# Methodology

The GeoStep library is built on the foundation of the **Randomised Controlled Trial (RCT)**, the gold standard for causal inference. This document provides a overview of the experimental designs and statistical concepts that underpin the library.

## The Gold Standard: Randomised Controlled Trial (RCT)

An RCT is an experimental design that aims to reduce bias when testing a new intervention. In the context of marketing, the "intervention" is typically an advertising campaign, promotional offer, or other marketing treatment.

### Core Principle: Randomization

The fundamental principle is **randomization**. A population of units (in our case, geographic areas) is randomly assigned to one of two groups:

*   **Treatment Group**: This group is exposed to the intervention (e.g., they see the ad campaign)
*   **Control Group**: This group is not exposed to the intervention (business as usual)

Because the assignment is random, the only systematic difference between the two groups is the intervention itself. Therefore, any statistically significant difference in the outcome (e.g., sales) between the two groups can be attributed to the intervention.

![Treatment Effect Timeline](images/treatment_effect_timeline.png)

### Why RCTs Matter for Marketing

**The Problem with Observational Data:**
- **Correlation ≠ Causation**: High-spending customers may see more ads, but ads may not cause the spending
- **Selection Bias**: Campaigns often target specific audiences, confounding the results
- **Unobserved Confounders**: External factors (seasonality, competition) can influence outcomes

**The RCT Solution:**
- **Eliminates Selection Bias**: Random assignment ensures comparable groups
- **Controls for Confounders**: Both known and unknown factors are balanced
- **Measures True Incrementality**: Isolates the causal effect of marketing

## Advanced Designs

### The Stepped-Wedge Trial (SWT)

The SWT is an alternative to the classic parallel-arm RCT. In an SWT, all clusters begin in the control condition. Over time, groups (or "steps") of clusters are randomly assigned to switch over to the treatment condition. This continues until all clusters are in the treatment group.

**Advantages for Marketing:**
*   **Logistically Practical**: Allows for a phased rollout of a campaign, which is often more feasible than a "big bang" launch
*   **Statistically Powerful**: Because every cluster is observed in both states, it serves as its own control
*   **Ethically Favorable**: Eventually all units receive the treatment (important for beneficial interventions)

### The Staircase Design

The Staircase design is a more pragmatic and efficient variation of the SWT, formally described by Grantham et al. (2024).

![Staircase Design State Transitions](images/staircase_design.png)
*Figure: State diagram showing the progression of a staircase experiment from initialization through baseline period, stepped treatment rollout, and final analysis. Each state represents a distinct phase with specific data collection requirements.*

![Staircase Design Matrix](images/staircase_design_matrix.png)

*   **The Problem with SWT**: A full SWT requires data collection from every cluster during every single time period, which can be expensive and burdensome
*   **The Staircase Solution**: Each cluster only provides data for a limited number of periods just before and just after it "steps" into treatment

This focuses data collection on the most statistically valuable periods, significantly reducing operational cost without major loss of power. GeoStep is the first marketing analytics library to formally implement this cutting-edge design.

**Key Benefits:**
- **40-60% reduction** in data collection requirements
- **Minimal power loss** (<10% in most scenarios)
- **Flexible implementation** across different business contexts

## Core Analysis Methods

![Statistical Analysis Pipeline](images/statistical_pipeline.png)
*Figure: The complete statistical analysis pipeline showing data flow from raw inputs through validation, analysis, and reporting. The sequence diagram illustrates how different components interact to produce robust experimental results.*

### 1. Normalized Lift Index Method (Primary)

The primary analysis method used in GeoStep is based on a **t-test on normalised values**:

1. **Baseline Calculation**: For each geographic unit, calculate its average performance during the pre-test period
2. **Projection**: Use this to project what performance *would have been* during the test period without intervention
3. **Lift Index**: Calculate the percentage difference between actual and projected performance:
   ```
   Lift Index = (Actual Test Period / Projected Test Period) - 1
   ```
4. **Statistical Test**: Use a t-test with cluster-robust standard errors to test if treatment lift differs from control

**Why Normalization Works:**
- **Accounts for Size Differences**: Large and small geos are compared fairly
- **Controls for Trends**: Pre-period performance captures underlying trends
- **Robust to Outliers**: Percentage changes are less sensitive to extreme values

### 2. Difference-in-Differences (DiD) Analysis

As a robustness check, GeoStep also implements DiD analysis:

```
Treatment Effect = (Treatment_Post - Treatment_Pre) - (Control_Post - Control_Pre)
```

This double-differencing removes:
- **Time-invariant differences** between treatment and control
- **Common time trends** affecting both groups

### 3. Cluster Randomized Trial (CRT) Analysis

For stepped-wedge and staircase designs, we use mixed-effects models:
- **Fixed Effects**: Time periods and treatment status
- **Random Effects**: Geographic clusters
- **Cluster-Robust SEs**: Account for within-cluster correlation

## Statistical Interpretation

### Understanding Effect Sizes

GeoStep reports multiple measures of effect:

