The goal of supernova is to create ANOVA tables in the format used by
Judd, McClelland, and Ryan (2017, ISBN: 978-1138819832) in their
introductory textbook, Data Analysis: A Model Comparison Approach to
Regression, ANOVA, and Beyond (book
website).* These tables
include proportional reduction in error, a useful measure for teaching
the underlying concepts of ANOVA and regression, and formatting to ease
the transition between the book and R.
* Note: we are NOT affiliated with the authors or their institution.
In keeping with the approach in Judd, McClelland, and Ryan (2017), the ANOVA tables in this package are calculated using a model comparison approach that should be understandable given a beginner’s understanding of base R and the information from the book (so if you have these and don’t understand what is going on in the code, let us know because we are missing the mark!). Here is an explanation of how the tables are calculated for fully independent predictor variables (i.e. between-subjects designs):
-
The “Total” row is calculated by updating the model passed to an empty model. For example,
lm(mpg ~ hp * disp, data = mtcars)is updated tolm(mpg ~ NULL, data = mtcars). From this empty model, the sum of squares and df can be calculated. -
If there is at least one predictor in the model, the overall model row and the error row are calculated. In the vernacular of the book, the compact model is represented by the updated empty model from 1 above, and the augmented model is the original model passed to
supernova(). From these models the SSE(A) is calculated bysum(resid(null_model) ^ 2), the SSR is calculated by SSE(C) - SSE(A), the PRE for the overall model is extracted from the fit (r.squared), the df for the error row is extracted from thelm()fit (df.residuals). -
If there are more than one predictors, the single term deletions are computed using
drop1(). For the modely ~ a * b(which expands toy ~ a + b + a:b, wherea:bis the interaction ofaandb),drop1()essentially creates three models each with one term removed:y ~ a + a:b,y ~ b + a:b, andy ~ a + b. These models are considered the compact models which do not include the tested termsa,b, anda:b, respectively.drop1()computes the SSR (Sum Sq) and SSE(C) (RSS) for each of these augmented and compact model pairs, and these values are used to compute the SSR and PRE for each. -
Finally, the
MS(SS / df),F(MSR / MSE), andpcolumns are calculated from already-computed values in the table.
The following models are explicitly tested and supported by
supernova(), for independent samples (between-subjects) data only.
For these models, there is also support for datasets with missing or
unbalanced data.
- empty models:
y ~ NULL - simple regression:
y ~ a - multiple regression:
y ~ a + b - interactive regression:
y ~ a * b
Additionally, a subset of within-subjects designs are supported and
explicitly tested. To accommodate these models supernova() can accept
models fit via lmer() as in the Examples below. Only
models like those included in those examples have been tested for
within-subjects designs.
Anything not included above is not (yet) explicitly tested and may yield errors or incorrect statistics. This includes, but is not limited to
- one-sample t-tests
In addition to the ANOVA table provided by supernova(), the
supernova package provides some useful functions for teaching ANOVA
and pairwise comparisons:
Generate models: Generate the models that were compared to create
each row of an ANOVA table using generate_models(). This can be done
for each of the different SS Types as described in Using Different SS
Types below.
Pairwise comparisons: Test each categorical group in a model against
the others using pairwise(). This function supports Tukey and
Bonferroni corrections. See the Pairwise
Comparisons section below.
