# Using causalOT

#### Eric Dunipace

## Introduction

`causalOT` was developed to reproduce the methods in Optimal transport methods for causal inference. The functions in the package are built to construct weights to make distributions more same and estimate causal effects. We recommend using the Causal Optimal Transport methods since they are semi- to non-parametric. This document will describe some simple usages of the functions in the package and should be enough to get users started.

## Setting up the data

The main functions of the package, `calc_weight` and `estimate_effect`, take arguments `x`, a numeric `matrix` of covariates; `z`, a treatment indicator in `c(0,1)`; and `y`, a numeric vector with the outcome data.

If easy to do, we can supply the necessary data directly.

    # packages
    library(causalOT)
    library(torch)

    # reproducible seeds
    set.seed(1111)
    torch_manual_seed(3249)

    # generated some data
    hainmueller <- Hainmueller$new(n = 512)
    hainmueller$gen_data()

    x <- hainmueller$get_x()
    z <- hainmueller$get_z()
    y <- hainmueller$get_y()

    # NOT RUN
    # weights <- calc_weight(x = x, z = z, y = y)

Note that the calc_weight function will not use the outcome data `y` when calculating the weights and will not pass it to the internal data constructor. It must be passed to the `estimate_effect` function later.

However, sometimes we get data as a `data.frame` and users may not know how to turn these into the required objects. In this case, we’ve supplied the `df2dataHolder` function to create the required data object like so:

    df <- data.frame(y = y, z = z, x)

    df2dH <- df2dataHolder(treatment.formula = "z ~ .",
                           outcome.formula = "y ~ .",
                           data = df)
    # NOT RUN
    # weights <- calc_weight(x = df2dH, z = NULL, y = NULL)

In this case, we can pass the `dataHolder` object directly to the `calc_weight` function inside the `x` argument and ignore the others; the `dataHolder` object already contains the `x`, `y`, and `z` data internally. Note that this function will need both outcome and treatment formulae since it needs to know which columns are actually confounders for the purposes of calculating the weights!

Finally, if you so desire, you can create a dataHolder object directly.

    dH <- dataHolder(x = x, z = z, y = y)

    # NOT RUN
    # weights <- calc_weight(x = dH, z = NULL, y = NULL)

This may be useful if you plan on reusing the data object.

## Estimating weights

The weights can be estimated by using the `calc_weight` function in the package. We select optimal hyperparameters through our bootstrap-based algorithm and target the average treatment effect.

    weights <- calc_weight(x = x, z = z,
                           method = "COT",
                           estimand = "ATE",
                           options = list(lambda.bootstrap = Inf,
                                          nboot = 1000L)
                           )

These weights will balance distributions, making estimates of treatment effects unbiased.

## Estimating treatment effects

We can then estimate effects with

    tau_hat <- estimate_effect(causalWeights = weights,
                               y = y)

The estimator generated here is a simple weighted difference in observed outcomes between treatment groups. Moreover, note we must supply the outcome information in argumnet `y` since the `calc_weight` function does not store it when we pass data matrices.

The output of the `estimate_effect` function creates an object of class `causalEffect` which can be fed into the native `R` function `confint` to calculate asymptotic confidence intervals,

    ci_tau <- confint(object = tau_hat, level = 0.95)

or into `vcov` to calculate the variance of your estimate using the semiparametrically efficient variance formula:

    var_tau <- vcov(object = tau_hat)

This then gives the following treatment effect estimate, variance, and C.I.

    print(coef(tau_hat))
    #>    estimate
    #> 0.007949831
    print(var_tau)
    #>           estimate
    #> estimate 0.1887772
    print(ci_tau)
    #>               2.5 %    97.5 %
    #> estimate -0.8436251 0.8595248

### Model based estimates

The function `estimate_effect` can also use models to estimate the treatment effects. There are also several additional arguments that will be demonstrated below. These are:

- `model.function`: either a character or function with the model you want to run
- `estimate.separately`: TRUE or FALSE. Should the model be estimated separately on each treatment group (TRUE) or jointly on the full data (FALSE)
- `augment.estimate`: Should an augmented estimator be used to calculate the final treatment effect? (TRUE or FALSE)
- `normalize.weights`: Should the weights be normalized to sum to one in each treatment group before being used. For methods except “Logistic”, “Probit”, or “CBPS”, the weights are by definition normalized to sum to one so this option will not have an effect for most methods in the package.

