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

# 15 Experiments

## 15.1 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.

### 15.1.1 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`

.

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"))`

### 15.1.2 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, which we create 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 |

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.

###
15.1.3 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:

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
```

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
```

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
```

### 15.1.4 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.)

## 15.2 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:

- 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|) 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
```

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.

### 15.2.1 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.