Interpreting a Logistic Regression Output Table

Why Read the Whole Output Table?

A logistic regression table compresses several distinct quantities into a few columns. The coefficient describes a conditional change in log odds; its standard error describes estimation uncertainty; the p-value summarizes evidence against a zero coefficient under a specified test; and the confidence interval gives a range of plausible coefficient values under the model. Exponentiating a coefficient or its confidence limits changes the scale from log odds to odds ratios.

These columns are related, but they are not interchangeable. A small p-value does not state the probability that the coefficient is zero, and a coefficient of 0.6 does not mean that the event probability rises by 60 percentage points. A careful interpretation begins with the model, predictor coding, and units.

Model and output definitions: For a binary outcome $Y_i$, logistic regression models the conditional probability $p_i=P(Y_i=1\mid \mathbf{x}_i)$ as $$ \log\left(\frac{p_i}{1-p_i}\right)=\eta_i=\beta_0+\beta_1x_{i1}+\cdots+\beta_px_{ip}. $$ The left side is the logit, or log odds. An output table commonly reports an estimate $\hat{\beta}_j$, its standard error $SE(\hat{\beta}_j)$, a Wald statistic $z_j=\hat{\beta}_j/SE(\hat{\beta}_j)$, a two-sided p-value for testing $H_0:\beta_j=0$, and a confidence interval for $\beta_j$. Unless specified otherwise, confidence intervals below are 95% Wald intervals.

The usual interpretation assumes the binary responses are independent conditional on the predictors and that the specified logistic mean model is appropriate. Wald calculations also use a large-sample approximation: the estimated coefficient is treated as approximately normal with standard error $SE(\hat{\beta}_j)$. These approximations can be unreliable when estimates are poorly determined, for example in settings with separation or very sparse data.

From Coefficients to Conditional Odds Ratios

Holding all other model predictors fixed, increasing $x_j$ by one unit changes the linear predictor by $\beta_j$. Since the linear predictor is the log odds, the corresponding odds ratio is the exponential of that coefficient.

\frac{\text{odds}(Y=1\mid x_j+1,\ \text{other predictors fixed})} {\text{odds}(Y=1\mid x_j,\ \text{other predictors fixed})} =\exp(\beta_j).

For a change of $c$ units in a quantitative predictor, the odds ratio is $\exp(c\beta_j)$. For a binary indicator coded 1 versus 0, $\exp(\beta_j)$ compares the odds for the coded group with the reference group, conditional on the other predictors. A 95% Wald interval for the coefficient is $\hat{\beta}_j\pm1.96SE(\hat{\beta}_j)$. Because the exponential function is increasing, exponentiating both endpoints gives the corresponding odds-ratio interval.

Identify the outcome and predictor coding.
Confirm which outcome is coded 1, the predictor's units, and any reference category.

Read the coefficient on its stated scale.
A coefficient is a conditional log-odds change per predictor unit, or relative to the reference category.

Assess uncertainty and evidence.
Use the standard error, test statistic, p-value, and coefficient interval for their separate purposes.

Translate to odds when useful.
Exponentiate the coefficient and both confidence limits; scale the coefficient first when comparing a multi-unit change.

Worked Output: Reading Every Column

Worked Example: Prior Claims and Renewal

In a hypothetical study, the outcome is whether a policyholder renews a policy ($Y=1$ for renewal). A logistic model includes age in decades relative to age 40, enrollment in a hypothetical service program, and an indicator for at least one prior claim. The program and claim indicators are coded 1 for Yes and 0 for No. The following table reports estimates and Wald output.

Predictor	Estimate	SE	z	p-value	95% coefficient interval
Intercept	-1.200	0.300	-4.000	<0.0001	[-1.788, -0.612]
Age, per 10 years	0.300	0.120	2.500	0.0124	[0.0648, 0.5352]
Service program: Yes	0.700	0.280	2.500	0.0124	[0.1512, 1.2488]
Prior claim: Yes	0.620	0.250	2.480	0.01314	[0.130, 1.110]

Method and test calculation. For the prior-claim coefficient, divide the estimate by its standard error: $z=0.620/0.250=2.480$. The two-sided standard normal p-value is $2P(Z\geq2.480)$, approximately 0.01314. This is evidence against a zero conditional coefficient at conventional 5% significance levels; it is not the probability that the null hypothesis is true.