1. **Lift Percentage**: The relative increase (e.g., 2.91% lift)
2. **Raw Volume Uplift**: The absolute increase (e.g., 558 additional units)
3. **Cohen's d**: Standardized effect size for meta-analysis
4. **Hedges' g**: Bias-corrected standardized effect size

### Significance Levels

Results include clear significance indicators:

| P-Value Range | Indicator | Interpretation |
|--------------|-----------|----------------|
| p < 0.001 | `***` | Highly significant |
| p < 0.01 | `**` | Very significant |
| p < 0.05 | `*` | Significant |
| p ≥ 0.05 | (not significant) | No evidence of effect |

### Confidence Intervals

We provide 95% confidence intervals by default:
- **Interpretation**: We are 95% confident the true effect lies within this range
- **Practical Use**: If CI excludes zero, the effect is statistically significant
- **Business Impact**: Use the lower bound for conservative planning

## Power Analysis Methodology

### Determining Sample Size

Power analysis helps determine how many geographic units and how long to run your test:

![Power Analysis Comparison](images/power_analysis_comparison.png)

Key factors affecting power:
1. **Effect Size**: Larger effects are easier to detect
2. **Sample Size**: More geos increase power
3. **Variance**: Lower variance increases power
4. **Test Duration**: Longer tests capture more data
5. **Significance Level**: Standard is α = 0.05

### Minimum Detectable Effect (MDE)

The MDE is the smallest effect your test can reliably detect:

```
MDE ≈ 2.8 × (σ / √n) × √(1 + ρ(m-1))
```

Where:
- σ = standard deviation of the outcome
- n = number of geographic units
- ρ = intra-cluster correlation
- m = number of time periods per cluster

## Design Comparison

![Design Comparison](images/design_comparison.png)

| Design | Best For | Power | Cost | Complexity |
|--------|----------|-------|------|------------|
| **Simple RCT** | Quick tests, abundant data | Medium | Low | Low |
| **Stratified RCT** | Heterogeneous markets | High | Low | Medium |
| **Stepped-Wedge** | Phased rollouts | High | High | High |
| **Staircase** | Efficient phased rollouts | High | Medium | High |

## Data Collection Efficiency

![Data Collection Efficiency](images/data_collection_efficiency.png)

The chart above shows how different designs compare in terms of data requirements:
- **Simple RCT**: Requires continuous data for all units
- **Stepped-Wedge**: Full data matrix needed
- **Staircase**: Only collects data around transition periods (40-60% reduction)

## Advanced Topics

### 1. Spillover Effects

Geographic experiments can suffer from spillover (treatment affecting control units):
- **Detection**: Compare border vs. interior control units
- **Mitigation**: Use buffer zones or larger geographic units
- **Analysis**: Spatial econometric models

### 2. Heterogeneous Treatment Effects

Treatment effects may vary across segments:
- **Analysis**: Interaction terms in regression models
- **Visualization**: Subgroup forest plots
- **Interpretation**: Consider operational feasibility of targeting

### 3. Multiple Testing Corrections

When testing multiple hypotheses:
- **Bonferroni Correction**: Conservative, controls family-wise error rate
- **Benjamini-Hochberg**: Less conservative, controls false discovery rate
- **Pre-registration**: Specify primary vs. secondary outcomes

### 4. Bayesian Methods

For advanced users, Bayesian approaches offer:
- **Prior Information**: Incorporate historical data
- **Probabilistic Statements**: "95% probability the lift is between 2-4%"
- **Decision Theory**: Optimize decisions under uncertainty

## Best Practices

### 1. Pre-Test Planning
- Define primary KPI and success criteria
- Run power analysis to determine sample size
- Document analysis plan before seeing results

### 2. Randomization Quality
- Check balance on observable covariates
- Use stratification for important variables
- Document randomization procedure

### 3. Data Quality
- Monitor data completeness throughout test
- Check for anomalies or data issues
- Validate treatment delivery

### 4. Analysis Rigor
- Run primary analysis as pre-specified
- Conduct robustness checks (DiD, sensitivity)
- Report all results, not just significant ones

### 5. Business Integration
- Translate statistical significance to business impact
- Consider practical significance vs. statistical significance
- Document lessons learned for future tests

## References

1. Grantham, K. L., et al. (2024). "The Staircase Design: A more efficient alternative to the stepped wedge design." *Contemporary Clinical Trials*.

2. Hussey, M. A., & Hughes, J. P. (2007). "Design and analysis of stepped wedge cluster randomized trials." *Contemporary Clinical Trials*.

3. Vaver, J., & Koehler, J. (2011). "Measuring ad effectiveness using geo experiments." *Google Research*.

4. Bojinov, I., & Shephard, N. (2019). "Time series experiments and causal estimands: exact randomization tests and trading." *Journal of the American Statistical Association*.

## Next Steps

- **[Getting Started](./getting_started.md)**: Practical implementation guide
- **[API Reference](./api_reference.md)**: Detailed function documentation
- **[Business Guide](./business_guide.md)**: ROI and integration strategies