Randomization inference (RI) is procedure for conducting hypothesis tests in randomized experiments. RI is useful for calculating the the probability that:
That probability is sometimes called a \(p\)-value.
Randomization inference has a beautiful logic to it that may be more appealing to you than traditional hypothesis frameworks that rely on \(t\)- or \(F\)-tests. For a really lovely introduction to randomization inference, read Chapter 3 of Gerber and Green (2012).
The hard part of actually conducting randomization inference is the accounting. We need to enumerate all (or a random subset of all) the possible randomizations, correctly implement the null hypothesis, maybe figure out the inverse probability weights, then calculate the test statistics over and over. You can do this by hand with loops. The
ri2 package for r was written so that you don’t have to.1
Gerber and Green (2012) describe a hypothetical experiment in which 2 of 7 villages are assigned a female council head and the outcome is the share of the local budget allocated to water sanitation. Their table 2.2 describes one way the experiment could have come out.
In order to conduct randomization inference, we need to supply 1) a test statistic, 2) a null hypothesis, and 3) a randomization procedure.
formulaargument of the
conduct_rifunction is similar to the regression syntax of r’s built-in
lmfunction. Our test statistic is the coefficient on
Zfrom a regression of
Z, or more simply, the difference-in-means.
sharp_hypothesisargument of the
conduct_rifunction indicates that we are imagining a (hypothetical!) world in which the true difference in potential outcomes is exactly
0for all units.
declare_rafunction from the
randomizrpackage allows us to declare a randomization procedure. In this case, we are assigning
2units to treatment out of
## Loading required package: randomizr
## Loading required package: estimatr
## coefficient estimate two_tailed_p_value ## 1 Z 6.5 0.3809524
ri2 package has specific support for all the randomization procedures that can be described by the
See the randomizr vignette for specifics on each of these procedures.
By way of illustration, let’s take the blocked-and-clustered design from the
ri package help files as an example. The call to
conduct_ri is the same as it was before, but we need to change the random assignment declaration to accomodate the fact that clusters of two units are assigned to treatment and control (within three separate blocks). Note that in this design, the probabilities of assignment to treatment are not constant across units, but the
conduct_ri function by default incorporates inverse probability weights to account for this design feature.
dat <- data.frame( Y = c(8, 6, 2, 0, 3, 1, 1, 1, 2, 2, 0, 1, 0, 2, 2, 4, 1, 1), Z = c(1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0), cluster = c(1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9), block = c(rep(1, 4), rep(2, 6), rep(3, 8)) ) # clusters in blocks 1 and 3 have a 1/2 probability of treatment # but clusters in block 2 have a 2/3 probability of treatment with(dat, table(block, Z))
## Z ## block 0 1 ## 1 2 2 ## 2 2 4 ## 3 4 4
## Random assignment procedure: Blocked and clustered random assignment ## Number of units: 18 ## Number of blocks: 3 ## Number of clusters: 9 ## Number of treatment arms: 2 ## The possible treatment categories are 0 and 1. ## The probabilities of assignment are NOT constant across units. Your analysis strategy must account for differential probabilities of assignment, typically by employing inverse probability weights.
## coefficient estimate two_tailed_p_value ## 1 Z 2 0.1944444
A traditional ANOVA hypothesis testing framework (implicitly or explicitly) compares two models, a restricted model and an unrestricted model, where the restricted model can be said to “nest” within the unrestricted model. The difference between models is summarized as an \(F\)-statistic. We then compare the observed \(F\)-statistic to a hypothetical null distribution that, under some possibly wrong assumptions can be said to follow an \(F\)-distribution.
In a randomization inference framework, we’re happy to use the \(F\)-statistic, but we want to construct a null distribution that corresponds to the distribution of \(F\)-statistics that we would obtain if a particular (typically sharp) null hypothesis were true and we cycled through all the possible random assignments.
To do this in the
ri2 package, we need to supply model formulae to the
model_2 arguments of
In this example, we consider a three-arm trial. We want to conduct the randomization inference analogue of an \(F\)-test to see if any of the treatments influenced the outcome. We’ll consider the sharp null hypothesis that each unit would express exactly the same outcome regardless of which of the three arms it was assigned to.
