Interpreting Renewal Odds Ratios As Probability Changes

From Renewal Odds to Renewal Probability

A logistic renewal model is often reported using coefficients or odds ratios. These summaries describe multiplicative changes in odds, not fixed changes in the probability of renewal. To communicate a model result in probability terms, first identify the baseline renewal probability and the predictor contrast, then convert the resulting odds back to a probability.

This distinction matters in ratemaking and retention analysis: the same odds ratio can correspond to different probability-point changes for policyholders with different baseline renewal probabilities. The calculations below focus on translating a model contrast while other predictors are held fixed. This material is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Definition: For an event with probability $p$, its odds are $p/(1-p)$. The odds ratio comparing a probability $p_1$ with a probability $p_0$ is $$ \frac{p_1/(1-p_1)}{p_0/(1-p_0)}. $$ A probability is between 0 and 1; odds are nonnegative and are not themselves probabilities.

The Logistic Coefficient and the Odds Ratio

For a binary renewal outcome $Y$, let $p=P(Y=1\mid \mathbf{x})$ denote the modeled renewal probability given predictors $\mathbf{x}$. A logistic regression specifies the log odds as a linear predictor:

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

If predictor $x_j$ changes by $\Delta x_j$, while all other predictors remain fixed, the log odds change by $\beta_j\Delta x_j$. Exponentiating that change gives the odds ratio for this contrast:

\operatorname{OR}=\exp(\beta_j\Delta x_j).

For a one-unit increase in $x_j$, the odds ratio is $\exp(\beta_j)$. If the predictor changes by several units, use the full contrast $\Delta x_j$; do not apply the one-unit odds ratio just once unless the contrast is one unit. A positive coefficient gives an odds ratio greater than 1 for a positive increase, while a negative coefficient gives an odds ratio less than 1.

Key result: If the baseline renewal probability is $p_0$ and the odds ratio for a specified contrast is $\operatorname{OR}$, then the probability after the contrast is $$ p_1=\frac{\operatorname{OR}\,p_0}{1-p_0+\operatorname{OR}\,p_0}. $$ The probability-point change is $p_1-p_0$. The result applies to the logistic-model contrast with the other modeled predictors held fixed.

Why the Conversion Formula Holds

The conversion follows directly from the definition of odds. Let the baseline odds be $O_0=p_0/(1-p_0)$. Multiplying by the odds ratio gives the new odds $O_1=\operatorname{OR}\,O_0$. For any odds $O_1$, the corresponding probability is $O_1/(1+O_1)$, since $O_1=p_1/(1-p_1)$ implies $p_1=O_1/(1+O_1)$. Substituting the baseline odds gives:

p_1 =\frac{\operatorname{OR}\,p_0/(1-p_0)} {1+\operatorname{OR}\,p_0/(1-p_0)} =\frac{\operatorname{OR}\,p_0} {1-p_0+\operatorname{OR}\,p_0}.

The denominator includes both the baseline nonrenewal probability and the odds-adjusted renewal probability. Consequently, multiplying $p_0$ directly by the odds ratio is generally not the correct probability calculation.

Worked Example: A Decrease in Renewal Odds

In a hypothetical renewal model, a specified predictor contrast changes the linear predictor by $-0.54$. The baseline renewal probability at the original predictor values is $0.35$. Calculate the odds ratio, the updated renewal probability, and the probability-point change. Round probabilities and changes to four decimal places.

Method. Convert the change in the linear predictor to an odds ratio, multiply the baseline odds by that ratio, and convert the resulting odds back to probability.

Step 1: Calculate the odds ratio. $$ \operatorname{OR}=\exp(-0.54)\approx0.5827483. $$

Step 2: Calculate baseline odds. $$ O_0=\frac{0.35}{1-0.35} =\frac{0.35}{0.65} =\frac{7}{13} \approx0.5384615. $$

Step 3: Apply the odds ratio. $$ O_1=\frac{7}{13}\exp(-0.54) \approx0.3137875. $$

Step 4: Convert the new odds to probability. $$ p_1=\frac{O_1}{1+O_1} =\frac{0.35\exp(-0.54)} {0.65+0.35\exp(-0.54)} \approx0.2388419. $$

Step 5: Calculate the change. $$ p_1-p_0=0.2388419-0.35=-0.1111581. $$

Rounded to four decimal places, the updated probability is $0.2388$, and the change is $-0.1112$, a decrease of about 11.12 percentage points. The result is consistent with the odds ratio below 1: the modeled renewal odds and probability both decrease.

Common mistake avoided. Multiplying the probability directly gives $0.35(0.5827483)\approx0.2040$, which is not the updated probability. The odds ratio multiplies $0.35/0.65$, not $0.35$.

The Same Odds Ratio Can Give Different Probability Changes

The conversion depends on both the odds ratio and the baseline probability. For a fixed odds ratio greater than 1, the probability increases, but the size of the increase in percentage points is not constant across baseline risks. This is why an odds ratio should not be reported as if it meant, for example, “a 20-point increase in renewal probability.”

Worked Example: One Odds Ratio at Two Baseline Risks

A hypothetical fitted model gives an odds ratio of $2.5$ for a specified contrast. Calculate the updated probability and probability-point change at baseline renewal probabilities of $0.20$ and $0.70$. Round results to four decimal places.

Method. Apply the same odds ratio separately to each baseline probability; do not treat the odds ratio as a probability multiplier.

