Odds Ratios and Predictor Units

From a Log-Odds Coefficient to an Odds Ratio

A logistic-regression coefficient is measured on the log-odds scale, which is not usually the scale on which a practical comparison is requested. An analyst may want to compare the odds of an event for two applicants whose recorded predictor values differ by several units, or to express an effect per year rather than per month. Exponentiation translates a difference in log odds into a ratio of odds.

The interpretation is conditional: when a model contains several predictors, compare the two cases while holding the other predictors fixed. The resulting odds ratio describes the fitted model, not necessarily a causal effect.

Definition: In a logistic-regression model with binary response $Y$, let $p=P(Y=1\mid x_1,\ldots,x_p)$. The conditional odds of $Y=1$ are $p/(1-p)$. In a model with no interaction involving $x_j$, a change of $\Delta$ units in $x_j$, with all other predictors held fixed, has conditional odds ratio $\exp(\beta_j\Delta)$. For a one-unit increase, the odds ratio is $\exp(\beta_j)$.

An odds ratio greater than 1 means the fitted odds are multiplied by a factor greater than 1 for the stated comparison; an odds ratio below 1 means they are multiplied by a factor below 1. Reversing the comparison takes the reciprocal. These statements concern odds, not a fixed increase in probability.

Why Exponentiation Gives the Odds Ratio

For the basic additive logistic model, the conditional log odds are a linear predictor. This identity is what makes a coefficient difference translate directly to an odds ratio.

\log\left(\frac{p}{1-p}\right) =\beta_0+\beta_1x_1+\cdots+\beta_px_p =\eta.

Exponentiating both sides gives the conditional odds $p/(1-p)=\exp(\eta)$. Compare two cases that differ by $\Delta$ in $x_j$ and agree on every other predictor. Their linear predictors differ by $\beta_j\Delta$, so the odds ratio is

\frac{\exp(\eta+\beta_j\Delta)}{\exp(\eta)} =\exp(\beta_j\Delta).

The cancellation of $\exp(\eta)$ explains why the odds ratio does not depend on the other predictor values in this no-interaction model. If an interaction changes the coefficient associated with $x_j$ across cases, this simple expression must be replaced by the difference between the two complete linear predictors. The formula also presumes the same outcome coding and the same model are used for both cases.

Identify the coefficient and predictor units.
Read $\beta_j$ as the change in fitted log odds for a one-unit increase in the predictor as it is actually recorded.

Specify the comparison.
Write the signed change $\Delta=x_{j,2}-x_{j,1}$, and state that the other predictors are held fixed.

Exponentiate the full log-odds change.
Calculate $\exp(\beta_j\Delta)$, not $\exp(\beta_j)$ multiplied by the number of units.

Worked Examples

Worked Example: A Five-Year Difference in Experience

In a hypothetical logistic model for whether a submitted claim is selected for additional review, experience $x$ is measured in years. Suppose the fitted coefficient is $\hat\beta_x=0.08$. For two otherwise identical hypothetical claims, the claim handlers have 37 and 42 years of experience. If the fitted odds for the 37-year handler are 0.20, find the odds ratio and the fitted odds for the 42-year handler.

Method. Use the multiple-unit formula with $\Delta=42-37=5$. The five-year change in fitted log odds is $\hat\beta_x\Delta=0.08(5)=0.40$.

\text{Odds ratio}=\exp(0.40)=1.4918.

Thus, according to the model, the fitted odds for the 42-year handler are 1.4918 times the fitted odds for the 37-year handler, with other predictors fixed. The second handler's fitted odds are $0.20(1.4918)=0.29836$, or 0.2984 to four decimal places. The check is $0.29836/0.20=1.4918$ to the displayed precision.

Common mistake avoided: The value $0.08$ is a log-odds change per year, not an 8-percentage-point increase in probability. Exponentiating the full five-year log-odds difference gives the requested odds ratio.

Worked Example: A Change Measured in Thousands of Kilometres

A hypothetical logistic model predicts whether a vehicle policy receives a particular service request during a year. Annual distance is recorded in thousands of kilometres, and its fitted coefficient is $0.12$ per one thousand kilometres. Compare two otherwise identical policies with annual distances of 20,000 km and 27,500 km. Calculate the conditional odds ratio for the higher-distance policy.

Method. Because the predictor is recorded in thousands of kilometres, the two values in the model are $20$ and $27.5$, not $20{,}000$ and $27{,}500$. Therefore, $\Delta=27.5-20=7.5$ model units.

The corresponding change in log odds is $0.12(7.5)=0.90$. Exponentiating,

\text{Odds ratio}=\exp(0.90)=2.4596.

The model therefore gives the higher-distance policy 2.4596 times the fitted odds of the outcome, conditional on the other predictors. As a check, using kilometres directly requires converting the coefficient to $0.12/1{,}000=0.00012$ per kilometre. The difference is $7{,}500$ km, and $0.00012(7{,}500)=0.90$, the same log-odds change.

Common mistake avoided: Applying the coefficient $0.12$ to a change of $7{,}500$ mixes kilometres with thousands of kilometres and produces an enormous, incorrect exponent. The predictor difference and coefficient must use matching units.

Changing Predictor Units

Changing units changes the numerical size of a coefficient because a one-unit increase now represents a different amount of the underlying quantity. It does not change the fitted relationship when the predictor and coefficient are transformed consistently. Suppose $x$ is measured in months and $z=x/12$ is the same quantity measured in years. In a model with linear predictor $\beta_0+\beta_x x$, substitute $x=12z$:

\beta_0+\beta_x x =\beta_0+\beta_x(12z) =\beta_0+(12\beta_x)z.

