diff --git a/src/test/unit/analyze/mcmc/ess_basic_test.cpp b/src/test/unit/analyze/mcmc/ess_basic_test.cpp
index a010909140..101e9c793b 100644
--- a/src/test/unit/analyze/mcmc/ess_basic_test.cpp
+++ b/src/test/unit/analyze/mcmc/ess_basic_test.cpp
@@ -42,6 +42,7 @@ class EssBasic : public testing::Test {
 };
 
 TEST_F(EssBasic, test_basic_ess) {
+  // computed via R pkg posterior
   double ess_lp_expect = 1335.4137;
   double ess_theta_expect = 1377.503;
 
@@ -53,9 +54,9 @@ TEST_F(EssBasic, test_basic_ess) {
   auto old_ess_basic_theta
       = stan::analyze::compute_effective_sample_size(draws_theta, sizes);
 
-  EXPECT_NEAR(ess_lp_expect, ess_basic_lp, 0.0001);
-  EXPECT_NEAR(ess_theta_expect, ess_basic_theta, 0.0001);
+  EXPECT_NEAR(ess_lp_expect, ess_basic_lp, 1e-4);
+  EXPECT_NEAR(ess_theta_expect, ess_basic_theta, 1e-4);
 
-  EXPECT_NEAR(old_ess_basic_lp, ess_basic_lp, 0.00001);
-  EXPECT_NEAR(old_ess_basic_theta, ess_basic_theta, 0.00001);
+  EXPECT_NEAR(old_ess_basic_lp, ess_basic_lp, 1e-9);
+  EXPECT_NEAR(old_ess_basic_theta, ess_basic_theta, 1e-9);
 }
diff --git a/src/test/unit/analyze/mcmc/mcse_test.cpp b/src/test/unit/analyze/mcmc/mcse_test.cpp
index 45f4ce6ccc..92efc49b47 100644
--- a/src/test/unit/analyze/mcmc/mcse_test.cpp
+++ b/src/test/unit/analyze/mcmc/mcse_test.cpp
@@ -35,18 +35,19 @@ class MonteCarloStandardError : public testing::Test {
 };
 
 TEST_F(MonteCarloStandardError, test_mcse) {
-  double mcse_mean_lp_expect = 0.020164778;
-  double mcse_mean_theta_expect = 0.0032339916;
+  // computed via R pkg posterior
+  double mcse_mean_lp_expect = 0.02016;
+  double mcse_mean_theta_expect = 0.00323;
 
-  double mcse_sd_lp_expect = 0.0355305;
-  double mcse_sd_theta_expect = 0.0021642137;
-  EXPECT_NEAR(mcse_mean_lp_expect, stan::analyze::mcse_mean(chains_lp), 0.0001);
+  double mcse_sd_lp_expect = 0.03553;
+  double mcse_sd_theta_expect = 0.00216;
+  EXPECT_NEAR(mcse_mean_lp_expect, stan::analyze::mcse_mean(chains_lp), 1e-4);
   EXPECT_NEAR(mcse_mean_theta_expect, stan::analyze::mcse_mean(chains_theta),
-              0.0001);
+              1e-4);
 
-  EXPECT_NEAR(mcse_sd_lp_expect, stan::analyze::mcse_sd(chains_lp), 0.0001);
+  EXPECT_NEAR(mcse_sd_lp_expect, stan::analyze::mcse_sd(chains_lp), 1e-4);
   EXPECT_NEAR(mcse_sd_theta_expect, stan::analyze::mcse_sd(chains_theta),
-              0.0001);
+              1e-4);
 }
 
 TEST_F(MonteCarloStandardError, const_fail) {
diff --git a/src/test/unit/analyze/mcmc/rhat_basic_test.cpp b/src/test/unit/analyze/mcmc/rhat_basic_test.cpp
index 4e15262000..5a09b585dd 100644
--- a/src/test/unit/analyze/mcmc/rhat_basic_test.cpp
+++ b/src/test/unit/analyze/mcmc/rhat_basic_test.cpp
@@ -40,6 +40,7 @@ class RhatBasic : public testing::Test {
 };
 
 TEST_F(RhatBasic, test_basic_rhat) {
+  // computed via R pkg posterior
   double rhat_lp_basic_expect = 1.0001296;
   double rhat_theta_basic_expect = 1.0029197;
 
@@ -51,9 +52,9 @@ TEST_F(RhatBasic, test_basic_rhat) {
   auto old_rhat_basic_theta
       = stan::analyze::compute_potential_scale_reduction(draws_theta, sizes);
 
-  EXPECT_NEAR(rhat_lp_basic_expect, rhat_basic_lp, 0.00001);
-  EXPECT_NEAR(rhat_theta_basic_expect, rhat_basic_theta, 0.00001);
+  EXPECT_NEAR(rhat_lp_basic_expect, rhat_basic_lp, 1e-5);
+  EXPECT_NEAR(rhat_theta_basic_expect, rhat_basic_theta, 1e-5);
 
-  EXPECT_NEAR(old_rhat_basic_lp, rhat_basic_lp, 0.00001);
-  EXPECT_NEAR(old_rhat_basic_theta, rhat_basic_theta, 0.00001);
+  EXPECT_NEAR(old_rhat_basic_lp, rhat_basic_lp, 1e-9);
+  EXPECT_NEAR(old_rhat_basic_theta, rhat_basic_theta, 1e-9);
 }