From Odds Ratios to Probability Changes

From Odds Ratios to Predicted Probabilities

A logistic-regression coefficient describes a change on the log-odds scale. Exponentiating the coefficient gives an odds ratio, which is often easier to interpret, but it still does not directly state how much a predicted probability changes. To find that change, you need both the odds ratio and a starting probability.

This distinction matters when communicating a model result. Saying that a predictor multiplies the odds by a fixed amount is not the same as saying that it adds a fixed number of percentage points to the probability. The same odds ratio can correspond to different probability changes for two cases with different starting probabilities.

Definition: In a logistic-regression model for a binary outcome, let \(p\) be the probability of the outcome for a case with specified predictors. The odds are \(p/(1-p)\), and the logit is the natural logarithm of the odds. For a one-unit increase in a predictor \(x\), holding the other predictors fixed in a model with no relevant interaction or nonlinear change, the odds ratio is \(\exp(\beta_x)\). For a change of \(\Delta x\) units, the odds ratio is \(\exp(\beta_x\Delta x)\). This describes a multiplicative change in odds, not probability.

The conversion requires a baseline probability \(p_0\). Its odds are \(p_0/(1-p_0)\). Multiply those odds by the odds ratio to obtain the new odds, then convert the new odds back to a probability. This procedure compares two predictions while holding all other model inputs fixed.

\text{OR}=\exp(\beta_x\Delta x), \qquad p_1=\frac{\text{OR}\,p_0}{1-p_0+\text{OR}\,p_0}

Here \(p_0\) is the predicted probability before the predictor change, \(p_1\) is the predicted probability after it, and \(\text{OR}\) is the odds ratio for that change. The probability change is \(p_1-p_0\). A positive difference is an increase; a negative difference is a decrease. To express the difference in percentage points, multiply it by 100.

Why the Probability Change Depends on the Starting Risk

The formula follows directly from the definition of odds. If the initial odds are \(o_0=p_0/(1-p_0)\), applying an odds ratio gives new odds \(o_1=\text{OR}\,o_0\). Any odds \(o_1\) corresponds to probability \(o_1/(1+o_1)\), because solving \(o_1=p_1/(1-p_1)\) for \(p_1\) gives that expression. Substituting the initial odds yields the probability conversion formula:

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

Subtracting \(p_0\) gives a useful expression for the probability difference:

p_1-p_0 =\frac{p_0(1-p_0)(\text{OR}-1)} {1-p_0+\text{OR}\,p_0}

The odds ratio alone does not determine this difference: the expression also contains \(p_0\). For \(\text{OR}>1\), the difference is positive; for \(0<\text{OR}<1\), it is negative; and for \(\text{OR}=1\), it is zero. But its size varies with the baseline probability. In particular, probabilities close to zero or one often have less room to move than probabilities in the middle of the range.

Worked Example: A One-Unit Increase and a Low Baseline Probability

In a hypothetical model for whether a submitted application requires additional review, the coefficient for a numeric risk score is \(\beta_x=\ln(1.5)\) per score point. For an application with baseline predicted probability \(p_0=0.20\), calculate the predicted probability after a one-point increase in the score and the probability change. Use the fitted logistic model as given, holding other predictors fixed.

Method. First convert the coefficient to the odds ratio for the specified one-point change:

\text{OR}=\exp\bigl(\ln(1.5)\times 1\bigr)=1.5

The baseline odds are \(0.20/(1-0.20)=0.20/0.80=0.25\). Multiplying by the odds ratio gives new odds \(0.25\times1.5=0.375\). Convert those odds to a probability:

p_1=\frac{0.375}{1+0.375} =\frac{0.375}{1.375} =0.272727\ldots

The probability change is \(0.272727\ldots-0.20=0.072727\ldots\). Rounded only at the final step to four decimal places, the new probability is \(0.2727\), and the increase is \(0.0727\), or \(7.27\) percentage points. As a check, the new odds are \(0.272727\ldots/(1-0.272727\ldots)=0.375\), which are \(1.5\) times the baseline odds of \(0.25\).

Common mistake avoided. Multiplying \(0.20\) by \(1.5\) would give \(0.30\), but that incorrectly treats the odds ratio as a probability multiplier. The odds must be multiplied first, and then converted back to probability.

Worked Example: A Multi-Unit Change With Lower Odds

For a hypothetical binary claim outcome, a logistic model has coefficient \(\beta_x=\ln(0.8)\) per unit of a monitoring score. A case has baseline predicted probability \(p_0=0.50\). Calculate the probability after the score increases by two units, and state the change. Assume the model contains no interaction involving this predictor and hold all other predictors fixed.

Method. The coefficient applies per unit, so the two-unit change has odds ratio:

\text{OR}=\exp\bigl(\ln(0.8)\times 2\bigr)=0.8^2=0.64

The baseline odds are \(0.50/(1-0.50)=1\). Thus the new odds are \(1\times0.64=0.64\). Converting to a probability gives:

p_1=\frac{0.64}{1+0.64} =\frac{0.64}{1.64} =0.3902439\ldots

The probability change is \(0.3902439\ldots-0.50=-0.1097561\ldots\). Rounded only at the final step to four decimal places, the new probability is \(0.3902\), a decrease of \(0.1098\), or \(10.98\) percentage points. The check is that the new odds, \(0.3902439\ldots/(1-0.3902439\ldots)=0.64\), are \(0.64\) times the baseline odds of \(1\).

Common mistake avoided. Applying the per-unit odds ratio only once would give \(0.8\), not \(0.64\). A two-unit change requires multiplying the log-odds coefficient by two, or equivalently raising the per-unit odds ratio to the second power.