You can install the released version of supernova from CRAN with:
install.packages("supernova")Alternatively you can download the package directly from this repository
using remotes:
library(remotes)
install_github("UCLATALL/supernova")Here are some basic examples of the code and output for this package:
supernova(lm(mpg ~ NULL, data = mtcars))
#> Analysis of Variance Table (Type III SS)
#> Model: mpg ~ NULL
#>
#> SS df MS F PRE p
#> ----- --------------- | -------- --- ------ --- --- ---
#> Model (error reduced) | --- --- --- --- --- ---
#> Error (from model) | --- --- --- --- --- ---
#> ----- --------------- | -------- --- ------ --- --- ---
#> Total (empty model) | 1126.047 31 36.324supernova(lm(mpg ~ hp, data = mtcars))
#> Analysis of Variance Table (Type III SS)
#> Model: mpg ~ hp
#>
#> SS df MS F PRE p
#> ----- --------------- | -------- -- ------- ------ ----- -----
#> Model (error reduced) | 678.373 1 678.373 45.460 .6024 .0000
#> Error (from model) | 447.674 30 14.922
#> ----- --------------- | -------- -- ------- ------ ----- -----
#> Total (empty model) | 1126.047 31 36.324supernova(lm(mpg ~ hp + disp, data = mtcars))
#> Analysis of Variance Table (Type III SS)
#> Model: mpg ~ hp + disp
#>
#> SS df MS F PRE p
#> ----- --------------- | -------- -- ------- ------ ----- -----
#> Model (error reduced) | 842.554 2 421.277 43.095 .7482 .0000
#> hp | 33.665 1 33.665 3.444 .1061 .0737
#> disp | 164.181 1 164.181 16.795 .3667 .0003
#> Error (from model) | 283.493 29 9.776
#> ----- --------------- | -------- -- ------- ------ ----- -----
#> Total (empty model) | 1126.047 31 36.324supernova(lm(mpg ~ hp * disp, data = mtcars))
#> Analysis of Variance Table (Type III SS)
#> Model: mpg ~ hp * disp
#>
#> SS df MS F PRE p
#> ------- --------------- | -------- -- ------- ------ ----- -----
#> Model (error reduced) | 923.189 3 307.730 42.475 .8198 .0000
#> hp | 113.393 1 113.393 15.651 .3586 .0005
#> disp | 188.449 1 188.449 26.011 .4816 .0000
#> hp:disp | 80.635 1 80.635 11.130 .2844 .0024
#> Error (from model) | 202.858 28 7.245
#> ------- --------------- | -------- -- ------- ------ ----- -----
#> Total (empty model) | 1126.047 31 36.324supernova(lm(mpg ~ hp * disp, data = mtcars), verbose = FALSE)
#> Analysis of Variance Table (Type III SS)
#> Model: mpg ~ hp * disp
#>
#> SS df MS F PRE p
#> ------- | -------- -- ------- ------ ----- -----
#> Model | 923.189 3 307.730 42.475 .8198 .0000
#> hp | 113.393 1 113.393 15.651 .3586 .0005
#> disp | 188.449 1 188.449 26.011 .4816 .0000
#> hp:disp | 80.635 1 80.635 11.130 .2844 .0024
#> Error | 202.858 28 7.245
#> ------- | -------- -- ------- ------ ----- -----
#> Total | 1126.047 31 36.324First let’s load up lme4 which gives us lmer(), the function we will
use to fit within-subjects models. Additionally, install and load the
JMRData package which has some short datasets with non-independent
observations, and load dplyr and tidyr so that we can tidy the data.
# Run this line if you do not have the JMRData package
# remotes::install_github("UCLATALL/JMRData")
library(lme4)
library(tidyr)
library(dplyr)
simple_crossed <- JMRData::ex11.9 |>
gather(condition, puzzles_completed, -subject) |>
mutate_at(vars(subject, condition), as.factor)
multiple_crossed <- JMRData::ex11.17 |>
gather(condition, recall, -Subject) |>
separate(condition, c("type", "time"), -1) |>
mutate(across(c(Subject, type, time), as.factor))Fitting the simple_crossed data with lm() would ignore the
non-independence due to observations coming from the same subject.
