What is the dispersion parameter in gamma regression?

The dispersion parameter phi reflects the shape of the gamma distribution. Smaller values indicate tighter clustering around the predicted mean. The Pearson chi-square divided by residual degrees of freedom gives a standard estimate.

Gamma Regression Calculator – Free Online GLM Tool | Effect Size, P-Value & APA Results

📊 GLM · Statistical Models

Gamma Regression Calculator

Fit a generalized linear model for positive, right-skewed continuous outcomes. Get coefficients, deviance, p-values, AIC, dispersion estimates and APA-format results — instantly.

GLM Gamma Family Log Link Inverse Link Free Online APA Ready

📥 1. Input Your Data

Sample Dataset:

A dataset is pre-loaded so you can run the analysis immediately.

Predictor name (X):

Predictor values (X) — comma-separated:

Outcome name (Y > 0):

Outcome values (Y) — comma-separated, must be > 0:

—

Upload CSV or Excel file:

Supports .csv, .txt, .xlsx, .xls — headers detected automatically. Choose two numeric columns (Predictor X, Outcome Y).

Enter X and Y values one row at a time. Click Add Row for more.

⚙️ 2. Model Configuration

Link Function:

Significance Level (α):

Tail Type:

Max Iterations (IRLS):

📊 3. Analysis Results

Run the analysis to see results.

Summary Cards

Coefficient Table

Term	Estimate (β)	SE	z	p-value	exp(β)	95% CI

Model Fit Statistics

Statistic	Value	Description

Visualizations

① Fitted vs Observed (Y > 0)

② Deviance Residuals vs Fitted

③ Q-Q Plot of Deviance Residuals

④ Outcome Histogram + Gamma Density

🧠 4. Plain-Language Interpretation

📖 Subsection 1 — Detailed Interpretation Results

Run the analysis to see a full plain-language interpretation auto-filled with your data.

📝 Subsection 2 — How to Write Your Results in Research (5 Examples)

Five copy-ready reporting templates for the gamma regression results — APA 7, dissertation, plain-language, conference abstract, and pre-registered open-science formats. Each updates automatically after you run the analysis.

🎯 5. Conclusion

Run the analysis to generate a detailed conclusion.

The conclusion section synthesises your gamma regression findings, model adequacy, practical implications, and limitations into a publication-ready closing statement.

🎯 6. When to Use Gamma Regression

This free gamma regression calculator is designed for analysing positive, right-skewed continuous outcomes — for example healthcare costs, insurance claim amounts, reaction times, rainfall, or hospital length of stay. Use it whenever a normal-errors linear regression would predict negative or implausible values for a strictly positive outcome.

✅ Use Gamma Regression When:

Your outcome variable is strictly positive (Y > 0) and continuous.
The distribution of Y is right-skewed (long upper tail).
Variance increases with the mean (variance ∝ mean²).
You want multiplicative interpretation of effects (with the log link).
Linear regression of Y or log(Y) gives biased predictions or heteroscedastic residuals.
Observations are independent (no nested/repeated structure).

📌 Real-World Examples

Healthcare costsModelling annual medical expenditure as a function of age, comorbidities, and insurance type.

Insurance claim sizesPredicting the dollar amount of a non-zero auto insurance claim from driver age and vehicle category.

Reaction time studiesAnalysing positive-only response latencies in cognitive psychology experiments.

Hydrology / rainfallModelling daily rainfall amount on rainy days as a function of temperature, humidity, and pressure.

🌳 Decision Tree — Is Gamma Regression Right for My Data?

Is Y strictly positive (Y > 0)? ├── No → Use linear regression (Gaussian GLM) └── Yes → Is Y continuous? ├── No (counts) → Use Poisson or Negative Binomial regression └── Yes → Is Y right-skewed with variance ∝ mean²? ├── No (symmetric) → Use linear regression └── Yes → ✅ Use Gamma regression └── Are there exact zeros? ├── Yes → Use Tweedie GLM or two-part hurdle model └── No → ✅ Gamma regression is appropriate

✅ 7. Assumption Checks

PENDING Run the analysis to evaluate model assumptions automatically.

🧮 8. Technical Notes — Formulas & Derivations

Gamma GLM Specification

Y_i ~ Gamma(μ_i, φ)
g(μ_i) = β₀ + β₁ X_i (linear predictor η_i)
Var(Y_i) = φ · μ_i²

Where:
• μ_i = expected value of Y for observation i
• φ = dispersion parameter (estimated from residuals)
• g(·) = link function (log, inverse, or identity)
• β₀, β₁ = intercept and slope coefficients on the link scale

Common Link Functions

Log link: g(μ) = log(μ) → μ = exp(η)
Inverse link: g(μ) = 1/μ → μ = 1/η (canonical)
Identity link: g(μ) = μ → μ = η

The log link is most common because exp(η) is always positive, ensuring valid predictions.

