Meta-Analysis Fixed Effect

Meta-Analysis Fixed Effect#

Fixed-effect meta-analysis combines study-specific estimates under the assumption of a common fixed true effect, making it mathematically equivalent to analyzing all individuals as if they came from a single pooled dataset.

Graphical Summary#

Fig

Key Formula#

In meta-analysis, the weighted mean effect size for a fixed-effects model is calculated as:

\[\hat{\beta} = \frac{\sum_{k=1}^{K} w_k \hat{\beta}_k}{\sum_{k=1}^{K} w_k}\]

Where:

  • \(\hat{\beta}\) is the combined effect estimate across all studies

  • \(\hat{\beta}_i\) is the effect estimate from study \(k\)

  • \(w_k\) is the weight assigned to study \(k\)

  • \(K\) is the number of studies

Technical Details#

Why Do We Need Meta-Analysis?#

Individual studies have limitations:

  • Small sample sizes -> low statistical power

  • Population-specific effects -> limited generalizability

  • Random variation -> unreliable estimates

Meta-analysis provides:

  • Larger effective sample size -> higher power to detect true effects

  • More precise effect estimates -> narrower confidence intervals

  • Broader population representation -> better generalizability

How the Weighting Works#

The key insight is that not all studies should contribute equally. Studies with more information should have more influence on the final result.

Weight is based on precision:

\[ w_k = \frac{1}{\text{SE}_k^2} \]

Where \(\text{SE}_k\) is the standard error of study \(k\).

This means:

  • Large studies (small SE) get high weight: more influence

  • Small studies (large SE) get low weight: less influence

  • Precise studies contribute more to the final estimate

The Fixed-Effects Assumption#

Fixed-effects meta-analysis assumes that all studies estimate the same true effect. Any differences between studies are due to random sampling variation only.

When to use fixed-effects:

  • Studies are very similar (same populations, methods, designs)

  • Low heterogeneity between study results

  • Want to estimate the common effect size

When NOT to use fixed-effects:

  • Studies differ substantially in populations or methods

  • High heterogeneity between results

  • Different genetic architectures across populations

Important Limitations#

  • Garbage in, garbage out: Meta-analysis cannot fix poorly designed individual studies

  • Publication bias: Published studies may not represent all conducted research

  • Population differences: Genetic effects may genuinely differ across populations

Example#

This example demonstrates a meta-analysis of genetic variants across two European cohorts. We’ll analyze 3 genetic variants (SNPs) that have been genotyped in both cohorts and combine their effects using fixed-effects meta-analysis.

We first generate the summary statistics for 3 variants from two independent European cohorts with different sample sizes (N=8000 and N=5500), and assuming that they are from the same genetic ancestry so they can be meta-analyzed.

Then we perform fixed-effects meta-analysis combining results using:

  • Sample size weighting (weight proportional to \(\frac{1}{\sqrt{N}}\))

  • Inverse variance weighting (weight = \(\frac{1}{\text{SE}^2}\))

Lastly we plot the effect size and p-values for each variant and compare the results from separate cohorts and meta-analysis results. We also compare the variance based on the sample size and inverse variance.

rm(list=ls())
set.seed(17)

# 1. Simulate genotype data (0/1/2) for a single SNP
N1 <- 5000
N2 <- 8000
maf1 <- 0.3
maf2 <- 0.35

variant_pop1 <- rbinom(N1, 2, maf1)
variant_pop2 <- rbinom(N2, 2, maf2)

# 2. Simulate phenotype with fixed effect beta=1 and noise
beta <- 1
y_pop1 <- beta * variant_pop1 + rnorm(N1, 0, 3)
y_pop2 <- beta * variant_pop2 + rnorm(N2, 0, 3)

# 3. Run regression in each population
lm_pop1 <- lm(y_pop1 ~ variant_pop1)
lm_pop2 <- lm(y_pop2 ~ variant_pop2)

# Extract summary statistics
beta_pop1 <- coef(lm_pop1)["variant_pop1"]
se_pop1 <- summary(lm_pop1)$coefficients["variant_pop1", "Std. Error"]

beta_pop2 <- coef(lm_pop2)["variant_pop2"]
se_pop2 <- summary(lm_pop2)$coefficients["variant_pop2", "Std. Error"]

# 4. Fixed-effect meta-analysis
w1 <- 1 / se_pop1^2
w2 <- 1 / se_pop2^2

beta_meta <- (beta_pop1 * w1 + beta_pop2 * w2) / (w1 + w2)
se_meta <- sqrt(1 / (w1 + w2))
z_meta <- beta_meta / se_meta
p_meta <- 2 * pnorm(-abs(z_meta))

res_meta = data.frame(beta_meta, se_meta, z_meta, p_meta)
rownames(res_meta) = NULL
res_meta
A data.frame: 1 x 4
beta_metase_metaz_metap_meta
<dbl><dbl><dbl><dbl>
1.0152510.0401181525.306542.706622e-141

Alternatively we can simply combine all individuals into one and calculate the summary statistics for this variant.

# 5. Pooled analysis
variant_all <- c(variant_pop1, variant_pop2)
y_all <- c(y_pop1, y_pop2)

lm_all <- lm(y_all ~ variant_all)
beta_all <- coef(lm_all)["variant_all"]
se_all <- summary(lm_all)$coefficients["variant_all", "Std. Error"]
z_all <- beta_all / se_all
p_all <- 2 * pnorm(-abs(z_all))

res_merged = data.frame(beta_all, se_all, z_all, p_all)
rownames(res_merged) = NULL
res_merged
A data.frame: 1 x 4
beta_allse_allz_allp_all
<dbl><dbl><dbl><dbl>
1.0130510.0399850125.335781.289396e-141
 
# 6. Compare results

res_meta
res_merged
A data.frame: 1 x 4
beta_metase_metaz_metap_meta
<dbl><dbl><dbl><dbl>
1.0152510.0401181525.306542.706622e-141
A data.frame: 1 x 4
beta_allse_allz_allp_all
<dbl><dbl><dbl><dbl>
1.0130510.0399850125.335781.289396e-141

The results are almost identical (after numerical rounding).

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