Introduction to Multivariable Functional Mendelian Randomization

Example: Joint Estimation of Two Exposures

Step 1: Simulate Data

We generate genetic instruments and two longitudinal exposures:

set.seed(12345)

# Generate exposure data
sim_data <- getX_multi_exposure(
  N = 300,              # Sample size
  J = 25,               # Number of genetic instruments
  nSparse = 10          # Observations per subject
)

# Check dimensions
cat("Sample size:", nrow(sim_data$details$G), "\n")
#> Sample size: 300
cat("Number of instruments:", ncol(sim_data$details$G), "\n")
#> Number of instruments: 25

Step 2: Generate Outcome

We create an outcome with linear effect from X1 and quadratic effect from X2:

outcome_data <- getY_multi_exposure(
  sim_data,
  X1Ymodel = "2",     # Linear: β₁(t) = 0.02 * t
  X2Ymodel = "5",     # Late-age effect: β₂(t) = 0.1 * (t > 30)
  X1_effect = TRUE,
  X2_effect = TRUE,
  outcome_type = "continuous"
)

cat("Outcome summary:\n")
#> Outcome summary:
summary(outcome_data$Y)
#>     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
#> -111.070  -31.849   -5.190   -3.012   26.394  112.621

Step 3: Functional Principal Component Analysis

We apply FPCA to both exposures to extract principal components:

# FPCA for exposure 1
fpca1 <- FPCA(
  sim_data$X1$Ly_sim, 
  sim_data$X1$Lt_sim,
  list(dataType = 'Sparse', error = TRUE, verbose = FALSE)
)

# FPCA for exposure 2
fpca2 <- FPCA(
  sim_data$X2$Ly_sim, 
  sim_data$X2$Lt_sim,
  list(dataType = 'Sparse', error = TRUE, verbose = FALSE)
)

cat("FPCA completed:\n")
#> FPCA completed:
cat("  Exposure 1:", fpca1$selectK, "components selected\n")
#>   Exposure 1: 3 components selected
cat("  Exposure 2:", fpca2$selectK, "components selected\n")
#>   Exposure 2: 4 components selected

Step 4: Joint Multivariable Estimation

Now we perform joint estimation using mvfmr():

result_joint <- mvfmr(
  G = sim_data$details$G,
  fpca_results = list(fpca1, fpca2),
  Y = outcome_data$Y,
  outcome_type = "continuous",
  method = "gmm",
  max_nPC1 = 4,
  max_nPC2 = 4,
  improvement_threshold = 0.001,
  bootstrap = FALSE,
  n_cores = 1,
  true_effects = list(model1 = "2", model2 = "5"),
  X_true = list(X1_true = sim_data$details$X1, X2_true = sim_data$details$X2),
  verbose = FALSE
)

# View results
print(result_joint)
#> 
#> Functional Multivariable MR Result
#> ==================================
#> Exposures: 2 
#> Sample size: 300 
#> Outcome: continuous 
#> Method: gmm 
#> Components selected: nPC1 = 2 , nPC2 = 2 
#> 
#> Performance Metrics:
#>   Exposure 1 - MISE: 0.001899 , Coverage: 0.824 
#>   Exposure 2 - MISE: 0.001591 , Coverage: 0.922

Step 5: Visualize Time-Varying Effects

The estimated time-varying causal effects can be visualized using the built-in plot method:

# Plot both effects
plot(result_joint)

The solid colored lines show the estimated time-varying causal effects, with shaded bands representing 95% confidence intervals. The dashed red lines (when present) indicate the true time-varying effects used in the simulation.

Step 6: Extract Coefficients

# Estimated beta coefficients for basis functions
coef(result_joint)
#> [1] -3.7775154  1.5166093 -0.3552057  0.4624341

# Time-varying effects at each time point
head(result_joint$effects$effect1)
#> [1] 0.02665083 0.03882021 0.05180349 0.06570553 0.08061147 0.09654441
head(result_joint$effects$effect2)
#> [1] -0.04923079 -0.05503448 -0.06005135 -0.06404133 -0.06679739 -0.06817082