Compare this output with the following output where the model was fit
with lmer() and the specification of subject as a random factor:
simple_crossed |>
lm(puzzles_completed ~ condition, data = _) |>
supernova(verbose = FALSE)
#> Analysis of Variance Table (Type III SS)
#> Model: puzzles_completed ~ condition
#>
#> SS df MS F PRE p
#> ----- | ------ -- ----- ----- ----- -----
#> Model | 2.250 1 2.250 1.518 .0978 .2382
#> Error | 20.750 14 1.482
#> ----- | ------ -- ----- ----- ----- -----
#> Total | 23.000 15 1.533
# use lmer() to specify the non-independence
simple_crossed |>
lmer(puzzles_completed ~ condition + (1 | subject), data = _) |>
supernova()
#> Analysis of Variance Table (Type III SS)
#> Model: puzzles_completed ~ condition + (1 | subject)
#>
#> SS df MS F PRE p
#> ---------------------- | ------ -- ----- ----- ----- -----
#> Between Subjects |
#> Total | 18.000 7 2.571
#> ---------------------- | ------ -- ----- ----- ----- -----
#> Within Subjects |
#> condition | 2.250 1 2.250 5.727 .4500 .0479
#> Error | 2.750 7 0.393
#> Total | 5.000 8 0.625
#> ---------------------- | ------ -- ----- ----- ----- -----
#> Total | 23.000 15 1.533Here is another example like the previous, but here multiple variables
(time, type) and their interaction have been specified:
# fitting this with lm would ignore the non-independence due to Subject
multiple_crossed |>
lm(recall ~ type * time, data = _) |>
supernova(verbose = FALSE)
#> Analysis of Variance Table (Type III SS)
#> Model: recall ~ type * time
#>
#> SS df MS F PRE p
#> --------- | ------- -- ------ ----- ----- -----
#> Model | 85.367 5 17.073 2.791 .3677 .0400
#> type | 12.100 1 12.100 1.978 .0761 .1724
#> time | 44.400 2 22.200 3.629 .2322 .0420
#> type:time | 1.867 2 0.933 0.153 .0126 .8593
#> Error | 146.800 24 6.117
#> --------- | ------- -- ------ ----- ----- -----
#> Total | 232.167 29 8.006
# using lmer() we can specify the non-independence
multiple_crossed |>
lmer(recall ~ type * time + (1 | Subject) + (1 | type:Subject) + (1 | time:Subject), data = _) |>
supernova()
#> Analysis of Variance Table (Type III SS)
#> Model: recall ~ type * time + (1 | Subject) + (1 | type:Subject) + (1 | time:Subject)
#>
#> SS df MS F PRE p
#> ---------------------- | ------- -- ------ ------ ----- -----
#> Between Subjects |
#> Total | 131.000 4 32.750
#> ---------------------- | ------- -- ------ ------ ----- -----
#> Within Subjects |
#> type | 17.633 1 17.633 11.376 .7399 .0280
#> Error | 6.200 4 1.550
#> time | 65.867 2 32.933 29.939 .8821 .0002
#> Error | 8.800 8 1.100
#> type:time | 1.867 2 0.933 9.333 .7000 .0081
#> Error | 0.800 8 0.100
#> Total | 101.167 25 4.047
#> ---------------------- | ------- -- ------ ------ ----- -----
#> Total | 232.167 29 8.006In this example, each person in a group of three generates a rating. If
we fit the data with lm() that would ignore the non-independence due
to the people being in the same group. Compare this output with the
following output where the group is specified as a random factor.
