---
title: "Design and Analysis of Experiments with randomizr"
author: "Alexander Coppock"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Design and Analysis of Experiments with randomizr}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, echo=FALSE}
  set.seed(17760701)
  knitr::opts_chunk$set(
    collapse = TRUE,
    comment = "#>",
    message = FALSE,
    warning = FALSE,
    fig.width = 7,
    fig.height = 3.5
  )
  options(digits=2)
```

```{r}
library(randomizr)
library(dplyr)
library(tidyr)
library(purrr)
library(ggplot2)
library(estimatr)
```

**randomizr** is a small package for R that simplifies the design and analysis of randomized experiments. In particular, it makes the random assignment **procedure** transparent, flexible, and most importantly reproducible. By the time that many experiments are written up and made public, the process by which some units received treatments is lost or imprecisely described. The **randomizr** package makes it easy for even the most forgetful of researchers to generate error-free, reproducible random assignments.

A hazy understanding of the random assignment procedure leads to two main problems at the analysis stage. First, units may have different probabilities of assignment to treatment. Analyzing the data as though they have the same probabilities of assignment leads to biased estimates of the treatment effect. Second, units are sometimes assigned to treatment as a **cluster**. For example, all the students in a single classroom may be assigned to the same intervention together. If the analysis ignores the clustering in the assignments, estimates of average causal effects and the uncertainty attending them may be incorrect.

# A hypothetical experiment
Throughout this vignette, we'll pretend we're conducting an experiment among the 592 individuals in the built-in `HairEyeColor` dataset. As we'll see, there are many ways to randomly assign subjects to treatments. We'll step through five common designs, each associated with one of five `randomizr` functions: `simple_ra()`, `complete_ra()`, `block_ra()`, `cluster_ra()`, and `block_and_cluster_ra()`. A sixth function, `balanced_ra()`, is experimental. It draws assignment with tight targets and is illustrated briefly after the blocked design.

The dataset ships with R as a three-way contingency table (see `?HairEyeColor`); converting it to a data frame gives one row per **type** of subject, with a count of how many subjects are of that type.

```{r}
data(HairEyeColor)

HairEyeColor |>
  as.data.frame() |>
  as_tibble()
```

We first need to transform this into a dataset in which each row describes an individual subject.

```{r}
# uncount() repeats each row Freq times, which is exactly what we want
hec <-
  HairEyeColor |>
  as.data.frame() |>
  as_tibble() |>
  uncount(Freq) |>
  select(Hair, Eye, Sex)

N <- nrow(hec)

hec
```

Typically, researchers know some basic information about their subjects before deploying treatment. For example, they usually know how many subjects there are in the experimental sample (N), and they usually know some basic demographic information about each subject.

Our new dataset has `r N` subjects. We have three pretreatment covariates, `Hair`, `Eye`, and `Sex`, which describe the hair color, eye color, and gender of each subject. 

We now need to create simulated *potential outcomes*. We'll call the untreated outcome `Y0` and we'll call the treated outcome `Y1`. Imagine that in the absence of any intervention, the outcome (`Y0`) is correlated with our pretreatment covariates. Imagine further that the effectiveness of the program varies according to these covariates, i.e., the difference between `Y1` and `Y0` is correlated with the pretreatment covariates.

If we were really running an experiment, we would only observe either `Y0` or `Y1` for each subject, but since we are simulating, we generate both. Our inferential target is the average treatment effect (ATE), which is defined as the average difference between `Y0` and `Y1`.

