeans package can help.

The emmeans package allows us to take our model(s) and compute the estimated marginal means a.k.a. predicted model means or least squares means. The package includes functions to not only compute them but also plot them as well as make comparisons. We’ve already done that above for our model with no covariate noCOVmodel but let’s see what that looks like just as a baseline.

# first the means
emmeans::pmmeans(noCOVmodel, "cut", by = "color")
## color = E:
##  cut  pmmean  SE   df lower.CL upper.CL
##  Fair   3406 184 1302     3045     3768
##  Good   3425 184 1302     3063     3786
## 
## color = F:
##  cut  pmmean  SE   df lower.CL upper.CL
##  Fair   3392 184 1302     3031     3754
##  Good   3196 184 1302     2835     3557
## 
## color = G:
##  cut  pmmean  SE   df lower.CL upper.CL
##  Fair   3340 184 1302     2979     3701
##  Good   3517 184 1302     3156     3878
## 
## Confidence level used: 0.95
# then plot them
emmeans::emmip(noCOVmodel, cut ~ color, CIs = TRUE)

# pairwise comparisons
pairs(emmeans::pmmeans(noCOVmodel, "color", by = "cut"), adjust = "scheffe")
## cut = Fair:
##  contrast estimate  SE   df t.ratio p.value
##  E - F        14.0 260 1302  0.054  0.9985 
##  E - G        66.1 260 1302  0.254  0.9683 
##  F - G        52.1 260 1302  0.200  0.9802 
## 
## cut = Good:
##  contrast estimate  SE   df t.ratio p.value
##  E - F       228.9 260 1302  0.879  0.6794 
##  E - G       -92.6 260 1302 -0.356  0.9386 
##  F - G      -321.5 260 1302 -1.235  0.4665 
## 
## P value adjustment: scheffe method with dimensionality 2
# not done above you can easily specify just one factor
emmeans::pmmeans(noCOVmodel, "cut")
## NOTE: Results may be misleading due to involvement in interactions
##  cut  pmmean  SE   df lower.CL upper.CL
##  Fair   3380 106 1302     3171     3588
##  Good   3379 106 1302     3171     3588
## 
## Results are averaged over the levels of: color 
## Confidence level used: 0.95
# or the other factor
emmeans::pmmeans(noCOVmodel, "color")
## NOTE: Results may be misleading due to involvement in interactions
##  color pmmean  SE   df lower.CL upper.CL
##  E       3416 130 1302     3160     3671
##  F       3294 130 1302     3039     3549
##  G       3429 130 1302     3173     3684
## 
## Results are averaged over the levels of: cut 
## Confidence level used: 0.95

Controlling for carat

None of that information is what we’re after however. We have the other model COVmodel with carat added and what we need are the estimated means with carat controlled for, or partialled out. We want to know the predicted or estimated means for our 6 conditions as if size (carat) were controlled for.

emmeans::pmmeans(COVmodel, "carat")
##  carat pmmean   SE   df lower.CL upper.CL
##  0.805   3379 35.8 1301     3309     3450
## 
## Results are averaged over the levels of: cut, color 
## Confidence level used: 0.95
emmeans::pmmeans(COVmodel, "cut", by = "color")
## color = E:
##  cut  pmmean   SE   df lower.CL upper.CL
##  Fair   3263 87.7 1301     3091     3435
##  Good   3787 87.8 1301     3615     3959
## 
## color = F:
##  cut  pmmean   SE   df lower.CL upper.CL
##  Fair   3070 87.8 1301     2897     3242
##  Good   3666 88.0 1301     3494     3839
## 
## color = G:
##  cut  pmmean   SE   df lower.CL upper.CL
##  Fair   2811 88.0 1301     2638     2984
##  Good   3680 87.7 1301     3508     3852
## 
## Confidence level used: 0.95
emmip(COVmodel, cut ~ color, CIs = TRUE)

For me comparing the two plots tells a striking story about what role cut and color play if we separate out the effect of size. If you look t the tables of means you can see they are different but looking at the plot gives you a much better idea of just how much the pattern has changed.

We can also run the appropriate significance tests using the very conservative scheffe option.

pairs(emmeans::pmmeans(COVmodel, "cut", by = "color"), adjust = "scheffe")
## color = E:
##  contrast    estimate  SE   df t.ratio p.value
##  Fair - Good     -524 124 1301 -4.222  <.0001 
## 
## color = F:
##  contrast    estimate  SE   df t.ratio p.value
##  Fair - Good     -597 125 1301 -4.791  <.0001 
## 
## color = G:
##  contrast    estimate  SE   df t.ratio p.value
##  Fair - Good     -869 124 1301 -6.988  <.0001
pairs(emmeans::pmmeans(COVmodel, "color", by = "cut"), adjust = "scheffe")
## cut = Fair:
##  contrast estimate  SE   df t.ratio p.value
##  E - F       193.0 124 1301  1.556  0.2984 
##  E - G       451.7 124 1301  3.640  0.0014 
##  F - G       258.8 124 1301  2.087  0.1138 
## 
## cut = Good:
##  contrast estimate  SE   df t.ratio p.value
##  E - F       120.7 124 1301  0.974  0.6226 
##  E - G       106.8 124 1301  0.861  0.6903 
##  F - G       -13.9 124 1301 -0.112  0.9937 
## 
## P value adjustment: scheffe method with dimensionality 2
pairs(emmeans::pmmeans(COVmodel, "color"), adjust = "scheffe")
## NOTE: Results may be misleading due to involvement in interactions
##  contrast estimate   SE   df t.ratio p.value
##  E - F         157 87.7 1301 1.789   0.2022 
##  E - G         279 87.8 1301 3.182   0.0065 
##  F - G         122 87.8 1301 1.395   0.3781 
## 
## Results are averaged over the levels of: cut 
## P value adjustment: scheffe method with dimensionality 2
pairs(emmeans::pmmeans(COVmodel, "cut"), adjust = "scheffe")
## NOTE: Results may be misleading due to involvement in interactions
##  contrast    estimate   SE   df t.ratio p.value
##  Fair - Good     -664 72.3 1301 -9.181  <.0001 
## 
## Results are averaged over the levels of: color

