diff --git a/Project.toml b/Project.toml
index 0500951..7d5cd9c 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,11 +1,12 @@
 name = "MCMCDiagnosticTools"
 uuid = "be115224-59cd-429b-ad48-344e309966f0"
 authors = ["David Widmann"]
-version = "0.2.0"
+version = "0.2.1"
 
 [deps]
 AbstractFFTs = "621f4979-c628-5d54-868e-fcf4e3e8185c"
 DataAPI = "9a962f9c-6df0-11e9-0e5d-c546b8b5ee8a"
+DataStructures = "864edb3b-99cc-5e75-8d2d-829cb0a9cfe8"
 Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 MLJModelInterface = "e80e1ace-859a-464e-9ed9-23947d8ae3ea"
@@ -18,6 +19,7 @@ Tables = "bd369af6-aec1-5ad0-b16a-f7cc5008161c"
 [compat]
 AbstractFFTs = "0.5, 1"
 DataAPI = "1.6"
+DataStructures = "0.18.3"
 Distributions = "0.25"
 MLJModelInterface = "1.6"
 SpecialFunctions = "0.8, 0.9, 0.10, 1, 2"
diff --git a/src/MCMCDiagnosticTools.jl b/src/MCMCDiagnosticTools.jl
index 6aca454..9c995f6 100644
--- a/src/MCMCDiagnosticTools.jl
+++ b/src/MCMCDiagnosticTools.jl
@@ -2,6 +2,7 @@ module MCMCDiagnosticTools
 
 using AbstractFFTs: AbstractFFTs
 using DataAPI: DataAPI
+using DataStructures: DataStructures
 using Distributions: Distributions
 using MLJModelInterface: MLJModelInterface
 using SpecialFunctions: SpecialFunctions
@@ -22,6 +23,7 @@ export mcse
 export rafterydiag
 export rstar
 
+include("utils.jl")
 include("bfmi.jl")
 include("discretediag.jl")
 include("ess.jl")
diff --git a/src/rstar.jl b/src/rstar.jl
index 7eb61c9..820c3a0 100644
--- a/src/rstar.jl
+++ b/src/rstar.jl
@@ -4,7 +4,8 @@
         classifier::MLJModelInterface.Supervised,
         samples,
         chain_indices::AbstractVector{Int};
-        subset::Real=0.8,
+        subset::Real=0.7,
+        split_chains::Int=2,
         verbosity::Int=0,
     )
 
@@ -23,26 +24,25 @@ function rstar(
     classifier::MLJModelInterface.Supervised,
     x,
     y::AbstractVector{Int};
-    subset::Real=0.8,
+    subset::Real=0.7,
+    split_chains::Int=2,
     verbosity::Int=0,
 )
     # checks
     MLJModelInterface.nrows(x) != length(y) && throw(DimensionMismatch())
     0 < subset < 1 || throw(ArgumentError("`subset` must be a number in (0, 1)"))
 
+    ysplit = split_chain_indices(y, split_chains)
+
     # randomly sub-select training and testing set
-    N = length(y)
-    Ntrain = round(Int, N * subset)
-    0 < Ntrain < N ||
+    train_ids, test_ids = shuffle_split_stratified(rng, ysplit, subset)
+    0 < length(train_ids) < length(y) ||
         throw(ArgumentError("training and test data subsets must not be empty"))
-    ids = Random.randperm(rng, N)
-    train_ids = view(ids, 1:Ntrain)
-    test_ids = view(ids, (Ntrain + 1):N)
 
     xtable = _astable(x)
 
     # train classifier on training data
-    ycategorical = MLJModelInterface.categorical(y)
+    ycategorical = MLJModelInterface.categorical(ysplit)
     xtrain = MLJModelInterface.selectrows(xtable, train_ids)
     fitresult, _ = MLJModelInterface.fit(
         classifier, verbosity, xtrain, ycategorical[train_ids]
@@ -79,7 +79,8 @@ end
         rng::Random.AbstractRNG=Random.default_rng(),
         classifier::MLJModelInterface.Supervised,
         samples::AbstractArray{<:Real,3};
-        subset::Real=0.8,
+        subset::Real=0.7,
+        split_chains::Int=2,
         verbosity::Int=0,
     )
 
@@ -91,8 +92,10 @@ This implementation is an adaption of algorithms 1 and 2 described by Lambert an
 
 The `classifier` has to be a supervised classifier of the MLJ framework (see the
 [MLJ documentation](https://alan-turing-institute.github.io/MLJ.jl/dev/list_of_supported_models/#model_list)
-for a list of supported models). It is trained with a `subset` of the samples. The training
-of the classifier can be inspected by adjusting the `verbosity` level.
+for a list of supported models). It is trained with a `subset` of the samples from each
+chain. Each chain is split into `split_chains` separate chains to additionally check for
+within-chain convergence. The training of the classifier can be inspected by adjusting the
+`verbosity` level.
 
