======  Random-Effects Model ======


🧠 What Is a Random-Effects Model in Meta-Analysis?
A random-effects model is a statistical method used in meta-analyses when the included studies are not functionally identical, and true effect sizes are assumed to vary across studies.

📊 Definition
In contrast to the fixed-effects model (which assumes that all studies estimate the same true effect), the random-effects model assumes that:

Each study estimates a different, yet related, true effect size

Observed differences between study results are due to both random sampling error and true between-study heterogeneity

⚙️ How It Works
The model incorporates:

Within-study variance (sampling error)

Between-study variance (true heterogeneity, denoted as τ²)

The final result is a weighted average that gives less weight to larger studies compared to fixed-effects models, which may downweight smaller or outlier studies too strongly.

📌 When to Use
Use a random-effects model when:

There is clinical, methodological, or statistical heterogeneity

Studies vary in:

Population

Intervention type or intensity

Study design or setting

The I² statistic (a measure of heterogeneity) is moderate to high (> 25–50%)

🔬 Formula (Simplified)
For study i:

𝜃
^
𝑖
∼
𝑁
(
𝜃
,
𝜎
𝑖
2
+
𝜏
2
)
θ
^
  
i
​
 ∼N(θ,σ 
i
2
​
 +τ 
2
 )
Where:

𝜃
^
𝑖
θ
^
  
i
​
 : observed effect size

𝜎
𝑖
2
σ 
i
2
​
 : within-study variance

𝜏
2
τ 
2
 : between-study variance (estimated from the data)

✅ Advantages
More realistic when studies differ

Produces more conservative confidence intervals

Acknowledges true heterogeneity in effect sizes

❌ Disadvantages
Wider confidence intervals

Less statistical power

Estimation of τ² may be unstable with few studies

🧠 Clinical Application (e.g., Neurosurgery)
In neurosurgery meta-analyses (like those evaluating surgical vs. conservative management of pituitary apoplexy), where:

Treatment protocols differ by center

Imaging modalities evolve over time

Outcome definitions vary

👉 Random-effects models are almost always more appropriate than fixed-effects models.