Experiments: 2x2 designs and regression adjustment

Analyzing 2x2 Experiments with `marginaleffects`

A 2×2 factorial design is a type of experimental design that allows researchers to understand the effects of two independent variables (each with two levels) on a single dependent variable. The design is popular among academic researchers as well as in industry when running A/B tests.

To illustrate how to analyze these designs with marginaleffects, we will use the mtcars dataset. We’ll analyze fuel efficiency, mpg (miles per gallon), as a function of am (transmission type) and vs (engine shape).

vs is an indicator variable for if the car has a straight engine (1 = straight engine, 0 = V-shaped). am is an indicator variable for if the car has manual transmission (1 = manual transmission, 0=automatic transmission). There are then four types of cars (1 type for each of the four combinations of binary indicators).

Fitting a Model

Let’s start by creating a model for fuel efficiency. For simplicity, we’ll use linear regression and model the interaction between vs and am.

library(tidyverse)
library(marginaleffects)

# See ?mtcars for variable definitions
fit <- lm(mpg ~ vs + am + vs:am, data=mtcars) # equivalent to ~ vs*am

We can plot the predictions from the model using the plot_predictions function. From the plot below, we can see a few things:

Straight engines (vs=1) are estimated to have better expected fuel efficiency than V-shaped engines (vs=0).
Manual transmissions (am=1) are estimated to have better fuel efficiency for both V-shaped and straight engines.
For straight engines, the effect of manual transmissions on fuel efficiency seems to increase.

plot_predictions(fit, by = c("vs", "am"))

Evaluating Effects From The Model Summary

Since this model is fairly simple the estimated differences between any of the four possible combinations of vs and am can be read from summary(fit) with a little arithmetic. The estimated model is

\[ \mbox{mpg} = 20.743 + 5.693 \cdot \mbox{vs} + 4.700 \cdot \mbox{am} + 2.929 \cdot \mbox{vs} \cdot \mbox{am} \>. \]

The estimated differences in fuel efficiency are:

5.693 mpg between straight engines and V-shaped engines when the car has automatic transmission.
4.700 mpg between manual transmissions and automatic transmissions when the car has a V-shaped engine.
7.629 mpg between manual transmissions and automatic transmissions when the car has a straight engine.
13.322 mpg between manual transmissions with straight engines and automatic transmissions with V-shaped engines.

Reading off these differences from the model summary becomes more difficult as more variables are added (not to mention obtaining their estimated standard errors becomes nightmarish). To make the process easier on ourselves, we can leverage the avg_comparisons function to get the same estimates and their uncertainty.

Using `avg_comparisons` To Estimate All Differences

Note that the dot in the grey rectangle is the estimated fuel efficiency when vs=0 and am=0 (that is, for an automatic transmission car with V-shaped engine).

Let’s use avg_comparisons to get the difference between straight engines and V-shaped engines when the car has automatic transmission. The call to avg_comparisons is shown below and results in the same estimate we made directly from the model. The contrast corresponding to this estimate is shown in the plot below.

avg_comparisons(fit,
  newdata = datagrid(am = 0),
  variables = "vs")
#> 
#>  Term Contrast Estimate Std. Error    z Pr(>|z|) 2.5 % 97.5 %
#>    vs    1 - 0     5.69       1.65 3.45   <0.001  2.46   8.93
#> 
#> Columns: rowid, term, contrast, estimate, std.error, statistic, p.value, conf.low, conf.high, predicted, predicted_hi, predicted_lo

The next difference is between manual transmissions and automatic transmissions when the car has a V-shaped engine. Again, the call to avg_comparisons is shown below, and the corresponding contrast is indicated in the plot below using an arrow.

avg_comparisons(fit,
  newdata = datagrid(vs = 0),
  variables = "am")
#> 
#>  Term Contrast Estimate Std. Error    z Pr(>|z|) 2.5 % 97.5 %
#>    am    1 - 0      4.7       1.74 2.71  0.00678   1.3    8.1
#> 
#> Columns: rowid, term, contrast, estimate, std.error, statistic, p.value, conf.low, conf.high, predicted, predicted_hi, predicted_lo

The third difference we estimated was between manual transmissions and automatic transmissions when the car has a straight engine. The model call and contrast are

avg_comparisons(fit,
  newdata = datagrid(vs = 1),
  variables = "am")
#> 
#>  Term Contrast Estimate Std. Error    z Pr(>|z|) 2.5 % 97.5 %
#>    am    1 - 0     7.63       1.86 4.11   <0.001  3.99   11.3
#> 
#> Columns: rowid, term, contrast, estimate, std.error, statistic, p.value, conf.low, conf.high, predicted, predicted_hi, predicted_lo