 If the classifier is deterministic, i.e., if it predicts a class, the value of the ``R^*``
 statistic is returned (algorithm 1). If the classifier is probabilistic, i.e., if it outputs
diff --git a/src/utils.jl b/src/utils.jl
new file mode 100644
index 0000000..ca18608
--- /dev/null
+++ b/src/utils.jl
@@ -0,0 +1,99 @@
+"""
+    unique_indices(x) -> (unique, indices)
+
+Return the results of `unique(collect(x))` along with the a vector of the same length whose
+elements are the indices in `x` at which the corresponding unique element in `unique` is
+found.
+"""
+function unique_indices(x)
+    inds = eachindex(x)
+    T = eltype(inds)
+    ind_map = DataStructures.SortedDict{eltype(x),Vector{T}}()
+    for i in inds
+        xi = x[i]
+        inds_xi = get!(ind_map, xi) do
+            return T[]
+        end
+        push!(inds_xi, i)
+    end
+    unique = collect(keys(ind_map))
+    indices = collect(values(ind_map))
+    return unique, indices
+end
+
+"""
+    split_chain_indices(
+        chain_inds::AbstractVector{Int},
+        split::Int=2,
+    ) -> AbstractVector{Int}
+
+Split each chain in `chain_inds` into `split` chains.
+
+For each chain in `chain_inds`, all entries are assumed to correspond to draws that have
+been ordered by iteration number. The result is a vector of the same length as `chain_inds`
+where each entry is the new index of the chain that the corresponding draw belongs to.
+"""
+function split_chain_indices(c::AbstractVector{Int}, split::Int=2)
+    cnew = similar(c)
+    if split == 1
+        copyto!(cnew, c)
+        return cnew
+    end
+    _, indices = unique_indices(c)
+    chain_ind = 1
+    for inds in indices
+        ndraws_per_split, rem = divrem(length(inds), split)
+        # here we can't use Iterators.partition because it's greedy. e.g. we can't partition
+        # 4 items across 3 partitions because Iterators.partition(1:4, 1) == [[1], [2], [3]]
+        # and Iterators.partition(1:4, 2) == [[1, 2], [3, 4]]. But we would want
+        # [[1, 2], [3], [4]].
+        i = j = 0
+        ndraws_this_split = ndraws_per_split + (j < rem)
+        for ind in inds
+            cnew[ind] = chain_ind
+            if (i += 1) == ndraws_this_split
+                i = 0
+                j += 1
+                ndraws_this_split = ndraws_per_split + (j < rem)
+                chain_ind += 1
+            end
+        end
+    end
+    return cnew
+end
+
+"""
+    shuffle_split_stratified(
+        rng::Random.AbstractRNG,
+        group_ids::AbstractVector,
+        frac::Real,
+    ) -> (inds1, inds2)
+
+Randomly split the indices of `group_ids` into two groups, where `frac` indices from each
+group are in `inds1` and the remainder are in `inds2`.
+
+This is used, for example, to split data into training and test data while preserving the
+class balances.
+"""
+function shuffle_split_stratified(
+    rng::Random.AbstractRNG, group_ids::AbstractVector, frac::Real
+)
+    _, indices = unique_indices(group_ids)
+    T = eltype(eltype(indices))
+    N1_tot = sum(x -> round(Int, length(x) * frac), indices)
+    N2_tot = length(group_ids) - N1_tot
+    inds1 = Vector{T}(undef, N1_tot)
+    inds2 = Vector{T}(undef, N2_tot)
+    items_in_1 = items_in_2 = 0
+    for inds in indices
+        N = length(inds)
+        N1 = round(Int, N * frac)
+        N2 = N - N1
+        Random.shuffle!(rng, inds)
+        copyto!(inds1, items_in_1 + 1, inds, 1, N1)
+        copyto!(inds2, items_in_2 + 1, inds, N1 + 1, N2)
+        items_in_1 += N1
+        items_in_2 += N2
+    end
+    return inds1, inds2
+end
diff --git a/test/rstar.jl b/test/rstar.jl
index 928e50a..9270641 100644
--- a/test/rstar.jl
+++ b/test/rstar.jl
@@ -30,7 +30,7 @@ const xgboost_deterministic = Pipeline(XGBoostClassifier(); operation=predict_mo
                 @test dist isa LocationScale
                 @test dist.ρ isa PoissonBinomial
                 @test minimum(dist) == 0
-                @test maximum(dist) == 3
+                @test maximum(dist) == 6
             end
             @test mean(dist) ≈ 1 rtol = 0.2
             wrapper === Vector && break
@@ -48,7 +48,7 @@ const xgboost_deterministic = Pipeline(XGBoostClassifier(); operation=predict_mo
                 @test dist isa LocationScale
                 @test dist.ρ isa PoissonBinomial
                 @test minimum(dist) == 0
-                @test maximum(dist) == 4
+                @test maximum(dist) == 8
             end
             @test mean(dist) ≈ 1 rtol = 0.15
 
