Experiments

2x2 Experiments

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.

In this notebook, we illustrate how to analyze these designs with the marginaleffects package for R. As we will see, marginaleffects includes many convenient functions for analyzing both experimental and observational data, and for plotting our results.

Fitting a Model

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).

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)
library(modelsummary)

## See ?mtcars for variable definitions
fit <- lm(mpg ~ vs + am + vs:am, data=mtcars) # equivalent to ~ vs*am
import polars as pl
import statsmodels.formula.api as smf
from marginaleffects import *

mtcars = pl.read_csv("https://vincentarelbundock.github.io/Rdatasets/csv/datasets/mtcars.csv")

fit = smf.ols("mpg ~ vs + am + vs:am", data = mtcars).fit() # 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"))

plot_predictions(fit, by = ["vs", "am"]).draw(show=True)

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 the regression table:

We create the regression table using the modelsummary package:

modelsummary(fit, gof_map = c("r.squared", "nobs"))
(1)
(Intercept) 15.050
(1.002)
vs 5.693
(1.651)
am 4.700
(1.736)
vs × am 2.929
(2.541)
R2 0.700
Num.Obs. 32
print(fit.summary())
#>                             OLS Regression Results                            
#> ==============================================================================
#> Dep. Variable:                    mpg   R-squared:                       0.700
#> Model:                            OLS   Adj. R-squared:                  0.668
#> Method:                 Least Squares   F-statistic:                     21.81
#> Date:                Mon, 19 Feb 2024   Prob (F-statistic):           1.73e-07
#> Time:                        13:34:31   Log-Likelihood:                -83.098
#> No. Observations:                  32   AIC:                             174.2
#> Df Residuals:                      28   BIC:                             180.1
#> Df Model:                           3                                         
#> Covariance Type:            nonrobust                                         
#> ==============================================================================
#>                  coef    std err          t      P>|t|      [0.025      0.975]
#> ------------------------------------------------------------------------------
#> Intercept     15.0500      1.002     15.017      0.000      12.997      17.103
#> vs             5.6929      1.651      3.448      0.002       2.311       9.075
#> am             4.7000      1.736      2.708      0.011       1.144       8.256
#> vs:am          2.9286      2.541      1.153      0.259      -2.276       8.134
#> ==============================================================================
#> Omnibus:                        0.609   Durbin-Watson:                   1.520
#> Prob(Omnibus):                  0.738   Jarque-Bera (JB):                0.713
#> Skew:                          -0.223   Prob(JB):                        0.700
#> Kurtosis:                       2.421   Cond. No.                         6.32
#> ==============================================================================
#> 
#> Notes:
#> [1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

We can express the same results in the form of a linear equation:

\[ \mbox{mpg} = 15.050 + 5.693 \cdot \mbox{vs} + 4.700 \cdot \mbox{am} + 2.929 \cdot \mbox{vs} \cdot \mbox{am}.\]

With a little arithmetic, we can compute estimated differences in fuel efficiency between different groups:

  • 4.700 mpg between am=1 and am=0, when vs=0.
  • 5.693 mpg between vs=1 and vs=0, when am=0.
  • 7.629 mpg between am=1 and am=0, when vs=1.
  • 8.621 mpg between vs=1 and vs=0, when am=1.
  • 13.322 mpg between a car with am=1 and vs=1, and a car with am=0 and vs=0.

Reading off these differences from the model summary is relatively straightforward in very simple cases like this one. However, it becomes more difficult as more variables are added to the model, not to mention obtaining estimated standard errors becomes nightmarish. To make the process easier, we can leverage the avg_comparisons() function from the marginaleffects package to compute the appropriate quantities and standard errors.

Using avg_comparisons To Estimate All Differences

The grey rectangle in the graph below 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. In this call, the variables argument indicates that we want to estimate the effect of a change of 1 unit in the vs variable. The newdata=datagrid(am=0) determines the values of the covariates at which we want to evaluate the contrast.

avg_comparisons(fit,
  variables = "vs",
  newdata = datagrid(am = 0))
#> 
#>  Term Contrast Estimate Std. Error    z Pr(>|z|)    S 2.5 % 97.5 %
#>    vs    1 - 0     5.69       1.65 3.45   <0.001 10.8  2.46   8.93
#> 
#> Columns: rowid, term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted 
#> Type:  response

