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 & 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 NE, Day NE (1980). Statistical Methods in Cancer Research, Volume 1: The Analysis of Case-Control Studies. IARC.

McNemar Q (1947). Note on the sampling error of the difference between correlated proportions or percentages. Psychometrika 12(2):153-157.

Pike MC, Hill AP, Smith PG (1980). Bias and efficiency in logistic analyses of stratified case-control studies. International Journal of Epidemiology 9(1):89-95.