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In addition to ML-UMR, mlumr provides two frequentist methods:
These serve as important benchmarks alongside the Bayesian ML-UMR approach.
For binary outcomes, the naive method compares crude event rates without any covariate adjustment:
\[ \text{LOR}_{\text{naive}} = \text{logit}(\hat{p}_A) - \text{logit}(\hat{p}_B) \]
where \(\hat{p}_A = \bar{Y}_{\text{IPD}}\) and \(\hat{p}_B = r_B / n_B\) from the AgD. The standard error is computed via the delta method:
\[ \text{SE} = \sqrt{\frac{1}{n_A \hat{p}_A(1-\hat{p}_A)} + \frac{1}{n_B \hat{p}_B(1-\hat{p}_B)}} \]
The chunks below operate on a small toy dat. Run this
setup once first so naive(), stc(), and the
comparison block all have data to work with.
library(mlumr)
set.seed(2026)
trial_a_data <- data.frame(
trt = "Drug_A",
response = rbinom(300, 1, 0.55),
age_cat = rbinom(300, 1, 0.40),
sex = rbinom(300, 1, 0.55)
)
trial_b_data <- data.frame(
trt = "Drug_B",
n_total = 400,
n_events = 160,
age_cat_mean = 0.35,
sex_mean = 0.50
)
ipd <- set_ipd(trial_a_data, treatment = "trt", outcome = "response",
covariates = c("age_cat", "sex"))
agd <- set_agd(trial_b_data, treatment = "trt",
outcome_n = "n_total", outcome_r = "n_events",
cov_means = c("age_cat_mean", "sex_mean"),
cov_types = c("binary", "binary"))# Prepare data (no integration points needed)
dat <- combine_data(ipd, agd)
result <- naive(dat)
print(result)
#> Naive Unadjusted Indirect Comparison
#> =====================================
#>
#> Treatments: Drug_A vs Drug_B
#>
#> Event rates:
#> Index (IPD): 0.560 (168/300)
#> Comparator (AgD): 0.400 (160/400)
#>
#> Log Odds Ratio: 0.6466 (SE: 0.1547)
#> 95% CI: [0.3433, 0.9499]The naive LOR is biased when covariate distributions differ between the IPD and AgD populations. It provides a useful reference point:
The function includes continuity-correction-style boundary guards for
extreme proportions (0% or 100% event rates), ensuring finite log odds
ratios even at the boundaries. Event-probability confidence intervals
use Wald standard errors and are bounded to [0, 1].
STC uses parametric G-computation to adjust for covariate differences:
The standard error uses the delta method, propagating parameter uncertainty through the logit-of-mean transformation.
# Without integration points (uses AgD means)
result_stc <- stc(dat)
# With integration points (better marginalization)
dat <- add_integration(dat, n_int = 64,
age_cat = distr(qbern, prob = age_cat_mean),
sex = distr(qbern, prob = sex_mean))
result_stc <- stc(dat)
print(result_stc)
#> Simulated Treatment Comparison (G-computation)
#> ===============================================
#>
#> Treatments: Drug_A vs Drug_B
#>
#> Marginalized P(Y=1|index trt, comp pop): 0.5555
#> Observed P(Y=1|comp trt, comp pop): 0.4000
#>
#> Log Odds Ratio: 0.6285 (SE: 0.1549)
#> 95% CI: [0.3250, 0.9321]
#>
#> Outcome model coefficients:
#> (Intercept) age_cat sex
#> 0.0133 -0.1527 0.5697For binary outcomes, event-probability confidence intervals are
bounded to [0, 1]. For Poisson outcomes, STC uses a 0.5
continuity correction for the comparator log rate when the AgD event
count is zero; the reported comparator rate remains the observed
rate.
