Continuous Predictor Coefficients on the Log-Odds Scale

What a Continuous-Predictor Coefficient Measures

In logistic regression, a continuous predictor can represent a quantity such as time, age, or a measured score. Its coefficient describes how the model's conditional log-odds change as that predictor increases, with the other predictors held fixed. The coefficient is not, by itself, a change in probability.

Let $Y$ be a binary outcome, and let $p(x,\mathbf{z})=P(Y=1\mid x,\mathbf{z})$, where $x$ is a continuous predictor and $\mathbf{z}$ denotes the other predictors. A logistic regression model can be written as

\log\left(\frac{p(x,\mathbf{z})}{1-p(x,\mathbf{z})}\right) = \beta_0+\beta_1x+\sum_{j=1}^{k}\gamma_jz_j.

The expression on the left is the logit, or log-odds, of the conditional probability. The coefficient $\beta_1$ is the change in this quantity for a one-unit increase in $x$, when the other predictors do not change.

Definition: In a logistic regression model with a linear predictor, the coefficient $\beta_1$ of a continuous predictor $x$ is the conditional change in log-odds for a one-unit increase in $x$, holding all other predictors fixed. For a change of $\Delta x$ units, the corresponding change in log-odds is $\beta_1\Delta x$.

The units matter. If $x$ is recorded in days, $\beta_1$ is a change per day. If the same quantity is recorded in weeks, its coefficient is on a per-week scale. Always identify the variable's actual model units before interpreting the coefficient.

Why the Change Is the Coefficient Times the Unit Change

The result follows directly by comparing the linear predictors at two values of $x$. Write the log-odds at $x$ as $\eta(x,\mathbf{z})$. Increase $x$ by $\Delta x$, leaving every $z_j$ unchanged. Subtract the original linear predictor from the new one:

\begin{aligned} \eta(x+\Delta x,\mathbf{z})-\eta(x,\mathbf{z}) &= \left[\beta_0+\beta_1(x+\Delta x)+\sum_{j=1}^{k}\gamma_jz_j\right] -\left[\beta_0+\beta_1x+\sum_{j=1}^{k}\gamma_jz_j\right]\\ &=\beta_1\Delta x. \end{aligned}

The intercept and all terms involving the unchanged predictors cancel. This is why the interpretation is conditional on the other predictors, and why a change spanning several units is $\beta_1\Delta x$, not merely $\beta_1$. This calculation concerns log-odds; it does not say that probability changes by the same amount.

Worked Example: A One-Day Increase With Another Predictor Fixed

Suppose a hypothetical logistic model predicts whether a submitted claim requires additional review. Let $Y=1$ indicate additional review, let $x$ be the number of days between submission and an information request, and let $z=1$ indicate that the claim includes a particular document. The fitted model is

\log\left(\frac{p(x,z)}{1-p(x,z)}\right)=-2.4+0.08x+0.5z.

Interpret the coefficient of $x$, then compare the fitted probabilities for $x=10$ and $x=15$ when $z=1$.

Method. First read the predictor's units: $x$ is measured in days. The coefficient $0.08$ therefore means an increase of one day corresponds to an increase of $0.08$ in conditional log-odds, holding $z$ fixed. For a five-day increase, the log-odds change is

$$ 0.08(15-10)=0.08(5)=0.40. $$

At $x=10$ and $z=1$, the linear predictor is

\eta=-2.4+0.08(10)+0.5(1)=-2.4+0.8+0.5=-1.1.

The corresponding probability is $1/(1+e^{-\eta})$, so

p(10,1)=\frac{1}{1+e^{1.1}}\approx 0.2497.

At $x=15$, with $z$ still equal to 1,

\eta=-2.4+0.08(15)+0.5(1)=-2.4+1.2+0.5=-0.7, $$ $$ p(15,1)=\frac{1}{1+e^{0.7}}\approx 0.3318.

The fitted probability difference is approximately $0.3318-0.2497=0.0821$. The coefficient's interpretation remains a $0.08$ log-odds increase per day; the probability difference for this five-day comparison is about $0.0821$, not $0.08$ per day. Check: the log-odds at the second value exceed those at the first by $-0.7-(-1.1)=0.40$, as calculated from $0.08(5)$. This avoids treating the coefficient as a fixed probability increment.

Using the Predictor's Recorded Units

A predictor may be recorded in scaled units for convenience. For example, if $w$ is income measured in units of $10,000, then increasing $w$ by 1 means increasing income by $10,000, not by $1. A change in the original quantity must first be expressed in the units used by the model. The coefficient-times-change rule then applies without modification.

Worked Example: A Predictor Recorded in $10,000 Units

A hypothetical logistic model for a binary benefit outcome is

\log\left(\frac{p(w,a)}{1-p(w,a)}\right)=-1+0.30w+0.01(a-40),

where $w$ is annual income measured in $10,000 units and $a$ is age in years. Compare the log-odds and fitted probabilities for two people of age 40 whose incomes are $30,000 and $55,000.