The Same Odds Ratio Can Give Different Probability Changes

The next comparison isolates the role of the starting probability. Suppose a predictor change doubles the odds, so \(\text{OR}=2\). For each baseline probability, calculate the new probability using the same odds-ratio conversion. The model change is identical in all three cases; only the starting probability differs.

Baseline probability \(p_0\)	Baseline odds	New odds	New probability \(p_1\)	Change \(p_1-p_0\)
0.10	0.10/0.90 = 0.1111...	0.2222...	0.1818	0.0818
0.50	0.50/0.50 = 1	2	0.6667	0.1667
0.80	0.80/0.20 = 4	8	0.8889	0.0889

For example, with \(p_0=0.10\), the new probability is \(0.2222\ldots/(1+0.2222\ldots)=0.1818\ldots\). With \(p_0=0.50\), it is \(2/(1+2)=0.6667\ldots\). With \(p_0=0.80\), it is \(8/(1+8)=0.8889\ldots\). The corresponding probability differences are therefore about \(8.18\), \(16.67\), and \(8.89\) percentage points. Each calculation gives new odds exactly twice the baseline odds, but the probability differences are not equal.

Worked Example: Comparing Two Cases Under the Same Odds Ratio

A hypothetical logistic model estimates an odds ratio of \(2\) for a specified change in a policyholder feature. One policyholder has baseline probability \(0.20\), and another has baseline probability \(0.60\). Calculate each new probability and probability change. Do not interpret the comparison as a causal effect; it is a comparison of model predictions holding other inputs fixed.

Method. For the first policyholder, baseline odds are \(0.20/0.80=0.25\). New odds are \(0.25\times2=0.50\), so the new probability is \(0.50/(1+0.50)=0.3333\ldots\). The probability difference is \(0.3333\ldots-0.20=0.1333\ldots\).

For the second policyholder, baseline odds are \(0.60/0.40=1.5\). New odds are \(1.5\times2=3\), so the new probability is \(3/(1+3)=0.75\). The probability difference is \(0.75-0.60=0.15\).

Rounded only at the final step to four decimal places, the first prediction changes from \(0.2000\) to \(0.3333\), an increase of \(0.1333\), or \(13.33\) percentage points. The second changes from \(0.6000\) to \(0.7500\), an increase of \(0.1500\), or \(15.00\) percentage points. As a check, the new odds are \(0.50\) and \(3\), respectively, each exactly twice its baseline odds.

Common mistake avoided. Reporting the same probability increase for both cases because their odds ratio is the same confuses an odds multiplier with an additive probability effect. The baseline probability must be included when calculating a predicted probability change.

Practical Interpretation and Exam Traps

When interpreting a logistic coefficient for a predictor change, keep the scale and the comparison clear. The coefficient is on the log-odds scale; exponentiating the coefficient times the change gives the odds ratio. To report predicted probabilities or percentage-point changes, start from a stated baseline prediction and perform the odds-to-probability conversion. If the requested predictor change is several units, use the full change in the linear predictor rather than treating a per-unit odds ratio as the odds ratio for the whole change.

In a model with several predictors, “holding other predictors fixed” means comparing the two linear predictors with the other inputs unchanged. If the model includes a nonlinear term or an interaction involving the changing predictor, the difference in linear predictors may not equal a single coefficient multiplied by the predictor change. In that case, calculate the model’s linear predictor at both specified cases, take their difference, exponentiate it to obtain the odds ratio between the cases, and then convert from the baseline probability. The conversion still works because it uses the odds ratio between the two predictions.

Do not multiply probability by the odds ratio. Multiply odds by the odds ratio, then convert the resulting odds to probability.
Do not omit the size of the predictor change. A per-unit coefficient must be multiplied by the number of units changed before exponentiating.
Do not report an odds change as a percentage-point change. An odds ratio of \(1.5\) means odds increase by 50%, not that probability increases by 50 percentage points.
State the baseline. A response-scale change cannot be calculated from an odds ratio alone; it also requires a baseline probability or equivalent baseline odds.
Keep comparisons conditional. The calculation describes model predictions for specified predictor values, not by itself a causal effect.

Key takeaway: For a specified change, first find the odds ratio, then multiply the baseline odds by it, and finally convert the new odds to probability. A fixed odds ratio produces a probability change that depends on the starting probability.

Check Your Understanding

For each problem, calculate the requested response-scale quantity. Round probabilities and probability changes to four decimal places.

A hypothetical binary-outcome model gives an odds ratio of \(1.6\) for a one-unit predictor increase. If the baseline probability is \(0.25\), find the new probability and the probability change. (Answer: New probability \(0.3478\); increase \(0.0978\), or \(9.78\) percentage points.)
A model gives an odds ratio of \(0.5\) for a specified predictor change. Starting from probability \(0.70\), find the new probability and the probability change. (Answer: New probability \(0.5385\); decrease \(0.1615\), or \(16.15\) percentage points.)
A coefficient is \(\ln(1.2)\) per unit. A case has baseline probability \(0.10\), and its predictor increases by four units. Find the total odds ratio, new probability, and probability change. (Answer: Total odds ratio \(2.0736\); new probability \(0.1873\); increase \(0.0873\), or \(8.73\) percentage points.)
The same predictor change has odds ratio \(3\) for two cases, whose baseline probabilities are \(0.20\) and \(0.60\). Find each new probability and compare their probability changes. (Answer: New probabilities \(0.4286\) and \(0.8182\); increases \(0.2286\) and \(0.2182\), respectively.)
A predictor change has odds ratio \(2\). After the change, a case has predicted probability \(0.75\). Find the baseline probability and the probability increase. (Answer: Baseline probability \(0.6000\); increase \(0.1500\), or \(15.00\) percentage points.)

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