The underlying logistic regression model is stored in the result:
# Coefficients
coef(result_stc$glm_fit)
#> (Intercept) age_cat sex
#> 0.0133087 -0.1527204 0.5696630
# Full GLM summary
summary(result_stc$glm_fit)
#>
#> Call:
#> glm(formula = .stc_formula(cov_names, family), family = glm_family,
#> data = ipd)
#>
#> Coefficients:
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 0.01331 0.18715 0.071 0.9433
#> age_cat -0.15272 0.24043 -0.635 0.5253
#> sex 0.56966 0.23615 2.412 0.0159 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> (Dispersion parameter for binomial family taken to be 1)
#>
#> Null deviance: 411.56 on 299 degrees of freedom
#> Residual deviance: 405.50 on 297 degrees of freedom
#> AIC: 411.5
#>
#> Number of Fisher Scoring iterations: 4
# Predicted probabilities
fitted(result_stc$glm_fit)
#> 1 2 3 4 5 6 7 8
#> 0.6417509 0.5033271 0.6059337 0.5033271 0.6059337 0.6417509 0.4652034 0.5033271
#> 9 10 11 12 13 14 15 16
#> 0.6059337 0.4652034 0.6059337 0.6059337 0.6417509 0.4652034 0.5033271 0.4652034
#> 17 18 19 20 21 22 23 24
#> 0.5033271 0.5033271 0.4652034 0.6059337 0.6059337 0.6417509 0.6417509 0.4652034
#> 25 26 27 28 29 30 31 32
#> 0.5033271 0.6059337 0.4652034 0.4652034 0.6417509 0.4652034 0.6417509 0.6059337
#> 33 34 35 36 37 38 39 40
#> 0.6059337 0.4652034 0.6417509 0.4652034 0.5033271 0.5033271 0.5033271 0.6059337
#> 41 42 43 44 45 46 47 48
#> 0.5033271 0.5033271 0.5033271 0.6417509 0.6059337 0.5033271 0.5033271 0.6417509
#> 49 50 51 52 53 54 55 56
#> 0.5033271 0.6417509 0.5033271 0.4652034 0.5033271 0.6417509 0.6059337 0.6059337
#> 57 58 59 60 61 62 63 64
#> 0.4652034 0.6417509 0.5033271 0.4652034 0.6417509 0.6417509 0.6059337 0.6417509
#> 65 66 67 68 69 70 71 72
#> 0.6059337 0.6417509 0.6059337 0.4652034 0.6417509 0.4652034 0.4652034 0.5033271
#> 73 74 75 76 77 78 79 80
#> 0.6059337 0.6417509 0.6059337 0.5033271 0.4652034 0.5033271 0.5033271 0.5033271
#> 81 82 83 84 85 86 87 88
#> 0.4652034 0.6059337 0.5033271 0.6417509 0.4652034 0.6059337 0.6059337 0.6417509
#> 89 90 91 92 93 94 95 96
#> 0.6417509 0.6417509 0.4652034 0.6059337 0.5033271 0.6417509 0.5033271 0.4652034
#> 97 98 99 100 101 102 103 104
#> 0.6417509 0.6059337 0.5033271 0.6417509 0.5033271 0.6417509 0.6059337 0.6417509
#> 105 106 107 108 109 110 111 112
#> 0.5033271 0.6417509 0.6417509 0.5033271 0.6417509 0.6059337 0.6059337 0.5033271
#> 113 114 115 116 117 118 119 120
#> 0.5033271 0.6059337 0.6417509 0.5033271 0.6417509 0.6417509 0.6059337 0.5033271
#> 121 122 123 124 125 126 127 128
#> 0.5033271 0.6417509 0.6059337 0.5033271 0.6417509 0.5033271 0.6417509 0.6417509
#> 129 130 131 132 133 134 135 136
#> 0.5033271 0.6059337 0.4652034 0.6417509 0.6059337 0.6417509 0.6059337 0.5033271
#> 137 138 139 140 141 142 143 144
#> 0.4652034 0.6417509 0.4652034 0.5033271 0.5033271 0.6417509 0.5033271 0.6417509
#> 145 146 147 148 149 150 151 152
#> 0.5033271 0.6059337 0.6417509 0.5033271 0.5033271 0.5033271 0.6059337 0.6417509
#> 153 154 155 156 157 158 159 160
#> 0.5033271 0.4652034 0.6417509 0.6417509 0.6059337 0.6059337 0.4652034 0.6059337
#> 161 162 163 164 165 166 167 168
#> 0.5033271 0.6059337 0.4652034 0.5033271 0.6059337 0.6417509 0.4652034 0.5033271
#> 169 170 171 172 173 174 175 176
#> 0.6417509 0.6417509 0.6059337 0.5033271 0.5033271 0.6417509 0.6059337 0.6059337
#> 177 178 179 180 181 182 183 184
#> 0.6059337 0.5033271 0.4652034 0.4652034 0.6059337 0.5033271 0.5033271 0.6417509
#> 185 186 187 188 189 190 191 192
#> 0.6059337 0.4652034 0.4652034 0.6417509 0.6417509 0.6059337 0.6417509 0.6059337
#> 193 194 195 196 197 198 199 200
#> 0.5033271 0.6059337 0.6417509 0.5033271 0.4652034 0.4652034 0.6417509 0.6059337