Coefficient interval. The 95% Wald interval is $0.620\pm1.96(0.250)=0.620\pm0.490=[0.130,1.110]$. It is entirely above zero, consistent with the two-sided test at the 5% level.

Odds-ratio interpretation. Exponentiating gives $\exp(0.620)=1.8589$. Conditional on age and program status, the estimated renewal odds for policyholders with a prior claim are about 1.8589 times the odds for those without a prior claim, or about 85.89% higher odds. The interval on the odds-ratio scale is $[\exp(0.130),\exp(1.110)]\approx[1.1388,3.0340]$.

Check and common mistake avoided. The transformed interval lies wholly above 1, as expected from the coefficient interval lying above 0. Calling the coefficient a 62% increase in odds would be incorrect: 0.620 is a log-odds change, while the odds multiplier is $\exp(0.620)$.

Why Exponentiation Gives the Odds Ratio

The odds corresponding to probability $p$ are $p/(1-p)$. If the linear predictor is $\eta$, the model states that the log odds equal $\eta$, so the odds equal $\exp(\eta)$. Increasing one predictor by one unit changes $\eta$ from $\eta$ to $\eta+\beta_j$, with other predictors fixed. Therefore the ratio of the new odds to the old odds is

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

This also explains the condition in the interpretation: the other predictors are held fixed so that their contributions cancel in the ratio. If a model includes an interaction involving $x_j$, its conditional effect can depend on the interacting predictor; a single coefficient alone then does not describe the odds ratio at every value of that predictor.

Scaling Effects and Comparing Probabilities

Worked Example: A Ten-Year Age Difference and Program Enrollment

Use the renewal model above. Compare two otherwise identical hypothetical policyholders, both with no prior claim: one is age 40 and not enrolled in the program; the other is age 50 and enrolled. Find the odds ratio for the age difference, and calculate each fitted probability.

Age effect. Age is measured in decades relative to age 40, so the age-50 policyholder has $x_{\text{age}}=1$ and the age-40 policyholder has $x_{\text{age}}=0$. The conditional odds ratio for a 10-year increase is $\exp(1(0.300))=\exp(0.300)=1.3499$.

First fitted probability. For age 40, no program enrollment, and no prior claim, the linear predictor is $\eta=-1.200+0.300(0)+0.700(0)+0.620(0)=-1.200$. Thus $p=1/(1+\exp(1.200))\approx0.2315$.

Second fitted probability. For age 50, program enrollment, and no prior claim, $\eta=-1.200+0.300(1)+0.700(1)+0.620(0)=-0.200$. Thus $p=1/(1+\exp(0.200))\approx0.4502$.

Check and common mistake avoided. The two probabilities differ by about 0.2187, but the odds ratio between these profiles is not $\exp(0.300)$: program status also differs. Their full linear-predictor difference is $0.300+0.700=1.000$, giving an odds ratio of $\exp(1.000)=2.7183$. The age-only odds ratio applies when program status and the other predictors are held fixed.

Odds ratios do not generally equal probability ratios or probability differences. The logistic transformation is nonlinear, so the same odds ratio can correspond to different probability changes at different starting probabilities. When a question asks for probabilities, substitute the complete predictor profile into the linear predictor and then apply $p=1/(1+\exp(-\eta))$; do not interpret an odds ratio as a percentage-point change.

Worked Example: Rescaling a Continuous Predictor

A hypothetical model for a binary audit flag reports a score coefficient of $-0.080$, standard error $0.035$, and 95% Wald coefficient interval $[-0.1486,-0.0114]$. The score is measured in single points. Interpret a five-point increase, including its odds-ratio interval, and assess the evidence for a score effect.