Step 7: Performance Metrics

Since we simulated data with known truth, we can evaluate performance:

cat("Performance Metrics:\n")
#> Performance Metrics:
cat("\nExposure 1 (Linear effect):\n")
#> 
#> Exposure 1 (Linear effect):
cat("  MISE:", round(result_joint$performance$MISE1, 6), "\n")
#>   MISE: 0.001899
cat("  Coverage:", round(result_joint$performance$Coverage1, 3), "\n")
#>   Coverage: 0.824

cat("\nExposure 2 (Quadratic effect):\n")
#> 
#> Exposure 2 (Quadratic effect):
cat("  MISE:", round(result_joint$performance$MISE2, 6), "\n")
#>   MISE: 0.001591
cat("  Coverage:", round(result_joint$performance$Coverage2, 3), "\n")
#>   Coverage: 0.922

MISE (Mean Integrated Squared Error): measures accuracy of effect estimation
Coverage: proportion of time points where true effect falls within 95% CI

Comparison: Joint vs. Separate Estimation

We can compare joint (multivariable) with separate (univariable) estimation:

result_separate <- mvfmr_separate(
  G1 = sim_data$details$G,
  G2 = sim_data$details$G,
  fpca_results = list(fpca1, fpca2),
  Y = outcome_data$Y,
  outcome_type = "continuous",
  method = "gmm",
  max_nPC1 = 4,
  max_nPC2 = 4,
  n_cores = 1,
  true_effects = list(model1 = "2", model2 = "5"),
  verbose = FALSE
)
#> [1] "Processing X1"
#> [1] "Processing X2"

print(result_separate)
#> 
#> Separate Univariable MR Results
#> ================================
#> Exposures: 2 
#> Separate instruments: TRUE 
#> Outcome: continuous 
#> Method: gmm 
#> 
#> Exposure 1:
#>   Components: 2 
#>   MSE: 0.005231 
#>   Coverage: 1 
#> 
#> Exposure 2:
#>   Components: 2 
#>   MSE: 0.075436 
#>   Coverage: 0.706

Performance Comparison

comparison <- data.frame(
  Method = rep(c("Joint (MV-FMR)", "Separate (U-FMR)"), each = 2),
  Exposure = rep(c("X1", "X2"), times = 2),
  MISE = c(
    result_joint$performance$MISE1,
    result_joint$performance$MISE2,
    result_separate$exposure1$performance$MISE,
    result_separate$exposure2$performance$MISE
  ),
  Coverage = c(
    result_joint$performance$Coverage1,
    result_joint$performance$Coverage2,
    result_separate$exposure1$performance$Coverage,
    result_separate$exposure2$performance$Coverage
  )
)

print(comparison)
#>             Method Exposure        MISE  Coverage
#> 1   Joint (MV-FMR)       X1 0.001899365 0.8235294
#> 2   Joint (MV-FMR)       X2 0.001591153 0.9215686
#> 3 Separate (U-FMR)       X1 0.005230941 1.0000000
#> 4 Separate (U-FMR)       X2 0.075436010 0.7058824

Key Insight: Joint estimation (MV-FMR) typically performs better when exposures are correlated or share pleiotropic instruments.

Instrument Strength Diagnostics

Check instrument strength using F-statistics:

# Calculate F-statistics
K_total <- result_joint$nPC_used$nPC1 + result_joint$nPC_used$nPC2

fstats <- IS(
  J = ncol(sim_data$details$G),
  K = K_total,
  PC = 1:K_total,
  datafull = cbind(
    sim_data$details$G,
    cbind(fpca1$xiEst[, 1:result_joint$nPC_used$nPC1], 
          fpca2$xiEst[, 1:result_joint$nPC_used$nPC2])
  ),
  Y = outcome_data$Y
)

print(fstats)
#>      PC         RR        FF       cFF   Qvalue df       pvalue
#> [1,]  1 0.06929360 0.8160015 0.5010741 227.2324 21 1.468923e-36
#> [2,]  2 0.06516010 0.7639326 0.5788612 227.2324 21 1.468923e-36
#> [3,]  3 0.08536319 1.0228984 0.4612395 227.2324 21 1.468923e-36
#> [4,]  4 0.09989841 1.2164033 1.0989657 227.2324 21 1.468923e-36

FF: Standard F-statistic for instrument strength
cFF: Conditional F-statistic (accounts for other exposures)
Qvalue: Hansen’s J overidentification test statistic
pvalue: P-value for Q-test (high p-value = instruments are valid)

Rule of thumb: cFF > 10 indicates strong instruments.