Let’s use ggplot to combine the two manually into one plot. We’ll plot the original model with dashed lines and the new model with covariate in dark bold lines.

withCOV <- broom::tidy(emmeans::pmmeans(COVmodel, "cut", by = "color"))
noCOV <- broom::tidy(emmeans::pmmeans(noCOVmodel, "cut", by = "color"))
ggplot(data = withCOV, 
       aes(x = color, 
           y = estimate, 
           group = cut, 
           color = cut)) +
   geom_point(shape = 18, 
              size = 4) +
   geom_line(size = 2) +
   ggrepel::geom_label_repel(aes(label = round(estimate, 2)),
                             nudge_x = -.35,
                             color = "black") +
   geom_point(data = noCOV,
              aes(x = color, 
                  y = estimate, 
                  group = cut, 
                  color = cut)) +
   geom_line(data = noCOV,
              aes(x =color, 
                  y = estimate, 
                  group = cut, 
                  color = cut),
             linetype = 2) +
   ggrepel::geom_label_repel(data = noCOV,
                             aes(label = round(estimate, 2)),
                             nudge_x = .35,
                             color = "black") +
   labs(title = "Estimated Mean Diamond Price",
        subtitle = "Dashed line without carat as a covariate",
        color = "Cut") +
   ylab("Price") +
   xlab("Color") +
   expand_limits(y = c(2700,4200))

Measuring the effects

The final thing we’d like to do is to better our understanding of the “effect size” of color and cut when we control for carat. Earlier we ran sjstats::anova_stats(car::Anova(COVmodel, type = 2)) %>% select(1:7) but that isn’t quite what we want, since the information about carat is still influencing our computations. We want to remove carat from the calculations. The trick to doing that in R with aov is to make carat an Error term. It sounds strange in some ways if you ask me, but it is effective in getting what we want.

Under the hood aov will create a model with two strata, Stratum 1 is all about our covariate carat. It pulls out or controls for its influence so that Stratum 2 which is labelled “Within” now contains the ANOVA for the other variables controlling for carat. The Within label always puts me off a bit since it makes me want to think of a within subjects design (which this clearly isn’t). But it’s having just the impact we like when you inspect the output. cut and color do matter! There’s hope we’ll get published in the JODR yet.

COVmodelError <- aov(price ~ cut * color + Error(carat), diamonds2)
summary(COVmodelError)
## 
## Error: carat
##     Df    Sum Sq   Mean Sq
## cut  1 7.283e+09 7.283e+09
## 
## Error: Within
##             Df    Sum Sq   Mean Sq F value  Pr(>F)    
## cut          1 1.405e+08 140501272  83.859 < 2e-16 ***
## color        2 1.704e+07   8519272   5.085 0.00631 ** 
## cut:color    2 7.215e+06   3607434   2.153 0.11653    
## Residuals 1301 2.180e+09   1675439                    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
car::Anova(COVmodelError$Within, type = 2)
## Anova Table (Type II tests)
## 
## Response: price
##               Sum Sq   Df F value    Pr(>F)    
## cut        141145906    1 84.2441 < 2.2e-16 ***
## color       17038543    2  5.0848  0.006314 ** 
## cut:color    7214868    2  2.1531  0.116534    
## Residuals 2179746650 1301                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
broomExtra::glance(COVmodelError$Within)
## # A tibble: 1 x 11
##   r.squared adj.r.squared sigma statistic  p.value    df  logLik    AIC
##       <dbl>         <dbl> <dbl>     <dbl>    <dbl> <int>   <dbl>  <dbl>
## 1    0.0703        0.0674 1294.      24.6 1.22e-19     5 -11209. 22430.
## # … with 3 more variables: BIC <dbl>, deviance <dbl>, df.residual <int>
sjstats::anova_stats(car::Anova(COVmodelError$Within, type = 2))  %>% select(1:7)
##        term      sumsq    meansq   df statistic p.value etasq
## 1       cut  141145906 141145906    1    84.244   0.000 0.060
## 2     color   17038543   8519272    2     5.085   0.006 0.007
## 3 cut:color    7214868   3607434    2     2.153   0.117 0.003
## 4 Residuals 2179746650   1675439 1301        NA      NA    NA

What don’t we know?

1. That this example would apply to across the rest of the dataset. Remember that one of the steps we took early on was to select only certain levels of our factors color and cut and even after that to choose balance over completeness.

2. How badly we violated key assumptions about our data. We know we have some evidence of heteroskedasticity and non-normality. I also, for the sake of brevity, did not discuss interactions at the second order between the covariate and our factors.

Done!

I hope you’ve found this useful. I am always open to comments, corrections and suggestions.

Chuck (ibecav at gmail dot com)