As expected, the results produced by avg_comparisons() are exactly the same as those which we read from the model summary table. The contrast that we just computed corresponds to the change illustrasted by the arrow in this plot:

print(avg_comparisons(fit,
  variables = "vs",
  newdata = datagrid(am = 0)))
#> shape: (1, 9)
#> ┌──────┬──────────────────────────┬──────────┬───────────┬───┬──────────┬──────┬──────┬───────┐
#> │ Term ┆ Contrast                 ┆ Estimate ┆ Std.Error ┆ … ┆ P(>|z|)  ┆ S    ┆ 2.5% ┆ 97.5% │
#> │ ---  ┆ ---                      ┆ ---      ┆ ---       ┆   ┆ ---      ┆ ---  ┆ ---  ┆ ---   │
#> │ str  ┆ str                      ┆ str      ┆ str       ┆   ┆ str      ┆ str  ┆ str  ┆ str   │
#> ╞══════╪══════════════════════════╪══════════╪═══════════╪═══╪══════════╪══════╪══════╪═══════╡
#> │ vs   ┆ mean(True) - mean(False) ┆ 5.69     ┆ 1.65      ┆ … ┆ 0.000565 ┆ 10.8 ┆ 2.46 ┆ 8.93  │
#> └──────┴──────────────────────────┴──────────┴───────────┴───┴──────────┴──────┴──────┴───────┘
#> 
#> Columns: term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high

As expected, the results produced by avg_comparisons() are exactly the same as those which we read from the model summary table. The contrast that we just computed corresponds to the change illustrasted by the arrow in this plot:

The next difference that we compute 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,
  variables = "am",
  newdata = datagrid(vs = 0))
#> 
#>  Term Contrast Estimate Std. Error    z Pr(>|z|)   S 2.5 % 97.5 %
#>    am    1 - 0      4.7       1.74 2.71  0.00678 7.2   1.3    8.1
#> 
#> Columns: rowid, term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted 
#> Type:  response

print(avg_comparisons(fit,
  variables = "am",
  newdata = datagrid(vs = 0)))
#> shape: (1, 9)
#> ┌──────┬──────────────────────────┬──────────┬───────────┬───┬─────────┬──────┬──────┬───────┐
#> │ Term ┆ Contrast                 ┆ Estimate ┆ Std.Error ┆ … ┆ P(>|z|) ┆ S    ┆ 2.5% ┆ 97.5% │
#> │ ---  ┆ ---                      ┆ ---      ┆ ---       ┆   ┆ ---     ┆ ---  ┆ ---  ┆ ---   │
#> │ str  ┆ str                      ┆ str      ┆ str       ┆   ┆ str     ┆ str  ┆ str  ┆ str   │
#> ╞══════╪══════════════════════════╪══════════╪═══════════╪═══╪═════════╪══════╪══════╪═══════╡
#> │ am   ┆ mean(True) - mean(False) ┆ 4.7      ┆ 1.74      ┆ … ┆ 0.00678 ┆ 7.21 ┆ 1.3  ┆ 8.1   │
#> └──────┴──────────────────────────┴──────────┴───────────┴───┴─────────┴──────┴──────┴───────┘
#> 
#> Columns: term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high

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,
  variables = "am",
  newdata = datagrid(vs = 1))
#> 
#>  Term Contrast Estimate Std. Error    z Pr(>|z|)    S 2.5 % 97.5 %
#>    am    1 - 0     7.63       1.86 4.11   <0.001 14.6  3.99   11.3
#> 
#> Columns: rowid, term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted 
#> Type:  response

print(avg_comparisons(fit,
  variables = "am",
  newdata = datagrid(vs = 1)))
#> shape: (1, 9)
#> ┌──────┬──────────────────────────┬──────────┬───────────┬───┬──────────┬──────┬──────┬───────┐
#> │ Term ┆ Contrast                 ┆ Estimate ┆ Std.Error ┆ … ┆ P(>|z|)  ┆ S    ┆ 2.5% ┆ 97.5% │
#> │ ---  ┆ ---                      ┆ ---      ┆ ---       ┆   ┆ ---      ┆ ---  ┆ ---  ┆ ---   │
#> │ str  ┆ str                      ┆ str      ┆ str       ┆   ┆ str      ┆ str  ┆ str  ┆ str   │
#> ╞══════╪══════════════════════════╪══════════╪═══════════╪═══╪══════════╪══════╪══════╪═══════╡
#> │ am   ┆ mean(True) - mean(False) ┆ 7.63     ┆ 1.86      ┆ … ┆ 3.94e-05 ┆ 14.6 ┆ 3.99 ┆ 11.3  │
#> └──────┴──────────────────────────┴──────────┴───────────┴───┴──────────┴──────┴──────┴───────┘
#> 
#> Columns: term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high

The last difference and contrast between manual transmissions with straight engines and automatic transmissions with V-shaped engines. We call this a “cross-contrast” because we are measuring the difference between two groups that differ on two explanatory variables at the same time. To compute this contrast, we use the cross argument of avg_comparisons:

avg_comparisons(fit,
  variables = c("am", "vs"),
  cross = TRUE)
#> 
#>  Estimate Std. Error    z Pr(>|z|)    S 2.5 % 97.5 % C: am C: vs
#>      13.3       1.65 8.07   <0.001 50.3  10.1   16.6 1 - 0 1 - 0
#> 
#> Columns: term, contrast_am, contrast_vs, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high 
#> Type:  response

