HRS-CMS Contingency Table Analysis

1. Linked Data Set: HRS-CMS Example

The HRS-CMS (Health and Retirement Study linked with Centers for Medicare & Medicaid Services) dataset combines survey and administrative records to study health and socioeconomic outcomes among older adults. This linkage enables researchers to analyze the relationships between health indicators and conditions recorded in different data sources while accounting for potential errors in record linkage.

A subset of records is considered high-quality based on stricter linkage criteria, whereas the remaining records are obtained through automated matching procedures that may introduce linkage error. Consequently, discrepancies between variables recorded in the two sources can occur, motivating the use of statistical methods that adjust for misclassification.

For demonstration purposes, we examine a small aggregated subset of the HRS-CMS dataset. Each row represents a cell count for one combination of HRS self-report, CMS administrative record, and linkage mismatch status.

agg <- data.frame(
  NRSHOM     = as.integer(c(0, 1, 0, 1, 0, 0)),
  mds_nrshom = as.integer(c(0, 0, 1, 1, 0, 1)),
  mismatch   = as.integer(c(0, 0, 0, 0, 1, 1)),
  count      = as.integer(c(271, 9, 5, 15, 56, 3))
)
agg # Data input (aggregated counts only)

Aggregated HRS-CMS data
NRSHOM	mds_nrshom	mismatch	count
0	0	0	271
1	0	0	9
0	1	0	5
1	1	0	15
0	0	1	56
0	1	1	3

The dataset includes the following variables:

NRSHOM: indicator of nursing home residence based on HRS self-reports (1: yes, 0: no).
mds_nrshom: indicator of nursing home residence based on CMS administrative records (1: yes, 0: no).
mismatch: indicator of linkage mismatch status (1: mismatched link, 0: correctly linked record).
count: number of records in each aggregated cell.

The overall linkage mismatch rate is computed as a weighted proportion using the aggregated counts:

mm_rate <- sum(agg$mismatch * agg$count) / sum(agg$count)
data.frame(`Mismatch rate` = round(mm_rate, 4))

Overall linkage mismatch rate
Mismatch.rate
0.1643

The observed mismatch rate motivates the use of statistical methods that explicitly adjust for linkage error. Naive analyses that ignore these mismatches may underestimate or distort associations between variables. In later sections, we demonstrate the workflow for contingency table analysis under linkage error using the postlink package.

2. Naive Approach

We first use the observed aggregated data without accounting for potential linkage mismatches. This approach treats the linked data as if all records were correctly matched.

# Total number of records represented in the aggregated data
n <- sum(agg$count)

# Naive contingency table in counts
ctab <- xtabs(count ~ NRSHOM + mds_nrshom, data = agg)
as.data.frame.matrix(ctab)

Naive contingency table in counts
	0	1
0	327	8
1	9	15

# Naive contingency table in proportions
ctab_prop <- ctab / sum(ctab)
round(as.data.frame.matrix(ctab_prop), 4)

Naive contingency table in proportions
	0	1
0	0.9109	0.0223
1	0.0251	0.0418

The naive table summarizes the observed joint distribution of HRS self-reported nursing home residence (NRSHOM) and CMS administrative nursing home residence (mds_nrshom) without adjusting for linkage error.

3. Exact Approach

We next restrict the analysis to aggregated cells corresponding to correctly linked records, where mismatch = 0. This provides a comparison table based only on records without linkage mismatches.

# Keep only correctly linked records
agg_exact <- subset(agg, mismatch == 0)

# Exact-match contingency table in counts
ctab_exact <- xtabs(count ~ NRSHOM + mds_nrshom, 
                    data = agg_exact)
as.data.frame.matrix(ctab_exact)

Exact-match contingency table in counts
	0	1
0	271	5
1	9	15

# Exact-match contingency table in proportions
ctab_exact_prop <- ctab_exact / sum(ctab_exact)
round(as.data.frame.matrix(ctab_exact_prop), 4)

Exact-match contingency table in proportions
	0	1
0	0.9033	0.0167
1	0.0300	0.0500

This table shows the joint distribution of HRS self-reports and CMS administrative records among correctly linked records only. Comparing this table with the naive table illustrates how linkage mismatches may affect the observed relationship between the two variables.

