Matched case-control odds ratios

In a matched case-control study each case is matched to one or more controls sharing the values of the matching variable(s). The design-faithful analysis is conditional logistic regression (conditional maximum likelihood): it conditions on the matched-set totals, which removes the matching-variable nuisance parameters. Fitting matched-set indicators by ordinary logistic regression instead biases the odds ratio — for 1:1 matching its estimate converges to the squared OR (Breslow and Day 1980) — so matchatr never does that.

matchatr provides three tools on matched_cc():

• estimator = "clogit" — the conditional logistic OR for any matched-set composition (1:1, m:1, variable-ratio).
• estimator = "mcnemar" — the closed-form 1:1 matched-pair OR.
• effect_modifier = — stratum-specific conditional ORs from an exposure × modifier interaction.

library(matchatr)

Conditional logistic regression

We use the built-in infert data: a matched case-control study of secondary infertility and prior abortion, with cases matched to controls on age and parity (stratum identifies the matched sets).

fit <- matcha(
  infert,
  outcome = "case", exposure = "spontaneous",
  design = matched_cc(strata = "stratum"),
  confounders = ~ induced, estimator = "clogit"
)
fit
#> <matchatr_fit>
#>  Design:     Matched case-control
#>  Estimator:  clogit  (engine: clogit)
#>  Outcome:    case
#>  Exposure:   spontaneous
#>  Confounders: ~induced
#>  N:          248  (cases: 83, controls: 165)

Conditioning on the matched-set totals controls the matching variables (age, parity) implicitly, so only the exposure and adjustment ORs are reported:

contrast(fit, type = "or")
#> <matchatr_result>
#>  Estimator:  clogit  (engine: clogit)
#>  Estimand:   conditional OR
#>  Contrast:   Odds ratio
#>  CI method:  model
#>  N:          248
#> 
#> Contrasts:
#>     comparison estimate     se ci_lower ci_upper
#>         <char>    <num>  <num>    <num>    <num>
#> 1: spontaneous 7.285423 2.5677 3.651357 14.53635

The interval is the partial-likelihood information-matrix Wald interval. "clogit" is the design’s default estimator, so estimator = "clogit" may be omitted. m:1 and variable-ratio matching need no special handling — the conditional likelihood treats any matched-set composition uniformly.

McNemar’s 1:1 matched-pair OR

For genuine 1:1 pairs there is a closed form: with n10 pairs where only the case is exposed and n01 where only the control is exposed, the matched-pair odds ratio is OR = n10 / n01 with Var(log OR) = 1/n10 + 1/n01 (McNemar 1947). Pairs concordant on exposure carry no information and cancel.

We simulate matched pairs from the conditional likelihood with a known conditional OR of 3:

set.seed(4)
n_pairs <- 1200
OR_true <- 3
# Each pair is exposure-discordant with probability 0.4; among discordant pairs
# the case (rather than the control) is the exposed member with probability
# OR/(1 + OR), so n10/n01 estimates the conditional OR.
discordant <- rbinom(n_pairs, 1, 0.4)
p_case_exposed <- OR_true / (1 + OR_true)
case_exposed <- ifelse(discordant == 1, rbinom(n_pairs, 1, p_case_exposed),
                       rbinom(n_pairs, 1, 0.5))
ctrl_exposed <- ifelse(discordant == 1, 1L - case_exposed, case_exposed)

pairs <- data.frame(
  set    = rep(seq_len(n_pairs), each = 2),
  case   = rep(c(1L, 0L), times = n_pairs),
  smoker = as.vector(rbind(case_exposed, ctrl_exposed))
)

fit_mn <- matcha(
  pairs,
  outcome = "case", exposure = "smoker",
  design = matched_cc(strata = "set"), estimator = "mcnemar"
)
contrast(fit_mn, type = "or")
#> <matchatr_result>
#>  Estimator:  mcnemar  (engine: mcnemar)
#>  Estimand:   McNemar OR
#>  Contrast:   Odds ratio
#>  CI method:  model
#>  N:          2400
#> 
#> Contrasts:
#>    comparison estimate        se ci_lower ci_upper
#>        <char>    <num>     <num>    <num>    <num>
#> 1:     smoker 2.715447 0.2864004 2.208332 3.339014