simple_nested <- JMRData::ex11.1 |>
gather(id, value, starts_with("score")) |>
mutate(across(c(group, instructions, id), as.factor))
# fitting this with lm would ignore the non-independence due to group
simple_nested |>
lm(value ~ instructions, data = _) |>
supernova(verbose = FALSE)
#> Analysis of Variance Table (Type III SS)
#> Model: value ~ instructions
#>
#> SS df MS F PRE p
#> ----- | ------ -- ------ ------ ----- -----
#> Model | 12.500 1 12.500 12.500 .4386 .0027
#> Error | 16.000 16 1.000
#> ----- | ------ -- ------ ------ ----- -----
#> Total | 28.500 17 1.676
# using lmer() we can specify the non-independence
simple_nested |>
lmer(value ~ instructions + (1 | group), data = _) |>
supernova()
#> Analysis of Variance Table (Type III SS)
#> Model: value ~ instructions + (1 | group)
#>
#> SS df MS F PRE p
#> ---------------------- | ------ -- ------ ----- ----- -----
#> Between Subjects |
#> instructions | 12.500 1 12.500 4.687 .5396 .0963
#> Error | 10.667 4 2.667
#> Total | 23.167 5 4.633
#> ---------------------- | ------ -- ------ ----- ----- -----
#> Within Subjects |
#> Total | 5.333 12 0.444
#> ---------------------- | ------ -- ------ ----- ----- -----
#> Total | 28.500 17 1.676In this example, each person in heterosexual marriage generates a rating
of satisfaction. Additionally, these couples were chosen such that they
either have children or not, and have been married 15 vs. 30 years. If
we fit the data with lm() that would ignore the non-independence due
to the people being in the same couple. Compare this output with the
following output where the group is specified as a random factor.
complex_nested <- JMRData::ex11.22 |>
gather(sex, rating, Male, Female) |>
mutate(across(c(couple, children, sex, yearsmarried), as.factor))
# fitting this with lm would ignore the non-independence due to group
complex_nested |>
lm(rating ~ sex * yearsmarried * children, data = _) |>
supernova(verbose = FALSE)
#> Analysis of Variance Table (Type III SS)
#> Model: rating ~ sex * yearsmarried * children
#>
#> SS df MS F PRE p
#> ------------------------- | ------ -- ----- ------ ----- ------
#> Model | 26.500 7 3.786 5.345 .6092 .0009
#> sex | 0.000 1 0.000 0.000 .0000 1.0000
#> yearsmarried | 1.125 1 1.125 1.588 .0621 .2197
#> children | 2.000 1 2.000 2.824 .1053 .1059
#> sex:yearsmarried | 2.250 1 2.250 3.176 .1169 .0874
#> sex:children | 0.063 1 0.063 0.088 .0037 .7690
#> yearsmarried:children | 7.562 1 7.562 10.676 .3079 .0033
#> sex:yearsmarried:children | 0.500 1 0.500 0.706 .0286 .4091
#> Error | 17.000 24 0.708
#> ------------------------- | ------ -- ----- ------ ----- ------
#> Total | 43.500 31 1.403
# using lmer() we can specify the non-independence
complex_nested |>
lmer(rating ~ sex * yearsmarried * children + (1 | couple), data = _) |>
supernova()
#> Analysis of Variance Table (Type III SS)
#> Model: rating ~ sex * yearsmarried * children + (1 | couple)
#>
#> SS df MS F PRE p
#> --------------------------- | ------ -- ------ ----- ----- -----
#> Between Subjects |
#> yearsmarried | 10.125 1 10.125 9.529 .4426 .0094
#> children | 0.500 1 0.500 0.471 .0377 .5058
#> yearsmarried:children | 10.125 1 10.125 9.529 .4426 .0094
#> Error | 12.750 12 1.063
#> Total | 33.500 15 2.233
#> --------------------------- | ------ -- ------ ----- ----- -----
#> Within Subjects |
#> sex | 3.125 1 3.125 8.824 .4237 .0117
#> sex:yearsmarried | 2.000 1 2.000 5.647 .3200 .0350