4. Adjustment Approach

We adjust for potential mismatches between HRS self-reports (NRSHOM) and CMS records (mds_nrshom) using the observed mismatch rate.

library(postlink)

# Expand the aggregated counts only for compatibility with postlink
hrscms_clean <- agg[
  rep(seq_len(nrow(agg)), agg$count),
  c("NRSHOM", "mds_nrshom")
]

rownames(hrscms_clean) <- NULL

# Mismatch rate from aggregated counts
mm_rate <- sum(agg$mismatch * agg$count) / sum(agg$count)

# Create adjustment object
adj <- adjMixture(
  linked.data = hrscms_clean,
  m.rate = mm_rate
)

# Adjusted contingency table
adjusted_table <- plctable(
  formula = ~ NRSHOM + mds_nrshom,
  adjustment = adj
)

# Adjusted probability table
round(as.data.frame.matrix(adjusted_table$phat), 4)

Adjusted contingency table in proportions
	0	1
0	0.9139	0.0170
1	0.0197	0.0494

We also demonstrate an alternative lower mismatch rate (not all non-exact matches are mismatches):

# Alternative lower mismatch rate
lower_mm_rate <- 0.5 * mm_rate

adj_lower <- adjMixture(
  linked.data = hrscms_clean,
  m.rate = lower_mm_rate
)

adjusted_table_lower <- plctable(
  formula = ~ NRSHOM + mds_nrshom,
  adjustment = adj_lower
)

round(as.data.frame.matrix(adjusted_table_lower$phat), 4)

Adjusted contingency table using lower mismatch rate
	0	1
0	0.9132	0.0194
1	0.0222	0.0452

These adjusted tables estimate the joint distribution of the two nursing home residence indicators after accounting for possible linkage errors.

5. Evaluation

We evaluate the adjusted contingency table for the HRSCMS dataset, quantifying how well it recovers the joint distribution of nursing home residence indicators across HRS and CMS records while accounting for mismatches.

# Use the adjusted table from the main adjustment approach
phat_adj <- as.vector(t(adjusted_table_lower$phat)) 

# Use the exact-match table from the aggregated data
exactprobs <- t(ctab_exact_prop)

# MRAE
mrae <- mean(abs((phat_adj - exactprobs) / exactprobs))

# KLD
kld <- sum(exactprobs * log(exactprobs / phat_adj))

# GOF
gof <- n * sum((exactprobs - phat_adj)^2 / exactprobs)

# Association
adj_counts <- matrix(
  n * phat_adj,
  byrow = TRUE,
  nrow = 2,
  ncol = 2
)

c2 <- suppressWarnings(chisq.test(adj_counts))

# Kappa
p_o <- sum(diag(adjusted_table_lower$phat))
p_e <- sum(rowSums(adjusted_table_lower$phat) * colSums(adjusted_table_lower$phat))
k <- (p_o - p_e) / (1 - p_e)

# Combine results into table
data.frame(
  Metric = c("MRAE", "KLD", "GOF", "Chi-square", "Kappa"),
  Value  = round(c(mrae, kld, gof,
                   as.numeric(c2$statistic), 
                   k), 4)
)

Evaluation metrics for the adjusted contingency table
Metric	Value
MRAE	0.1334
KLD	0.0017
GOF	1.0995
Chi-square	147.0581
Kappa	0.6625

These metrics summarize the behavior of the postlink adjustment procedure relative to the exact-match contingency table. MRAE quantifies the average relative difference in joint cell probabilities, KLD measures distributional divergence, GOF summarizes the overall discrepancy between adjusted and exact-match probabilities, the chi-square statistic measures association between the HRS and CMS indicators, and Kappa evaluates agreement beyond chance.

References

Centers for Medicare & Medicaid Services (CMS). 2026. CMS Data, U.S. Department of Health & Human Services, Washington, DC, US.

Health and Retirement Study (HRS). 2026. HRS Dataset, University of Michigan, Ann Arbor, MI, US.