Group Comparisons with Extra Sum-of-Squares F-Test

Brent Kaplan

2026-03-02

Introduction

When you have multiple groups (e.g., treatment vs. control, different populations), it is often useful to compare whether separate demand curves are preferred over a single curve. The Extra Sum-of-Squares F-test addresses this by fitting a single curve to all data, computing the residual deviations, and comparing those residuals to those obtained from separate group-level curves.

For background on fitting individual demand curves, see vignette("beezdemand"). For mixed-effects approaches to group comparisons, see vignette("mixed-demand").

Note: For formal statistical inference on group differences, mixed-effects models (vignette("mixed-demand")) are the preferred modern approach, offering random effects and estimated marginal means. ExtraF() remains useful for quick, traditional group comparisons.

Setting Up Groups

We will use the built-in apt dataset and manufacture random groupings for demonstration purposes.

## setting the seed initializes the random number generator so results will be
## reproducible
set.seed(1234)

## manufacture random grouping
apt$group <- NA
apt[apt$id %in% sample(unique(apt$id), length(unique(apt$id))/2), "group"] <- "a"
apt$group[is.na(apt$group)] <- "b"

## take a look at what the new groupings look like in long form
knitr::kable(apt[1:20, ])
id x y group
19 0.0 10 a
19 0.5 10 a
19 1.0 10 a
19 1.5 8 a
19 2.0 8 a
19 2.5 8 a
19 3.0 7 a
19 4.0 7 a
19 5.0 7 a
19 6.0 6 a
19 7.0 6 a
19 8.0 5 a
19 9.0 5 a
19 10.0 4 a
19 15.0 3 a
19 20.0 2 a
30 0.0 3 b
30 0.5 3 b
30 1.0 3 b
30 1.5 3 b

Running the Extra Sum-of-Squares F-Test

The ExtraF() function will determine whether a single \(\alpha\) or a single \(Q_0\) is better than multiple \(\alpha\)s or \(Q_0\)s. A resulting F statistic will be reported along with a p value.

## in order for this to run, you will have had to run the code immediately
## preceeding (i.e., the code to generate the groups)
ef <- ExtraF(dat = apt, equation = "koff", k = 2, groupcol = "group", verbose = TRUE)

A summary table (broken up here for ease of display) will be created when the option verbose = TRUE. This table can be accessed as the dfres object resulting from ExtraF. In the example above, we can access this summary table using ef$dfres:

Fitted Measures
Group Q0d K R2 Alpha
Shared NA NA NA NA
a 8.489634 2 0.6206444 0.0040198
b 5.848119 2 0.6206444 0.0040198
Not Shared NA NA NA NA
a 8.503442 2 0.6448801 0.0040518
b 5.822075 2 0.5242825 0.0039376
Uncertainty and Model Information
Group N AbsSS SdRes
Shared NA NA NA
a 160 387.0945 1.570213
b 160 387.0945 1.570213
Not Shared NA NA NA
a 80 249.2764 1.787695
b 80 137.7440 1.328890
Derived Measures
Group EV Omaxd Pmaxd
Shared NA NA NA
a 0.8795301 22.63159 8.453799
b 0.8795301 22.63159 12.272265
Not Shared NA NA NA
a 0.8725741 22.45260 8.373320
b 0.8978945 23.10414 12.584550
Convergence and Summary Information
Group Omaxa Notes
Shared NA NA
a 22.63190 converged
b 22.63190 converged
Not Shared NA NA
a 22.45291 converged
b 23.10445 converged

Visualizing Group Comparisons

When verbose = TRUE, objects from the result can be used in subsequent graphing. The following code generates a plot of our two groups. We can use the predicted values already generated from the ExtraF function by accessing the newdat object. In the example above, we can access these predicted values using ef$newdat. Note that we keep the linear scaling of y given we used Koffarnus et al. (2015)’s equation fitted to the data.

## be sure that you've loaded the tidyverse package (e.g., library(tidyverse))
ggplot(apt, aes(x = x, y = y, group = group)) +
  ## the predicted lines from the sum of squares f-test can be used in subsequent
  ## plots by calling data = ef$newdat
  geom_line(aes(x = x, y = y, group = group, color = group),
            data = ef$newdat[ef$newdat$x >= .1, ]) +
  stat_summary(fun.data = mean_se, aes(width = .05, color = group),
               geom = "errorbar") +
  stat_summary(fun = mean, aes(fill = group), geom = "point", shape = 21,
               color = "black", stroke = .75, size = 4) +
  scale_x_log10(limits = c(.4, 50), breaks = c(.1, 1, 10, 100)) +
  scale_color_discrete(name = "Group") +
  scale_fill_discrete(name = "Group") +
  labs(x = "Price per Drink", y = "Drinks Purchased") +
  theme(legend.position = c(.85, .75)) +
  ## theme_apa is a beezdemand function used to change the theme in accordance
  ## with American Psychological Association style
  theme_apa()

See Also