diff --git a/R/nested_rhat.R b/R/nested_rhat.R
index f2eebff5..bfca55c6 100644
--- a/R/nested_rhat.R
+++ b/R/nested_rhat.R
@@ -16,7 +16,7 @@
 #' rhat_nested(mu, superchain_ids = c(1,1,2,2))
 #'
 #' d <- as_draws_rvars(example_draws("multi_normal"))
-#' rhat(d$Sigma, superchain_ids = c(1,1,2,2))
+#' rhat_nested(d$Sigma, superchain_ids = c(1,1,2,2))
 #'
 #' @export
 rhat_nested <- function(x, superchain_ids, ...) UseMethod("rhat_nested")
@@ -33,38 +33,56 @@ rhat_nested.rvar <- function(x, superchain_ids, ...) {
   summarise_rvar_by_element_with_chains(x, rhat_nested, superchain_ids = superchain_ids, ...)
 }
 
-
 .rhat_nested <- function(x, superchain_ids, ...) {
 
-  array_dims <- dim(x)
-  ndraws <- array_dims[1]
+  x <- as.matrix(x)
+  niterations <- NROW(x)
   nchains_per_superchain <- max(table(superchain_ids))
-  nsuperchains <- length(unique(superchain_ids))
+  superchains <- unique(superchain_ids)
+
+  # mean and variance of chains calculated as in rhat
+  chain_mean <- matrixStats::colMeans2(x)
+  chain_var <- matrixStats::colVars(x, center = chain_mean)
 
-  superchain_mean <- sapply(unique(superchain_ids), function(k) mean(x[, which(superchain_ids == k)]))
+  # mean of superchains calculated by only including specified chains
+  # (equation 15 in Margossian et al. 2023)
+  superchain_mean <- sapply(
+    superchains,
+    function(k) mean(x[, which(superchain_ids == k)])
+  )
 
-  chain_mean <- matrix(matrixStats::colMeans2(x), nrow = 1)
-  chain_var <- matrixStats::colVars(x, center=chain_mean)
-  
+  # overall mean (as defined in equation 16 in Margossian et al. 2023)
   overall_mean <- mean(superchain_mean)
 
+  # between-chain variance estimate (B_k in equation 18 in Margossian et al. 2023)
   if (nchains_per_superchain == 1) {
     var_between_chain <- 0
   } else {
-    var_between_chain <- sapply(unique(superchain_ids), function(k) var(chain_mean[, which(superchain_ids == k)]))
+    var_between_chain <- sapply(
+      superchains,
+      function(k) var(chain_mean[which(superchain_ids == k)])
+    )
   }
-  if (ndraws == 1) {
+
+  # within-chain variance estimate (W_k in equation 18 in Margossian et al. 2023)
+  if (niterations == 1) {
     var_within_chain <- 0
   } else {
-    var_within_chain <- sapply(unique(superchain_ids), function(k) mean(chain_var[which(superchain_ids == k)]))
+    var_within_chain <- sapply(
+      superchains,
+      function(k) mean(chain_var[which(superchain_ids == k)])
+    )
   }
-  
+
+  # between-superchain variance (nB in equation 17 in Margossian et al. 2023)
   var_between_superchain <- matrixStats::colVars(
     as.matrix(superchain_mean),
     center = overall_mean
   )
-  
+
+  # within-superchain variance (nW in equation 18 in Margossian et al. 2023)
   var_within_superchain <- mean(var_within_chain + var_between_chain)
 
-  sqrt(1 + var_between_superchain / var_within_superchain)  
+  # nested Rhat (nRhat in equation 19 in Margossian et al. 2023)
+  sqrt(1 + var_between_superchain / var_within_superchain)
 }
diff --git a/man/rhat_nested.Rd b/man/rhat_nested.Rd
index d741cd7c..5988b91a 100644
--- a/man/rhat_nested.Rd
+++ b/man/rhat_nested.Rd
@@ -49,7 +49,7 @@ mu <- extract_variable_matrix(example_draws(), "mu")
 rhat_nested(mu, superchain_ids = c(1,1,2,2))
 
 d <- as_draws_rvars(example_draws("multi_normal"))
-rhat(d$Sigma, superchain_ids = c(1,1,2,2))
+rhat_nested(d$Sigma, superchain_ids = c(1,1,2,2))
 
 }
 \references{
diff --git a/tests/testthat/test-rhat_nested.R b/tests/testthat/test-rhat_nested.R
new file mode 100644
index 00000000..1d71ea9e
--- /dev/null
+++ b/tests/testthat/test-rhat_nested.R
@@ -0,0 +1,11 @@
+test_that("rhat_nested returns reasonable values", {
+  tau <- extract_variable_matrix(example_draws(), "tau")
+
+  rhat <- rhat_nested(tau, superchain_ids = c(1, 2, 1, 2))
+  expect_true(rhat > 0.99 & rhat < 1.05)
+
+  rhat <- rhat_nested(tau, superchain_ids = c(1, 2, 1, 2))
+  expect_true(rhat > 0.99 & rhat < 1.05)
+})
+
+