“Factors associated with” is the most common objective in health research theses: factors associated with institutional delivery, with stunting, with modern contraceptive use. The analysis is almost always the same: binary logistic regression, reported as crude and adjusted odds ratios. This guide goes through it step by step, with the checks reviewers ask about.
1. When to use it
Use binary logistic regression when the outcome has two categories (delivered at a facility or at home; stunted or not) and you want to know which factors go with it. The factors can be categories (education, residence) or numbers (age, distance). For an outcome with three or more categories, or a count, or a measurement, a different model fits: multinomial or ordinal logistic, Poisson, or linear regression.
2. Step 1: each factor alone (bivariable analysis)
Fit a model with the outcome and one factor at a time. Each gives a crude odds ratio (COR): the association before taking anything else into account.
Most studies then carry forward factors with p < 0.25 into the multivariable model. The cut-off comes from purposeful selection (Hosmer and Lemeshow; Bursac et al., 2008): it is deliberately loose, so that a factor whose effect only shows once others are adjusted for is not dropped too early. It is a convention, not a law. A factor known from earlier studies or theory to matter, such as a likely confounder, can go in whatever its crude p. Say so in the methods.
3. A worked crude odds ratio
Place of delivery by number of antenatal care (ANC) visits, among 600 women:
| ANC visits | Health facility | Home | Total |
|---|---|---|---|
| 4 or more | 180 (a) | 70 (b) | 250 |
| Fewer than 4 | 150 (c) | 200 (d) | 350 |
COR = (a × d) / (b × c) = (180 × 200) / (70 × 150) = 3.43
The 95% confidence interval is worked on the log scale (Woolf, 1955):
SE = √(1/a + 1/b + 1/c + 1/d) = √(1/180 + 1/70 + 1/150 + 1/200) = 0.178
exp(ln 3.43 ± 1.96 × 0.178) = 2.42 to 4.86
So: COR 3.43 (95% CI 2.42 to 4.86). Women with four or more visits had 3.43 times the odds of delivering at a facility. The interval does not include 1, so the association is statistically significant. A logistic regression with this one factor gives exactly the same result.
4. Step 2: the multivariable model
Put the selected factors into one model together. Each now gets an adjusted odds ratio (AOR): its association with the outcome when the other factors in the model are held constant. Women with more ANC visits may also be more educated and live closer to a facility; the AOR separates the part that belongs to ANC itself.
Expect the AOR to differ from the COR. If it falls a lot, part of the crude association was due to the other factors. Factors with p < 0.05 in this model are usually called significantly associated.
5. Checks before you report
- Enough events. A common rule is at least 10 outcome events (in the smaller outcome group) for each parameter in the model (Peduzzi et al., 1996). A factor with 4 categories uses 3 parameters. With 120 women in the smaller outcome group, that is 12 parameters at most.
- Multicollinearity. Two factors that measure nearly the same thing (such as education and literacy) make both estimates unstable. Check the variance inflation factor (VIF); values above 10, or by stricter rules above 5, are a warning.
- Model fit. The Hosmer–Lemeshow test compares predicted and observed outcomes. Here a p above 0.05 is what you want: no evidence of poor fit.
- Empty or tiny cells. A category with almost no events gives huge odds ratios with enormous intervals (an AOR of 45 with a CI from 2 to 900). Merge categories that make sense together.
6. When the outcome is common, odds ratios mislead
In the example, 55% of women delivered at a facility. Among women with four or more visits it was 72%; with fewer, 43%. That is a prevalence ratio of 0.72 / 0.43 = 1.68. The odds ratio was 3.43.
Both are correct, but they answer different questions. When the outcome is rare (under about 10%), the odds ratio is close to the prevalence or risk ratio. When it is common, the odds ratio is much further from 1 (Zhang & Yu, 1998), and “3.43 times more likely” would be wrong: they were 1.68 times as likely. Write “3.43 times the odds”, or, better for a common outcome, use modified Poisson regression (Poisson regression with robust standard errors; Zou, 2004), which gives adjusted prevalence ratios directly.
7. The table and the methods paragraph
One table carries both steps. Here is how it looks (the adjusted numbers are made up for illustration):
| Variable | Facility | Home | COR (95% CI) | AOR (95% CI) |
|---|---|---|---|---|
| ANC visits | ||||
| 4 or more | 180 | 70 | 3.43 (2.42–4.86) | 2.61 (1.78–3.83) |
| Fewer than 4 | 150 | 200 | 1 | 1 |
Give the counts in each outcome group, mark the reference category with 1, show every factor that entered the model (not only the significant ones), and put the significant AORs in bold. Then the methods:
“Binary logistic regression was used to identify factors associated with institutional delivery. Each factor was first fitted alone (bivariable analysis), and factors with p < 0.25 were entered into a multivariable model. Crude (COR) and adjusted odds ratios (AOR) are reported with 95% confidence intervals; p < 0.05 was taken as statistically significant. Multicollinearity was checked with the variance inflation factor, and model fit with the Hosmer–Lemeshow test.”
8. Common mistakes
- “Times more likely” for an odds ratio of a common outcome. Say “times the odds”, or report prevalence ratios.
- “Risk factor” and “effect” in a cross-sectional study. It shows association, not cause; say “associated with”.
- p values without confidence intervals. The interval shows how precise the estimate is; always give it.
- No reference category, or one with very few people, which makes every other estimate unstable.
- Ignoring the sampling design. With cluster sampling, ordinary standard errors are too small. Use survey-design (weighted, clustered) analysis or a multilevel model.
Do this in Researchite
Pick the outcome and the factors, and Researchite fits each alone, carries forward those under your cut-off, and builds the COR and AOR table with Hosmer–Lemeshow, ready for Word. It also fits modified Poisson, survey-design and multilevel models, writes the methods paragraph, and gives you the Stata, R and SPSS commands to check it.
References
- Bursac Z, Gauss CH, Williams DK, Hosmer DW. Purposeful selection of variables in logistic regression. Source Code Biol Med. 2008;3:17.
- Hosmer DW, Lemeshow S, Sturdivant RX. Applied Logistic Regression. 3rd ed. Hoboken: John Wiley & Sons; 2013.
- Peduzzi P, Concato J, Kemper E, Holford TR, Feinstein AR. A simulation study of the number of events per variable in logistic regression analysis. J Clin Epidemiol. 1996;49(12):1373–9.
- Woolf B. On estimating the relation between blood group and disease. Ann Hum Genet. 1955;19(3):251–3.
- Zhang J, Yu KF. What’s the relative risk? A method of correcting the odds ratio in cohort studies of common outcomes. JAMA. 1998;280(19):1690–1.
- Zou G. A modified Poisson regression approach to prospective studies with binary data. Am J Epidemiol. 2004;159(7):702–6.
Tesfahun Mulatu
Epidemiologist and biostatistician with more than 10 years in academia and research, including multi-country and mixed-methods studies with local and international research institutions, and monitoring, evaluation and learning across Ethiopia’s development and humanitarian sectors on projects funded by UN agencies and international donors. Tesfahun’s work spans maternal and child health, sexual and reproductive health, gender-based violence and mental health, with over 19 publications.
Epidemiologist and biostatistician, founder of Researchite · LinkedIn