```{r}
# Set a seed for reproducibility
set.seed(343)

# Create untreated and treated outcomes for all subjects
hec <-
  hec |>
  mutate(
    Y0 = rnorm(n = N,
               mean = 2 * as.numeric(Hair) - 4 * as.numeric(Eye) - 6 * as.numeric(Sex),
               sd = 5),
    Y1 = Y0 + 6 * as.numeric(Hair) + 4 * as.numeric(Eye) + 2 * as.numeric(Sex)
  )

# Calculate true ATE
hec |> summarize(ATE = mean(Y1 - Y0))
```

We are now ready to allocate treatment assignments to subjects. Let's start by contrasting simple and complete random assignment.

## Simple random assignment

Simple random assignment assigns all subjects to treatment with an equal probability by flipping a (weighted) coin for each subject. The main trouble with simple random assignment is that the number of subjects assigned to treatment is itself a random number: depending on the random assignment, a different number of subjects might be assigned to each group.

The `simple_ra()` function has one required argument `N`, the total number of subjects.  If no other arguments are specified, `simple_ra()` assumes a two-group design and a 0.50 probability of assignment.

```{r echo=TRUE, results="hide"}
Z <- simple_ra(N = N)

table(Z)
```
```{r echo=FALSE}
knitr::kable(t(as.matrix(table(Z))))
```

To change the probability of assignment, specify the `prob` argument:

```{r echo=TRUE, results="hide"}
Z <- simple_ra(N = N, prob = 0.30)

table(Z)
```
```{r echo=FALSE}
knitr::kable(t(as.matrix(table(Z))))
```

If you specify `num_arms` without changing `prob_each`, `simple_ra()` will assume equal probabilities across all arms.

```{r echo=TRUE, results="hide"}
Z <- simple_ra(N = N, num_arms = 3)

table(Z)
```
```{r echo=FALSE}
knitr::kable(t(as.matrix(table(Z))))
```

You can also just specify the probabilities of your multiple arms. The probabilities must sum to 1. 

```{r echo=TRUE, results="hide"}
Z <- simple_ra(N = N, prob_each = c(0.2, 0.2, 0.6))

table(Z)
```
```{r echo=FALSE}
knitr::kable(t(as.matrix(table(Z))))
```

You can also name your treatment arms.

```{r echo=TRUE, results="hide"}
Z <- simple_ra(N = N,
               prob_each = c(0.2, 0.2, 0.6),
               conditions = c("control", "placebo", "treatment"))

table(Z)
```
```{r echo=FALSE}
knitr::kable(t(as.matrix(table(Z))))
```

## Complete random assignment
Complete random assignment is very similar to simple random assignment, except that the researcher can specify *exactly* how many units are assigned to each condition. 

The syntax for `complete_ra()` is very similar to that of `simple_ra()`. The argument `m` is the number of units assigned to treatment in two-arm designs; it is analogous to `simple_ra()`'s `prob`. Similarly, the argument `m_each` is analogous to `prob_each`.

If you only specify `N`, `complete_ra()` assigns exactly half of the subjects to treatment.

```{r echo=TRUE, results="hide"}
Z <- complete_ra(N = N)

table(Z)
```
```{r echo=FALSE}
knitr::kable(t(as.matrix(table(Z))))
```

To change the number of units assigned, specify the `m` argument:

```{r echo=TRUE, results="hide"}
Z <- complete_ra(N = N, m = 200)

table(Z)
```
```{r echo=FALSE}
knitr::kable(t(as.matrix(table(Z))))
```

If you specify multiple arms, `complete_ra()` will assign an equal (within rounding) number of units to treatment.

```{r echo=TRUE, results="hide"}
Z <- complete_ra(N = N, num_arms = 3)

table(Z)
```
```{r echo=FALSE}
knitr::kable(t(as.matrix(table(Z))))
```

You can also specify exactly how many units should be assigned to each arm. The total of `m_each` must equal `N`. 

```{r echo=TRUE, results="hide"}
Z <- complete_ra(N = N, m_each = c(100, 200, 292))

table(Z)
```
```{r echo=FALSE}
knitr::kable(t(as.matrix(table(Z))))
```

You can also name your treatment arms.