The model functions we can use need to have a few components

1. Have a `formula` argument
2. Have a `data` argument that accepts a `data.frame`
3. Have a `weights` argument
4. Have a predict method that accepts a `newdata` argument.

One such function we could use is `lm`.

#### Linear models with `lm`

    tau_hat_lm <- estimate_effect(causalWeights = weights,
                               y = y,
                               model.function = lm,
                               estimate.separately = TRUE,
                               augment.estimate = FALSE,
                               normalize.weights = TRUE)

In this case, separate models will be fit to treated and controls and the predictions from the model will be used to estimate treatment effects. We can also calculate the augmented (aka doubly robust) estimate with argument `augment.estimate`.

    tau_hat_dr <- estimate_effect(causalWeights = weights,
                               y = y,
                               model.function = lm,
                               estimate.separately = TRUE,
                               augment.estimate = TRUE,
                               normalize.weights = TRUE)

We can also fit a weighted OLS by specifying `estimate.separately = FALSE`:

    tau_hat_wols <- estimate_effect(causalWeights = weights,
                               y = y,
                               model.function = lm,
                               estimate.separately = FALSE,
                               augment.estimate = FALSE,
                               normalize.weights = TRUE)

This fits a single weighted OLS model on the entire data.

#### Barycentric projections

An outcome model that is particular to this package is the function `barycentric_projection` that estimates, as the name implies, barycentric projections of the outcome data. To use this function, there are a couple of steps. Unlike the case for linear models with `lm`, we need to think carefully about what sample the data arise from.

To use this function outside of the main causalOT functions, we would do

    df <- data.frame(z = z, y = y, x)
    bp <- barycentric_projection(formula = "y ~ x + z",
            data = df,
            weights = weights,
            separate.samples.on = "z",
            penalty = 0.01,
            cost_function = NULL,
            p = 2,
            debias = FALSE,
            cost.online = "auto",
            diameter = NULL,
            niter = 1000,
            tol = 1e-7)

This will run the optimal transport problem between the samples denoted by “z” and get the dual potentials for the Sinkhorn Divergence problem.

Then, we can run a `predict` function to see what the outcomes would be if the samples had arisen from a different distribution. Let’s say that everyone had actually been treated

    newdf   <- df
    newdf$z <- 1L
    preds   <- predict(object = bp,
                       newdata = newdf,
                       source.sample = df$z)
    head(preds)
    #> [1] -2.80749441  0.52369862  5.79605734 -1.55036673  0.01254699
    #> [6]  1.55753697

The argument `source.sample` should be a vector that denotes the original treatment group of the samples. This allows the function to use the appropriate dual potentials to calculate the expected outcome.

In the context of the estimate_effect function, we need to supply some extra arguments in the `...` argument.

    tau_hat_bp <- estimate_effect(causalWeights = weights,
                               y = y,
                               model.function = barycentric_projection,
                               estimate.separately = FALSE,
                               augment.estimate = TRUE,
                               normalize.weights = TRUE,
                          # special args for barycentric_projection
                               separate.samples.on = "z",
                               penalty = 0.01,
                               cost_function = NULL,
                               p = 3,
                               debias = FALSE,
                               cost.online = "tensorized",
                               diameter = NULL,
                               niter = 1000L,
                               tol = 1e-7,
                               line_search_fn = "strong_wolfe"
                               )
    print(tau_hat_bp@estimate)
    #> [1] -0.02549428