N <- 100 # three-arm trial, treat 33, 33, 34 or 33, 34, 33, or 34, 33, 33 declaration <- declare_ra(N = N, num_arms = 3) Z <- conduct_ra(declaration) Y <- .9 * .2 * (Z == "T2") + -.1 * (Z == "T3") + rnorm(N) dat <- data.frame(Y, Z) ri2_out <- conduct_ri( model_1 = Y ~ 1, # restricted model model_2 = Y ~ Z, # unrestricted model declaration = declaration, sharp_hypothesis = 0, data = dat ) plot(ri2_out)
## coefficient estimate two_tailed_p_value ## 1 F-statistic 4.386123 0.018
## Analysis of Variance Table ## ## Model 1: Y ~ 1 ## Model 2: Y ~ Z ## Res.Df RSS Df Sum of Sq F Pr(>F) ## 1 99 89.814 ## 2 97 82.365 2 7.4488 4.3861 0.01501 * ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Oftentimes in an experiment, we’re interested the difference in average treatment effects by subgroups defined by pre-treatment characteristics. For example, we might want to know if the average treatment effect is larger for men or women. If we want to conduct a formal hypothesis test, we’re not interested in testing against the sharp null of no effect for any unit – we want to test against the null hypothesis of constant effects. In the example below, we test using the null hypothesis that all units have a constant effect equal to the estimated ATE. See Gerber and Green (2012) Chapter 9 for more information on this procedure.
## Z ## -0.06336663
## coefficient estimate two_tailed_p_value ## 1 F-statistic 2.997533 0.091
## Analysis of Variance Table ## ## Model 1: Y ~ Z + X ## Model 2: Y ~ Z + X + Z * X ## Res.Df RSS Df Sum of Sq F Pr(>F) ## 1 97 105.71 ## 2 96 102.51 1 3.2008 2.9975 0.0866 . ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
A major benefit of randomization inference is we can specify any scalar test statistic, which means we can conduct hypothesis tests for estimators beyond the narrow set for which statisticians have derived the variance. The
ri2 package accommodates this with the
test_function argument of
conduct_ri. You supply a function that takes a
data.frame as its only argument and returns a scalar;
conduct_ri does the rest!
For example, we can conduct a difference-in-variances test against the sharp null of no effect for any unit:
##  0.1968688
## coefficient estimate two_tailed_p_value ## 1 Custom Test Statistic 0.1968688 0.001
Researchers sometimes conduct balance tests as a randomization check. Rather than conducting separate tests covariate-by-covariate, we might be interested in conducting an omnibus test.
Imagine we’ve got three covariates
X3. We’ll get a summary balance stat (the \(F\)-statistic in this case), but it really could be anything!
## value ## 0.09382125
## Warning in data.frame(est_sim = test_stat_sim, est_obs = test_stat_obs, : ## row names were found from a short variable and have been discarded
## coefficient estimate two_tailed_p_value ## 1 Custom Test Statistic 0.09382125 0.958
## ## Call: ## lm(formula = Z ~ X1 + X2 + X3, data = dat) ## ## Residuals: ## Min 1Q Median 3Q Max ## -0.56176 -0.49916 -0.01695 0.50157 0.56252 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 0.475400 0.110128 4.317 3.85e-05 *** ## X1 -0.020211 0.050006 -0.404 0.687 ## X2 0.002240 0.103098 0.022 0.983 ## X3 0.009942 0.029755 0.334 0.739 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 0.5096 on 96 degrees of freedom ## Multiple R-squared: 0.002923, Adjusted R-squared: -0.02824 ## F-statistic: 0.09382 on 3 and 96 DF, p-value: 0.9633
All of the randomization inference procedures had to, somehow or other, provide three pieces of information:
Randomization inference is a useful tool because we can conduct hypothesis tests without making additional assumptions about the distributions of outcomes or estimators. We can also do tests for arbitrary test statistics – we’re not just restricted to the set for which statisticians have worked out analytic hypothesis testing procedures.
A downside is that RI can be a pain to set up – the
ri2 package is designed to make this part easier.
ri2 package is the successor to the
ri package, written by Peter Aronow and Cyrus Samii.
ri was lightweight, fast, and correct for the set of tasks it handled.
ri2 hopes to be all that and more.↩