```{r echo=TRUE, results="hide"}
Z <- complete_ra(N = N,
                 m_each = c(100, 200, 292),
                 conditions = c("control", "placebo", "treatment"))

table(Z)
```
```{r echo=FALSE}
knitr::kable(t(as.matrix(table(Z))))
```

## Simple and complete random assignment compared
If the number of units is known beforehand, `complete_ra()` is preferred, for two reasons:

1. Researchers can plan exactly how many treatments will be deployed.
2. The standard errors associated with complete random assignment are generally smaller, increasing experimental power.

Since you need to know `N` beforehand in order to use `simple_ra()`, it may seem like a useless function. Sometimes, however, the random assignment isn't directly in the researcher's control. For example, when deploying a survey experiment on a platform like Qualtrics, simple random assignment is the only possibility due to the inflexibility of the built-in random assignment tools. When reconstructing the random assignment for analysis after the experiment has been conducted, `simple_ra()` provides a convenient way to do so.

To compare the two designs, let's conduct a small simulation with our `HairEyeColor` dataset. We draw a fresh assignment many times over, estimate the ATE each time, and collect the estimates. The spread of those estimates *is* the sampling distribution.

```{r}
sims <- 1000

simulate_once <- function(i) {
  hec <-
    hec |>
    mutate(
      # Conduct both kinds of random assignment
      Z_simple = simple_ra(N = N),
      Z_complete = complete_ra(N = N),
      # Reveal observed potential outcomes
      Y_simple = if_else(Z_simple == 1, Y1, Y0),
      Y_complete = if_else(Z_complete == 1, Y1, Y0)
    )

  fit_simple <- difference_in_means(Y_simple ~ Z_simple, data = hec)
  fit_complete <- difference_in_means(Y_complete ~ Z_complete, data = hec)

  bind_rows(
    tidy(fit_simple) |> filter(term == "Z_simple") |> mutate(design = "Simple"),
    tidy(fit_complete) |> filter(term == "Z_complete") |> mutate(design = "Complete")
  )
}

estimates <- map(1:sims, simulate_once) |> list_rbind()
```

The standard error of an estimate is defined as the standard deviation of the sampling distribution of the estimator. When standard errors are estimated (i.e., by using the `summary()` command on a model fit), they are estimated using some approximation. This simulation allows us to measure the standard error directly, since `estimates` describes the sampling distribution of each design.

```{r}
estimates |>
  group_by(design) |>
  summarize(empirical_se = sd(estimate))
```

Plotting the two sampling distributions side by side shows how similar they are:

```{r}
gg_df <-
  estimates |>
  mutate(design = factor(design, levels = c("Simple", "Complete")))

ggplot(gg_df, aes(x = estimate)) +
  geom_histogram(bins = 40) +
  geom_vline(xintercept = mean(hec$Y1 - hec$Y0), linetype = "dashed") +
  facet_wrap(~design) +
  labs(x = "ATE estimate", y = "Count",
       title = "Sampling distributions under simple and complete random assignment",
       subtitle = "Dashed line is the true ATE") +
  theme_bw() +
  theme(legend.position = "none")
```

Both designs are unbiased: each distribution is centered on the true ATE. Their spreads are also nearly identical, which may be surprising given that complete random assignment is supposed to be the more precise design.

It is more precise, but by very little at this sample size. At `N = 592` the true reduction in sampling variance is about 0.2%, which is far too small to see in 1,000 simulations: the simulation error in this comparison is several percentage points, so whichever design comes out ahead in the printed standard errors above, it came out ahead by chance.

The advantage comes from one thing. Simple random assignment does not fix the number of treated units: `m` is a binomial draw that lands near `N/2` but wanders around it. Complete random assignment fixes `m` exactly, and the variance that removes shrinks like `1/N`, which is why it is invisible at `N = 592`. With only 10 subjects, the wandering is plain to see.