#> 201 202 203 204 205 206 207 208
#> 0.5033271 0.6059337 0.4652034 0.5033271 0.6417509 0.5033271 0.6417509 0.4652034
#> 209 210 211 212 213 214 215 216
#> 0.6059337 0.5033271 0.6417509 0.5033271 0.5033271 0.5033271 0.4652034 0.5033271
#> 217 218 219 220 221 222 223 224
#> 0.5033271 0.6059337 0.6417509 0.5033271 0.4652034 0.6059337 0.6059337 0.6059337
#> 225 226 227 228 229 230 231 232
#> 0.5033271 0.4652034 0.5033271 0.6417509 0.5033271 0.6417509 0.6417509 0.6417509
#> 233 234 235 236 237 238 239 240
#> 0.5033271 0.5033271 0.4652034 0.5033271 0.4652034 0.5033271 0.6417509 0.5033271
#> 241 242 243 244 245 246 247 248
#> 0.4652034 0.4652034 0.5033271 0.4652034 0.5033271 0.5033271 0.6059337 0.5033271
#> 249 250 251 252 253 254 255 256
#> 0.6059337 0.6417509 0.4652034 0.6417509 0.6059337 0.6417509 0.6417509 0.5033271
#> 257 258 259 260 261 262 263 264
#> 0.5033271 0.4652034 0.6417509 0.6417509 0.6059337 0.6059337 0.4652034 0.6417509
#> 265 266 267 268 269 270 271 272
#> 0.5033271 0.5033271 0.6417509 0.6417509 0.4652034 0.5033271 0.6417509 0.4652034
#> 273 274 275 276 277 278 279 280
#> 0.6417509 0.6417509 0.6417509 0.5033271 0.6059337 0.5033271 0.5033271 0.6059337
#> 281 282 283 284 285 286 287 288
#> 0.4652034 0.4652034 0.5033271 0.6417509 0.6059337 0.6417509 0.6059337 0.6059337
#> 289 290 291 292 293 294 295 296
#> 0.5033271 0.6417509 0.6059337 0.6417509 0.5033271 0.6417509 0.6417509 0.5033271
#> 297 298 299 300
#> 0.5033271 0.6059337 0.6417509 0.6059337The delta method SE for STC accounts for uncertainty in:
The gradient of \(\text{logit}(\text{mean}(\sigma(\mathbf{X}\boldsymbol{\beta})))\) with respect to \(\boldsymbol{\beta}\) is computed analytically:
\[ \frac{\partial}{\partial \boldsymbol{\beta}} \text{logit}\left(\frac{1}{N}\sum_j \sigma(\mathbf{x}_j^T\boldsymbol{\beta})\right) = \frac{1}{\bar{p}(1-\bar{p})} \cdot \frac{1}{N} \sum_j \sigma'(\eta_j) \mathbf{x}_j \]
where \(\sigma(\eta) = 1/(1 + e^{-\eta})\) and \(\sigma'(\eta) = \sigma(\eta)(1 - \sigma(\eta))\).
The same naive() and stc() functions
support normal and Poisson outcomes. For normal outcomes,
naive() compares the IPD mean against the inverse- variance
weighted AgD mean, while stc() fits a Gaussian GLM and
G-computes the index-treatment mean in the comparator population. For
Poisson outcomes, both methods report log rate ratios; zero observed
event counts use a 0.5 continuity correction on the log-rate scale so
estimates remain finite.
# Normal-family benchmark
ipd_normal <- set_ipd(
data.frame(
trt = "Drug_A",
score = rnorm(120, mean = 3.0, sd = 1.0),
age_cat = rbinom(120, 1, 0.40)
),
treatment = "trt",
outcome = "score",
covariates = "age_cat",
family = "normal"
)
agd_normal <- set_agd(
data.frame(trt = "Drug_B", y_mean = 2.7, se = 0.12, age_cat_mean = 0.35),
treatment = "trt",
family = "normal",
outcome_mean = "y_mean",
outcome_se = "se",
cov_means = "age_cat_mean",
cov_types = "binary"
)
dat_normal <- combine_data(ipd_normal, agd_normal)
naive(dat_normal)
#> Naive Unadjusted Indirect Comparison
#> =====================================
#>
#> Treatments: Drug_A vs Drug_B
#>
#> Mean outcomes:
#> Index (IPD): 3.1649
#> Comparator (AgD): 2.7000
#>
#> Mean Difference: 0.4649 (SE: 0.1532)
#> 95% CI: [0.1647, 0.7652]
stc(dat_normal)
#> Simulated Treatment Comparison (G-computation)
#> ===============================================
#>
#> Treatments: Drug_A vs Drug_B
#>
#> Marginalized E[Y|index trt, comp pop]: 3.1777
#> Observed E[Y|comp trt, comp pop]: 2.7000
#>
#> Mean Difference: 0.4777 (SE: 0.1536)
#> 95% CI: [0.1768, 0.7787]
#>
#> Outcome model coefficients:
#> (Intercept) age_cat
#> 3.2545 -0.2194# Poisson-family benchmark