#> sex:children | 0.125 1 0.125 0.353 .0286 .5635
#> sex:yearsmarried:children | 0.500 1 0.500 1.412 .1053 .2577
#> Error | 4.250 12 0.354
#> Total | 10.000 16 0.625
#> --------------------------- | ------ -- ------ ----- ----- -----
#> Total | 43.500 31 1.403supernova(lm(mpg ~ hp * disp, data = mtcars))
#> Analysis of Variance Table (Type III SS)
#> Model: mpg ~ hp * disp
#>
#> SS df MS F PRE p
#> ------- --------------- | -------- -- ------- ------ ----- -----
#> Model (error reduced) | 923.189 3 307.730 42.475 .8198 .0000
#> hp | 113.393 1 113.393 15.651 .3586 .0005
#> disp | 188.449 1 188.449 26.011 .4816 .0000
#> hp:disp | 80.635 1 80.635 11.130 .2844 .0024
#> Error (from model) | 202.858 28 7.245
#> ------- --------------- | -------- -- ------- ------ ----- -----
#> Total (empty model) | 1126.047 31 36.324These are equivalent to the above:
supernova(lm(mpg ~ hp * disp, data = mtcars), type = 3)
supernova(lm(mpg ~ hp * disp, data = mtcars), type = "III")
supernova(lm(mpg ~ hp * disp, data = mtcars), type = "orthogonal")supernova(lm(mpg ~ hp * disp, data = mtcars), type = 1)
#> Analysis of Variance Table (Type I SS)
#> Model: mpg ~ hp * disp
#>
#> SS df MS F PRE p
#> ------- --------------- | -------- -- ------- ------ ----- -----
#> Model (error reduced) | 923.189 3 307.730 42.475 .8198 .0000
#> hp | 678.373 1 678.373 93.634 .7698 .0000
#> disp | 164.181 1 164.181 22.661 .4473 .0001
#> hp:disp | 80.635 1 80.635 11.130 .2844 .0024
#> Error (from model) | 202.858 28 7.245
#> ------- --------------- | -------- -- ------- ------ ----- -----
#> Total (empty model) | 1126.047 31 36.324These are equivalent to the above:
supernova(lm(mpg ~ hp * disp, data = mtcars), type = "I")
supernova(lm(mpg ~ hp * disp, data = mtcars), type = "sequential")supernova(lm(mpg ~ hp * disp, data = mtcars), type = 2)
#> Analysis of Variance Table (Type II SS)
#> Model: mpg ~ hp * disp
#>
#> SS df MS F PRE p
#> ------- --------------- | -------- -- ------- ------ ----- -----
#> Model (error reduced) | 923.189 3 307.730 42.475 .8198 .0000
#> hp | 33.665 1 33.665 4.647 .1423 .0399
#> disp | 164.181 1 164.181 22.661 .4473 .0001
#> hp:disp | 80.635 1 80.635 11.130 .2844 .0024
#> Error (from model) | 202.858 28 7.245
#> ------- --------------- | -------- -- ------- ------ ----- -----
#> Total (empty model) | 1126.047 31 36.324These are equivalent to the above:
supernova(lm(mpg ~ hp * disp, data = mtcars), type = "II")
supernova(lm(mpg ~ hp * disp, data = mtcars), type = "hierarchical")This package is based on a model comparison approach to understanding
regression and ANOVA. As such it is useful to know which models are
being compared for any given term in an ANOVA table. The
generate_models() function accepts a linear model and the desired type
of SS and returns a list of the models that should be compared to
appropriately evaluate each term in the full model.
generate_models(lm(mpg ~ hp * disp, data = mtcars), type = 2)
#>
#> ── Comparison Models for Type II SS ────────────────────────────────────────────
#>
#> ── Full Model
#> complex: mpg ~ hp + disp + hp:disp
#> simple: mpg ~ NULL
#>
#> ── hp
#> complex: mpg ~ hp + disp
#> simple: mpg ~ disp
#>
#> ── disp
#> complex: mpg ~ hp + disp
#> simple: mpg ~ hp
#>
#> ── hp:disp
#> complex: mpg ~ hp + disp + hp:disp
#> simple: mpg ~ hp + dispThe pairwise() function takes a linear model and performs the
requested pairwise comparisons on the categorical terms in the model.