Binary Outcomes

MV-FMR also supports binary outcomes using the control function approach:

# Generate binary outcome
outcome_binary <- getY_multi_exposure(
  sim_data,
  X1Ymodel = "2",
  X2Ymodel = "5",
  outcome_type = "binary"
)

# Estimate with control function
result_binary <- mvfmr(
  G = sim_data$details$G,
  fpca_results = list(fpca1, fpca2),
  Y = outcome_binary$Y,
  outcome_type = "binary",
  method = "cf",  # Control function for binary
  max_nPC1 = 3,
  max_nPC2 = 3,
  n_cores = 1,
  verbose = FALSE
)

print(result_binary)

Next Steps

Learn More

Univariable FMR: See vignette("univariable-fmr") for single exposure analysis
Advanced examples: Check inst/examples/test_MV-FMR.R for complete scenarios
Manuscript: See paper for methodological details

Citation

If you use this package, please cite:

Fontana, N., Ieva, F., Zuccolo, L., Di Angelantonio, E., & Secchi, P. (2025). Unraveling time-varying causal effects of multiple exposures: integrating Functional Data Analysis with Multivariable Mendelian Randomization. arXiv preprint arXiv:2512.19064.

Session Info

sessionInfo()
#> R version 4.3.0 (2023-04-21)
#> Platform: aarch64-apple-darwin20 (64-bit)
#> Running under: macOS Ventura 13.6
#> 
#> Matrix products: default
#> BLAS:   /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib 
#> LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.11.0
#> 
#> locale:
#> [1] C/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
#> 
#> time zone: Europe/Rome
#> tzcode source: internal
#> 
#> attached base packages:
#> [1] stats     graphics  grDevices utils     datasets  methods   base     
#> 
#> other attached packages:
#> [1] ggplot2_3.5.2 fdapace_0.5.9 mvfmr_0.1.0  
#> 
#> loaded via a namespace (and not attached):
#>  [1] gtable_0.3.6        shape_1.4.6         xfun_0.41          
#>  [4] bslib_0.6.1         htmlwidgets_1.6.4   lattice_0.21-8     
#>  [7] numDeriv_2016.8-1.1 vctrs_0.6.5         tools_4.3.0        
#> [10] generics_0.1.3      parallel_4.3.0      tibble_3.2.1       
#> [13] fansi_1.0.6         highr_0.10          cluster_2.1.4      
#> [16] pkgconfig_2.0.3     Matrix_1.6-4        data.table_1.14.10 
#> [19] checkmate_2.3.1     lifecycle_1.0.4     farver_2.1.1       
#> [22] compiler_4.3.0      stringr_1.5.1       progress_1.2.3     
#> [25] munsell_0.5.0       codetools_0.2-19    htmltools_0.5.7    
#> [28] sass_0.4.8          yaml_2.3.8          glmnet_4.1-8       
#> [31] htmlTable_2.4.2     Formula_1.2-5       pracma_2.4.4       
#> [34] pillar_1.9.0        crayon_1.5.3        jquerylib_0.1.4    
#> [37] MASS_7.3-58.4       cachem_1.0.8        Hmisc_5.1-1        
#> [40] iterators_1.0.14    rpart_4.1.19        foreach_1.5.2      
#> [43] tidyselect_1.2.0    digest_0.6.33       stringi_1.8.3      
#> [46] dplyr_1.1.4         labeling_0.4.3      splines_4.3.0      
#> [49] fastmap_1.1.1       grid_4.3.0          colorspace_2.1-0   
#> [52] cli_3.6.2           magrittr_2.0.3      base64enc_0.1-3    
#> [55] survival_3.5-7      utf8_1.2.4          withr_2.5.2        
#> [58] foreign_0.8-84      prettyunits_1.2.0   scales_1.3.0       
#> [61] backports_1.4.1     rmarkdown_2.25      nnet_7.3-18        
#> [64] gridExtra_2.3       hms_1.1.3           evaluate_0.23      
#> [67] knitr_1.45          doParallel_1.0.17   rlang_1.1.6        
#> [70] Rcpp_1.0.11         glue_1.6.2          pROC_1.18.5        
#> [73] rstudioapi_0.15.0   jsonlite_1.8.8      R6_2.5.1           
#> [76] plyr_1.8.9