```{r echo=TRUE, results="hide"}
# how many of 10 subjects end up treated, over many draws
m_simple <- replicate(sims, sum(simple_ra(N = 10)))
m_complete <- replicate(sims, sum(complete_ra(N = 10)))

table(m_simple)

table(m_complete)
```
```{r echo=FALSE}
knitr::kable(t(as.matrix(table(m_simple))))

knitr::kable(t(as.matrix(table(m_complete))))
```

That spread is the source of the extra variance, and it shrinks as `N` grows. The true reduction in sampling variance from using `complete_ra()` instead of `simple_ra()` on these potential outcomes is about 13% at `N = 10`, 3.6% at `N = 30`, and 0.2% at `N = 592`. Repeating the sampling-distribution exercise on a small sample measures that gain directly. We use the first 30 subjects: few enough that the gain is visible, and many enough that no draw is so lopsided that a standard error cannot be calculated from it. A 3.6% difference in variance is only a 1.8% difference in the standard error, so it takes more draws to resolve than the `N = 592` comparison did.

```{r}
set.seed(20260824)

sims_small <- 10000

hec_small <- hec |> slice(1:30)

simulate_once_small <- function(i) {
  hec_small <-
    hec_small |>
    mutate(
      # Conduct both kinds of random assignment
      Z_simple = simple_ra(N = 30),
      Z_complete = complete_ra(N = 30),
      # Reveal observed potential outcomes
      Y_simple = if_else(Z_simple == 1, Y1, Y0),
      Y_complete = if_else(Z_complete == 1, Y1, Y0)
    )

  fit_simple <- difference_in_means(Y_simple ~ Z_simple, data = hec_small)
  fit_complete <- difference_in_means(Y_complete ~ Z_complete, data = hec_small)

  bind_rows(
    tidy(fit_simple) |> filter(term == "Z_simple") |> mutate(design = "Simple"),
    tidy(fit_complete) |> filter(term == "Z_complete") |> mutate(design = "Complete")
  )
}

estimates_small <- map(1:sims_small, simulate_once_small) |> list_rbind()

estimates_small |>
  group_by(design) |>
  summarize(empirical_se = sd(estimate))
```

The simulated gap is a little wider than the true one, which is what a percent or two of remaining simulation error looks like. That is the practical case for `complete_ra()`: not that it buys a lot of precision in a large sample, but that it removes an avoidable source of variability and guarantees the number of units you can afford to treat.

## Block random assignment

Block random assignment (sometimes known as stratified random assignment) is a powerful tool when used well. In this design, subjects are sorted into blocks (strata) according to their pre-treatment covariates, and then complete random assignment is conducted within each block. For example, a researcher might block on gender, assigning exactly half of the men and exactly half of the women to treatment. 

There are two main reasons to block. The first is to signal to future readers that treatment effect heterogeneity may be of interest: is the treatment effect different for men versus women? Of course, such heterogeneity could be explored if complete random assignment had been used, but blocking on a covariate defends a researcher (somewhat) against claims of data dredging. The second reason is to increase precision. If the blocking variables are predictive of the outcome (i.e., they are correlated with the outcome), then blocking may help to decrease sampling variability. It's important, however, not to overstate these advantages. The gains from a blocked design can often be realized through covariate adjustment alone.

Blocking can also produce complications for estimation, because it can give different subjects different probabilities of assignment. This complication is typically addressed in one of two ways: "controlling for blocks" in a regression context, or inverse probability weights (IPW), in which units are weighted by the inverse of the probability that the unit is in the condition that it is in.  

The only required argument to `block_ra()` is `blocks`, which is a vector of length `N` that describes which block a unit belongs to. `blocks` can be a factor, character, or numeric variable. If no other arguments are specified, `block_ra()` assigns an approximately equal proportion of each block to treatment. 

```{r echo=TRUE, results="hide"}
Z <- block_ra(blocks = hec$Hair)

table(Z, hec$Hair)
```
```{r echo=FALSE}
knitr::kable(table(Z, hec$Hair))
```

For multiple treatment arms, use the `num_arms` argument, with or without the `conditions` argument

```{r echo=TRUE, results="hide"}
Z <- block_ra(blocks = hec$Hair, num_arms = 3)

table(Z, hec$Hair)
```
```{r echo=FALSE}
knitr::kable(table(Z, hec$Hair))
```

```{r echo=TRUE, results="hide"}
Z <- block_ra(blocks = hec$Hair,
              conditions = c("Control", "Placebo", "Treatment"))

table(Z, hec$Hair)
```
```{r echo=FALSE}
knitr::kable(table(Z, hec$Hair))
```

`block_ra()` provides a number of ways to adjust the number of subjects assigned to each condition. The `prob_each` argument describes what proportion of each block should be assigned to each treatment arm. Note, of course, that `block_ra()` still uses complete random assignment within each block; the appropriate number of units to assign to treatment within each block is automatically determined.

