Comparing Cumulative Logit and Probit Scales

Why Compare the Scales?

Cumulative logit and cumulative probit models describe ordinal responses in much the same structural way: each cumulative probability is determined by a threshold and a linear predictor. Their links differ, however. The logistic and standard-normal cumulative distribution functions have different shapes and scales, so a coefficient of a given numerical size does not represent the same change under both models. Comparing coefficients without accounting for that difference can be misleading.

The distinction matters even when the same predictors and response categories are used. If thresholds and predictors are held numerically fixed, the two links can produce noticeably different category probabilities. If instead the thresholds and slopes are rescaled to put the links on roughly comparable scales, the fitted probabilities may be close—but the match is approximate, not an identity between logistic and normal distributions.

Two Cumulative Links

Let $Y$ have ordered categories $1,\ldots,K$, with ordered thresholds $\theta_1<\cdots<\theta_{K-1}$, and let $x$ denote a predictor vector. We use the convention that the cumulative probability of category $j$ or below is the link CDF evaluated at the threshold minus the linear predictor.

Definition: The cumulative logit model and cumulative probit model are, respectively, $$ P(Y\leq j\mid x)=\Lambda(\theta_j-x^\mathsf{T}\beta_{\mathrm{logit}}), \qquad P(Y\leq j\mid x)=\Phi(\theta_j-x^\mathsf{T}\beta_{\mathrm{probit}}), $$ for $j=1,\ldots,K-1$, where $\Lambda(t)=1/(1+e^{-t})$ is the standard logistic CDF and $\Phi(t)$ is the standard normal CDF. The standard normal density is $\phi(t)=(2\pi)^{-1/2}e^{-t^2/2}$.

The sign convention has a useful interpretation: a positive slope makes the probability of being at or below a given category decrease as that predictor increases. The category probabilities follow by differencing cumulative probabilities. In particular, for three categories they are $F_1$, $F_2-F_1$, and $1-F_2$, where $F_j=P(Y\leq j\mid x)$. Thus, the choice of link affects all category probabilities, not just the cumulative probabilities considered separately.

The thresholds and slopes must be considered together. Multiplying every threshold and slope by the same positive constant changes the numerical scale of the linear arguments. For a single link family, an associated rescaling of its scale gives an exactly equivalent model. Switching from logistic to normal CDFs is different: no constant rescaling makes the two CDFs identical for every argument.

Worked Example: Same Thresholds and Predictor, Different Fitted Probabilities

Suppose there are three categories and, at a particular predictor value, the two cumulative arguments are $-0.9$ and $0.6$. Use those same arguments for a logit model and a probit model, then calculate the fitted category probabilities under each link.

For the logit model, $F_1=\Lambda(-0.9)=1/(1+e^{0.9})\approx0.2890505$, and $F_2=\Lambda(0.6)=1/(1+e^{-0.6})\approx0.6456563$. The category probabilities are therefore $F_1\approx0.2891$, $F_2-F_1\approx0.6456563-0.2890505=0.3566058$, and $1-F_2\approx0.3543437$.

For the probit model, $F_1=\Phi(-0.9)\approx0.1840601$ and $F_2=\Phi(0.6)\approx0.7257469$. The category probabilities are $0.1840601$, $0.7257469-0.1840601=0.5416868$, and $1-0.7257469=0.2742531$.

Rounded to four decimal places, the logit probabilities are $(0.2891,0.3566,0.3543)$, while the probit probabilities are $(0.1841,0.5417,0.2743)$. Each vector sums to one before rounding. These differences arise because the same numerical arguments are being passed to two different CDFs; the thresholds and predictor contribution have not been calibrated to a common scale.

Why Coefficients Use Different Scales

A link coefficient is measured in units of that link's argument. To see the consequence, imagine that a probit model has cumulative argument $z=\theta_j-x^\mathsf{T}\beta_{\mathrm{probit}}$. Multiplying $z$ by a constant changes the rate at which the argument moves as a predictor changes. A logit slope and a probit slope that yield similar changes in cumulative probabilities will therefore generally have different numerical values.

One transparent way to compare the links near their centers is to match their derivatives at zero. The logistic derivative is $\Lambda'(t)=\Lambda(t)\{1-\Lambda(t)\}$, so $\Lambda'(0)=1/4$. The standard normal derivative is $\Phi'(t)=\phi(t)$, so $\Phi'(0)=1/\sqrt{2\pi}$.

Result: To match the central slopes of $\Lambda(kz)$ and $\Phi(z)$, choose $$ \frac{k}{4}=\frac{1}{\sqrt{2\pi}}, \qquad k=\frac{4}{\sqrt{2\pi}}\approx1.5958. $$ Thus, a common central-slope approximation is $\beta_{\mathrm{logit}}\approx1.6\beta_{\mathrm{probit}}$, with the thresholds rescaled by the same factor. This is a local calibration around argument zero, not an exact conversion for all predictor values.

The calculation follows by differentiating at zero. By the chain rule, the derivative of $\Lambda(kz)$ with respect to $z$ at zero is $k\Lambda'(0)=k/4$. The derivative of $\Phi(z)$ there is $\phi(0)=1/\sqrt{2\pi}$. Equating these derivatives gives the stated $k$. Since both CDFs equal $1/2$ at zero, the calibration matches their value and their first derivative at that point. It does not make their tails or their values elsewhere identical. $\square$

A different common calibration matches the variances of the underlying standard distributions. A standard logistic random variable has variance $\pi^2/3$, whereas a standard normal random variable has variance $1$. Matching those variances gives the factor $\sqrt{\pi^2/3}=\pi/\sqrt{3}\approx1.8138$. This is not the same as the central-slope factor $1.5958$: the two procedures answer different questions. Neither factor is a universal conversion that makes coefficients interchangeable.

