Choosing And Interpreting A Link: Worked Problems

A Practical Way to Choose and Interpret a Link

A link choice determines both which mean values a model can represent and how changes in the linear predictor translate to the mean scale. A reliable worked solution therefore does two things: it checks that the inverse link can produce the required kind of mean, then it translates the predictor change using that inverse link. The numerical change in the predictor is not generally the numerical change in the mean.

Write $\mu=E[Y\mid x]$ for the conditional mean at predictor values $x$, and write $\eta=x^\mathsf{T}\beta$ for the linear predictor. A link $g$ relates them by $g(\mu)=\eta$; equivalently, $\mu=g^{-1}(\eta)$ when the inverse is defined. The mean’s required range is a useful first guide: an unrestricted real-valued mean, a strictly positive mean, and a mean strictly between zero and one suggest different standard choices.

Definition: The admissible mean range of a specified link is the range of its inverse function $g^{-1}$. For a finite linear predictor, the model mean must lie in this range. The identity link has inverse range $\mathbb{R}$; the log link has inverse range $(0,\infty)$; and the logit link has inverse range $(0,1)$.

These are useful matches, not a rule that every problem has only one possible link. The response distribution and scientific meaning still matter. For the worked problems here, use the identity link when any real mean is acceptable, the log link when the mean must be positive, and the logit link when the mean is a probability strictly between zero and one.

Check the Range Before Doing the Arithmetic

The inverse link is the function that converts a predictor value back to a mean. For the three links used in this tutorial, the conversions are

\begin{aligned} g(\mu)=\mu &\quad\Longrightarrow\quad g^{-1}(\eta)=\eta,\\ g(\mu)=\log(\mu) &\quad\Longrightarrow\quad g^{-1}(\eta)=\exp(\eta),\\ g(\mu)=\log\left(\frac{\mu}{1-\mu}\right) &\quad\Longrightarrow\quad g^{-1}(\eta)=\frac{\exp(\eta)}{1+\exp(\eta)}. \end{aligned}

The identity inverse can produce any real number. The exponential is strictly positive for every finite real input. The logistic function is strictly between zero and one for every finite real input. These range checks are part of the model interpretation, not an optional afterthought: a calculated value outside the required range signals that the proposed link or predictor is unsuitable for that case.

Identify the mean’s range.
Decide whether the conditional mean may be any real number, must be positive, or must be between zero and one.

Choose a compatible inverse link.
For these standard choices, use the identity, exponential, or logistic inverse, respectively.

Translate the predictor change.
Compute the mean before and after the change, or use the exact multiplicative or odds interpretation when applicable.

Check and report the scale.
Verify that each mean is admissible, then state whether the effect is an additive mean difference, a multiplicative factor, or an odds ratio.

Worked Problems

Worked Example: An Unrestricted Mean With the Identity Link

Suppose a response has a mean that can be any real number, and the model is $\mu(x)=2.4+0.7x$. Find the mean at $x=1$ and $x=3$, and interpret the change.

The response mean is unrestricted, so the identity link is compatible. Here $\eta(x)=2.4+0.7x$ and $g^{-1}(\eta)=\eta$. At $x=1$,

\mu(1)=2.4+0.7(1)=3.1.

At $x=3$,

\mu(3)=2.4+0.7(3)=4.5.

Thus the change in the mean is $4.5-3.1=1.4$. The predictor increased by $0.7(3-1)=1.4$, and under the identity link this is exactly the increase in the mean. Both computed means are real, so both satisfy the stated unrestricted range. This result depends on the identity link and should not be assumed for other links.

Worked Example: A Positive Mean With the Log Link

Suppose the baseline mean is $8$, and a predictor change increases the linear predictor by $0.15$. Use the log link to find the new mean, its absolute increase, and its percentage increase.

For the log link, $\eta=\log(\mu)$, so increasing $\eta$ by $0.15$ gives

\mu_1=\exp\{\log(8)+0.15\} =8\exp(0.15).

Since $\exp(0.15)\approx1.16183424$,

\mu_1\approx 8(1.16183424)=9.29467392\approx9.295.

The absolute increase is $9.29467392-8=1.29467392$, or approximately $1.295$. The percentage increase is

100\left(\frac{9.29467392-8}{8}\right)\% =100\{\exp(0.15)-1\}\% \approx16.1834\%.

The mean remains positive, as required by the log link. Notice that the predictor change of $0.15$ is neither an increase of $0.15$ in the mean nor an increase of $15\%$. It multiplies the mean by $\exp(0.15)$, giving an increase of about $16.18\%$.

Worked Example: A Probability Mean With the Logit Link

A binary-outcome model has baseline probability $p_0=0.20$. A specified predictor change increases the linear predictor by $0.70$. Find the new probability and describe the effect in terms of odds and probability.

For the logit link, $\eta=\log\{p/(1-p)\}$. The baseline odds are $0.20/(1-0.20)=0.25$. Increasing the logit by $0.70$ multiplies the odds by $\exp(0.70)\approx2.0137527$. The new odds are therefore

0.25\exp(0.70)\approx0.5034382.

Convert these odds back to a probability by dividing by one plus the odds:

p_1 =\frac{0.5034382}{1+0.5034382} \approx0.334858 \approx0.3349.

The probability increases by approximately $0.334858-0.20=0.134858$, or 13.49 percentage points. The odds, in contrast, are multiplied by about $2.014$. The new probability is admissible because it is strictly between zero and one. It would be incorrect to describe the predictor increase of $0.70$ as a probability increase of $0.70$.