exposure <- runif(120, 0.5, 2.0)
ipd_poisson <- set_ipd(
data.frame(
trt = "Drug_A",
events = rpois(120, exp(0.2) * exposure),
person_years = exposure,
age_cat = rbinom(120, 1, 0.40)
),
treatment = "trt",
outcome = "events",
covariates = "age_cat",
family = "poisson",
exposure = "person_years"
)
agd_poisson <- set_agd(
data.frame(trt = "Drug_B", n_events = 40, person_years = 180,
age_cat_mean = 0.35),
treatment = "trt",
family = "poisson",
outcome_r = "n_events",
outcome_E = "person_years",
cov_means = "age_cat_mean",
cov_types = "binary"
)
dat_poisson <- combine_data(ipd_poisson, agd_poisson)
naive(dat_poisson)
#> Naive Unadjusted Indirect Comparison
#> =====================================
#>
#> Treatments: Drug_A vs Drug_B
#>
#> Rates:
#> Index (IPD): 1.2726
#> Comparator (AgD): 0.2222
#>
#> Log Rate Ratio: 1.7451 (SE: 0.1747)
#> 95% CI: [1.4027, 2.0875]
stc(dat_poisson)
#> Simulated Treatment Comparison (G-computation)
#> ===============================================
#>
#> Treatments: Drug_A vs Drug_B
#>
#> Marginalized rate (index trt, comp pop): 1.2667
#> Observed rate (comp trt, comp pop): 0.2222
#>
#> Log Rate Ratio: 1.7405 (SE: 0.1751)
#> 95% CI: [1.3972, 2.0837]
#>
#> Outcome model coefficients:
#> (Intercept) age_cat
#> 0.2142 0.0633# Fit all three methods
naive_result <- naive(dat)
stc_result <- stc(dat)
mlumr_result <- mlumr(
dat, model = "spfa",
chains = 2, iter = 500, warmup = 250,
seed = 42, refresh = 0, verbose = FALSE
)
# Extract LORs for comparison
le_naive <- naive_result$link_effect
le_stc <- stc_result$link_effect
lor_mlumr <- marginal_effects(mlumr_result, effect = "lor",
population = "comparator")
cat("Method comparison (LOR in comparator population):\n")
#> Method comparison (LOR in comparator population):
cat(sprintf(" Naive: %.3f [%.3f, %.3f]\n",
naive_result$link_effect, naive_result$ci_lower, naive_result$ci_upper))
#> Naive: 0.647 [0.343, 0.950]
cat(sprintf(" STC: %.3f [%.3f, %.3f]\n",
stc_result$link_effect, stc_result$ci_lower, stc_result$ci_upper))
#> STC: 0.629 [0.325, 0.932]
cat(sprintf(" ML-UMR: %.3f [%.3f, %.3f]\n",
lor_mlumr$mean, lor_mlumr$q2.5, lor_mlumr$q97.5))
#> ML-UMR: 0.630 [0.320, 0.928]Both naive() and stc() accept a
conf_level parameter:
# 90% confidence intervals
naive_90 <- naive(dat, conf_level = 0.90)
stc_90 <- stc(dat, conf_level = 0.90)
print(naive_90)
#> Naive Unadjusted Indirect Comparison
#> =====================================
#>
#> Treatments: Drug_A vs Drug_B
#>
#> Event rates:
#> Index (IPD): 0.560 (168/300)
#> Comparator (AgD): 0.400 (160/400)
#>
#> Log Odds Ratio: 0.6466 (SE: 0.1547)
#> 90% CI: [0.3921, 0.9012]
print(stc_90)
#> Simulated Treatment Comparison (G-computation)
#> ===============================================
#>
#> Treatments: Drug_A vs Drug_B
#>
#> Marginalized P(Y=1|index trt, comp pop): 0.5555
#> Observed P(Y=1|comp trt, comp pop): 0.4000
#>
#> Log Odds Ratio: 0.6285 (SE: 0.1549)
#> 90% CI: [0.3738, 0.8833]
#>
#> Outcome model coefficients:
#> (Intercept) age_cat sex
#> 0.0133 -0.1527 0.5697| Feature | STC | Naive | ML-UMR |
|---|---|---|---|
| Covariate adjustment | Outcome model | None | Joint model |
| Population weighting | G-computation | None | QMC integration |
| Uncertainty | Delta method | Delta method | Posterior |
| Effect modification | Not captured | N/A | Relaxed model |
| Speed | Instant | Instant | Minutes |
| No. parameters | p+1 | 0 | 2+p (SPFA) or 2+2p (Relaxed) |
STC is faster but makes stronger modeling assumptions. ML-UMR jointly models both data sources and naturally propagates all sources of uncertainty through the posterior.
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
Health stats visible at Monitor.