```{r echo=TRUE, results="hide"}
Z <- block_ra(blocks = hec$Hair, prob_each = c(0.3, 0.7))

table(Z, hec$Hair)
```
```{r echo=FALSE}
knitr::kable(table(Z, hec$Hair))
```

For finer control, use the `block_m_each` argument, which takes a matrix with as many rows as there are blocks, and as many columns as there are treatment conditions.  Remember that the rows are in the same order as `sort(unique(blocks))`, a command that is good to run before constructing a `block_m_each` matrix. 

```{r echo=TRUE}
sort(unique(hec$Hair))

block_m_each <- rbind(c(78, 30),
                      c(186, 100),
                      c(51, 20),
                      c(87, 40))

block_m_each
```

```{r echo=TRUE, results="hide"}
Z <- block_ra(blocks = hec$Hair, block_m_each = block_m_each)

table(Z, hec$Hair)
```
```{r echo=FALSE}
knitr::kable(table(Z, hec$Hair))
```

In the example above, the different blocks have different probabilities of assignment to treatment. In this case, people with Black hair have a 30/108 = 27.8% chance of being treated, those with Brown hair have 100/286 = 35.0% chance, etc. Left unaddressed, this discrepancy could bias treatment effects. We can see this directly with the `declare_ra()` function.

```{r echo=TRUE, results="hide"}
declaration <-
  declare_ra(blocks = Hair, block_m_each = block_m_each, data = hec)

# show the probability that each unit is assigned to each condition
head(declaration$probabilities_matrix)
```
```{r echo=FALSE}
knitr::kable(head(declaration$probabilities_matrix))
```

```{r echo=TRUE, results="hide"}
# Show that the probability of treatment is different within block
table(hec$Hair, round(declaration$probabilities_matrix[, 2], 3))
```
```{r echo=FALSE}
knitr::kable(table(hec$Hair, round(declaration$probabilities_matrix[, 2], 3)))
```

There are two common ways to address this problem: LSDV (Least-Squares Dummy Variable, also known as "control for blocks") or IPW (Inverse-probability weights).

The following code snippet shows how to use either the LSDV approach or the IPW approach. A note for scrupulous readers: the estimands of these two approaches are subtly different from one another. The LSDV approach estimates the average **block-level** treatment effect. The IPW approach estimates the average **individual-level** treatment effect. They can be different. Since the average block-level treatment effect is not what most people have in mind when thinking about causal effects, analysts using this approach should present both. The `obtain_condition_probabilities()` function used to calculate the probabilities of assignment is explained below.

```{r echo=TRUE, results="hide"}
hec <-
  hec |>
  mutate(
    Z_blocked = block_ra(blocks = Hair, block_m_each = block_m_each),
    Y_blocked = if_else(Z_blocked == 1, Y1, Y0),
    cond_prob = obtain_condition_probabilities(declaration, Z_blocked),
    IPW_weights = 1 / cond_prob
  )

fit_LSDV <- lm_robust(Y_blocked ~ Z_blocked + Hair, data = hec)
fit_IPW <- lm_robust(Y_blocked ~ Z_blocked, weights = IPW_weights, data = hec)

tidy(fit_LSDV)
```
```{r echo=FALSE}
knitr::kable(tidy(fit_LSDV))
```

```{r echo=TRUE, results="hide"}
tidy(fit_IPW)
```
```{r echo=FALSE}
knitr::kable(tidy(fit_IPW))
```

Blocks can be built by hand from the covariates. In the `HairEyeColor` dataset, we could make a block for each unique combination of hair color, eye color, and sex.

