How To Calculate Maximum Rescaled R Squared For Each Variable

Estimate Cox-Snell and maximum rescaled R squared values for multiple predictors from logistic model fit statistics. Enter your sample size, null model fit, and each variable’s one-predictor model fit to compare explanatory contribution side by side.

Calculator Inputs

What the calculator computes

Cox-Snell R squared 1 – exp((2/n)(LL0 – LL1))

Maximum possible Cox-Snell 1 – exp((2/n)LL0)

Maximum rescaled R squared Cox-Snell / Max Possible

Use case Compare one-variable logistic models

Interpretation tip: the maximum rescaled R squared is often referred to as Nagelkerke R squared. It rescales Cox-Snell so the upper bound can approach 1.00, making comparisons across candidate predictors easier.

How to Calculate Maximum Rescaled R Squared for Each Variable

Maximum rescaled R squared is a pseudo-R squared measure used most often in logistic regression and related maximum likelihood models. If you are used to ordinary least squares regression, you already know the familiar R squared measures the proportion of variance explained by a linear model. In logistic regression, that exact interpretation does not carry over because the dependent variable is categorical and the model is estimated by likelihood rather than minimizing squared residuals. That is why analysts rely on pseudo-R squared metrics such as Cox-Snell, McFadden, and Nagelkerke, with maximum rescaled R squared being one of the most practical when you want a value that is easier to compare across predictors.

When people ask how to calculate maximum rescaled R squared for each variable, they usually mean one of two things. First, they may want to fit a separate one-predictor logistic model for each candidate variable and compare the resulting values. Second, they may want to evaluate the incremental contribution of each predictor inside a larger model. The calculator above focuses on the first and most transparent scenario: you provide the null model fit statistic, then a model fit statistic for each variable’s one-variable logistic model, and the tool computes Cox-Snell and the maximum rescaled version for each predictor.

Why maximum rescaled R squared matters

Cox-Snell R squared has a useful likelihood-based foundation, but it has a limitation: its maximum possible value is often less than 1.00. That makes interpretation awkward, especially for applied users comparing multiple candidate variables. Nagelkerke solved this by dividing Cox-Snell R squared by its theoretical maximum. The result is often labeled maximum rescaled R squared. This does not magically make it identical to ordinary R squared, but it does make the metric easier to read. A variable with a larger maximum rescaled R squared generally gives more improvement over the null model than a variable with a smaller value, all else equal.

Core idea: the stronger the variable improves model likelihood relative to the intercept-only model, the larger the pseudo-R squared. The maximum rescaled version simply adjusts Cox-Snell to a 0 to 1 style scale.

The formula step by step

Suppose you have:

• n = sample size
• LL0 = log-likelihood for the null model
• LL1 = log-likelihood for the model containing one predictor

The Cox-Snell R squared is:

Cox-Snell R squared = 1 – exp((2/n)(LL0 – LL1))

The maximum possible Cox-Snell value, given the null model, is:

Max possible = 1 – exp((2/n)LL0)

Then maximum rescaled R squared is:

Maximum rescaled R squared = Cox-Snell R squared / Max possible

If your software outputs -2 Log Likelihood instead of log-likelihood, convert it first. Because -2LL = -2 × LL, then:

• LL = -0.5 × (-2LL value)
• Example: if -2LL = 346.42, then LL = -173.21

Worked example

Assume your null model log-likelihood is -173.21 and your sample size is 250. You fit separate logistic models with one predictor at a time and obtain the following model log-likelihoods:

Variable	Model Log-Likelihood	Likelihood Improvement vs Null	Practical Reading
Age	-165.40	7.81	Modest improvement over null model
Income	-160.10	13.11	Stronger standalone predictor
Education	-168.90	4.31	Small but nonzero gain
Credit Score	-150.25	22.96	Largest contribution in this set
Tenure	-158.70	14.51	Competitive explanatory power

Because the null log-likelihood is negative, the maximum possible Cox-Snell value is less than 1 before rescaling. That is exactly why the maximum rescaled version is helpful. In many software packages, this value is what users informally call Nagelkerke R squared.

How to calculate it manually for one variable

1. Record the sample size n.
2. Fit the intercept-only model and record its log-likelihood LL0.
3. Fit a model with one predictor and record the predictor model log-likelihood LL1.
4. Compute Cox-Snell R squared using 1 – exp((2/n)(LL0 – LL1)).
5. Compute the maximum possible Cox-Snell using 1 – exp((2/n)LL0).
6. Divide Cox-Snell by the maximum possible value.
7. Repeat for each candidate variable and compare the results.

