A Nested Case-Control Study

1:4 matching with a variable ratio, simulated data

The study

These data are simulated, so that the true odds ratios are known. They mimic a case-control study nested in a cohort. For each of 200 people who developed a disease (the cases), four controls were drawn from the members of the cohort of the same sex and age group who were still free of it. In some sets only two or three suitable controls were found. Download nested.csv.

Variable	Content
`set`	matched set, 1–200
`case`	1 = case, 0 = control
`sex`, `agegrp`	the matching factors
`smoking`	never, former, current
`coffee`	cups per day (missing for 8 subjects)
`active`	physically active: yes, no

The data were generated with true odds ratios of 1.5 for former and 2.5 for current smoking (both against never), 0.6 for being active, and 1 for coffee. But smokers drink more coffee (on average 2.8 cups a day for current smokers against 1.6 for never smokers), so smoking confounds the effect of coffee.

Step 1: the reference level of a text factor

Analyses ▸ Regression ▸ Logistic Regression ▸ Conditional (Matched). case to Dependent Variable, set to Matching Variable, coffee to Covariates, smoking and active to Factors.

jamovi orders the levels of a text variable alphabetically, so smoking starts with current, and that becomes the reference level:

Model Coefficients - case
		OR	Lower 95% CI	Upper 95% CI	p
smoking	current	1.000			< .001
smoking	former	0.663	0.455	0.966	.032
smoking	never	0.393	0.264	0.585	< .001

These odds ratios are correct, but they compare the wrong way round for this question. Open Reference Levels and set never for smoking (and no for active).

Step 2: blocks to see confounding

In the Model Builder, keep coffee in block 1. Click Add New Block and move smoking and active into block 2.
In Model Coefficients, tick Likelihood ratio tests. The odds ratios, with their 95% confidence intervals, are shown by default.

Matched Sets
Per set = Cases	Controls	Sets	Observations
0	4	2	8
1	2	16	48
1	3	34	136
1	4	148	740

Note. 2 uninformative sets (no case or no control left)

Model Comparisons
Model		Model	χ²	df	p
1	-	2	20.288	3	< .001

Model Coefficients - case
		OR	Lower 95% CI	Upper 95% CI	p
coffee	per unit	1.197	1.036	1.384	.015

Model Coefficients - case
		OR	Lower 95% CI	Upper 95% CI	p
coffee	per unit	1.039	0.883	1.222	.648
smoking	never	1.000			< .001
smoking	current	2.333	1.491	3.650	< .001
smoking	former	1.690	1.095	2.608	.018
active	no	1.000
active	yes	0.662	0.477	0.920	.014

Alone (model 1), each daily cup of coffee seems to raise the odds of disease by 20% (OR 1.20, p = .015). Once smoking and activity are in the model (model 2), the odds ratio for coffee is 1.04 (0.88–1.22): the crude association was smoking. The smoking odds ratios, 2.33 for current and 1.69 for former smokers, and 0.66 for activity, are compatible with the values used to simulate the data. The model comparison (χ² = 20.3, 3 df, p < .001) tests both added terms together.

Step 3: what the matched-set table tells

The table above shows the variable matching ratio: 148 sets with four controls, 34 with three and 16 with two. The conditional likelihood handles any mix of ratios: no weighting or special option is needed.

It also shows 2 sets with no case. In those sets the case has no coffee value, so the case’s row is dropped. The controls left behind have nobody to be compared with, and those sets contribute nothing to the estimates. Whenever the table shows sets with 0 cases or 0 controls, check the missing values of the case and controls in those sets. If the variable with missing values is not needed, leaving it out of the analysis brings those sets back.

Step 4: collinearity

Tick Collinearity statistics in Assumption Checks:

Collinearity Statistics
	VIF	Tolerance
coffee	1.238	0.808
smoking	1.239	0.807
active	1.002	0.998

coffee and smoking are correlated, which is the confounding seen above, but their VIFs are far from the values (5 or 10) usually taken as a warning. smoking has two coefficients, so its VIF is the generalised VIF of the factor as a whole.