@@ -58,7 +58,7 @@ const xgboost_deterministic = Pipeline(XGBoostClassifier(); operation=predict_mo
                 100 .* cos.(1:N) 100 .* sin.(1:N)
             ])
             chain_indices = repeat(1:2; inner=N)
-            dist = rstar(classifier, samples, chain_indices)
+            dist = rstar(classifier, samples, chain_indices; split_chains=1)
 
             # Mean of the statistic should be close to 2, i.e., the classifier should be able to
             # learn an almost perfect decision boundary between chains.
@@ -71,6 +71,17 @@ const xgboost_deterministic = Pipeline(XGBoostClassifier(); operation=predict_mo
                 @test maximum(dist) == 2
             end
             @test mean(dist) ≈ 2 rtol = 0.15
+
+            # Compute the R⋆ statistic for identical chains that individually have not mixed.
+            samples = ones(sz)
+            samples[div(N, 2):end, :] .= 2
+            chain_indices = repeat(1:4; outer=div(N, 4))
+            dist = rstar(classifier, samples, chain_indices; split_chains=1)
+            # without split chains cannot distinguish between chains
+            @test mean(dist) ≈ 1 rtol = 0.15
+            dist = rstar(classifier, samples, chain_indices)
+            # with split chains can learn almost perfect decision boundary
+            @test mean(dist) ≈ 2 rtol = 0.15
         end
         wrapper === Vector && continue
 
diff --git a/test/runtests.jl b/test/runtests.jl
index 63fdade..af19e7b 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -10,6 +10,10 @@ using Test
 Random.seed!(1)
 
 @testset "MCMCDiagnosticTools.jl" begin
+    @testset "utils" begin
+        include("utils.jl")
+    end
+
     @testset "Bayesian fraction of missing information" begin
         include("bfmi.jl")
     end
diff --git a/test/utils.jl b/test/utils.jl
new file mode 100644
index 0000000..de374e8
--- /dev/null
+++ b/test/utils.jl
@@ -0,0 +1,62 @@
+using MCMCDiagnosticTools
+using Test
+using Random
+
+@testset "unique_indices" begin
+    @testset "indices=$(eachindex(inds))" for inds in [
+        rand(11:14, 100), transpose(rand(11:14, 10, 10))
+    ]
+        unique, indices = @inferred MCMCDiagnosticTools.unique_indices(inds)
+        @test unique isa Vector{Int}
+        if eachindex(inds) isa CartesianIndices{2}
+            @test indices isa Vector{Vector{CartesianIndex{2}}}
+        else
+            @test indices isa Vector{Vector{Int}}
+        end
+        @test issorted(unique)
+        @test issetequal(union(indices...), eachindex(inds))
+        for i in eachindex(unique, indices)
+            @test all(inds[indices[i]] .== unique[i])
+        end
+    end
+end
+
+@testset "split_chain_indices" begin
+    c = [2, 2, 1, 3, 4, 3, 4, 1, 2, 1, 4, 3, 3, 2, 4, 3, 4, 1, 4, 1]
+    @test @inferred(MCMCDiagnosticTools.split_chain_indices(c, 1)) == c
+
+    cnew = @inferred MCMCDiagnosticTools.split_chain_indices(c, 2)
+    @test issetequal(Base.unique(cnew), 1:maximum(cnew))  # check no indices skipped
+    unique, indices = MCMCDiagnosticTools.unique_indices(c)
+    uniquenew, indicesnew = MCMCDiagnosticTools.unique_indices(cnew)
+    for (i, inew) in enumerate(1:2:7)
+        @test length(indicesnew[inew]) ≥ length(indicesnew[inew + 1])
+        @test indices[i] == vcat(indicesnew[inew], indicesnew[inew + 1])
+    end
+
+    cnew = MCMCDiagnosticTools.split_chain_indices(c, 3)
+    @test issetequal(Base.unique(cnew), 1:maximum(cnew))  # check no indices skipped
+    unique, indices = MCMCDiagnosticTools.unique_indices(c)
+    uniquenew, indicesnew = MCMCDiagnosticTools.unique_indices(cnew)
+    for (i, inew) in enumerate(1:3:11)
+        @test length(indicesnew[inew]) ≥
+            length(indicesnew[inew + 1]) ≥
+            length(indicesnew[inew + 2])
+        @test indices[i] ==
+            vcat(indicesnew[inew], indicesnew[inew + 1], indicesnew[inew + 2])
+    end
+end
+
+@testset "shuffle_split_stratified" begin
+    rng = Random.default_rng()
+    c = rand(1:4, 100)
+    unique, indices = MCMCDiagnosticTools.unique_indices(c)
+    @testset "frac=$frac" for frac in [0.3, 0.5, 0.7]
+        inds1, inds2 = @inferred(MCMCDiagnosticTools.shuffle_split_stratified(rng, c, frac))
+        @test issetequal(vcat(inds1, inds2), eachindex(c))
+        for inds in indices
+            common_inds = intersect(inds1, inds)
+            @test length(common_inds) == round(frac * length(inds))
+        end
+    end
+end