Comparison of major pseudo-R squared measures

Different pseudo-R squared measures emphasize different properties. Maximum rescaled R squared is popular because it gives many applied readers a more intuitive scale. Still, you should not treat it as interchangeable with OLS R squared.

Measure	Typical Formula Basis	Nominal Range	Common Use	Interpretation Note
McFadden R squared	1 – (LL1 / LL0)	0 to below 1	Model comparison in discrete choice and logistic models	Values around 0.20 to 0.40 are often considered strong in practice
Cox-Snell R squared	Likelihood ratio transformation	0 to below 1, but upper bound often less than 1	Likelihood-based fit summary	Cannot always reach 1.00
Maximum rescaled or Nagelkerke R squared	Cox-Snell divided by its maximum	0 to 1	Readable pseudo-R squared for applied reports	Easiest to compare across single-predictor models

Real benchmark statistics commonly cited in applied work

Analysts often want a frame of reference for model strength. While there is no universal threshold that defines a good pseudo-R squared, logistic model literature frequently treats moderate values as meaningful because binary outcomes are inherently noisy. The table below summarizes commonly used practical guideposts drawn from widely cited applied modeling conventions, especially for McFadden and related fit summaries.

Statistic	Illustrative Range	Context	What it often suggests
McFadden R squared	0.20 to 0.40	Discrete choice and logistic models	Often described as excellent fit in many applied settings
Maximum rescaled R squared	0.10 to 0.30	Many health, education, and social science logistic models	Often practically useful, especially with rare or difficult outcomes
Maximum rescaled R squared	Above 0.40	Stronger classification structure or cleaner signal	Usually indicates a relatively powerful predictor set

These ranges are not hard rules. A low pseudo-R squared can still accompany a highly valuable model if calibration, discrimination, and inference are strong. Likewise, a high value does not guarantee robustness, transportability, or absence of overfitting.

How to interpret results for each variable

Suppose your calculator output shows Credit Score with a maximum rescaled R squared of 0.24 and Education with 0.05. The natural interpretation is that Credit Score produces a much larger improvement over the null model when used alone. This does not necessarily mean Education is unimportant in a multivariable model. Predictors can become more or less useful once considered jointly because of confounding, mediation, suppression, and collinearity.

• A larger value means better standalone fit improvement.
• Values near zero imply little gain over the intercept-only model.
• Comparisons are most meaningful when the sample size and outcome definition are the same across variables.
• One-variable rankings are screening tools, not final causal conclusions.

Common mistakes to avoid

1. Mixing up log-likelihood and -2 log likelihood. This is the most common input error. Always convert properly if needed.
2. Using different samples for different variables. Missing data can shrink or change the analytic sample, which breaks fair comparisons.
3. Comparing across different outcomes. Pseudo-R squared values depend on the underlying outcome and base rates.
4. Treating pseudo-R squared like OLS variance explained. It is a fit index, not a literal variance decomposition.
5. Ignoring model diagnostics. A respectable maximum rescaled R squared does not replace calibration, discrimination, residual checks, or external validation.

Best practice for reporting

In a professional report, present the null model fit statistic, the sample size, and the pseudo-R squared formula family you used. Because software packages can report multiple pseudo-R squared measures, specify that your value is the maximum rescaled or Nagelkerke version. If you compare variables, show the variables in descending order by value and note that the models are one-predictor logistic models fit on the same analytic sample.

A strong reporting pattern is:

• Outcome definition
• Sample size used for all comparisons
• Null model fit statistic
• Predictor model fit statistic for each variable
• Cox-Snell and maximum rescaled R squared
• Any caveats about missingness, coding, or nonlinearity

Authoritative references and learning resources

For deeper technical grounding, review these high-quality resources:

• Penn State Eberly College of Science statistical course materials
• UCLA Statistical Methods and Data Analytics resources
• NIST Engineering Statistics Handbook

Final takeaway

If you need to calculate maximum rescaled R squared for each variable, the cleanest workflow is simple: use the same sample, fit the same outcome definition, collect the intercept-only log-likelihood, fit one-variable models for each predictor, and apply the Nagelkerke rescaling formula. The result gives you a practical, comparable ranking of which variables contribute the most standalone improvement in model fit. Used carefully, this is an excellent screening tool before you build a full multivariable logistic regression model.