This is exact, not a model fit. And because the conditional likelihood reduces to McNemar’s for 1:1 binary data, clogit on the same pairs gives an identical point estimate and interval:

fit_cl <- matcha(
  pairs,
  outcome = "case", exposure = "smoker",
  design = matched_cc(strata = "set"), estimator = "clogit"
)
contrast(fit_cl, type = "or")
#> <matchatr_result>
#>  Estimator:  clogit  (engine: clogit)
#>  Estimand:   conditional OR
#>  Contrast:   Odds ratio
#>  CI method:  model
#>  N:          2400
#> 
#> Contrasts:
#>    comparison estimate        se ci_lower ci_upper
#>        <char>    <num>     <num>    <num>    <num>
#> 1:     smoker 2.715447 0.2864004 2.208332 3.339014

"mcnemar" applies only to genuine 1:1 pairs with a binary exposure: a richer matched set (m:1) is rejected with a pointer to "clogit", as is a non-binary exposure.

Stratum-specific effect modification

To ask whether the exposure effect differs across the levels of a categorical modifier, pass effect_modifier =. The conditional logistic engine then fits outcome ~ exposure * modifier + confounders + strata(set), and contrast(type = "or") reports the exposure’s conditional OR within each modifier level — the main coefficient at the reference level and the linear combination β_x + β_{x:level} elsewhere — with Wald intervals from the joint partial-likelihood variance (so each per-level SE accounts for the main–interaction covariance).

Here we split infert by maternal age and ask whether the spontaneous-abortion OR differs between younger and older women:

infert2 <- infert
infert2$age_grp <- factor(ifelse(infert2$age < 31, "younger", "older"))

fit_em <- matcha(
  infert2,
  outcome = "case", exposure = "spontaneous",
  design = matched_cc(strata = "stratum"), estimator = "clogit",
  effect_modifier = "age_grp"
)
contrast(fit_em, type = "or")
#> <matchatr_result>
#>  Estimator:  clogit  (engine: clogit)
#>  Estimand:   stratum-specific conditional OR
#>  Contrast:   Odds ratio
#>  CI method:  model
#>  N:          248
#> 
#> Contrasts:
#>                         comparison estimate        se ci_lower  ci_upper
#>                             <char>    <num>     <num>    <num>     <num>
#> 1:   spontaneous | age_grp = older 6.672340 2.9883692 2.773620 16.051270
#> 2: spontaneous | age_grp = younger 1.943721 0.5588702 1.106347  3.414888

The modifier may coincide with a matching variable — that is the canonical use (does the exposure effect vary across the matching factor?). Effect modification is supported for a single-coefficient exposure (binary, continuous, or two-level factor) crossed with a categorical modifier; a 3+-level factor exposure or a numeric modifier is rejected with a classed error.

What the design does and does not identify

As with the unmatched design, only the odds ratio is identified; risk differences and risk ratios need a source-population prevalence q0 and a case-control-weighted estimator. The conditional fit reports the information-matrix interval only, so ci_method = "sandwich" / "bootstrap" are declined for these engines.

Covered combinations

Legend. ✅ truth-pinned in tests · ⛔ rejected with an informative error.

Matching	Estimator	Estimand	Status
1:1 / m:1 / variable-ratio	clogit	conditional OR	✅
1:1 (binary exposure)	mcnemar	matched-pair OR	✅
any	clogit + effect_modifier	stratum-specific OR	✅
m:1 (or richer)	mcnemar	—	⛔ use clogit
non-binary exposure	mcnemar	—	⛔ use clogit
any	clogit / mcnemar	RD / RR	⛔ need q0

See FEATURE_COVERAGE_MATRIX.md for the authoritative status of every combination.

References

Breslow, Norman E., and Nicholas E. Day. 1980. Statistical Methods in Cancer Research, Volume 1: The Analysis of Case-Control Studies. International Agency for Research on Cancer (IARC Scientific Publications No. 32).

McNemar, Quinn. 1947. “Note on the Sampling Error of the Difference Between Correlated Proportions or Percentages.” Psychometrika 12 (2): 153–57. https://doi.org/10.1007/BF02295996.

Pike, Malcolm C., Adrian P. Hill, and Peter G. Smith. 1980. “Bias and Efficiency in Logistic Analyses of Stratified Case-Control Studies.” International Journal of Epidemiology 9 (1): 89–95. https://doi.org/10.1093/ije/9.1.89.