This method currently doesn’t have a variance estimator accounting for the weight uncertainty but we can use the asymptotic variance estimator of Hahn (1998):

    vcov(tau_hat_bp)
    #>             estimate
    #> estimate 0.008051722
    vcov(tau_hat_wols)
    #>            estimate
    #> estimate 0.01705504
    vcov(tau_hat)
    #>           estimate
    #> estimate 0.1887772

#### Note

In neither of these cases did we feed data or a formula to the model function. By default, the `estimate_effect` function will regress the outcome in argument `y` on all of the covariates from the `calc_weight` function and adjust for the treatment indicator as appropriate given the selected options. If you want to change the covariates for the outcome model from the weighting estimating function, you can provide new covariates in an argument `x`:

    estimate_effect(causalWeights = weights,
                               x = x_new,
                               y = y)

Note this data must have the same observation order as the previous data and must also be an object of class `matrix`.

## Diagnostics

Diagnostics are also an important part of deciding whether the weights perform well. There are several areas that we will explore:

1. Effective sample size
2. Mean balance
3. Distributional balance

#### 1. Effective sample size

Typically, estimated samples sizes with weights are calculated as \(\sum_i 1/w_i^2\) and gives us a measure of how much information is in the sample. The lower the effective sample size (ESS), the higher the variance, and the lower the sample size, the more weight a few individuals have. Of course, we can calculate this in `causalOT`!

    ESS(weights)
    #>   Control   Treated
    #> 116.66625  84.32589

Of course, this measure has problems because it can fail to diagnose problems with variable weights. In response, Vehtari et al. use Pareto smoothed importance sampling. We offer some shell code to adapt the class `causalWeights` to the `loo` package:

    raw_psis <- PSIS(weights)

This will also return the Pareto smoothed weights and log weights.

If we want to easily examine the PSIS diagnostics, we can pull those out too

    PSIS_diag(raw_psis)
    #> $w0
    #> $w0$pareto_k
    #> [1] -0.2320877
    #>
    #> $w0$n_eff
    #> [1] 115.7568
    #>
    #>
    #> $w1
    #> $w1$pareto_k
    #> [1] 0.2478191
    #>
    #> $w1$n_eff
    #> [1] 82.99825

We can see that all of the \(k\) values are below the recommended 0.5, indicating finite variance and that the central limit theorem holds. Note the estimated sample sizes are a bit lower than the `ESS` method above.

#### 2. Mean balance

Many authors consider the standardized absolute mean balance as a marker for important balance: see Stuart (2010). That is \[ \frac{|\overline{X}_c - \overline{X}_t| }{\sigma_{\text{pool}}},\] where \(\overline{X}_c\) is the mean in the controls, \(\overline{X}_t\) is the mean in the treated, and \(\sigma_{\text{pool}}\) is the pooled standard deviation. We offer such checks in `causalOT` as well.

First, we consider pre-weighting mean balance between treatment groups

    mean_balance(x = hainmueller)
    #>        X1        X2        X3        X4        X5        X6
    #> 1.0889178 0.9320099 0.9322327 0.3741322 0.2798986 0.2644929

and after weighting mean balance between treatment groups

    mean_balance(x = hainmueller, weights = weights)
    #>          X1          X2          X3          X4          X5          X6
    #> 0.036900048 0.006577199 0.003137536 0.003228830 0.002445118 0.060551316

Pretty good! However, mean balance doesn’t ensure distributional balance.

#### 3. Distributional balance

Ultimately, distributional balance is what we care about in causal inference. Fortunately, we can also measure that too. We consider the 2-Sinkhorn divergence of Genevay et al. since it metrizes the convergence in distribution.