```{r echo=TRUE, results="hide"}
block_id <- paste(hec$Hair, hec$Eye, hec$Sex, sep = "_")

Z <- block_ra(blocks = block_id)

head(table(block_id, Z))
```
```{r echo=FALSE}
knitr::kable(head(table(block_id, Z)))
```

An alternative is to use the [blockTools](https://CRAN.R-project.org/package=blockTools) package, which constructs matched pairs, trios, quartets, etc. from pretreatment covariates. Two of its functions do the work here: `blockTools::block()` builds the blocks, and `blockTools::createBlockIDs()` turns them into a blocking variable of length `N`.

```{r eval=FALSE, results="hide"}
library(blockTools)

# blockTools requires that all variables be numeric
numeric_mat <- model.matrix(~ Hair + Eye + Sex, data = hec)[, -1]

# blockTools also requires an id variable
df_forBT <- data.frame(id_var = 1:nrow(numeric_mat), numeric_mat)

# Conducting the actual blocking: let's make trios
out <- blockTools::block(df_forBT,
                         n.tr = 3,
                         id.vars = "id_var",
                         block.vars = colnames(df_forBT)[-1])

# Extract the block_ids
hec <- hec |>
  mutate(block_id = blockTools::createBlockIDs(out, df_forBT, id.var = "id_var"))

# Conduct actual random assignment with randomizr
Z_blocked <- block_ra(blocks = hec$block_id, num_arms = 3)

head(table(hec$block_id, Z_blocked))
```

A note for `blockTools` users: that package also has an assignment function. My preference is to extract the blocking variable, then conduct the assignment with `block_ra()`, so that fewer steps are required to reconstruct the random assignment or generate new random assignments for a randomization inference procedure.

## Balanced random assignment

`balanced_ra()` is a new addition to the suite and still experimental. It is used for settings where randomization is constrained to hit tight targets.

To illustrate, say there are six units, two blocks of three, half assigned to treatment. `complete_ra(N = 6, m = 3)` treats three units every time, but a block can receive zero or three. `block_ra` ensures that 1 or 2 are treated in each block; but overall there is no guarantee that 3 will be treated. The twin targets of 3 overall and 1 to 2 in each block cannot be hit by either of these.

```{r echo=TRUE, results="hide"}
blocks <- rep(1:2, each = 3)
Z <- balanced_ra(blocks = blocks)
table(blocks, Z)
```
```{r echo=FALSE}
blocks <- rep(1:2, each = 3)
Z <- balanced_ra(blocks = blocks)
knitr::kable(table(blocks, Z))
```

With `balanced_ra` however we see that every draw treats three units overall and one or two in each block. Declare the design with `declare_ra(..., ra_type = "balanced")`. The [balanced_ra vignette](balanced_ra.html) has the details.

For a still simpler example, consider two units with target assignment probabilities of 0.6, and 0.9 and want the number assigned to be close to 0.6 + 0.9 = 1.5. This cannot be achieved by `simple_ra` or `complete_ra`, but `balanced_ra` handles it easily, in particular by making sure that the two units are never both assigned to control at the same time.



## Clustered assignment

Clustered assignment is unfortunate. If you can avoid assigning subjects to treatments by cluster, you should. Sometimes, clustered assignment is unavoidable. Some common situations include:

1. Housemates in households: whole households are assigned to treatment or control
2. Students in classrooms: whole classrooms are assigned to treatment or control
3. Residents in towns or villages: whole communities are assigned to treatment or control

Clustered assignment decreases the effective sample size of an experiment. In the extreme case when outcomes are perfectly correlated with clusters, the experiment has an effective sample size equal to the number of clusters. When outcomes are perfectly uncorrelated with clusters, the effective sample size is equal to the number of subjects. Almost all cluster-assigned experiments fall somewhere in the middle of these two extremes. 

The only required argument for the `cluster_ra()` function is the `clusters` argument, which is a vector of length `N` that indicates which cluster each subject belongs to. Let's pretend that for some reason, we have to assign treatments according to the unique combinations of hair color, eye color, and gender. 