Thus the coefficient per year is $12\beta_x$, while the coefficient per month is $\beta_x$. The intercept is unchanged because this is a pure change of scale, with zero months corresponding to zero years. More generally, if $z=cx$, then $x=z/c$, and the coefficient on $z$ must be $\beta_x/c$. A change that also shifts the zero point, such as replacing $x$ with $z=x-a$, changes the intercept as well.

Key result: If $z=cx$ for nonzero $c$, rewrite the predictor coefficient as $\beta_z=\beta_x/c$. Then $\beta_xx=\beta_zz$ for every observation. The linear predictor, fitted log odds, fitted odds, and fitted probabilities are consequently unchanged; only the coefficient's unit changes.

Worked Example: Converting Age From Months to Years

Consider a hypothetical model for whether a member uses a preventive service. Let age $m$ be measured in months and let $w$ be an indicator for a specified member characteristic, held fixed in the comparison. Suppose $\operatorname{logit}(p)=-2.4+0.015m+0.5w$. For a member with $w=0$ and age 20 years, (a) find the fitted probability using months, (b) rewrite the model in years and verify the probability, and (c) find the odds ratio for a one-year age increase.

Method. Convert 20 years to $m=20(12)=240$ months. Substitution gives the linear predictor $-2.4+0.015(240)+0.5(0)=-2.4+3.6=1.2$. The fitted probability is therefore

p=\frac{\exp(1.2)}{1+\exp(1.2)} =\frac{3.3201}{1+3.3201} \approx 0.7685.

The value is rounded to four decimal places. For the years version, set $y=m/12$, so $m=12y$. The predictor becomes $-2.4+0.015(12y)+0.5w=-2.4+0.18y+0.5w$. At $y=20,w=0$, it is $-2.4+0.18(20)=1.2$, exactly as before; it therefore gives the same fitted probability, 0.7685.

For a one-year increase, the log-odds change in the years model is $0.18(1)=0.18$, so the odds ratio is $\exp(0.18)=1.1972$. In the months model, one year is 12 months, giving $\exp(0.015(12))=\exp(0.18)=1.1972$. The two calculations agree.

Common mistake avoided: The coefficient $0.015$ applies to one month, not one year. Treating it as a per-year coefficient would give the wrong one-year odds ratio. The check is that both model versions produce the same linear predictor at the same age.

Interpreting and Reporting Odds Ratios

A clear report includes the event, the predictor change, the direction of comparison, the units, and the conditioning statement. For example: “Holding the other modeled predictors fixed, a 5-year increase in experience multiplies the fitted odds of additional review by 1.4918.” This wording says exactly which contrast is being made and avoids treating a one-unit coefficient as an effect over an unspecified range.

For a decrease, retain the signed change rather than silently reversing the comparison. If $\beta_j=0.08$ and the predictor decreases by five units, then $\Delta=-5$, so the odds ratio is $\exp(0.08(-5))=\exp(-0.40)=1/1.4918$, approximately $0.6703$. This is consistent with the reverse comparison: the odds for the lower value are about 0.6703 times those for the higher value.

An odds ratio does not itself give a probability difference. For example, the same odds multiplier can correspond to different probability comparisons at different starting probabilities. The odds-ratio calculation is still valid; probability comparisons require the relevant fitted probabilities. This distinction is important when communicating model results and is aligned with the SOA Exam SRM (Statistics for Risk Modeling) syllabus.

Common Mistakes

Reporting the coefficient as the odds ratio. A coefficient $\beta_j$ is a log-odds change per model unit; exponentiate it for a one-unit odds ratio.
Failing to multiply by the stated change. For a $\Delta$-unit contrast, calculate $\exp(\beta_j\Delta)$, not just $\exp(\beta_j)$.
Multiplying an odds ratio by the number of units. The model adds log odds across units, so exponentiation gives $\exp(\beta_j\Delta)$; it does not give $\Delta\exp(\beta_j)$.
Mixing predictor units. If a coefficient is per thousand kilometres, express the predictor difference in thousands of kilometres or convert the coefficient first.
Calling an odds change a probability change. An odds ratio is a multiplier of odds. It is not a percentage-point change in probability.
Ignoring the comparison direction. Reversing the two cases takes the reciprocal of the odds ratio.
Changing units without changing the coefficient. A unit conversion must preserve the product of coefficient and predictor value, and hence the linear predictor.

Key takeaway: State the predictor units and the contrast first, hold the other predictors fixed, and exponentiate the coefficient multiplied by the signed change. Under a consistent rescaling, the coefficient changes but the model's predictions do not.

Check Your Understanding

For each question, state the relevant units and keep the direction of comparison explicit. Round odds ratios to four decimal places.

A hypothetical logistic model has coefficient $0.06$ per year of membership. What is the conditional odds ratio for a member with four more years than an otherwise identical member? (Answer: $\exp(0.24)=1.2712$.)
A fitted coefficient is $-0.20$ per five-point increase in a risk score, where the score is recorded in five-point units. Find the odds ratio for an increase of 15 points. (Answer: $\exp(-0.60)=0.5488$.)
A model uses benefit amount in dollars and has coefficient $0.0003$ per dollar. Find the odds ratio for a $2,000 increase, with other predictors fixed. (Answer: $\exp(0.60)=1.8221$.)
A predictor is recorded as the number of ten-unit blocks, with coefficient $0.35$ per block. Rewrite the coefficient per single raw unit and find the odds ratio for a six-unit increase. (Answer: coefficient $0.035$ per unit; odds ratio $\exp(0.21)=1.2337$.)
For a continuous predictor with coefficient $0.04$ per year, compare fitted odds for values five years apart in each direction. (Answer: higher versus lower value, $\exp(0.20)=1.2214$; lower versus higher value, $\exp(-0.20)=0.8187$, the reciprocal to rounding.)

What You'll Learn