Continuous-Predictor Odds Ratios and Units

From a Logistic Slope to an Odds Ratio

A logistic regression coefficient for a continuous predictor is a change in log odds per unit of the predictor. That description is mathematically direct, but often the question of interest uses a different increment: five years of age, ten units of a laboratory measurement, or one meter rather than one centimeter. The model gives a simple exact conversion from any such increment to an odds ratio.

Consider a binary outcome $Y$, with $Y=1$ denoting the event of interest, and a continuous predictor $x$. In the single-predictor logistic regression model, the probability of the event at predictor value $x$ is $p(x)=P(Y=1\mid x)$, and the log odds are linear in $x$.

Definition: The single-predictor logistic regression model is $$ \log\left(\frac{p(x)}{1-p(x)}\right)=\alpha+\beta x, $$ where $0<p(x)<1$, $\alpha$ is the intercept, and $\beta$ is the coefficient for the continuous predictor $x$. The odds at $x$ are $p(x)/(1-p(x))$. For a comparison from $x$ to $x+\Delta$, the odds ratio is the odds at $x+\Delta$ divided by the odds at $x$, provided both values are in the predictor domain under consideration.

The coefficient $\beta$ describes the change in log odds for a one-unit increase in $x$. Exponentiating converts that log-odds change to a multiplicative change in odds. The same reasoning applies to an increment $\Delta$ of any size.

The Odds Ratio for an Arbitrary Increment

Write $O(x)=p(x)/(1-p(x))$ for the odds at $x$. Exponentiating the model gives $O(x)=\exp(\alpha+\beta x)$. Comparing two predictor values therefore involves dividing their odds, so the intercept cancels.

Theorem: Under the single-predictor logistic regression model, for any two predictor values $x$ and $x+\Delta$ in the domain of comparison, the odds ratio is $$ \frac{O(x+\Delta)}{O(x)}=\exp(\beta\Delta). $$ In particular, a one-unit increase has odds ratio $\exp(\beta)$, and a $k$-unit increase has odds ratio $\exp(k\beta)$.

Proof. By the definition of the logistic model,

O(x)=\exp(\alpha+\beta x) \quad\text{and}\quad O(x+\Delta)=\exp\bigl(\alpha+\beta(x+\Delta)\bigr).

Divide the second odds by the first. Using the laws of exponents,

\frac{O(x+\Delta)}{O(x)} = \frac{\exp\bigl(\alpha+\beta x+\beta\Delta\bigr)} {\exp(\alpha+\beta x)} = \exp(\beta\Delta).

This holds for every permitted $x$ and increment $\Delta$; in this model, the ratio depends on the size and direction of the increment, not on the starting value. $\square$

For a positive increment, $\beta>0$ gives an odds ratio greater than 1, while $\beta<0$ gives an odds ratio less than 1. A negative increment reverses the comparison: moving down by $\Delta$ units has odds ratio $\exp(-\beta\Delta)$, the reciprocal of moving up by $\Delta$ units. These statements concern odds, not a fixed additive change in probability.

Worked Example: A Five-Year Increase in Age

Suppose a fitted logistic model for an outcome has age $x$, measured in years, as its continuous predictor, with $\alpha=-5$ and $\beta=0.08$. Find the odds ratio for a five-year increase, and check it by comparing the model odds at ages 50 and 55.

Five-year odds ratio. Here $\Delta=5$, so

\exp(\beta\Delta)=\exp(0.08\times 5) =\exp(0.40)\approx 1.4918.

Thus, according to this model, the odds at an age five years greater are about 1.49 times the odds at the starting age, or about 49.2% greater. The percentage is calculated as $(1.4918-1)\times 100\%\approx49.2\%$.

Check using the specified ages. At age 50, the linear predictor is $-5+0.08(50)=-1$, so the odds are $\exp(-1)$. At age 55, it is $-5+0.08(55)=-0.6$, so the odds are $\exp(-0.6)$. Their ratio is

\frac{\exp(-0.6)}{\exp(-1)} =\exp(-0.6+1) =\exp(0.4) \approx1.4918.

The direct comparison agrees with the general formula. The starting age changes the two odds, but it does not change their ratio in this model.

Choosing the Increment and the Units

The phrase “one-unit increase” only has meaning after the predictor’s units are specified. If $x$ is measured in milligrams per deciliter, one unit means one milligram per deciliter; if it is measured in centimeters, one unit means one centimeter. A coefficient cannot be interpreted correctly without naming those units.

For an increment of $k$ original units, multiply the coefficient by $k$ before exponentiating. Do not multiply the one-unit odds ratio by $k$: odds ratios combine multiplicatively, whereas their logarithms combine additively. For example, a two-unit increase has odds ratio $\exp(2\beta)=[\exp(\beta)]^2$.

Worked Example: Ten Units of a Laboratory Measurement

Suppose $x$ is a laboratory measurement recorded in units of 1 mg/dL, and a fitted logistic model has coefficient $\beta=-0.015$ per mg/dL. Calculate the odds ratios for increases of 10 and 20 mg/dL.

Increase of 10 mg/dL. The increment is $\Delta=10$, so

\exp(-0.015\times10)=\exp(-0.15)\approx0.8607.

The model therefore gives odds about 0.861 times as large after a 10 mg/dL increase, a decrease of approximately $(1-0.8607)\times100\%=13.93\%$ in odds.