Worked Example: Rescaling a Probit Model for a Local Comparison

Consider a three-category probit model with thresholds $\theta_1=-0.5$, $\theta_2=0.5$, and no predictor contribution at the predictor value of interest. Its cumulative arguments are $-0.5$ and $0.5$. Apply the central-slope factor $k=1.5958$ to both thresholds to form a locally calibrated logit comparison.

The rescaled logit thresholds are approximately $k(-0.5)=-0.7979$ and $k(0.5)=0.7979$. The probit cumulative probabilities are $\Phi(-0.5)\approx0.3085$ and $\Phi(0.5)\approx0.6915$, giving category probabilities approximately $(0.3085,0.3829,0.3085)$.

For the rescaled logit, $\Lambda(-0.7979)\approx0.3105$ and $\Lambda(0.7979)\approx0.6895$. The category probabilities are approximately $(0.3105,0.3790,0.3105)$. These are close to the probit probabilities near the center, as the derivative matching suggests, but they are not exactly equal.

For a model with predictors, the same comparison rescales every threshold and every slope: $\theta_{j,\mathrm{logit}}\approx k\theta_{j,\mathrm{probit}}$ and $\beta_{\mathrm{logit}}\approx k\beta_{\mathrm{probit}}$. Rescaling the thresholds but not the slopes would not rescale the full cumulative argument.

Coefficients and Interpretation

The numerical coefficient conversion is only an approximate comparison of link scales. The model-specific interpretation of a coefficient remains tied to its link. In a cumulative logit model under the convention used here, a positive coefficient $\beta_r$ multiplies the odds of being above any fixed cutpoint by $e^{\beta_r}$ for a one-unit increase in predictor $x_r$, with other predictors held fixed. A probit coefficient instead describes a shift in the standard-normal cumulative argument; it is not itself a log odds ratio.

Worked Example: Interpreting a Logit Slope and Comparing Its Size

Suppose a cumulative logit model has slope $\beta_{\mathrm{logit}}=0.8$ for a predictor measured in one-unit increments. For a fixed cutpoint, the odds of being above that cutpoint are $\{1-P(Y\leq j\mid x)\}/P(Y\leq j\mid x)$. From the model, these odds equal $\exp(x^\mathsf{T}\beta_{\mathrm{logit}}-\theta_j)$. Increasing the predictor by one unit multiplies them by $e^{0.8}\approx2.2255$.

For a rough central-slope comparison, the corresponding probit coefficient would be $0.8/1.5958\approx0.5013$. This does not mean the probit coefficient has an odds-ratio interpretation: it is a change on the probit argument scale. Nor does the approximate conversion imply identical fitted probabilities at every value of the predictors.

Exact Rescaling Within a Link Family

There is an important distinction between changing a scale and changing a link. For any CDF $F$ and any positive constant $a$, define a rescaled CDF $G(u)=F(u/a)$. If a cumulative model using $F$ has thresholds $\theta_j$ and slopes $\beta$, then the model using $G$, thresholds $a\theta_j$, and slopes $a\beta$ has exactly the same cumulative probabilities. Indeed,

G(a\theta_j-x^\mathsf{T}a\beta) =F\!\left(\frac{a\theta_j-x^\mathsf{T}a\beta}{a}\right) =F(\theta_j-x^\mathsf{T}\beta).

The equality uses $a>0$, so the rescaling does not reverse the ordering of the thresholds. This is an exact identity for a CDF and its rescaled version. It does not establish an exact identity between the logistic and standard-normal CDFs, which have different shapes.

Common Mistake: Treating the Conversion as Exact

Wrong reasoning: “Logit coefficients are about 1.6 times probit coefficients, so multiplying every probit coefficient by 1.6 must give the identical logit model.”
Correction: The factor $1.6$ comes from matching derivatives at zero. It is a useful local approximation, and thresholds must be rescaled along with slopes for a scale comparison. The two links still differ away from the center, so their fitted probabilities need not agree.
Another pitfall: Comparing coefficients while ignoring the thresholds. The cumulative probabilities depend on $\theta_j-x^\mathsf{T}\beta$; a scale comparison must account for both terms.

In practice, report which link was used and interpret its coefficients on that link's scale. When comparing fitted probabilities, calculate them using the fitted thresholds, predictor values, and chosen link. A scale conversion can help describe approximate coefficient magnitudes across models, but it should not replace probability calculations or the link-specific interpretation.

Check Your Understanding

Use the cumulative-link convention in this tutorial.

For a three-category response, write the category probabilities in terms of the two cumulative probabilities.
What central-slope factor relates $\Lambda(kz)$ to $\Phi(z)$, and what feature of the links does this factor match?
Why is the variance-matching factor $\pi/\sqrt{3}$ different from the central-slope factor?
If all logit arguments in a model are multiplied by a positive constant, which parameters must be rescaled to preserve the cumulative probabilities?
A positive cumulative-logit slope is $0.4$. What is the multiplier for the odds of being above a fixed cutpoint for a one-unit increase in that predictor?