Calculator Inputs
Example Data Table
These examples show how interaction strength and model complexity change the needed analyzable sample.
| Scenario | Base R² | ΔR² | Tested terms | Full predictors | Alpha | Power | Estimated N |
|---|---|---|---|---|---|---|---|
| Single small interaction | 0.20 | 0.02 | 1 | 4 | 0.05 | 0.80 | 309 |
| Single medium interaction | 0.25 | 0.06 | 1 | 4 | 0.05 | 0.80 | 93 |
| Two-term interaction block | 0.30 | 0.05 | 2 | 7 | 0.05 | 0.90 | 168 |
Formula Used
Full model R²: R²full = R²base + ΔR²
Effect size: f² = ΔR² / (1 − R²full)
Numerator degrees of freedom: df1 = number of tested interaction terms
Denominator degrees of freedom: df2 = N − Pfull − 1
Critical value: Fcritical = F−1(1 − α; df1, df2)
Noncentrality: λ = f² × N
Power: 1 − CDF of the noncentral F distribution at Fcritical
The calculator searches upward until the estimated power reaches the target. It then inflates the analyzable sample for attrition and design effect.
How to Use This Calculator
- Enter the target statistical power, such as 0.80 or 0.90.
- Set the alpha level, usually 0.05 for standard tests.
- Choose whether to use ΔR² or direct Cohen f².
- Enter the expected base R² from prior research.
- Add the expected variance increase from the moderation term.
- Count all predictors in the final regression model.
- Enter the number of interaction terms being tested.
- Add attrition and design effect when planning recruitment.
- Press the calculate button and review the result above the form.
- Use the CSV or PDF buttons to save the estimate.
Article: Planning Sample Size for Moderation Models
Why A Priori Planning Matters
A moderation model asks whether one variable changes the effect of another variable. In physics research, this structure appears in experimental design, instrumentation, simulations, and measurement studies. A material property may moderate a force response. A temperature setting may change an efficiency relationship. These questions often depend on interaction terms. Interaction effects are usually smaller than main effects. They also need more data. That is why a priori planning is important.
What the Calculator Estimates
This calculator estimates the analyzable sample needed before data collection starts. It focuses on the added value of the interaction block. The user enters expected base model R² and interaction ΔR². The tool converts that change into Cohen f². It then tests whether the interaction block can be detected with the selected power and alpha. The final result gives a raw analyzable sample. It also gives a recruitment target after attrition.
Why Predictor Count Matters
Moderation models include main effects, interaction terms, and possible covariates. Each added predictor uses degrees of freedom. A large control set can raise the required sample. A block with several interaction terms also raises the numerator degrees of freedom. These details should match the planned final model. Otherwise, the estimate can be too optimistic.
Using Effect Sizes Carefully
The hardest input is the expected interaction effect. Prior studies are useful. Pilot data can help. Domain knowledge can guide a conservative value. Small interaction effects are common in complex systems. If the effect is uncertain, test several scenarios. Compare small, medium, and optimistic values. This gives a useful planning range.
Interpreting the Result
The output is a planning estimate, not a guarantee. Real results depend on measurement error, model assumptions, sampling quality, and missing data. Use the adjusted recruitment target when dropout or unusable observations are expected. Keep a record of assumptions. This makes the study plan clearer and more defensible.
FAQs
1. What is a moderation model?
A moderation model tests whether one variable changes the relationship between another predictor and an outcome. The key test is usually an interaction term, such as X multiplied by M.
2. What does a priori sample size mean?
It means estimating sample size before collecting data. This helps researchers plan recruitment, budget, timing, and statistical power before the study begins.
3. What is interaction ΔR²?
Interaction ΔR² is the added explained variance after interaction terms enter the model. It measures how much the moderation block improves prediction beyond the base model.
4. What is Cohen f²?
Cohen f² is a regression effect size. For moderation, it can be computed from interaction ΔR² and the full model R².
5. Should I use one tested interaction term?
Use one term for a simple single interaction. Use more than one when testing a block, such as several product terms or dummy-coded interaction terms.
6. Why does attrition increase the recruitment target?
Attrition reduces usable observations. The calculator inflates the required analyzable sample so the final dataset can still meet the planned power target.
7. What design effect should I use?
Use 1 for independent sampling. Use a larger value when clustering, repeated settings, or grouped sampling reduces the effective information in the data.
8. Is this calculator a substitute for expert review?
No. It is a planning tool. Complex designs, nonlinear models, nonnormal outcomes, and clustered data may need simulation or specialist statistical review.