mathematics

Pseudo-R-squared

Pseudo-R-squared is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Pseudo-R-squared rather than just read about it. In short: In statistics, pseudo-R-squared values are used when the outcome variable is nominal or ordinal such that the coefficient of determination R2 cannot be applied as a measure for goodness of fit and when a likelihood function is used to fit a model. In linear regression, the squared multiple correlation, R2 is used to assess goodness of fit as it represents the proportion of variance in the criterion that is explained…

Key takeaways

Pseudo-R-squared belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Pseudo-R-squared to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Pseudo-R-squared from memory before moving on to harder problems.

Reference excerpt

In statistics, pseudo-R-squared values are used when the outcome variable is nominal or ordinal such that the coefficient of determination R2 cannot be applied as a measure for goodness of fit and when a likelihood function is used to fit a model. In linear regression, the squared multiple correlation, R2 is used to assess goodness of fit as it represents the proportion of variance in the criterion that is explained by the predictors. In logistic regression analysis, there is no agreed upon analogous measure, but there are several competing measures each with limitations. Some commonly used indices are examined in this article:

Likelihood ratio R2L Unadjusted and adjusted geometric mean squared improvement R2M and R2N Tjur R2T

R2L by McFadden The pseudo R2 credited to McFadden (sometimes called likelihood ratio index or log-likelihood ratio R2) is defined as

R L 2 = 1 − ln ⁡ ( L M ) ln ⁡ ( L 0 ) , {\displaystyle R_{\text{L}}^{2}=1-{\frac {\ln(L_{M})}{\ln(L_{0})}},}

or expressed using deviance rather than likelihood

R L 2 = D 0 − D M D 0 {\displaystyle R_{\text{L}}^{2}={\frac {D_{\text{0}}-D_{\text{M}}}{D_{\text{0}}}}}

and is preferred over R2M by Paul D. Allison. The two expressions R2L and R2M are then related respectively by,

R M 2 = 1 − ( 1 L 0 ) 2 ( R L 2 ) n R L 2 = − n 2 ⋅ ln ⁡ ( 1 − R M 2 ) ln ⁡ L 0 {\displaystyle {\begin{matrix}R_{\text{M}}^{2}=1-\left({\dfrac {1}{L_{0}}}\right)^{\frac {2(R_{\text{L}}^{2})}{n}}\\[1.5em]R_{\text{L}}^{2}=-{\dfrac {n}{2}}\cdot {\dfrac {\ln(1-R_{\text{M}}^{2})}{\ln L_{0}}}\end{matrix}}}

This is the most analogous index to the squared multiple correlations in linear regression. It represents the proportional reduction in the deviance wherein the deviance is treated as a measure of variation analogous but not identical to the variance in linear regression analysis. One limitation of the likelihood ratio R2 is that it is not monotonically related to the odds ratio, meaning that it does not necessarily increase as the odds ratio increases and does not necessarily decrease as the odds ratio decreases.

Adjusted The adjusted version includes the number of predictors K as a penalisation term

R L 2 = 1 − ln ⁡ ( L M ) − K ln ⁡ ( L 0 ) , {\displaystyle R_{\text{L}}^{2}=1-{\frac {\ln(L_{M})-K}{\ln(L_{0})}},}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Pseudo-R-squared

Start with the simplest possible case. Write down what Pseudo-R-squared claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Pseudo-R-squared before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Pseudo-R-squared ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Pseudo-R-squared

In research: Pseudo-R-squared appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Pseudo-R-squared in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Pseudo-R-squared is common in secondary-school and first-year university syllabi. It links to neighbouring topics Regression diagnostics, Statistical ratios, so understanding it makes those chapters shorter.
In everyday life: Look for Pseudo-R-squared outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Pseudo-R-squared in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Pseudo-R-squared means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Pseudo-R-squared out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Pseudo-R-squared in simple terms?

Why does Pseudo-R-squared matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Pseudo-R-squared?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Pseudo-R-squared.