Calculator Inputs
Example Data Tables
Example: Cochran's Q
| Subject | Treatment 1 | Treatment 2 | Treatment 3 | Treatment 4 |
|---|---|---|---|---|
| 1 | 1 | 0 | 1 | 1 |
| 2 | 1 | 1 | 1 | 0 |
| 3 | 0 | 0 | 1 | 1 |
| 4 | 1 | 1 | 0 | 1 |
Formula Used
1) Cochran's Q
Q = (k - 1) × [k × Σ(Gj²) - T²] / [k × T - Σ(Li²)]
Gj is the column total for treatment j, Li is the row total for subject i, T is the grand total, and df = k - 1.
2) Meta-analysis heterogeneity Q
Q = Σ[wi × (θi - θ̄)²]
θ̄ = Σ(wi × θi) / Σwi
The calculator also reports I² = max(0, (Q - df) / Q) × 100 and Tau² using the DerSimonian-Laird estimator.
3) Box-Pierce and Ljung-Box Q
Box-Pierce: Q = n × Σ(rk²)
Ljung-Box: Q = n(n + 2) × Σ[rk² / (n - k)]
rk is the sample autocorrelation at lag k, n is sample size, and df usually equals lags minus fitted model parameters.
How to Use This Calculator
- Select the Q statistic family that matches your problem: matched binary responses, meta-analysis heterogeneity, or residual autocorrelation.
- Set the significance level and decimal precision.
- Paste your data in the format shown beside each input area.
- For portmanteau testing, enter sample size and any model degrees-of-freedom adjustment.
- Click Calculate Q Statistic to place the results above the form.
- Review the summary table, detailed table, and Plotly chart.
- Use the CSV and PDF buttons to export your outputs.
- Compare the p value against alpha before making an inferential decision.
FAQs
1) What does the Q statistic measure?
The meaning depends on the selected method. It can compare matched binary treatments, test effect-size heterogeneity across studies, or detect residual autocorrelation across lags.
2) When should I use Cochran's Q?
Use it when the same subjects are measured under three or more matched binary conditions, such as pass or fail, yes or no, or event and no event.
3) What is a significant heterogeneity Q in meta-analysis?
A significant result suggests the study effects vary more than sampling error alone would explain. That often motivates checking moderators, outliers, or random-effects modeling.
4) Why does the calculator report I² and Tau²?
These help interpret heterogeneity magnitude. I² estimates the percent of total variability due to heterogeneity, while Tau² estimates the between-study variance scale.
5) Should I prefer Ljung-Box or Box-Pierce?
Ljung-Box is usually preferred in practice because it performs better in smaller samples. Box-Pierce is simpler, but it can under-adjust finite-sample behavior.
6) Why is my p value missing or invalid?
This usually means the degrees of freedom are not positive, the sample size is too small for the requested lag structure, or the denominator became zero.
7) Can I paste data with spaces instead of commas?
Yes for Cochran and portmanteau inputs. The meta-analysis block expects comma-separated values because it may include study labels alongside effect and weight columns.
8) Does this calculator replace full statistical software?
No. It is a strong decision-support tool, but final reporting should still check assumptions, data quality, model fit, and any domain-specific analysis requirements.