For simple one-way models where a single categorical variable predicts
and outcome, you will get output similar to other methods of computing
pairwise comparisons (e.g. TukeyHSD() or t.test()). Essentially, the
differences on the outcome between each of the groups defined by the
categorical variable are compared with the requested test, and their
confidence intervals and p-values are adjusted by the requested
correction.
However, when more than two variables are entered into the model, the outcome will diverge somewhat from other methods of computing pairwise comparisons. For traditional pairwise tests you need to estimate an error term, usually by pooling the standard deviation of the groups being compared. This means that when you have other predictors in the model, their presence is ignored when running these tests. For the functions in this package, we instead compute the pooled standard error by using the mean squared error (MSE) from the full model fit.
Let’s take a concrete example to explain that. If we are predicting a
car’s miles-per-gallon (mpg) based on whether it has an automatic or
manual transmission (am), we can create that linear model and get the
pairwise comparisons like this:
pairwise(lm(mpg ~ factor(am), data = mtcars))
#>
#> ── Tukey's Honestly Significant Differences ────────────────────────────────────
#> Model: mpg ~ factor(am)
#> factor(am)
#> Levels: 2
#> Family-wise error-rate: 0.05
#>
#> group_1 group_2 diff pooled_se q df lower upper p_adj
#> <chr> <chr> <dbl> <dbl> <dbl> <int> <dbl> <dbl> <dbl>
#> 1 1 0 7.245 1.248 5.807 30 3.642 10.848 .0003The output of this code will is one table showing the comparison of manual and automatic transmissions with regard to miles-per-gallon. The pooled standard error is the same as the square root of the MSE from the full model.
In these data the am variable did not have any other values than
automatic and manual, but we can imagine situations where the
predictor has more than two levels. In these cases, the pooled SD would
be calculated by taking the MSE of the full model (not of each group)
and then weighting it based on the size of the groups in question
(divide by n).
To improve our model, we might add the car’s displacement (disp) as a
quantitative predictor:
pairwise(lm(mpg ~ factor(am) + disp, data = mtcars))
#>
#> ── Tukey's Honestly Significant Differences ────────────────────────────────────
#> Model: mpg ~ factor(am) + disp
#> factor(am)
#> Levels: 2
#> Family-wise error-rate: 0.05
#>
#> group_1 group_2 diff pooled_se q df lower upper p_adj
#> <chr> <chr> <dbl> <dbl> <dbl> <int> <dbl> <dbl> <dbl>
#> 1 1 0 7.245 0.819 8.846 29 4.876 9.614 .0000Note that the output still only has a table for am. This is because we
can’t do a pairwise comparison using disp because there are no groups
to compare. Most functions will drop or not let you use this variable
during pairwise comparisons. Instead, pairwise() uses the same
approach as in the 3+ groups situation: we use the MSE for the full
model and then weight it by the size of the groups being compared.
Because we are using the MSE for the full model, the effect of disp is
accounted for in the error term even though we are not explicitly
comparing different displacements. Importantly, the interpretation
of the outcome is different than in other traditional t-tests. Instead
of saying, “there is a difference in miles-per-gallon based on the type
of transmission,” we must add that this difference is found “after
accounting for displacement.”
Finally, the output can be plotted either by using plot() on the
returned object, or specifying plot = TRUE:
output <- pairwise(lm(mpg ~ factor(am) + disp, data = mtcars))
plot(output)
This would produce an identical plot:
pairwise(lm(mpg ~ factor(am) + disp, data = mtcars), plot = TRUE)If you see an issue, problem, or improvement that you think we should know about, or you think would fit with this package, please let us know on our issues page. Alternatively, if you are up for a little coding of your own, submit a pull request:
- Fork it!
- Create your feature branch:
git checkout -b my-new-feature - Commit your changes:
git commit -am 'Add some feature' - Push to the branch:
git push origin my-new-feature - Submit a pull request :D