Estimation — Iteratively Reweighted Least Squares (IRLS)

β^(t+1) = (XᵀW X)⁻¹ XᵀW z
W = diag(1 / [Var(μ) · g'(μ)²])
z = η + (y − μ) · g'(μ)

Iterations continue until the deviance change drops below 1e-8. The Fisher information at convergence gives the standard errors for β.

Deviance & Goodness of Fit

D = 2 · Σ [ (y_i − μ_i)/μ_i − log(y_i / μ_i) ]
Pearson χ² = Σ (y_i − μ_i)² / μ_i²
φ̂ = Pearson χ² / (n − p)

Residual deviance approximates a chi-square distribution with (n − p) degrees of freedom under correct model specification. Smaller deviance = better fit.

Pseudo R² (McFadden)

R²_McFadden = 1 − (D_full / D_null)

D_null = deviance of intercept-only model. Values 0.2–0.4 indicate excellent fit.

AIC (Akaike Information Criterion)

AIC = −2 · log L + 2k

k = number of estimated parameters. Lower AIC = better model. Used to compare nested or non-nested gamma GLMs.

📘 9. How to Use This Gamma Regression Calculator

A complete walkthrough — using the built-in Healthcare Costs dataset as a worked example.

1Enter Your Data. Use the Type/Paste tab for comma-separated values, the Upload tab for CSV/Excel files, or the Manual Entry tab for small datasets row by row.

Example: Predictor X = "52, 48, 55, 61, 47" (Age in years), Outcome Y = "2400, 1850, 3120, 4800, 1620" (Annual cost in $).

2Choose a Sample Dataset. Five built-in datasets cover healthcare costs, insurance claims, reaction times, rainfall, and hospital length of stay.

Example: Selecting "Healthcare Costs" loads age (X) and annual cost (Y) for 30 patients.

3Configure the Model. Pick your link function (log is recommended), alpha level, tail type, and IRLS iterations.

Example: Log link, α = 0.05, two-tailed, 50 iterations.

4Run the Analysis. Click the green "Run Gamma Regression Analysis" button. IRLS converges in 4–8 iterations for well-behaved data.

Example: The button turns the page into a full report within ~50 ms.

5Read the Summary Cards. Six colour-coded cards show n, residual deviance, dispersion, AIC, slope p-value, and significance.

Example: A green "Significant" card indicates p < 0.05 for the X coefficient.

6Inspect the Coefficient Table. Each row gives β, SE, z, p, exp(β) for log-link models, and 95% CI.

Example: β₁ = 0.043, exp(β₁) = 1.044 means a 1-year increase in age multiplies expected cost by ~4.4%.

7Examine Both Charts. The fitted-vs-observed scatter shows model adequacy; the deviance residuals plot reveals structural problems if a pattern is visible.

Example: A funnel-shaped residual plot suggests heteroscedasticity beyond what the gamma family handles.

8Check Assumptions. Green badges = pass; yellow = inspect; red = consider alternative model.

Example: A red "All Y > 0" badge means at least one observation has Y ≤ 0 — switch to Tweedie or Gaussian.

9Read the Interpretation. Section 4 gives a 5-paragraph plain-language summary plus 5 ready-to-use reporting templates.

Example: Copy the APA 7 paragraph directly into your manuscript with correct symbols and decimal precision.

10Export Your Results. Use "Download Report (TXT)" for plain-text submission packages or "Download PDF" for archival print.

Example: The PDF includes all 8 sections, page footer, and STATS UNLOCK branding for clean printing.

❓ 10. Frequently Asked Questions

Q1. What is gamma regression and when should I use it?

Gamma regression is a generalized linear model (GLM) for continuous, strictly positive, right-skewed outcomes such as healthcare costs, insurance claims, reaction times, or rainfall amounts. Use it when your outcome is always greater than zero and the variance grows with the mean — situations where ordinary linear regression would predict implausible negative values or violate the constant-variance assumption.

Q2. What is the difference between gamma regression and linear regression?

Linear regression assumes a normally distributed outcome with constant variance. Gamma regression assumes a gamma-distributed outcome with variance proportional to the mean squared, which is far more realistic for skewed positive data. Gamma regression also uses a non-identity link function (typically log), guaranteeing positive predictions.

Q3. What does the log link function do in gamma regression?

The log link sets log(μ) = β₀ + β₁X, so predicted means μ = exp(β₀ + β₁X) are always positive. Coefficients become multiplicative: exp(β₁) is the factor by which the mean changes per unit increase in X. The log link is the most common choice in practice.

Q4. How do I interpret gamma regression coefficients?

With the log link, exp(β) is a multiplicative effect on the mean: exp(β) = 1.05 means a 5% increase in expected outcome per unit X. With the inverse link, the effect is on the reciprocal scale and is harder to interpret directly. Always exponentiate log-link coefficients when reporting practical effects.

