Derivation of adjusted summary statistics: Meta-Reductive Approach (MRA)

We analyzed two distinct sets of summary statistics:

1. A compilation from n datasets meta-analyzed using an inverse variance-based approach [7].
2. A specific dataset of interest that was also part of the meta-analysis.

For these datasets:

• B and SE symbolize the effect size and standard error, respectively, from the aggregate meta-analysis across n datasets.
• β_i and se_i specify the effect size and standard error for the individual cohort i.

Our primary aim was to compute a summary statistic that eliminates the influence of the dataset of interest, providing a clearer perspective on the overarching genetic structure.

1. Inverse-variance-weighted effect-size estimation. The inverse variance method gives more weight to studies with smaller variance because they offer more precise estimates. The weight, w_i, is the inverse of the variance, or squared standard error, of the effect size, β_i.
   Given,
   where the
   Expanding this:
   This is the weighted sum of the effect sizes across all datasets, including the one of interest.
   Now, to remove the effect of the specific dataset, β_n, we rearrange:
   Which yields:
   This equation essentially adjusts the overall effect size, B, by subtracting the influence of the dataset of interest.
2. Standard error derivation. The standard error (SE) offers a measure of the statistical accuracy of an estimate. Here, we adjust the SE based on the weights of all datasets excluding the one of interest.
   Using:
   We derive:
   This equation gives the combined weight of all datasets, excluding the dataset of interest.
3. Adjusted effect size and standard error. Post removing the influence of the dataset of interest, the modified effect size is given by:
   This adjusted beta, B_adj, having nullified the contribution of the specific dataset n.
   Additionally, the adjusted standard error is:
   This adjustment ensures that the standard error reflects the precision of our new effect size estimate, free from the influence of the specific dataset.

Ethical approval was not required for this study as it utilized publicly available summary statistics.

Validation using real data

To validate our methodological approach, we utilized summary statistics from four publicly accessible Alzheimer disease studies: Kunkle et al. [8], Kunkle et al. [9] AA, Bellinguez et al. [10], and Moreno-Grau S. et al. [11] From these studies, 100,000 markers were selected to conduct a meta-analysis using the METASOFT software [12].

Following the initial meta-analysis, we applied a systematic "leave-one-out" strategy. For each iteration, we excluded the summary statistics from one dataset and conducted a meta-analysis of the remaining three. The results from this procedure served as our individual-level data for the three datasets in question.

For the final step of validation, we calculated the adjusted B_adj and values based on MRA and compared them against the individual-level data summary statistics derived from the "leave-one-out" FEMA. Our results showed that the summary statistics of the FEMA and MRA approaches yielded the exact same values. To demonstrate this, we plotted the betas and standard errors (Fig 1). The graphical representation illustrates that both beta and standard error values from the “leave-one-out”“FEMA and MRA give the same results.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Comparison between the adjusted results from the Meta-Reductive Analysis (MRA) (Beta_adj and SE2_adj) approach and the "leave-one-out" inverse variance-based fixed effect meta-analysis (FEMA) (Beta_LOO and SE2_LOO).

The MRA-adjusted values show identical results with the FEMA calculation for both Beta values (A) and Standard Error (B).

https://doi.org/10.1371/journal.pone.0296207.g001

Additionally, we conducted a validation analysis for the APOE4 allele, utilizing data from a multi-ancestry study by Rajabli et al. [13], which included four population-based studies: African American, East Asian, Hispanic, and non-Hispanic Whites (S1 Table). We applied a "leave-one-out" strategy by removing one population at a time and performing the validation on the remaining studies. We followed the same steps as described previously, and the results were exact same across all tests, as detailed in Table 1.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Validation analysis for APOE4 allele using summary statistics from Rajabli et al. study.

"MRA-Beta" and "MRA-SE" denote beta and standard error values derived using the MRA approach, respectively. "Traditional-Beta" and "Traditional-SE" refer to beta and standard error values obtained from the meta-analysis of three studies.

https://doi.org/10.1371/journal.pone.0296207.t001

Simulation

We simulated Beta coefficients and their corresponding SEs across ten studies, each containing 10,000 markers (using R programming language.) We generated random Beta coefficients utilizing the “rnorm” function, under the assumption of a normal distribution, characterized by a mean of zero and a standard deviation of one. We used “runif” function to produce random SEs values from a uniform distribution, with specified minimum and maximum limits of 0.1 and 0.5, respectively. The simulated Betas and SEs for each marker within a study were then organized into dedicated columns within a data frame.

We applied the “rma” function from the metaphor [14] package to facilitate a fixed-effects meta-analysis on the generated Betas and SEs. Then we implemented "leave-one-out" strategy, mirroring the methodology applied to real data. We calculated the adjusted B_adj and values employing our proposed method and compared it with individual-level data derived from the "leave-one-out" meta-analyses. The outcomes revealed that the summary statistics were identical, similar to the findings from real data analysis. The simulation script is provided with the MRA function here: https://github.com/hihg-um/MRA.