Getting Started with MAIVE

Overview

MAIVE (Meta-Analysis Instrumental Variable Estimator) addresses a fundamental problem in meta-analysis of observational research: spurious precision. Traditional meta-analysis assigns more weight to studies with lower standard errors, assuming higher precision. However, in observational research, precision must be estimated and is vulnerable to manipulation through practices like p-hacking to achieve statistical significance.

For a guided, interactive workflow, visit https://www.easymeta.org.

This manipulation can invalidate:

• Inverse-variance weighting schemes
• Bias-correction methods like funnel plots
• Traditional publication bias corrections

MAIVE introduces an instrumental variable approach to limit bias caused by spurious precision in meta-analysis.

The Problem: Spurious Precision

In observational research, researchers can inadvertently or deliberately manipulate their analyses to achieve statistically significant results. This includes:

• Selective reporting of specifications
• Outcome switching
• Sample trimming
• Selective controls inclusion

These practices create spuriously precise estimates that appear more reliable than they actually are. Traditional meta-analysis methods that weight by inverse variance will overweight these manipulated studies, leading to biased conclusions.

The MAIVE Solution

MAIVE uses instrumental variables to correct for spurious precision:

1. First-stage regression: Instruments the potentially manipulated standard errors using inverse sample sizes (which researchers cannot easily manipulate)
2. Second-stage regression: Uses the instrumented standard errors in meta-regression models

This approach provides:

• Robust meta-estimates that account for spurious precision
• Hausman-type tests comparing IV and OLS estimates
• Anderson-Rubin confidence intervals for weak instruments
• Publication bias tests based on instrumented standard errors

Installation

# Install from CRAN (once published)
install.packages("MAIVE")

# Or install development version from GitHub
install.packages("devtools")
devtools::install_github("PetrCala/MAIVE")

library(MAIVE)

Data Structure

The maive() function accepts either the default column names or custom mappings:

Custom column names: You can map your own column names using:

• estimate for the estimate column
• se for the standard error column
• n for the sample size column
• study_id for study identifiers (optional)

Example:

custom_dat <- data.frame(
  my_est = c(0.50, 0.60, 0.40, 0.55, 0.45, 0.52, 0.48, 0.58, 0.42, 0.50, 0.47, 0.56),
  my_se = c(0.20, 0.18, 0.25, 0.22, 0.24, 0.19, 0.23, 0.17, 0.26, 0.21, 0.24, 0.18),
  my_n = c(80, 120, 95, 110, 90, 130, 100, 140, 85, 105, 88, 125),
  my_study = c("A", "A", "B", "B", "C", "C", "D", "D", "A", "B", "C", "D")
)

result <- maive(
  dat = custom_dat,
  estimate = "my_est",
  se = "my_se",
  n = "my_n",
  study_id = "my_study",
  method = 3,
  weight = 0,
  instrument = 1,
  studylevel = 2,
  SE = 3,
  AR = 1
)
#> Registered S3 method overwritten by 'clubSandwich':
#>   method    from    
#>   bread.mlm sandwich

result$beta
#> [1] 0.7628903

Validation rules still apply: required columns must be numeric, non-missing, finite, and there must be at least 4 observations after removing completely empty rows. With a study identifier, there must be at least as many rows as unique studies plus three (the fixture above has 12 rows for 4 studies).

Study identifier fallback: when study_id is not supplied, a column named study_id is used wherever it sits. If there is no such column and the data frame has four or more columns, the fourth column is used as the study identifier and a warning names it. If column four is a moderator or a year rather than a study identifier, drop it or name the correct column explicitly, because the identifier drives the study dummies and clustering at every studylevel other than 0.

