# Causal Conditional Distance Correlation

#### Eric W. Bridgeford

#### 2025-01-07

    require(causalBatch)
    require(ggplot2)
    require(tidyr)
    n = 200

To start, we will begin with a simulation example, similar to the ones we were working in for the simulations, which you can access from:

    vignette("cb.simulations", package="causalBatch")

Let’s regenerate our working example data with some plotting code:

    # a function for plotting a scatter plot of the data
    plot.sim <- function(Ys, Ts, Xs, title="",
                         xlabel="Covariate",
                         ylabel="Outcome (1st dimension)") {
      data = data.frame(Y1=Ys[,1], Y2=Ys[,2],
                        Group=factor(Ts, levels=c(0, 1), ordered=TRUE),
                        Covariates=Xs)

      data %>%
        ggplot(aes(x=Covariates, y=Y1, color=Group)) +
        geom_point() +
        labs(title=title, x=xlabel, y=ylabel) +
        scale_x_continuous(limits = c(-1, 1)) +
        scale_color_manual(values=c(`0`="#bb0000", `1`="#0000bb"),
                           name="Group/Batch") +
        theme_bw()
    }

Next, we will generate a simulation:

    sim = cb.sims.sim_sigmoid(n=n, eff_sz=1, unbalancedness=1.5)

    plot.sim(sim$Ys, sim$Ts, sim$Xs, title="Sigmoidal Simulation")

Despite the fact that the covariate distributions for each group/batch do not overlap perfectly (note that `unbalancedness` is not \(1\)), it looks like the two batches still appear to be slightly different. We can test this using the causal conditional distance correlation, like so:

    result <- cb.detect.caus_cdcorr(sim$Ys, sim$Ts, sim$Xs, R=100)

Here, we set the number of null replicates `R` to \(100\) to make the simulation run faster, but in practice you should typically use at least \(1000\) null replicates. To make this faster, we would suggest setting `num.threads` to be close to the maximum number of cores available on your machine. You can identify the number of cores available on your machine using `parallel::detectCores()`.

With the \(\alpha\) of the test at \(0.05\), we see that the \(p\)-value is:

    print(sprintf("p-value: %.4f", result$Test$p.value))
    #> [1] "p-value: 0.0099"

Since the \(p\)-value is \(< \alpha\), we reject the null hypothesis in favor of the alternative; that is, that the group/batch causes a difference in the outcome variable.

We could optionally have pre-computed a distance matrix for the outcomes, like so:

    # compute distance matrix for outcomes
    DY = dist(sim$Ys)

In your use-cases, you could substitute this distance function for any distance function of your choosing, and pass a distance matrix directly to the detection algorithm, by specifying that `distance=TRUE`:

    result <- cb.detect.caus_cdcorr(DY, sim$Ts, sim$Xs, distance=TRUE, R=100)