Before weighting, distributional balance looks poor:

    # controls
    ot_distance(x1 = hainmueller$get_x0(), x2 = hainmueller$get_x(),
                  a = NULL, b = rep(1/512,512),
                  p = 2, penalty = 1e3, debias = TRUE)
    #> [1] 0.5311378
    #treated
    ot_distance(x1 = hainmueller$get_x1(), x2 = hainmueller$get_x(),
                  a = NULL, b = rep(1/512,512),
                  p = 2, penalty = 1e3, debias = TRUE)
    #> [1] 0.4612781

But after weighting, it looks much better!

    # controls
    ot_distance(x1 = hainmueller$get_x0(), x2 = hainmueller$get_x(),
                  a = weights@w0, b = rep(1/512,512),
                  p = 2, penalty = 1e3, debias = TRUE)
    #> [1] 0.003670398
    # treated
    ot_distance(x1 = hainmueller$get_x1(), x2 = hainmueller$get_x(),
                  a = weights@w1, b = rep(1/512,512),
                  p = 2, penalty = 1e3, debias = TRUE)
    #> [1] 0.002123788

After Causal Optimal Transport, the distributions are much more similar. We can also simply feed the output of calc_weight directly into the `ot_distance` function:

    ot_distance(x1 = weights, p = 2, penalty = 1e3, debias = TRUE)
    #> $pre
    #>   control   treated
    #> 0.5311378 0.4612781
    #>
    #> $post
    #>     control     treated
    #> 0.003670398 0.002123788

and the S4 deployment takes care of the rest.

Finally, we can construct a summary of the optimal transport distances, Pareto k statistics, effective sample size, and mean balance using the summary method:

    summarized_cw <- summary(weights, penalty = 1000)

We can then print the object to the screen:

    summarized_cw
    #> Diagnostics for causalWeights for estimand ATE
    #> Control group
    #>                                pre          post
    #> OT distance              0.5311378   0.003670398
    #> Pareto k                        NA  -0.232087710
    #> N eff                  247.0000000 115.756836449
    #> Avg. std. mean balance   0.3067516   0.017646355
    #>
    #> Treated group
    #>                                pre         post
    #> OT distance              0.4612781 2.123788e-03
    #> Pareto k                        NA 2.478191e-01
    #> N eff                  265.0000000 8.299825e+01
    #> Avg. std. mean balance   0.2859156 9.871006e-04

or we can make some diagnostic plots too!

    plot(summarized_cw)
    #> Plotting diagnostics for causalWeights for estimand ATE

## Other methods

The calc weight function can also handle other methods. We have implemented methods for logistic or probit regression, the covariate balancing propensity score (CBPS), stable balancing weights (SBW), entropy balancing weights (EntropyBW), and the synthetic control method (SCM).

    calc_weight(x = hainmueller, method = "Logistic",
                           estimand = "ATE")

    calc_weight(x = hainmueller, method = "Probit",
                           estimand = "ATE")

    calc_weight(x = hainmueller, method = "CBPS",
                           estimand = "ATE")

    calc_weight(x = hainmueller, method = "SBW")

    calc_weight(x = hainmueller, method = "EntropyBW")

    calc_weight(x = hainmueller, method = "SCM")

The function also accepts methods “EnergyBW”, for Energy Balancing Weights of Hainmueller and Mak (2020), and “NNM”, for nearest neighbor matching with replacement, but these are special cases of COT with the penalty parameter \(\lambda\) forced to be \(\infty\) and \(0\), respectively.

## Further information

### Options?

The argument `options` is a little vague. So we also have a function `cotOptions` which is avaible to help. The documentation provides more details. The other optimization methods “SBW”, “EntropyBW”, and “SCM” provide their own options function. The options for “Logistic” and “Probit” pass arguments to `glm` and “CBPS” will pass arguments to the `CBPS` function in the package of the same name.

## More complicated models via object-oriented functions

The package also provides more flexible optimal transport weights and modeling via some object-oriented programming via the `R6` package. These functions don’t have as many safeguards and everything is done by reference so that you have to be more careful about what you do. However, it also gives you more flexibility on the types of problems you can solve. To learn more, see the vignette on object-oriented solvers.