Coming from metafor: if your effect sizes are already in a metafor::escalc() data frame or an rma() fit, maive_from_metafor() builds this data frame for you. It uses sqrt(vi) as the standard error and takes the sample sizes from the object (ni, the ni attribute that escalc() records, or n1i + n2i), never inferring them from the variance. Rows of an rma.uni fit pass through the fit’s subset and missing-value masks, so a study_id vector for the original data stays aligned.

smd <- metafor::escalc(
  measure = "SMD",
  m1i = c(2.1, 2.5, 1.9, 2.4, 2.2, 2.6, 2.3, 2.0),
  sd1i = c(1.0, 1.2, 0.9, 1.1, 1.0, 1.3, 1.1, 1.0),
  n1i = c(40, 50, 30, 45, 60, 35, 55, 48),
  m2i = c(1.8, 2.0, 1.7, 2.1, 1.9, 2.2, 2.0, 1.8),
  sd2i = c(1.0, 1.1, 1.0, 1.0, 1.1, 1.2, 1.0, 1.1),
  n2i = c(42, 55, 33, 48, 58, 37, 52, 50)
)

maive_from_metafor(smd)
#>          bs      sebs  Ns
#> 1 0.2971772 0.2221442  82
#> 2 0.4321116 0.1976638 105
#> 3 0.2071041 0.2529363  63
#> 4 0.2834756 0.2085368  93
#> 5 0.2837736 0.1850656 118
#> 6 0.3166657 0.2372653  72
#> 7 0.2829629 0.1943881 107
#> 8 0.1885831 0.2025211  98

Basic Usage

Let’s create a simple example dataset:

# Simulated meta-analysis data: 50 estimates from 10 studies, with reported
# precision driven by sample size (as the first stage assumes)
set.seed(123)
n_estimates <- 50
Ns <- sample(100:1000, n_estimates, replace = TRUE)

data <- data.frame(
  bs = rnorm(n_estimates, mean = 0.3, sd = 0.2),
  sebs = 2 / sqrt(Ns) * runif(n_estimates, min = 0.8, max = 1.2),
  Ns = Ns,
  study_id = rep(1:10, each = 5)
)

head(data)
#>          bs       sebs  Ns study_id
#> 1 0.2409857 0.09350867 514        1
#> 2 0.4790251 0.08013701 562        1
#> 3 0.4756267 0.11301618 278        1
#> 4 0.4643162 0.08107801 625        1
#> 5 0.4377281 0.12785564 294        1
#> 6 0.4107835 0.05867782 917        2

# Run MAIVE with defaults
result <- maive(
  dat = data,
  method = 3,      # PET-PEESE (default)
  weight = 0,      # No weights (default)
  instrument = 1,  # Instrument SEs (default)
  studylevel = 2,  # Cluster-robust (default)
  SE = 3,          # Wild bootstrap (default)
  AR = 1           # Anderson-Rubin CI (default)
)

# View key results
cat("MAIVE Estimate:", round(result$beta, 3), "\n")
#> MAIVE Estimate: 0.335
cat("MAIVE SE:", round(result$SE, 3), "\n")
#> MAIVE SE: 0.05
cat("Standard Estimate:", round(result$beta_standard, 3), "\n")
#> Standard Estimate: 0.348
cat("Hausman Test:", round(result$Hausman, 3), "\n")
#> Hausman Test: 0.001
cat("First-stage F-test:", round(result$`F-test`, 3), "\n")
#> First-stage F-test: 373.181

Method Options

MAIVE supports multiple meta-regression methods:

result_pet <- maive(
  dat = data,
  method = 1,  # FAT-PET
  weight = 0,
  instrument = 1,
  studylevel = 2,
  SE = 3,
  AR = 1
)

cat("PET Estimate:", round(result_pet$beta, 3), "\n")

result_peese <- maive(
  dat = data,
  method = 2,  # PEESE
  weight = 0,
  instrument = 1,
  studylevel = 2,
  SE = 3,
  AR = 1
)

cat("PEESE Estimate:", round(result_peese$beta, 3), "\n")

result_petpeese <- maive(
  dat = data,
  method = 3,  # PET-PEESE (default)
  weight = 0,
  instrument = 1,
  studylevel = 2,
  SE = 3,
  AR = 1
)

cat("PET-PEESE Estimate:", round(result_petpeese$beta, 3), "\n")

result_ek <- maive(
  dat = data,
  method = 4,  # EK
  weight = 0,
  instrument = 1,
  studylevel = 2,
  SE = 3,
  AR = 0  # AR not available for EK
)

cat("EK Estimate:", round(result_ek$beta, 3), "\n")