And the last difference and contrast between manual transmissions with straight engines and automatic transmissions with V-shaped engines is

avg_comparisons(fit,
  newdata = datagrid("vs", "am"),
  variables = c("am", "vs"),
  cross = TRUE)
#> 
#>  C: am C: vs Estimate Std. Error    z Pr(>|z|) 2.5 % 97.5 %
#>  1 - 0 1 - 0     13.3       1.65 8.07   <0.001  10.1   16.6
#> 
#> Columns: rowid, term, contrast_am, contrast_vs, estimate, std.error, statistic, p.value, conf.low, conf.high, predicted, predicted_hi, predicted_lo

Conclusion

The 2x2 design is a very popular design, and when using a linear model, the estimated differences between groups can be directly read off from the model summary, if not with a little arithmetic. However, when using models with a non-identity link function, or when seeking to obtain the standard errors for estimated differences, things become considerably more difficult. This vignette showed how to use avg_comparisons to specify contrasts of interests and obtain standard errors for those differences. The approach used applies to all generalized linear models and effects can be further stratified using the by argument (although this is not shown in this vignette.)

Regression adjustment in experiments

Many analysts who conduct and analyze experiments wish to use regression adjustment with a linear regression model to improve the precision of their estimate of the treatment effect. Unfortunately, regression adjustment can introduce small-sample bias and other undesirable properties (Freedman 2008). Lin (2013) proposes a simple strategy to fix these problems in sufficiently large samples:

Center all predictors by subtracting each of their means.
Estimate a linear model in which the treatment is interacted with each of the covariates.

The estimatr package includes a convenient function to implement this strategy:

library(estimatr)
library(marginaleffects)
lalonde <- read.csv("https://vincentarelbundock.github.io/Rdatasets/csv/MatchIt/lalonde.csv")

mod <- lm_lin(
    re78 ~ treat,
    covariates = ~ age + educ + race,
    data = lalonde,
    se_type = "HC3")
summary(mod)
#> 
#> Call:
#> lm_lin(formula = re78 ~ treat, covariates = ~age + educ + race, 
#>     data = lalonde, se_type = "HC3")
#> 
#> Standard error type:  HC3 
#> 
#> Coefficients:
#>                    Estimate Std. Error t value  Pr(>|t|) CI Lower CI Upper  DF
#> (Intercept)         6488.05     356.71 18.1885 2.809e-59  5787.50   7188.6 604
#> treat                489.73     878.52  0.5574 5.774e-01 -1235.59   2215.0 604
#> age_c                 85.88      35.42  2.4248 1.561e-02    16.32    155.4 604
#> educ_c               464.04     131.51  3.5286 4.495e-04   205.77    722.3 604
#> racehispan_c        2775.47    1155.40  2.4022 1.660e-02   506.38   5044.6 604
#> racewhite_c         2291.67     793.30  2.8888 4.006e-03   733.71   3849.6 604
#> treat:age_c           17.23      76.37  0.2256 8.216e-01  -132.75    167.2 604
#> treat:educ_c         226.71     308.43  0.7350 4.626e-01  -379.02    832.4 604
#> treat:racehispan_c -1057.84    2652.42 -0.3988 6.902e-01 -6266.92   4151.2 604
#> treat:racewhite_c  -1205.68    1805.21 -0.6679 5.045e-01 -4750.92   2339.6 604
#> 
#> Multiple R-squared:  0.05722 ,   Adjusted R-squared:  0.04317 
#> F-statistic: 4.238 on 9 and 604 DF,  p-value: 2.424e-05

We can obtain the same results by fitting a model with the standard lm function and using the comparisons() function:

mod <- lm(re78 ~ treat * (age + educ + race), data = lalonde)
avg_comparisons(
    mod,
    variables = "treat",
    vcov = "HC3")
#> 
#>   Term Contrast Estimate Std. Error     z Pr(>|z|) 2.5 % 97.5 %
#>  treat    1 - 0      490        879 0.557    0.577 -1232   2212
#> 
#> Columns: term, contrast, estimate, std.error, statistic, p.value, conf.low, conf.high

Notice that the treat coefficient and associate standard error in the lm_lin regression are exactly the same as the estimates produced by the comparisons() function.

References

Freedman, David A. “On Regression Adjustments to Experimental Data.” Advances in Applied Mathematics 40, no. 2 (February 2008): 180–93.
Lin, Winston. “Agnostic Notes on Regression Adjustments to Experimental Data: Reexamining Freedman’s Critique.” Annals of Applied Statistics 7, no. 1 (March 2013): 295–318. https://doi.org/10.1214/12-AOAS583.

Analyzing 2x2 Experiments with marginaleffects