# Not implemented yet

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

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:

  1. Center all predictors by subtracting each of their means.
  2. 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
lalonde = pl.read_csv("https://vincentarelbundock.github.io/Rdatasets/csv/MatchIt/lalonde.csv", infer_schema_length=200)

lalonde = lalonde.to_dummies('race')
lalonde = lalonde.with_columns(
  pl.col('age').sub(pl.col('age').mean()),
  pl.col('educ').sub(pl.col('educ').mean()),
  pl.col('race_hispan').sub(pl.col('race_hispan').mean()),
  pl.col('race_white').sub(pl.col('race_white').mean()),
)

mod = smf.ols("re78 ~ treat + age + educ + race_hispan + race_white + treat*(age + educ + race_hispan + race_white)", data = lalonde.to_pandas()).fit()

print(mod.get_robustcov_results(cov_type='HC3').summary())
#>                             OLS Regression Results                            
#> ==============================================================================
#> Dep. Variable:                   re78   R-squared:                       0.057
#> Model:                            OLS   Adj. R-squared:                  0.043
#> Method:                 Least Squares   F-statistic:                     4.238
#> Date:                Mon, 19 Feb 2024   Prob (F-statistic):           2.42e-05
#> Time:                        13:34:35   Log-Likelihood:                -6328.8
#> No. Observations:                 614   AIC:                         1.268e+04
#> Df Residuals:                     604   BIC:                         1.272e+04
#> Df Model:                           9                                         
#> Covariance Type:                  HC3                                         
#> =====================================================================================
#>                         coef    std err          t      P>|t|      [0.025      0.975]
#> -------------------------------------------------------------------------------------
#> Intercept          6488.0506    356.712     18.188      0.000    5787.505    7188.597
#> treat               489.7253    878.517      0.557      0.577   -1235.594    2215.045
#> age                  85.8798     35.417      2.425      0.016      16.324     155.436
#> educ                464.0448    131.509      3.529      0.000     205.774     722.315
#> race_hispan        2775.4733   1155.400      2.402      0.017     506.384    5044.563
#> race_white         2291.6714    793.301      2.889      0.004     733.708    3849.634
#> treat:age            17.2262     76.367      0.226      0.822    -132.751     167.203
#> treat:educ          226.7111    308.434      0.735      0.463    -379.023     832.445
#> treat:race_hispan -1057.8406   2652.417     -0.399      0.690   -6266.921    4151.240
#> treat:race_white  -1205.6827   1805.205     -0.668      0.504   -4750.925    2339.559
#> ==============================================================================
#> Omnibus:                      195.576   Durbin-Watson:                   1.856
#> Prob(Omnibus):                  0.000   Jarque-Bera (JB):              755.710
#> Skew:                           1.434   Prob(JB):                    7.94e-165
#> Kurtosis:                       7.616   Cond. No.                         94.5
#> ==============================================================================
#> 
#> Notes:
#> [1] Standard Errors are heteroscedasticity robust (HC3)

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|)   S 2.5 % 97.5 %
#>  treat    1 - 0      490        879 0.557    0.577 0.8 -1232   2212
#> 
#> Columns: term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high 
#> Type:  response
print(avg_comparisons(
    mod,
    variables = "treat",
    vcov = "HC3"))
#> shape: (1, 9)
#> ┌───────┬──────────────┬──────────┬───────────┬───┬─────────┬───────┬───────────┬──────────┐
#> │ Term  ┆ Contrast     ┆ Estimate ┆ Std.Error ┆ … ┆ P(>|z|) ┆ S     ┆ 2.5%      ┆ 97.5%    │
#> │ ---   ┆ ---          ┆ ---      ┆ ---       ┆   ┆ ---     ┆ ---   ┆ ---       ┆ ---      │
#> │ str   ┆ str          ┆ str      ┆ str       ┆   ┆ str     ┆ str   ┆ str       ┆ str      │
#> ╞═══════╪══════════════╪══════════╪═══════════╪═══╪═════════╪═══════╪═══════════╪══════════╡
#> │ treat ┆ mean(True) - ┆ 490      ┆ 879       ┆ … ┆ 0.577   ┆ 0.793 ┆ -1.23e+03 ┆ 2.21e+03 │
#> │       ┆ mean(False)  ┆          ┆           ┆   ┆         ┆       ┆           ┆          │
#> └───────┴──────────────┴──────────┴───────────┴───┴─────────┴───────┴───────────┴──────────┘
#> 
#> Columns: term, contrast, estimate, std_error, statistic, p_value, s_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.