Horvitz-Thompson estimator for two-armed trialsSource:
Horvitz-Thompson estimators that are unbiased for designs in which the randomization scheme is known
horvitz_thompson( formula, data, blocks, clusters, simple = NULL, condition_prs, condition_pr_mat = NULL, ra_declaration = NULL, subset, condition1 = NULL, condition2 = NULL, se_type = c("youngs", "constant", "none"), ci = TRUE, alpha = 0.05, return_condition_pr_mat = FALSE )
an object of class formula, as in
lm, such as
Y ~ Zwith only one variable on the right-hand side, the treatment.
An optional bare (unquoted) name of the block variable. Use for blocked designs only. See details.
An optional bare (unquoted) name of the variable that corresponds to the clusters in the data; used for cluster randomized designs. For blocked designs, clusters must be within blocks.
logical, optional. Whether the randomization is simple (TRUE) or complete (FALSE). This is ignored if
blocksare specified, as all blocked designs use complete randomization, or either
condition_pr_matare passed. Otherwise, it defaults to
An optional bare (unquoted) name of the variable with the condition 2 (treatment) probabilities. See details. May also use a single number for the condition 2 probability if it is constant.
An optional 2n * 2n matrix of marginal and joint probabilities of all units in condition1 and condition2. See details.
An object of class
"ra_declaration", from the
declare_rafunction in the randomizr package. This is the third way that one can specify a design for this estimator. Cannot be used along with any of
condition_pr_mat. See details.
An optional bare (unquoted) expression specifying a subset of observations to be used.
value in the treatment vector of the condition to be the control. Effects are estimated with
condition1as the control and
condition2as the treatment. If unspecified,
condition1is the "first" condition and
condition2is the "second" according to levels if the treatment is a factor or according to a sortif it is a numeric or character variable (i.e if unspecified and the treatment is 0s and 1s,
condition1will by default be 0 and
condition2will be 1). See the examples for more.
value in the treatment vector of the condition to be the treatment. See
can be one of
c("youngs", "constant", "none")and corresponds the estimator of the standard errors. Default estimator uses Young's inequality (and is conservative) while the other uses a constant treatment effects assumption and only works for simple randomized designs at the moment. If "none" then standard errors will not be computed which may speed up run time if only the point estimate is required.
logical. Whether to compute and return p-values and confidence intervals, TRUE by default.
The significance level, 0.05 by default.
logical. Whether to return the condition probability matrix. Returns NULL if the design is simple randomization, FALSE by default.
Returns an object of class
The post-estimation commands functions
return results in a
data.frame. To get useful data out of the return,
you can use these data frames, you can use the resulting list directly, or
you can use the generic accessor functions
An object of class
"horvitz_thompson" is a list containing at
least the following components:
the estimated difference in totals
the estimated standard error
the estimated degrees of freedom
the p-value from a two-sided z-test using
the lower bound of the
1 - alphapercent confidence interval
the upper bound of the
1 - alphapercent confidence interval
a character vector of coefficient names
the significance level specified by the user
the number of observations used
the name of the outcome variable
the condition probability matrix if
This function implements the Horvitz-Thompson estimator for treatment effects for two-armed trials. This estimator is useful for estimating unbiased treatment effects given any randomization scheme as long as the randomization scheme is known.
In short, the Horvitz-Thompson estimator essentially reweights each unit by the probability of it being in its observed condition. Pivotal to the estimation of treatment effects using this estimator are the marginal condition probabilities (i.e., the probability that any one unit is in a particular treatment condition). Pivotal to the estimating the variance variance whenever the design is more complicated than simple randomization, are the joint condition probabilities (i.e., the probabilities that any two units have a particular set of treatment conditions, either the same or different). The estimator we provide here considers the case with two treatment conditions.
Users interested in more details can see the mathematical notes for more information and references, or see the references below.
There are three distinct ways that users can specify the design to the
function. The preferred way is to use
declare_ra function in the randomizr
package. This function takes several arguments, including blocks, clusters,
treatment probabilities, whether randomization is simple or not, and more.
Passing the outcome of that function, an object of class
"ra_declaration" to the
ra_declaration argument in this function,
will lead to a call of the
function which generates the condition probability matrix needed to
estimate treatment effects and standard errors. We provide many examples
below of how this could be done.
The second way is to pass the names of vectors in your
clusters. You can further
specify whether the randomization was simple or complete using the
argument. Note that if
blocks are specified the randomization is
always treated as complete. From these vectors, the function learns how to
build the condition probability matrix that is used in estimation.
In the case
condition_prs is specified, this function assumes those
probabilities are the marginal probability that each unit is in condition2
and then uses the other arguments (
simple) to learn the rest of the design. If users do not pass
condition_prs, this function learns the probability of being in
condition2 from the data. That is, none of these arguments are specified,
we assume that there was a simple randomization where the probability
of each unit being in condition2 was the average of all units in condition2.
Similarly, we learn the block-level probability of treatment within
blocks by looking at the mean number of units in condition2 if
condition_prs is not specified.
