A paper states: “Students with part-time jobs are 2.5 times more likely to drop out.” The 2.5 is an odds ratio from a logistic regression. Looking at the raw data, the dropout rate is 30% among working students and 15% among others. The actual risk is twice as high, not 2.5 times. That sentence overclaims, and the same slip appears in countless theses.
Logistic regression is the standard tool when the outcome has two values. Running it is easy; reading it correctly is where most errors happen, and they are errors of interpretation rather than software.
Why not linear regression for a yes/no outcome?
If you code dropout as 1 and staying as 0 and run ordinary regression (the linear probability model), three problems follow: predicted values can fall below 0 or above 1, the error variance is structurally unequal, and the true relationship is usually S-shaped rather than straight. The linear probability model is still used in some fields, notably economics, with robust standard errors. But if your supervisor and journal expect logistic regression, use it.
Three quantities: probability, odds, log-odds
| Quantity | Formula | Example with probability 0.30 |
|---|---|---|
| Probability (p) | cases / total | 0.30 |
| Odds | p / (1 − p) | 0.30 / 0.70 ≈ 0.43 |
| Log-odds (logit) | ln(odds) | ln(0.43) ≈ −0.85 |
Logistic regression models the log-odds as a linear function of the predictors. The B coefficients your software prints are therefore on the log-odds scale, which is hard to interpret directly. Exponentiate B, shown as Exp(B) or e to the power B, and you get the odds ratio.
Reading an odds ratio
- OR = 1: no difference in odds.
- OR > 1: higher odds of the outcome. OR = 1.8 means odds 80% higher.
- OR < 1: lower odds. OR = 0.6 means odds 40% lower.
For a continuous predictor, the OR is the change in odds per one-unit increase. If the unit is tiny, such as income in single currency units, the OR sits close to 1 and looks meaningless. Rescale (thousands, decades of age) or standardise before fitting.
For categorical predictors, every OR is relative to the reference category. Choose a meaningful, reasonably large reference group and state it in the table.
Odds ratios are not risk ratios
A risk ratio compares two probabilities; an odds ratio compares two odds. When the outcome is rare (roughly below 10%), they are close. When it is common, the OR lies noticeably further from 1 than the RR.
Back to the opening example: working students p = 0.30, odds ≈ 0.43; others p = 0.15, odds ≈ 0.18. OR ≈ 0.43 / 0.18 ≈ 2.4, while RR = 0.30 / 0.15 = 2.0. Writing “2.4 times the risk” is wrong. Say “about 2.4 times the odds”, or, better for most readers, report predicted probabilities for each group.
Predicted probabilities: numbers readers understand
Most readers, reviewers included, grasp “a predicted dropout probability of 28% versus 16%” far faster than “OR = 2.1”. Most software can compute predicted probabilities for groups with other variables held at typical values, or average marginal effects. A table or plot of predicted probabilities next to the OR table makes a paper noticeably stronger.
How many predictors is too many?
In logistic regression, information is limited by the number of cases in the smaller outcome group, not the total sample. If 60 of 1,000 students drop out, your data are only as strong as those 60 cases. The traditional rule of thumb was about 10 events per parameter; more recent methodological work argues that the right number depends on context and that sample size should be calculated more carefully. Either way, putting 15 predictors into a model with 60 events will almost certainly give unstable, inflated ORs and very wide confidence intervals.
Warning signs: enormous ORs (tens or hundreds), intervals running from 0.1 to 500, or convergence warnings. These often signal complete separation, where a predictor perfectly predicts the outcome in a small subgroup.
Checking the model
- Linearity in the logit for continuous predictors: try quadratic terms or splines, or group the variable to inspect the trend.
- Multicollinearity: check as in linear regression.
- Discrimination: the area under the ROC curve (AUC or C-statistic) shows how well the model separates the two groups.
- Calibration: do predicted probabilities match observed rates? The Hosmer–Lemeshow test is popular but sensitive to sample size; a calibration plot is more informative.
- Pseudo R-squared (Nagelkerke, McFadden) does not mean what R-squared means in linear regression; do not make it your headline measure.
In practice: a reporting template
- State the outcome, its coding and the number of cases in each group.
- List the predictors and why they were chosen (theory, prior work), not by automatic stepwise selection.
- Table: OR, 95% confidence interval and p-value for each predictor, with reference categories and units stated.
- One sentence interpreting the main result as odds, plus predicted probabilities.
- Model information: number of observations, AUC, calibration check.
Sample sentence: “After adjusting for gender, year of study and entry grades, students working more than 20 hours a week had higher odds of dropping out (OR = 2.1; 95% CI 1.3–3.4). The predicted dropout probability was 24% compared with 13% for students without jobs.”
Next step: open the logistic regression table in your most recent thesis or paper, find every sentence containing “risk” or “times more likely”, and check whether the number beside it really is a risk ratio. If the outcome is common, add predicted probabilities.
Câu hỏi thường gặp
When should I use logistic regression instead of linear regression?
When your outcome has two values, such as pass/fail or yes/no. Linear regression can predict probabilities outside 0–1 and does not capture the S-shaped relationship.
Is an odds ratio the same as a risk ratio?
No. An odds ratio compares two odds; a risk ratio compares two probabilities. When the outcome is common, the odds ratio is further from 1 and should not be described as “n times the risk”.
What does Exp(B) mean in SPSS output?
It is the exponentiated logistic coefficient, which is the odds ratio. B is on the log-odds scale; Exp(B) tells you how many times the odds change.
What sample size does logistic regression need?
Information is limited by the number of cases in the smaller outcome group. The old rule was about 10 events per parameter, but a context-specific sample size calculation is better.
How do I know if my logistic model is any good?
Check discrimination with the AUC, calibration with a plot or test, and whether coefficients are plausible. Do not rely on pseudo R-squared as the main measure.