Test evidence. The Wald statistic is $z=-0.080/0.035=-2.2857$, giving a two-sided p-value of approximately 0.0223. The coefficient interval excludes zero, so it agrees with the test at the 5% level.

Five-point odds ratio. A five-point increase changes the linear predictor by $5(-0.080)=-0.400$. The odds ratio is $\exp(-0.400)=0.6703$. Thus, holding the other predictors fixed, the odds of the audit flag are estimated to be multiplied by 0.6703 for a five-point increase, a decrease of about 32.97% in the odds.

Interval and check. Multiply both coefficient limits by 5 to obtain $[-0.743,-0.057]$, then exponentiate: $[\exp(-0.743),\exp(-0.057)]\approx[0.4758,0.9446]$. The entire interval is below 1, which matches the negative coefficient interval. The mistake to avoid is reporting $\exp(-0.080)$ as the five-point odds ratio; that is the multiplier for a one-point increase.

What the Table Does and Does Not Establish

A coefficient’s sign gives the direction of its conditional association with the modeled outcome: positive estimates correspond to higher fitted odds as the predictor increases, and negative estimates correspond to lower fitted odds. The magnitude is most naturally communicated using an odds ratio for a clearly stated change in the predictor. For a factor, always name the comparison and reference category.

Estimate: the fitted conditional log-odds change, not a probability change.
Standard error: an estimate of sampling uncertainty on the coefficient scale; it is not the standard deviation of individual outcomes.
Wald p-value: evidence against the tested zero-coefficient null under the approximation and model assumptions; it is not an effect-size measure.
Confidence interval: a range on the scale shown in the table. Transform coefficient endpoints by exponentiation before calling them an odds-ratio interval.
Intercept: the fitted log odds when quantitative predictors equal zero and indicators are at their reference levels. Its practical meaning depends on whether that profile is meaningful.

A significant coefficient does not establish causation, and a non-significant coefficient does not prove that its effect is exactly zero. Interpretation remains conditional on the modeled predictors and the data-generating assumptions. Check units, coding, outcome definition, and interval scale before drawing a substantive conclusion. The output-reading skills here are aligned with the SOA Exam SRM (Statistics for Risk Modeling) syllabus.

Key takeaway: Read an output table in sequence: establish the coding and units, interpret the estimate on the log-odds scale, use the standard error and p-value for inference, and exponentiate the estimate and interval to describe conditional odds ratios.

Check Your Understanding

For each question, distinguish coefficient-scale interpretation from odds-scale interpretation and keep predictor units explicit.

A logistic output gives a coefficient of $-0.400$ with standard error $0.200$ for a one-unit predictor. Find the Wald statistic, two-sided p-value (to four decimals), and 95% coefficient interval. (Answer: $z=-2.000$, $p=0.0455$, interval $[-0.792, -0.008]$.)
A continuous predictor is measured in five-point units, and its estimated coefficient is $0.180$. Find the conditional odds ratio for a ten-point increase. (Answer: $\exp(0.360)=1.4333$.)
A binary indicator has estimate $0.500$ and standard error $0.250$, with 0 as the reference category. Find the Wald statistic, two-sided p-value, 95% coefficient interval, and odds ratio. (Answer: $z=2.000$, $p=0.0455$, interval $[0.010,0.990]$, odds ratio $1.6487$.)
A coefficient for a hypothetical service indicator is $0.700$, where 1 means enrolled and 0 means not enrolled. State the conditional odds ratio and explain what is held fixed. (Answer: $\exp(0.700)=2.0138$; compare enrolled with not enrolled at the same values of all other included predictors.)
A predictor has coefficient $0.040$ per unit and standard error $0.020$. Calculate the odds ratio for a 20-unit increase and its 95% Wald interval. (Answer: estimate $=\exp(0.800)=2.2255$; coefficient interval per unit $[0.0008,0.0792]$, so the 20-unit odds-ratio interval is approximately $[1.0161,4.8731]$.)

What You'll Learn