The third way is to pass a
condition_pr_mat directly. One can
see more about this object in the documentation for
permutations_to_condition_pr_mat. Essentially, this 2n * 2n
matrix allows users to specify marginal and joint marginal probabilities
of units being in conditions 1 and 2 of arbitrary complexity. Users should
only use this option if they are certain they know what they are doing.
Aronow, Peter M, and Joel A Middleton. 2013. "A Class of Unbiased Estimators of the Average Treatment Effect in Randomized Experiments." Journal of Causal Inference 1 (1): 135-54. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.712.5830&rep=rep1&type=pdf.
Aronow, Peter M, and Cyrus Samii. 2017. "Estimating Average Causal Effects Under Interference Between Units." Annals of Applied Statistics, forthcoming. https://arxiv.org/abs/1305.6156v3.
Middleton, Joel A, and Peter M Aronow. 2015. "Unbiased Estimation of the Average Treatment Effect in Cluster-Randomized Experiments." Statistics, Politics and Policy 6 (1-2): 39-75. doi:10.1515/spp-2013-0002 .
# Set seed set.seed(42) # Simulate data n <- 10 dat <- data.frame(y = rnorm(n)) library(randomizr) #---------- # 1. Simple random assignment #---------- dat$p <- 0.5 dat$z <- rbinom(n, size = 1, prob = dat$p) # If you only pass condition_prs, we assume simple random sampling horvitz_thompson(y ~ z, data = dat, condition_prs = p) #> Estimate Std. Error t value Pr(>|t|) CI Lower CI Upper DF #> z -0.2532128 0.609167 -0.4156706 0.677651 -1.447158 0.9407325 NA # Assume constant effects instead horvitz_thompson(y ~ z, data = dat, condition_prs = p, se_type = "constant") #> Estimate Std. Error t value Pr(>|t|) CI Lower CI Upper DF #> z -0.2532128 0.6038814 -0.4193088 0.6749904 -1.436799 0.930373 NA # Also can use randomizr to pass a declaration srs_declaration <- declare_ra(N = nrow(dat), prob = 0.5, simple = TRUE) horvitz_thompson(y ~ z, data = dat, ra_declaration = srs_declaration) #> Estimate Std. Error t value Pr(>|t|) CI Lower CI Upper DF #> z -0.2532128 0.609167 -0.4156706 0.677651 -1.447158 0.9407325 NA #---------- # 2. Complete random assignment #---------- dat$z <- sample(rep(0:1, each = n/2)) # Can use a declaration crs_declaration <- declare_ra(N = nrow(dat), prob = 0.5, simple = FALSE) horvitz_thompson(y ~ z, data = dat, ra_declaration = crs_declaration) #> Estimate Std. Error t value Pr(>|t|) CI Lower CI Upper DF #> z -0.2312303 0.5310475 -0.435423 0.6632554 -1.272064 0.8096037 NA # Can precompute condition_pr_mat and pass it # (faster for multiple runs with same condition probability matrix) crs_pr_mat <- declaration_to_condition_pr_mat(crs_declaration) horvitz_thompson(y ~ z, data = dat, condition_pr_mat = crs_pr_mat) #> Estimate Std. Error t value Pr(>|t|) CI Lower CI Upper DF #> z -0.2312303 0.5310475 -0.435423 0.6632554 -1.272064 0.8096037 NA #---------- # 3. Clustered treatment, complete random assigment #----------- # Simulating data dat$cl <- rep(1:4, times = c(2, 2, 3, 3)) dat$prob <- 0.5 clust_crs_decl <- declare_ra(N = nrow(dat), clusters = dat$cl, prob = 0.5) dat$z <- conduct_ra(clust_crs_decl) # Easiest to specify using declaration ht_cl <- horvitz_thompson(y ~ z, data = dat, ra_declaration = clust_crs_decl) # Also can pass the condition probability and the clusters ht_cl_manual <- horvitz_thompson( y ~ z, data = dat, clusters = cl, condition_prs = prob, simple = FALSE ) ht_cl #> Estimate Std. Error t value Pr(>|t|) CI Lower CI Upper DF #> z 0.0482231 0.230729 0.2090032 0.8344458 -0.4039975 0.5004437 NA ht_cl_manual #> Estimate Std. Error t value Pr(>|t|) CI Lower CI Upper DF #> z 0.0482231 0.230729 0.2090032 0.8344458 -0.4039975 0.5004437 NA # Blocked estimators specified similarly #---------- # More complicated assignment #---------- # arbitrary permutation matrix possible_treats <- cbind( c(1, 1, 0, 1, 0, 0, 0, 1, 1, 0), c(0, 1, 1, 0, 1, 1, 0, 1, 0, 1), c(1, 0, 1, 1, 1, 1, 1, 0, 0, 0) ) arb_pr_mat <- permutations_to_condition_pr_mat(possible_treats) # Simulating a column to be realized treatment dat$z <- possible_treats[, sample(ncol(possible_treats), size = 1)] horvitz_thompson(y ~ z, data = dat, condition_pr_mat = arb_pr_mat) #> Estimate Std. Error t value Pr(>|t|) CI Lower CI Upper DF #> z 0.7576713 0.7715048 0.9820695 0.3260656 -0.7544502 2.269793 NA