```{r echo=TRUE, results="hide"}
hec <-
  hec |>
  mutate(cluster_id = paste(Hair, Eye, Sex, sep = "_"))

Z_clust <- cluster_ra(clusters = hec$cluster_id)

head(table(hec$cluster_id, Z_clust))
```
```{r echo=FALSE}
knitr::kable(head(table(hec$cluster_id, Z_clust)))
```

The table shows that each cluster is either assigned to treatment or control. No two units within the same cluster are assigned to different conditions.

As with all functions in `randomizr`, you can specify multiple treatment arms in a variety of ways:

```{r echo=TRUE, results="hide"}
Z_clust <- cluster_ra(clusters = hec$cluster_id, num_arms = 3)

head(table(hec$cluster_id, Z_clust))
```
```{r echo=FALSE}
knitr::kable(head(table(hec$cluster_id, Z_clust)))
```

... or using `conditions`

```{r echo=TRUE, results="hide"}
Z_clust <- cluster_ra(clusters = hec$cluster_id,
                      conditions = c("Control", "Placebo", "Treatment"))

head(table(hec$cluster_id, Z_clust))
```
```{r echo=FALSE}
knitr::kable(head(table(hec$cluster_id, Z_clust)))
```

... or using `m_each`, which describes how many clusters should be assigned to each condition.  `m_each` must sum to the number of clusters.

```{r echo=TRUE, results="hide"}
Z_clust <- cluster_ra(clusters = hec$cluster_id, m_each = c(5, 15, 12))

head(table(hec$cluster_id, Z_clust))
```
```{r echo=FALSE}
knitr::kable(head(table(hec$cluster_id, Z_clust)))
```

## Blocked and clustered assignment

The power of clustered experiments can sometimes be improved through blocking. In this scenario, whole clusters are members of a particular block: imagine villages nested within discrete regions, or classrooms nested within discrete schools.

As an example, let's group our clusters into blocks by size using `dplyr`.

```{r echo=TRUE, results="hide"}
cluster_level_df <-
  hec |>
  group_by(cluster_id) |>
  summarize(cluster_size = n()) |>
  arrange(cluster_size) |>
  mutate(block_id = paste0("block_", sprintf("%02d", rep(1:16, each = 2))))

hec <- left_join(hec, cluster_level_df, by = "cluster_id")

Z <- block_and_cluster_ra(clusters = hec$cluster_id, blocks = hec$block_id)

head(table(hec$cluster_id, Z))

head(table(hec$block_id, Z))
```
```{r echo=FALSE}
knitr::kable(head(table(hec$cluster_id, Z)))

knitr::kable(head(table(hec$block_id, Z)))
```

## Calculating probabilities of assignment

All of the random assignment functions in `randomizr` assign units to treatment with known (if sometimes complicated) probabilities. The `declare_ra()` and `obtain_condition_probabilities()` functions calculate these probabilities according to the parameters of your experimental design.

Let's take a look at the block random assignment we used before.

```{r echo=TRUE, results="hide"}
block_m_each <- rbind(c(78, 30),
                      c(186, 100),
                      c(51, 20),
                      c(87, 40))

Z <- block_ra(blocks = hec$Hair, block_m_each = block_m_each)

table(Z, hec$Hair)
```
```{r echo=FALSE}
knitr::kable(table(Z, hec$Hair))
```

In order to calculate the probabilities of assignment, we call the `declare_ra()` function with the same design arguments we used for the `block_ra()` call. The `declaration` object contains a matrix of probabilities of assignment:

```{r echo=TRUE, results="hide"}
declaration <-
  declare_ra(blocks = Hair, block_m_each = block_m_each, data = hec)

prob_mat <- declaration$probabilities_matrix

head(prob_mat)
```
```{r echo=FALSE}
knitr::kable(head(prob_mat))
```

The `prob_mat` object has `N` rows and as many columns as there are treatment conditions, in this case 2.

In order to use inverse-probability weights, we need to know the probability of each unit being in the **condition that it is in**.  For each unit, we need to pick the appropriate probability. The `obtain_condition_probabilities()` function handles this bookkeeping automatically.