Why the Transformations Work

The calculations above follow from two general facts. First, a change on the predictor scale must be passed through the inverse link to obtain a mean-scale change. Second, particular inverse links give especially convenient interpretations: the log link produces a mean ratio, while the logit link produces an odds ratio. The derivations make clear both what those interpretations say and what they do not say.

Result: Let $g$ be a one-to-one link, let $\eta_0=g(\mu_0)$, and let $\eta_1=\eta_0+\Delta$, where both predictor values are in the domain of $g^{-1}$. Then $$ \mu_1-\mu_0 =g^{-1}(\eta_0+\Delta)-g^{-1}(\eta_0). $$ If $g$ is differentiable near $\mu_0$ and $g'(\mu_0)\ne0$, then for a small predictor change $\Delta$, $$ \mu_1-\mu_0 \approx \frac{\Delta}{g'(\mu_0)}. $$

Derivation. By the definition of the inverse link, $\mu_0=g^{-1}(\eta_0)$ and $\mu_1=g^{-1}(\eta_0+\Delta)$. Subtracting gives the exact equality. For the local approximation, the inverse function theorem gives $(g^{-1})'(\eta_0)=1/g'(\mu_0)$ when $g'(\mu_0)\ne0$. A first-order Taylor expansion at $\eta_0$ then yields

g^{-1}(\eta_0+\Delta) =g^{-1}(\eta_0)+(g^{-1})'(\eta_0)\Delta+o(\Delta) =\mu_0+\frac{\Delta}{g'(\mu_0)}+o(\Delta).

Thus the local mean change depends on the starting mean through $g'(\mu_0)$, except in the identity-link case. The approximation is local; for a substantial predictor change, calculate the exact inverse-link difference instead.

For a log link, let $\eta_1=\eta_0+\Delta$ and $\eta_j=\log(\mu_j)$. Then

\frac{\mu_1}{\mu_0} =\frac{\exp(\eta_0+\Delta)}{\exp(\eta_0)} =\exp(\Delta).

So the exact mean ratio is $\exp(\Delta)$, and the absolute difference is $\mu_0\{\exp(\Delta)-1\}$. For a logit link, define the odds $o_j=\mu_j/(1-\mu_j)$. Since $\eta_j=\log(o_j)$,

\frac{o_1}{o_0} =\frac{\exp(\eta_0+\Delta)}{\exp(\eta_0)} =\exp(\Delta).

This proves that $\exp(\Delta)$ is an odds ratio under the logit link. It is not, in general, a probability ratio or a probability difference. In both derivations the predictor increment is assumed to be a fixed finite number and the means are in the inverse link’s domain.

Admissibility and Interpretation Checks

For finite $\eta$, the identity inverse yields a real number; the exponential inverse yields a strictly positive number; and the logistic inverse yields a number strictly between zero and one. These statements follow directly from the respective inverse functions. In particular, a finite logit predictor cannot yield exactly zero or exactly one. Such boundary probabilities require limiting, not finite, predictor values.

A mathematically defined mean is not automatically scientifically sensible. For example, the identity model in the first example would give $\mu(-5)=2.4-3.5=-1.1$. That is admissible for an unrestricted real-valued mean, but not for a nonnegative response whose mean must be nonnegative. The link must be judged against the actual mean restrictions in the application.

Worked Example: Rejecting an Inadmissible Identity-Link Prediction

Suppose a count response is assigned the identity-link model $\mu(x)=1.2-0.5x$. Check the predicted mean at $x=3$, and decide whether the identity link is appropriate for that prediction.

Substitution gives $\mu(3)=1.2-0.5(3)=1.2-1.5=-0.3$. A count has a nonnegative mean, so $-0.3$ is inadmissible. The arithmetic is correct, but this model prediction violates the response’s mean restriction. For this predictor value, the identity-link specification is therefore unsuitable; a link with a positive inverse range, such as the log link, would respect positivity for every finite predictor value.

Common Mistakes

Treating a predictor change as a mean change. This is correct for an identity link, but not generally. Apply the inverse link to the old and new predictors.
Calling a log-link coefficient a percentage change equal to the coefficient. A predictor increment $\Delta$ gives a mean ratio $\exp(\Delta)$; the percentage change is $100\{\exp(\Delta)-1\}\%$.
Interpreting a logit increment as a probability increment. It multiplies the odds by $\exp(\Delta)$. Convert the resulting odds to a probability if the question asks about probabilities.
Checking only that a calculation can be performed. Also check that the predicted mean lies in the range required by the response and link.
Rounding too early. Keep sufficient precision in the inverse-link calculation, then round the final mean and difference. For example, $8\exp(0.15)\approx9.295$, so the increase from 8 is approximately $1.295$, not $1.294$.

Key takeaway: Select a link whose inverse range matches the mean restrictions, then interpret changes through that inverse link. Identity gives additive mean changes, the log link gives multiplicative mean changes, and the logit link gives multiplicative odds changes.

Check Your Understanding

For each question, state the scale of the quantity you report and check that the mean is admissible.

A model uses the identity link and predicts means $2.5$ and $4.0$ at two covariate settings. What is the mean difference?
A positive-mean model has baseline mean $5$ and predictor increment $0.2$ under the log link. Write the exact new mean and mean ratio.
A binary-outcome model has baseline odds $1/3$ and a logit increment of $-0.4$. What is the odds ratio, and in which direction do the odds change?
Why can a finite predictor under the logit link not give a predicted probability of exactly one?
An identity-link model for a nonnegative response predicts a mean of $-0.1$ at one covariate setting. Is the prediction admissible, and what does the result indicate?