Baseline $p_0=0.20$. The baseline odds are $0.20/0.80=0.25$. The new odds are $2.5(0.25)=0.625$. Thus, $$ p_1=\frac{0.625}{1+0.625} =\frac{0.625}{1.625} \approx0.3846154. $$ The probability-point change is $0.3846154-0.20=0.1846154$, or $+0.1846$ when rounded to four decimals.

Baseline $p_0=0.70$. The baseline odds are $0.70/0.30=7/3$. The new odds are $2.5(7/3)=35/6$. Therefore, $$ p_1=\frac{35/6}{1+35/6} =\frac{35}{41} \approx0.8536585. $$ The probability-point change is $0.8536585-0.70=0.1536585$, or $+0.1537$ when rounded to four decimals.

The same $2.5$ odds ratio produces an increase of about 18.46 percentage points from the $0.20$ baseline, but about 15.37 percentage points from the $0.70$ baseline. Both are increases, yet they are not the same size.

Common mistake avoided. It would be wrong to report $2.5(0.20)=0.50$ or $2.5(0.70)=1.75$ as updated probabilities. A probability cannot exceed 1, and an odds ratio acts on odds rather than probabilities.

Using a Coefficient for a Multi-Unit Contrast

A coefficient is tied to the predictor's unit of measurement. For a contrast of $\Delta x$ units, the change in log odds is $\beta\Delta x$, so the odds ratio is $\exp(\beta\Delta x)$. This remains true whether the predictor is measured in dollars, percentage points, or another unit, provided the contrast uses the same units as the fitted model.

Worked Example: Applying a Coefficient Across Two Units

In a hypothetical model, a predictor has coefficient $\beta=-0.40$ per unit. For a policyholder with baseline renewal probability $0.50$, the predictor increases by 2 units and all other predictors are unchanged. Find the odds ratio, the new probability, and the probability-point change. Round the probability and change to four decimal places.

Method. Multiply the coefficient by the full two-unit contrast before exponentiating.

Step 1: Find the log-odds change and odds ratio. $$ \Delta\eta=(-0.40)(2)=-0.80, \qquad \operatorname{OR}=\exp(-0.80)\approx0.4493290. $$

Step 2: Find the baseline odds. $$ O_0=\frac{0.50}{1-0.50}=1. $$

Step 3: Find the new odds and probability. The new odds are $O_1=1(0.4493290)=0.4493290$, so $$ p_1=\frac{0.4493290}{1+0.4493290} \approx0.3100255. $$

Step 4: Find the probability-point change. $$ p_1-p_0=0.3100255-0.50=-0.1899745. $$

The updated probability is $0.3100$, and the change is $-0.1900$, a decrease of about 19.00 percentage points. The odds ratio is below 1 because the two-unit contrast produces a negative change in log odds.

Common mistake avoided. Using $\exp(-0.40)$ would give only the one-unit odds ratio. For a two-unit contrast, the correct odds ratio is $\exp(2(-0.40))$, not $\exp(-0.40)$.

Interpreting and Communicating the Result

A complete interpretation identifies the contrast, baseline probability, updated probability, and probability-point change. For example: “For this two-unit contrast, with the other predictors fixed and a baseline renewal probability of 0.50, the model gives an updated probability of 0.3100, a decrease of 19.00 percentage points.” This is more informative than saying only that the odds decrease by a certain percentage.

A percentage-point change is the arithmetic difference between two probabilities. A relative percentage change instead divides that difference by the baseline probability. In the preceding example, the probability-point change is $-0.1900$, while the relative change is approximately $-0.1900/0.50=-38.0\%$. These quantities answer different questions and should be labelled clearly.

The interpretation is conditional on the model and on holding the other predictors fixed. It is not, by itself, evidence that changing a predictor will cause renewal behavior to change. In particular, an estimated association in observational data may reflect other differences among policyholders. The conversion formula translates the model's stated contrast; it does not establish a causal effect.

Exam trap: An odds ratio is not a percentage-point change, and it is not a multiplier for the probability. First compute $\operatorname{OR}=\exp(\beta\Delta x)$, then use $$ p_1=\frac{\operatorname{OR}\,p_0}{1-p_0+\operatorname{OR}\,p_0}, $$ and finally subtract $p_0$ if the question asks for a probability-point change.

Check Your Understanding

For each question, distinguish the odds change from the probability change. Round probabilities and changes to four decimal places where appropriate.

A hypothetical renewal model has baseline probability $0.25$ and an odds ratio of $1.8$ for a contrast. Find the updated probability and probability-point change. (Answer: $p_1=0.3750$; change $=+0.1250$, or $+12.50$ percentage points.)
A policyholder has baseline renewal probability $0.60$. A specified contrast has odds ratio $0.70$. Find the updated probability and change. (Answer: $p_1\approx0.5122$; change $\approx-0.0878$, or $-8.78$ percentage points.)
A model coefficient is $0.25$ per unit. The predictor increases by 3 units from a baseline renewal probability of $0.40$. Find the odds ratio and updated probability. (Answer: $\operatorname{OR}=\exp(0.75)\approx2.1170$; $p_1\approx0.5853$, an increase of about $0.1853$, or 18.53 percentage points.)
A specified contrast has odds ratio $4$ for a group with baseline renewal probability $0.15$. Find the updated probability and probability-point change. (Answer: $p_1=0.4138$; change $=+0.2638$, or $+26.38$ percentage points.)
The same odds ratio, $0.60$, applies to two modeled baselines, $0.10$ and $0.70$. Calculate each updated probability and change, and state whether the changes match. (Answer: from $0.10$, $p_1=0.0625$, change $=-0.0375$; from $0.70$, $p_1\approx0.5833$, change $\approx-0.1167$. The probability-point changes do not match.)

What You'll Learn