```{r echo=TRUE, results="hide"}
cond_prob <- obtain_condition_probabilities(declaration, Z)

table(round(cond_prob, 2), Z)
```
```{r echo=FALSE}
knitr::kable(table(round(cond_prob, 2), Z))
```

# Best practices

## Random assignment procedure = Random assignment **function**

Random assignment procedures are often described as a series of steps that are manually carried out by the researcher. A procedure written that way can only be carried out by hand, and two readers of the same description will often carry it out differently. In order to make the procedure reproducible, these steps need to be translated into a **function** that returns a different random assignment each time it is called. Once the procedure is a function, it can be run again by anyone, its probabilities of assignment can be recovered by simulation, and the randomization distribution needed for randomization inference can be generated by calling it a few thousand times.

For example, consider the following procedure for randomly allocating school vouchers.

1. Every eligible student's name is put on a list
2. Each name is assigned a random number
3. Balls with the numbers associated with all students are put in an urn.
4. Then the urn is "shuffled" 
5. Students' names are drawn one by one from the urn until all slots are given out.
6. If one sibling in a family wins, all other siblings automatically win too.

If we write such a procedure into a function, it might look like this:

```{r echo=TRUE, results="hide"}
# 400 families have 1 child in the lottery, 100 families have 2
family_id <- c(sprintf("%03d", 1:500), sprintf("%03d", 1:100))

school_ra <- function(m) {
  N <- length(family_id)
  random_number <- sample(1:N, replace = FALSE)
  Z <- rep(0, N)
  i <- 1
  while (sum(Z) < m) {
    Z[family_id == family_id[random_number[i]]] <- 1
    i <- i + 1
  }
  return(Z)
}

Z <- school_ra(200)

table(Z)
```
```{r echo=FALSE}
knitr::kable(t(as.matrix(table(Z))))
```

This assignment procedure is complicated by the sibling rule, which has two effects: first, students are cluster-assigned by family, and second, the probability of assignment varies student to student. Obviously, families who have two children in the lottery have a higher probability of winning the lottery because they effectively have two "tickets." There may be better ways of running this assignment procedure (for example, with `cluster_ra()`), but the purpose of this example is to show how complicated *real-world* procedures can be written up in a simple function. With this function, the random assignment procedure can be reproduced exactly, the complicated probabilities of assignment can be calculated, and the analysis is greatly simplified.

## Check probabilities of assignment directly

For many designs, the probability of assignment to treatment can be calculated analytically.  For example, in a completely randomized design with 200 units, 60 of which are assigned to treatment, the probability is exactly 0.30 for all units. However, in more complicated designs (such as the schools example described above), analytic probabilities are difficult to calculate. In such a situation, an easy way to obtain the probabilities of assignment is through simulation.  

1. Call your random assignment function an approximately infinite number of times (about 10,000 for most purposes).
2. Count how often each unit is assigned to each treatment arm.

```{r}
Z_matrix <- replicate(1000, school_ra(200))

gg_df <- tibble(student = seq_len(nrow(Z_matrix)),
                prob = rowMeans(Z_matrix))

ggplot(gg_df, aes(x = student, y = prob)) +
  geom_point(size = 0.8) +
  labs(x = "Student", y = "Estimated probability of assignment") +
  theme_bw()
```

The plot shows that the students who have a sibling in the lottery have a higher probability of assignment. The more simulations, the more precise the estimate of the probability of assignment.

## Save your random assignment
Whenever you conduct a random assignment for use in an experiment, save it! At a minimum, the random assignment should be saved with an id variable in a csv.

```{r,eval=FALSE}
hec <-
  hec |>
  mutate(
    Z_complete = complete_ra(N = N,
                             m_each = c(100, 200, 292),
                             conditions = c("control", "placebo", "treatment")),
    id_var = row_number()
  )

hec |>
  select(id_var, Z_complete) |>
  readr::write_csv("MyRandomAssignment.csv")
```