Formula Used
The planned linear contrast is calculated as:
L = sum(cᵢ × μᵢ)
The required total analyzable sample is estimated as:
N = ((zcrit + zpower)² × σ² × sum(cᵢ² / wᵢ)) / Δ²
Here, cᵢ is the contrast coefficient. μᵢ is the planned group mean. σ² is the common variance. wᵢ is the allocation weight. Δ is the target contrast effect. zcrit is based on alpha and test direction. zpower is based on desired power.
When unusable readings are included, enrolled sample size is inflated by dividing analyzable sample size by the retention rate.
How to Use This Calculator
- Enter contrast coefficients for each planned physics condition.
- Enter the expected mean for each matching condition.
- Leave manual target contrast blank, or enter a fixed effect.
- Choose standard deviation or variance as the variation input.
- Enter alpha, desired power, tails, and adjustment method.
- Add allocation ratios and unusable reading rate.
- Press Calculate to view the powered sample plan.
- Use CSV or PDF buttons to save the report.
Example Data Table
| Physics Condition | Coefficient | Expected Mean | Allocation Ratio |
|---|---|---|---|
| Baseline detector setting | -1 | 10.0 | 1 |
| Intermediate detector setting | 0 | 10.4 | 1 |
| Enhanced detector setting | 1 | 11.0 | 1 |
Article: Sample Size Power for Linear Contrast
Purpose
Physics experiments often compare more than two conditions. A linear contrast turns that planned comparison into one number. It may compare detector settings, beam strengths, material treatments, or calibration stages. The contrast uses chosen coefficients. The coefficients weight each planned mean. A useful contrast usually sums to zero. That makes it a comparison, not a simple total.
Power Planning
Power planning asks a direct question. How many observations are needed to detect the contrast? The answer depends on the target contrast, common variation, alpha level, desired power, and allocation. Small effects need larger samples. High variance also increases the requirement. Strict alpha adjustments raise the sample size too.
Calculation Logic
This calculator uses a normal approximation for the planned contrast test. It first estimates the contrast effect from the entered means. You may also enter a manual target effect. Then it computes the variance factor from the contrast coefficients and group allocation. Unequal allocation is allowed. This is helpful when one physics condition is costly, slow, or limited.
Advanced Options
The tool also supports one sided and two sided tests. It can adjust alpha for multiple planned contrasts. Bonferroni is conservative. Sidak is often slightly less strict. A dropout or unusable reading rate can be added. This inflates the enrolled sample before data collection begins.
Interpretation
Use the result as a planning guide. It is not a substitute for a full experimental protocol. Confirm assumptions with pilot data when possible. Check whether normal errors, independent measurements, and a shared variance model are reasonable. In many laboratory studies, those assumptions are practical. In complex detector, particle, thermal, or optical work, simulation may be needed.
Reporting
The group table helps translate the total sample into real runs. It shows the planned allocation, enrolled counts, and expected analyzable counts. The achieved power is recalculated after rounding. This matters because rounded group sizes may slightly change precision. Export the report for records. Save the CSV for spreadsheets. Save the PDF for quick review. Good sample planning protects equipment time, material budgets, and researcher effort. It also makes results easier to explain later.
Final Check
Before final enrollment, review the smallest scientific effect worth detecting. Enter that value honestly. Oversized guesses create weak studies. Conservative planning gives the experiment a fairer test in practice today.
FAQs
What is a linear contrast?
A linear contrast is a planned weighted comparison of group means. The coefficients define the comparison. In many designs, the coefficients sum to zero, which makes the result a focused comparison rather than a total.
Can this calculator handle unequal allocation?
Yes. Enter allocation ratios in the same order as the coefficients. The calculator converts them into allocation weights and uses those weights inside the variance factor.
What does target contrast mean?
Target contrast is the effect size for the planned comparison. It can be computed from group means and coefficients. You may also enter a manual target when you already know the desired detectable effect.
Should I use one sided or two sided testing?
Use a one sided test only when the opposite direction is not scientifically useful. Use a two sided test when either direction matters. Two sided testing usually needs a larger sample.
What does alpha adjustment do?
Alpha adjustment controls error when several planned contrasts are tested. Bonferroni divides alpha by the number of contrasts. Sidak uses a related probability correction.
Why add unusable readings?
Physics measurements may fail because of noise, sensor issues, material loss, or invalid runs. Adding an unusable reading percentage inflates enrolled sample size before the experiment starts.
Why is achieved power recalculated?
Sample sizes must be whole numbers. Rounding can change group balance and precision. The calculator recalculates achieved power using the rounded group counts.
Is this suitable for final protocol approval?
It is useful for planning and documentation. For final approval, confirm assumptions with pilot data, simulations, or a statistician, especially for complex physics systems.