Q5. What is the dispersion parameter (φ) in gamma regression?

The dispersion parameter φ controls the shape of the gamma distribution. Smaller φ values mean tighter clustering of outcomes around the predicted mean. The estimate φ̂ = Pearson χ² / (n − p) is standard. Values much larger than 1 may indicate model mis-specification, while very small values suggest near-deterministic relationships.

Q6. What assumptions does gamma regression require?

Five core assumptions: (1) outcomes are strictly positive (Y > 0); (2) the conditional distribution of Y is gamma; (3) observations are independent; (4) the link function is correctly specified; and (5) variance is proportional to the mean squared. Violations of (1) or (5) are most damaging.

Q7. How do I check goodness of fit in gamma regression?

Look at residual deviance versus residual degrees of freedom (a ratio close to 1 suggests adequate fit), AIC for model comparison, McFadden's pseudo R², and diagnostic plots of deviance residuals against fitted values. A funnel-shaped residual plot signals lingering heteroscedasticity.

Q8. Can gamma regression handle zeros in the outcome?

No. The gamma distribution has support on (0, ∞), so exact zeros are invalid. For semi-continuous data with exact zeros, fit a Tweedie GLM (with 1 < p < 2) or a two-part hurdle model (logistic for zero-vs-positive, then gamma for positive values).

Q9. How is gamma regression reported in APA 7th edition format?

Report the model fit (residual deviance, df, AIC, dispersion), exponentiated coefficients with 95% confidence intervals, and exact p-values. Use italics for all statistical symbols. Section 4 of this tool gives a fully filled-in APA 7 example you can copy directly.

Q10. What if my gamma regression p-value is not significant?

A non-significant predictor means the data don't provide sufficient evidence of a non-zero effect at your chosen alpha level. Possible causes: small sample size, low effect size, mis-specified link function, omitted variables, or the outcome doesn't actually follow a gamma distribution. Inspect diagnostic plots before concluding "no effect".

📚 11. References

The following peer-reviewed sources support the methodology, formulas, and interpretation guidelines used in this gamma regression calculator — a generalized linear model with multiplicative interpretation and built-in p-value reporting for right-skewed positive outcomes.

Nelder, J. A., & Wedderburn, R. W. M. (1972). Generalized linear models. Journal of the Royal Statistical Society: Series A, 135(3), 370–384. https://doi.org/10.2307/2344614
McCullagh, P., & Nelder, J. A. (1989). Generalized Linear Models (2nd ed.). London: Chapman & Hall/CRC. https://doi.org/10.1201/9780203753736
Hardin, J. W., & Hilbe, J. M. (2018). Generalized Linear Models and Extensions (4th ed.). College Station, TX: Stata Press.
Dobson, A. J., & Barnett, A. G. (2018). An Introduction to Generalized Linear Models (4th ed.). Boca Raton, FL: CRC Press. https://doi.org/10.1201/9781315182780
Manning, W. G., Basu, A., & Mullahy, J. (2005). Generalized modeling approaches to risk adjustment of skewed outcomes data. Journal of Health Economics, 24(3), 465–488. https://doi.org/10.1016/j.jhealeco.2004.09.011
Manning, W. G., & Mullahy, J. (2001). Estimating log models: To transform or not to transform? Journal of Health Economics, 20(4), 461–494. https://doi.org/10.1016/S0167-6296(01)00086-8
de Jong, P., & Heller, G. Z. (2008). Generalized Linear Models for Insurance Data. Cambridge University Press. https://doi.org/10.1017/CBO9780511755408
Faraway, J. J. (2016). Extending the Linear Model with R (2nd ed.). Boca Raton, FL: Chapman & Hall/CRC. https://doi.org/10.1201/9781315382722
Fox, J. (2016). Applied Regression Analysis and Generalized Linear Models (3rd ed.). Thousand Oaks, CA: Sage.
Zuur, A. F., Ieno, E. N., Walker, N. J., Saveliev, A. A., & Smith, G. M. (2009). Mixed Effects Models and Extensions in Ecology with R. Springer. https://doi.org/10.1007/978-0-387-87458-6
Wood, S. N. (2017). Generalized Additive Models: An Introduction with R (2nd ed.). CRC Press. https://doi.org/10.1201/9781315370279
Akaike, H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19(6), 716–723. https://doi.org/10.1109/TAC.1974.1100705
R Core Team (2024). R: A language and environment for statistical computing. R Foundation for Statistical Computing. https://www.R-project.org/
Pinheiro, J. C., & Bates, D. M. (2000). Mixed-Effects Models in S and S-PLUS. Springer. https://doi.org/10.1007/b98882
American Psychological Association. (2020). Publication Manual of the American Psychological Association (7th ed.). https://doi.org/10.1037/0000165-000