Weighting Schemes

result_noweight <- maive(
  dat = data,
  method = 3,
  weight = 0,  # No weights
  instrument = 1,
  studylevel = 2,
  SE = 3,
  AR = 1
)

result_ivweight <- maive(
  dat = data,
  method = 3,
  weight = 1,  # Inverse-variance weights
  instrument = 1,
  studylevel = 2,
  SE = 3,
  AR = 0  # AR not available with weights
)

result_maiveweight <- maive(
  dat = data,
  method = 3,
  weight = 2,  # MAIVE-adjusted weights
  instrument = 1,
  studylevel = 2,
  SE = 3,
  AR = 1
)

Study-Level Correlation

Control for study-level correlation when you have multiple estimates per study:

# No study-level adjustment
result_none <- maive(data, method = 3, weight = 0, instrument = 1, 
                     studylevel = 0, SE = 0, AR = 1)

# Study fixed effects (demeaned)
result_fe <- maive(data, method = 3, weight = 0, instrument = 1,
                   studylevel = 1, SE = 1, AR = 0)  # AR not available with FE

# Cluster-robust standard errors
result_cluster <- maive(data, method = 3, weight = 0, instrument = 1,
                        studylevel = 2, SE = 3, AR = 1)

# Both fixed effects and clustering
result_both <- maive(data, method = 3, weight = 0, instrument = 1,
                     studylevel = 3, SE = 3, AR = 0)

Standard Error Options

# CR0 (Huber-White)
result_cr0 <- maive(data, method = 3, weight = 0, instrument = 1,
                    studylevel = 2, SE = 0, AR = 1)

# CR1 (Standard empirical correction)
result_cr1 <- maive(data, method = 3, weight = 0, instrument = 1,
                    studylevel = 2, SE = 1, AR = 1)

# CR2 (Bias-reduced estimator)
result_cr2 <- maive(data, method = 3, weight = 0, instrument = 1,
                    studylevel = 2, SE = 2, AR = 1)

# Wild bootstrap (recommended, default)
result_boot <- maive(data, method = 3, weight = 0, instrument = 1,
                     studylevel = 2, SE = 3, AR = 1)

First-Stage Specification

MAIVE allows two functional forms for the first-stage regression:

result_levels <- maive(data, method = 3, weight = 0, instrument = 1,
                       studylevel = 2, SE = 3, AR = 1, first_stage = 0)

cat("First-stage (levels) F-test:", round(result_levels$`F-test`, 3), "\n")

result_log <- maive(data, method = 3, weight = 0, instrument = 1,
                    studylevel = 2, SE = 3, AR = 1, first_stage = 1)

cat("First-stage (log) F-test:", round(result_log$`F-test`, 3), "\n")

WAIVE: More Aggressive Correction

WAIVE (Weighted Adjusted Instrumental Variable Estimator) provides a more aggressive correction for p-hacking and spurious precision by combining variance instrumentation with exponential-decay downweighting of studies with spurious precision or extreme outliers:

For technical details and methodology, see the WAIVE slides.

result_waive <- waive(
  dat = data,
  method = 3,
  weight = 0,
  instrument = 1,
  studylevel = 2,
  SE = 3,
  AR = 1
)

cat("WAIVE Estimate:", round(result_waive$beta, 3), "\n")
cat("WAIVE SE:", round(result_waive$SE, 3), "\n")

WAIVE is particularly useful when:

• You need a more aggressive correction for p-hacking beyond standard MAIVE
• You suspect extreme outliers in your data
• Standard errors may be severely manipulated
• You want automatic downweighting of both spuriously precise estimates and outliers

Interpretation Guidelines

References

Irsova, Z., Bom, P.R.D., Havranek, T., & Rachinger, H. (2025). Spurious precision in meta-analysis of observational research. Nature Communications, 16, 8454. https://doi.org/10.1038/s41467-025-63261-0

Keane, M., & Neal, T. (2023). Instrument strength in IV estimation and inference: A guide to theory and practice. Journal of Econometrics, 235(2), 1625-1653.

See Also

• ?maive for detailed parameter documentation
• ?waive for the robust WAIVE estimator
• Project website: https://meta-analysis.cz/maive/