Increase of 20 mg/dL. Now $\Delta=20$, giving

\exp(-0.015\times20)=\exp(-0.30)\approx0.7408.

This is also the square of the 10-unit odds ratio: $0.8607^2\approx0.7408$, with the small difference, if any, due to rounding. A 20-unit increase is two successive 10-unit increases, so their multiplicative odds changes multiply.

The ratio below 1 indicates lower odds for the higher measurement under this fitted model. It is not a statement that the probability itself falls by 13.93 or 25.92 percentage points.

Changing the recorded units changes the numerical coefficient, but it need not change the model’s fitted odds. The coefficient transforms so that the linear predictor remains the same.

Theorem: Let $x$ be measured in original units and define a new predictor $z=x/c$, where $c>0$. If the original model is $\operatorname{logit}(p)=\alpha+\beta x$, then the same model written in terms of $z$ is $\operatorname{logit}(p)=\alpha+\beta^*z$, where $\beta^*=c\beta$. Consequently, a one-unit increase in $z$—equivalent to an increase of $c$ original units—has odds ratio $\exp(c\beta)$.

Proof. Since $z=x/c$, we have $x=cz$. Substitution into the original linear predictor gives

\alpha+\beta x =\alpha+\beta(cz) =\alpha+(c\beta)z.

Thus the coefficient of $z$ is $\beta^*=c\beta$, and the linear predictor is unchanged for every corresponding pair $x,z$. The one-unit odds ratio for $z$ is $\exp(\beta^*)=\exp(c\beta)$, as claimed. $\square$

Worked Example: Converting a Coefficient from Centimeters to Meters

Suppose a logistic model uses a distance $x$ measured in centimeters and has coefficient $\beta=0.004$ per centimeter. Rewrite the predictor in meters and calculate the odds ratio for an increase of one meter.

Let $z=x/100$, so $z$ is distance in meters and $c=100$. The coefficient per meter is

\beta^*=100(0.004)=0.4.

The odds ratio for a one-meter increase is therefore

\exp(0.4)\approx1.4918.

This is the same as calculating the odds ratio for 100 centimeters in the original units: $\exp(0.004\times100)=\exp(0.4)$. For example, a distance of 250 cm is $z=2.5$ m. Its contribution to the linear predictor is $0.004(250)=1$ in the centimeter model and $0.4(2.5)=1$ in the meter model. The fitted linear predictor, and hence the fitted odds, agree.

Interpreting Fitted Coefficients Carefully

When a model is fitted to data, the coefficient is an estimate, often written $\hat{\beta}$. The corresponding estimated odds ratio for an increment $\Delta$ is $\exp(\Delta\hat{\beta})$. This reports the model-based point estimate for that comparison; it does not by itself describe uncertainty or establish a causal effect.

A useful interpretation states four things: the direction of the comparison, the size of the increment, the predictor’s units, and the outcome whose odds are being compared. For example: “For a 10 mg/dL increase in the measurement, the fitted model estimates the odds of the event to be multiplied by 0.861.” This avoids leaving “one unit” ambiguous and distinguishes an odds multiplier from an absolute probability change.

The constancy of the odds ratio across starting values follows from the model’s linear predictor in $x$. It is a feature of this specified model, not a guarantee about every real relationship. If the model does not adequately represent how log odds vary with $x$, then a single coefficient may not summarize the relationship well over the full range of observed values. Interpret comparisons within the predictor range that the data and model support.

Common Mistakes

Using the coefficient as the odds ratio. The coefficient $\beta$ is a change in log odds per unit. The one-unit odds ratio is $\exp(\beta)$, not $\beta$.
Forgetting to scale the coefficient for a larger increment. For $k$ units, calculate $\exp(k\beta)$. Multiplying $\exp(\beta)$ by $k$ is not the correct conversion.
Dividing an odds ratio by the number of units. If a five-unit odds ratio is 1.5, the one-unit odds ratio is $1.5^{1/5}$, not $1.5/5$. Indeed, $1.5^{1/5}=\exp(\log(1.5)/5)\approx1.0845$, and $1.0845^5\approx1.5$.
Leaving the units unstated. “A one-unit increase” is incomplete unless readers know whether the unit is a year, centimeter, or another measurement unit.
Calling an odds ratio a probability ratio or probability difference. The formula compares odds. It does not say that probability is multiplied by the same factor or changes by a fixed number of percentage points.

Key takeaway: In a single-predictor logistic model, an increase of $\Delta$ units multiplies the odds by $\exp(\beta\Delta)$. State $\Delta$ in the predictor’s actual units, and transform the coefficient consistently whenever those units change.

Check Your Understanding

For each question, make the increment and units explicit in your reasoning.

A logistic model has coefficient $0.06$ per year. What is the odds ratio for a three-year increase?
A fitted odds ratio for a 10-unit increase is 0.80. What is the odds ratio for a 20-unit increase under the same model?
A predictor is changed from grams to kilograms by defining the new value as the original value divided by 1,000. How does its coefficient change?
A five-unit odds ratio is 1.4. Explain why dividing 1.4 by 5 does not give the one-unit odds ratio, and write the correct expression.
In one sentence, state why a logistic odds ratio for a 10-unit increase should not be described as a 10-unit probability increase.