Infertility after Abortion

1:2 matched case-control study, real data

The study

Trichopoulos and colleagues studied whether induced and spontaneous abortions increase the risk of secondary infertility (Trichopoulos et al. 1976). Each of 83 infertile women (the cases) was matched with two fertile women (the controls) of the same age, parity and education. One set has a single control. The data are R’s infert, and they ship with the module: in jamovi, Open ▸ Data Library ▸ Conditional Logistic Regression ▸ Infertility after abortion. Or download infert.csv.

Variable	Content
`stratum`	matched set, 1–83
`case`	1 = infertile (case), 0 = fertile (control)
`spontaneous`, `induced`	number of previous spontaneous / induced abortions (0, 1, 2 or more)
`age`, `parity`, `education`	the matching factors
`pooled_stratum`	a coarser grouping of the sets, not used here

The basic analysis

Open the data and choose Analyses ▸ Regression ▸ Logistic Regression ▸ Conditional (Matched).
Move case to Dependent Variable, stratum to Matching Variable, and spontaneous and induced to Covariates.
In Model Fit, tick Overall model test. The odds ratios, with their 95% confidence intervals, are in the coefficient table by default.

Matched Sets
Per set = Cases	Controls	Sets	Observations
1	1	1	2
1	2	82	246

Model Fit Measures
Model	Deviance	AIC	R²-McF	χ²	df	p
1	128.404	132.404	0.293	53.154	2	< .001

Model Coefficients - case
		OR	Lower 95% CI	Upper 95% CI	p
spontaneous	per unit	7.285	3.651	14.536	< .001
induced	per unit	4.092	2.018	8.298	< .001

The matched-set table confirms the design: 82 sets of a case and two controls, one set with a single control, all informative. Each spontaneous abortion multiplies the odds of infertility by 7.29 (95% CI 3.65–14.54), and each induced abortion by 4.09 (2.02–8.30). Both effects hold among women of the same age, parity and education.

A count as a factor

Entering the number of abortions as a covariate assumes the same odds ratio for each additional one. To relax that for spontaneous abortions, enter it as a factor. In jamovi, spontaneous is a nominal integer, so it can go into Factors. Put induced in block 1 and the factor in block 2 of the Model Builder, so that the Model Comparisons table tests its addition. Then tick Likelihood ratio tests for the omnibus test of the factor.

Model Comparisons
Model		Model	χ²	df	p
1	-	2	53.079	2	< .001

Omnibus Likelihood Ratio Tests
Predictor	χ²	df	p
induced	19.149	1	< .001
spontaneous_f	53.079	2	< .001

Model Coefficients - case
		OR	Lower 95% CI	Upper 95% CI	p
induced	per unit	4.076	2.008	8.274	< .001
spontaneous_f	0	1.000			< .001
spontaneous_f	1	7.744	3.200	18.736	< .001
spontaneous_f	2	51.082	12.362	211.088	< .001

Against women with no spontaneous abortion, one multiplies the odds by 7.7 and two or more by 51.1 (12.4–211.1). The covariate coding implied 7.3 for one and 7.3² ≈ 53 for two. The factor agrees with it, so the simpler per-abortion odds ratio describes these data well.

Matching factors and effect modification

Age was matched on. Add age to Covariates and to the model, and also the interaction induced * age: select both variables in the Model Builder and choose Interaction from the arrow’s menu.

Model Coefficients - case
		OR	Lower 95% CI	Upper 95% CI	p
spontaneous	per unit	7.121	3.567	14.217	< .001
induced	per unit	1.508	0.042	54.462	.822
age	per unit
induced × age	per unit	1.032	0.923	1.154	.579

Note. Not estimable: age. A predictor that does not vary within the matched sets (e.g. a matching variable) or that is collinear with others cannot be estimated by conditional logistic regression.

Omnibus Likelihood Ratio Tests
Predictor	χ²	df	p
spontaneous	51.069	1	< .001
induced	0.050	1	.822
age
induced:age	0.309	1	.579

Note. A term with no estimable coefficient (e.g. a matching factor) is not tested

age has no coefficient and no test. It is constant within every matched set, so it cancels from the conditional likelihood, and the note says so. The interaction is still estimable, because it asks whether the effect of induced abortions differs between sets of different ages, and that is a within-set contrast. Here there is no evidence of it (p = .579). With the interaction in the model, the induced row is the odds ratio at age 0, which has no practical meaning. Look at the interaction test, not at that row.

What if the matching is ignored?

The same coefficients from ordinary logistic regression (R’s glm) of the same data:

Analysis	spontaneous OR	induced OR
Conditional (this module)	7.29	4.09
Unconditional, matching ignored	3.31	1.52
Unconditional, one indicator per set	25.29	8.94
Unconditional, adjusted for age, parity and education	7.74	3.63

Ignoring the matching shrinks both odds ratios to less than half, because the matching factors are related to the exposures. An indicator per matched set inflates them several-fold, which is the bias described in the method. Adjusting an unconditional model for the matching factors comes closer here, because the matching was on a few broad categories. The conditional analysis needs no such luck.

References

Trichopoulos, D., N. Handanos, J. Danezis, Anna Kalandidi, and Victoria Kalapothaki. 1976. “Induced Abortion and Secondary Infertility.” British Journal of Obstetrics and Gynaecology 83 (8): 645–50. https://doi.org/10.1111/j.1471-0528.1976.tb00904.x.