# Effect Models in Meta-Analysis

## Overview

Effect models determine how individual study effects are combined to produce a pooled estimate.

## Fixed Effect Model

### Concept

Assumes all studies estimate the **same underlying true effect**. Differences between studies are due only to sampling error.

### Mathematical Framework

```
θ_i = θ + ε_i

Where:
- θ_i = observed effect in study i
- θ = true common effect (unknown)
- ε_i ~ N(0, SE_i²) sampling error
```

### Weighting

Studies weighted by inverse variance:

```
w_i = 1 / SE_i²

θ_pooled = Σ(w_i × θ_i) / Σw_i
Var(θ_pooled) = 1 / Σw_i
```

### When to Use

- Studies are very similar (identical protocol)
- Heterogeneity is negligible (I² < 25%)
- Inference limited to included studies only
- Combining results from single large study

### TSA Java Code Reference

```java
// From MetaAnalysis.java
public static final int FIXED = 10;
// Uses standard inverse variance weighting
```

## Random Effects Model

### Concept

Assumes each study estimates a **different true effect**, and these effects come from a distribution.

### Mathematical Framework

```
θ_i = μ + u_i + ε_i

Where:
- μ = mean of effect distribution (what we estimate)
- u_i ~ N(0, τ²) between-study variance
- ε_i ~ N(0, SE_i²) within-study variance
```

### Weighting

Modified weights incorporating τ²:

```
w_i* = 1 / (SE_i² + τ²)

θ_pooled = Σ(w_i* × θ_i) / Σw_i*
Var(θ_pooled) = 1 / Σw_i*
```

### Common Estimators for τ²

#### DerSimonian-Laird (DL)
- Most widely used
- Method of moments estimator
- Can underestimate τ² with few studies

```java
// From MetaAnalysis.java
public static final int RANDOM_DL = 11;
```

#### Sidik-Jonkman (SJ)
- Alternative estimator
- Better performance with few studies
- Less biased than DL

```java
public static final int RANDOM_SJ = 16;
```

#### Bayesian (BT)
- Uses prior distribution for τ
- Incorporates uncertainty in τ² estimate
- Better coverage properties

```java
public static final int RANDOM_BT = 12;
```

### When to Use

- Clinical/methodological diversity expected
- Heterogeneity observed (I² > 25%)
- Generalizing beyond included studies
- Most real-world meta-analyses

## Hybrid Models

### Concept

Combines features of fixed and random effects based on heterogeneity.

```java
public static final int HYBRID_DL = 14;
public static final int HYBRID_BT = 15;
```

### Implementation

- If I² = 0: uses fixed effect (τ² = 0)
- If I² > 0: uses random effects with estimated τ²
- Provides adaptive weighting

## Effect Measures

### Odds Ratio (OR)

```java
public static final int ODDS_RATIO = 3;
```

**Properties:**
- Range: 0 to ∞ (1 = no effect)
- Symmetrical when log-transformed
- Works in case-control studies
- Overestimates RR for common events

### Relative Risk (RR)

```java
public static final int RELATIVE_RISK = 1;
```

**Properties:**
- Range: 0 to ∞ (1 = no effect)
- More intuitive than OR
- Cannot use in case-control studies
- Can calculate NNT directly

### Risk Difference (RD)

```java
public static final int RISK_DIFFERENCE = 2;
```

**Properties:**
- Range: -1 to +1 (0 = no effect)
- Absolute measure
- Directly shows clinical impact
- Varies with baseline risk

### Mean Difference (MD)

```java
public static final int MEAN_DIFFERENCE = 4;
```

**Properties:**
- For continuous outcomes
- Natural units
- Requires same measurement scale
- Can use SMD if scales differ

### Peto Odds Ratio

```java
public static final int PETO_ODDS_RATIO = 6;
```

**Properties:**
- For rare events
- No continuity correction needed
- Handles zero cells well
- Biased if OR ≠ 1 or groups unbalanced

## Model Selection Guidelines

### Use Fixed Effect When:
1. Studies are truly identical
2. I² < 25% and Q-test non-significant
3. Only inferring about these specific studies
4. Very few studies (2-3) with similar precision

### Use Random Effects When:
1. Any clinical/methodological diversity
2. I² ≥ 25% or Q-test significant
3. Generalizing to broader population
4. Default choice for most analyses

### Model Comparison

| Aspect | Fixed Effect | Random Effects |
|--------|--------------|----------------|
| Assumption | Single true effect | Distribution of effects |
| CI width | Narrower | Wider |
| Small study weight | Less | More |
| Generalizability | Limited | Broader |
| With heterogeneity | Inappropriate | Appropriate |

## TSA Implications

- Fixed effect: OIS based on assumed common effect
- Random effects: OIS adjusted for heterogeneity
- Model choice affects boundary calculations
- High heterogeneity increases required sample size