Method. Convert each income to the model's units. For $30,000, $w=30{,}000/10{,}000=3$. For $55,000, $w=55{,}000/10{,}000=5.5$. The predictor change is therefore $5.5-3=2.5$ model units, with age fixed at 40. The log-odds change is

$$ 0.30(2.5)=0.75. $$

At $w=3$ and $a=40$, the age term is $0.01(40-40)=0$. Thus

\eta=-1+0.30(3)+0=-1+0.90=-0.10, $$ $$ p=\frac{1}{1+e^{0.10}}\approx 0.4750.

At $w=5.5$ and $a=40$,

\eta=-1+0.30(5.5)+0=-1+1.65=0.65, $$ $$ p=\frac{1}{1+e^{-0.65}}\approx 0.6570.

The log-odds difference is $0.65-(-0.10)=0.75$, which matches the calculation from the coefficient and the 2.5-unit change. The fitted probability difference is approximately $0.6570-0.4750=0.1820$. Check: a $25,000 increase equals 2.5 units of $10,000, so using $0.30(25{,}000)$ would use the wrong units. This example avoids that scaling error.

Negative Coefficients and Multi-Unit Changes

A negative coefficient means that, holding other predictors fixed, increasing the predictor decreases the model's log-odds. For a multi-unit increase, multiply the signed coefficient by the signed change in the predictor. A decrease in $x$ has $\Delta x<0$, so it reverses the direction of the change relative to an increase.

Worked Example: A Negative Coefficient Over Several Units

In a hypothetical logistic model for whether a device triggers an alert, let $x$ be operating temperature in degrees Celsius and let $z$ be a sensor score. Suppose

\log\left(\frac{p(x,z)}{1-p(x,z)}\right)=1.2-0.05x+0.4z.

Compare the model for $x=8$ and $x=14$ when $z=2$. Interpret the coefficient and calculate both fitted probabilities.

Method. The coefficient of temperature is $-0.05$ per degree Celsius. With $z$ held at 2, increasing temperature by $14-8=6$ degrees changes the log-odds by

$$ (-0.05)(6)=-0.30. $$

At $x=8,z=2$,

\eta=1.2-0.05(8)+0.4(2)=1.2-0.4+0.8=1.6, $$ $$ p=\frac{1}{1+e^{-1.6}}\approx 0.8320.

At $x=14,z=2$,

\eta=1.2-0.05(14)+0.4(2)=1.2-0.7+0.8=1.3, $$ $$ p=\frac{1}{1+e^{-1.3}}\approx 0.7858.

The log-odds difference is $1.3-1.6=-0.30$, as expected. The fitted probability difference is approximately $0.7858-0.8320=-0.0462$. Check: the negative coefficient and positive six-degree change give a negative log-odds change. The coefficient does not mean a decrease of $0.05$ probability units per degree; the probability difference here depends on the starting predictor values. This avoids confusing a coefficient with a probability decrement.

Common Interpretation Traps

Reporting a probability change. A coefficient is on the log-odds scale. Do not describe $0.08$, for example, as an eight-percentage-point increase in probability.
Ignoring the predictor's units. If a model uses units of $10,000, a one-unit increase represents $10,000. Translate real-world changes into model units first.
Forgetting to hold other predictors fixed. The coefficient gives a conditional comparison. If another predictor changes too, its contribution must also be included when comparing the linear predictors.
Dropping the sign or the size of the change. For a change from $x_1$ to $x_2$, use $\beta_1(x_2-x_1)$, preserving both signs.

For a general comparison, if the predictor changes from $x_1$ to $x_2$ and other predictors change from $z_j^{(1)}$ to $z_j^{(2)}$, the difference in linear predictors is

\beta_1(x_2-x_1)+\sum_{j=1}^{k}\gamma_j\left(z_j^{(2)}-z_j^{(1)}\right).

When the goal is to interpret $\beta_1$ alone, set every other-predictor difference to zero. Keeping that comparison clear makes the coefficient's meaning precise and keeps units and direction consistent. This treatment is aligned with the SOA Exam SRM (Statistics for Risk Modeling) syllabus.

Check Your Understanding

For each problem, interpret or calculate the requested change on the log-odds scale. Keep all stated other predictors fixed.

A model contains $0.06x$, where $x$ is measured in days. Find the log-odds change when $x$ increases from 5 to 9. (Answer: $0.24$)
A model contains $0.12w$, where $w$ is income in $1,000 units. Find the log-odds change for an income increase from $42,000 to $49,500. (Answer: $0.90$)
A model's linear predictor is $-2+0.04x+0.5z$. For $z=1$, calculate the log-odds at $x=20$ and at $x=30$, then give the change. (Answer: $-0.70$ and $-0.30$; change $=0.40$)
A predictor $q$ is measured in five-unit increments, and its coefficient is $-0.20$. Calculate the log-odds change for an increase of 12 units in the original measurement. (Answer: $-0.48$)
A continuous predictor has coefficient $0.025$ per day. Calculate the log-odds change when its value decreases from 14 days to 2 days, with all other predictors fixed. (Answer: $-0.30$)

What You'll Learn