Meta-Analysis Exercise: Standardising Regression Coefficients

Introduction

Each study below estimates the effect of temperature on insect abundance using a simple linear regression.

You’ll extract and standardise the regression coefficients so they are comparable, then run a meta-analysis using metafor.

Load packages

library(dplyr)
library(metafor)

Formula Reminder

To standardise a regression slope β:

\[ \beta_{\text{standardised}} = \beta \cdot \frac{\text{SD}_X}{\text{SD}_Y} \]

And for the standard error:

\[SE std = SE raw * \frac{\text{SD}_X}{\text{SD}_Y}\]

Study 1: Temperate Forest

A study in New Hampshire found that insect abundance increased by 2.3 individuals per °C (SE = 0.6). The standard deviation of temperature was 3.8°C, and the SD of abundance was 18.5.

Study 2: Agricultural Field

Regression slope: 1.1 (SE = 0.3). SD of temp = 4.2; SD of abundance = 12.3

Study 3: Tropical Rainforest

β = 0.9 (SE = 0.2); SD_temp = 2.5, SD_abundance = 6.0

Study 4: Urban Garden Network

The slope of the regression was 3.4 individuals per °C (SE = 0.9). SD of temp = 5.1, SD of abundance = 20.1.

Study 5: Alpine Meadow

Insects were more sensitive in colder climates: β = 1.7 (SE = 0.5), with SD of temp = 3.0 and SD of abundance = 10.2.

Combine Your Results

# Example structure
dat <- data.frame(
  study = paste0("Study_", 1:5),
  beta_raw = c(, , , , ),
  se_raw = c(, , , , ),
  sd_temp = c(, , , , ),
  sd_abund = c(, , , , )
)

dat <- dat |>
  mutate(
    beta_std = beta_raw * (sd_temp / sd_abund),
    se_std = se_raw * (sd_temp / sd_abund),
    vi_std = se_std^2
  )

Meta-Analysis and Forest Plot

res <- rma(yi = beta_std, vi = vi_std, data = dat) # change this to the correct model type
forest(res, slab = dat$study,
       xlab = "Standardised Effect (SD change in abundance per SD temp)")

Interpretation

• You are now meta-analysing standardised slopes

• Unitless: interpretable as “effect per SD increase in temperature”

• Useful when raw units differ across studies