Reading the Output

Every table, and how its numbers are computed

The tables on this page come from one analysis of the infert data. Model 1 has the number of induced abortions; model 2 adds the number of spontaneous abortions, entered as a factor (0, 1, 2 or more).

Notation: \(\ell_0\) is the conditional log-likelihood at \(\boldsymbol\beta = 0\), \(\ell_1\) its maximum, \(k\) the number of estimated coefficients, \(n\) the number of observations analysed and \(d\) the number of cases.

Matched Sets

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

One row for each composition of a set (cases, controls), with the number of sets and the observations in them. This table is the first thing to check:

a matching ratio you did not expect usually means the Matching Variable is wrong (for example, a variable that is not the set identifier);
rows with 0 cases or 0 controls are uninformative sets. Usually missing values have removed every case or every control of the set. They do not enter the estimates, and the note gives how many sets are informative.

The counts are taken after removing the rows with a missing value in any variable of the analysis, as the models are fitted.

Descriptives

Descriptives ▸ Descriptive table (off by default).

Descriptives
		case = 0	case = 1	Total
case	n (%)	165 (66.5%)	83 (33.5%)	248 (100.0%)
induced	Mean ± SD	0.56 ± 0.73	0.59 ± 0.75	0.57 ± 0.74
spontaneous_f	0	113 (68.5%)	28 (33.7%)	141 (56.9%)
spontaneous_f	1	40 (24.2%)	31 (37.3%)	71 (28.6%)
spontaneous_f	2	12 (7.3%)	24 (28.9%)	36 (14.5%)

A first table for a paper:

the dependent variable: the numbers of controls and cases, with their percentages of the total;
each covariate: mean ± SD in controls, in cases and in total;
each factor: n (%) for every level, with percentages of the known values within each column;
a Missing row, n (% of the column), under a variable only if it has missing values (see the endometrial cancer use case).

The table describes every row whose case/control status and matched set are known, and includes every variable in Covariates and Factors, whether or not it is in a model block. The models use fewer rows when there are missing values: only those complete for every variable.

Model Fit Measures

Model Fit Measures
Model	Deviance	AIC	BIC	R²-McF	R²-CS	R²-N	χ²	df	p
1	181.436	183.436	185.854	0.001	0.000	0.001	0.123	1	.726
2	128.357	134.357	141.613	0.293	0.193	0.372	53.202	3	< .001

Column	Definition
Deviance	\(-2\ell_1\)
AIC	\(-2\ell_1 + 2k\)
BIC	\(-2\ell_1 + k\log d\): the number of cases is the sample size, as `BIC()` uses for a Cox model
R²_McF	McFadden: \(1 - \ell_1/\ell_0\)
R²_CS	Cox & Snell: \(1 - \exp\{2(\ell_0 - \ell_1)/n\}\)
R²_N	Nagelkerke: R²_CS divided by its maximum, \(1 - \exp(2\ell_0/n)\)
Overall Model Test	likelihood ratio test of all coefficients against zero: \(\chi^2 = 2(\ell_1 - \ell_0)\) on \(k\) df

The null model is \(\boldsymbol\beta = 0\): within each set, every subject is equally likely to be the case. The pseudo R² values therefore measure what the predictors explain within matched sets. They are not comparable with those of an unmatched logistic regression, whose null model has a single intercept.

Model Comparisons

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

Likelihood ratio test between consecutive blocks: \(\chi^2 = 2(\ell_1^{(2)} - \ell_1^{(1)})\), on as many df as coefficients were added. Every model is fitted to the same observations (those complete for all the variables in any block), so the models are nested.

Omnibus Likelihood Ratio Tests

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

One test per term. The model is refitted without that term’s columns, and the fit is compared with the full model. A factor with several levels gets a single test on several df. With an interaction in the model, the test of a main effect depends on the reference levels, as in jmv.

In jmv’s logistic regression these tests come from car::Anova. Here they are computed by the module, because car substitutes Wald tests for any model with strata.

Model Coefficients

Model Coefficients - case
		OR	Lower	Upper	p	p trend	OR	Lower Multivariable	Upper Multivariable	p	p trend	Estimate	SE	Z
induced	per unit	1.077	0.714	1.624	0.725		4.076	2.008	8.274	< .001		1.405	0.361	3.891
spontaneous_f	0	1.000			0.000	0.000	1.000			< .001	0.000
spontaneous_f	1	3.776	1.838	7.758	0.000		7.744	3.200	18.736	< .001		2.047	0.451	4.540
spontaneous_f	2	9.719	3.805	24.825	0.000		51.082	12.362	211.088	< .001		3.933	0.724	5.434

One row per level, with the variable named once on the left (the layout of the prognosis module’s Cox regression):

a factor starts with its reference level, OR = 1, set in the Reference Levels box (default: the first level), then one row for each other level, labelled by the level alone and compared with the reference;
a covariate has one row, per unit: the OR for a one-unit increase;
an interaction has one row per coefficient, labelled by the levels involved;
there is no intercept row (see the method).

OR is the odds ratio of being a case (the non-reference level of the dependent variable) rather than a control, within the matched sets. Its confidence interval is a Wald interval, exponentiated (\(\exp(\hat\beta \pm z_{1-\alpha/2}\,\mathrm{SE})\), as confint() gives for clogit), at the level set in Confidence interval (95% by default), and p is the Wald test. On the reference row of a factor with three or more levels, p is the likelihood ratio test of the whole factor, the same as in the omnibus table.

Covariate Scaling, a box like Reference Levels, sets the unit of the odds ratio of each covariate: per unit (the default), per 1 SD (the standard deviation on the rows of the model, shown in the label), or per 10, per 100 or per 1000 units. The covariate is divided by that amount before fitting, so the OR becomes \(\mathrm{OR}_{\text{unit}}^{\,c}\) for a scale \(c\), with its interval scaled alike. The p-values, the likelihood ratio tests and the fit measures do not change. Univariable odds ratios, interactions and the forest plot use the same units.

The options of Model Coefficients:

Univariable and multivariable adds the odds ratios of each variable alone (Univariable) beside those of the model (Multivariable). Both are estimated on the same rows, those complete for every variable of the model, so the difference between them is the adjustment alone.
Trend test (factors with 3+ levels) adds p trend on the reference row: the factor replaced by scores 1, 2, 3, … in the order of its levels (Wald test of the score). The univariable trend uses the factor alone, and the multivariable trend keeps the other variables of the model. The scores follow the order of the levels in the data, whichever level is the reference, so order the levels before testing. There is no trend test for a factor that is part of an interaction.
Log odds ratio, SE and Z adds the estimate \(\hat\beta\) (the log odds ratio), its standard error and \(Z = \hat\beta / \mathrm{SE}\) at the right.

Forest Plot

Forest plot draws the odds ratios of each model, one line per row of the table. Reference levels are open diamonds at 1. With Univariable and multivariable, both estimates are drawn, univariable in grey.

Notes

The notes that explain each table (how the odds ratios are defined, the number of rows used, what the matched-set table means) are hidden by default. Tick Show notes in Options to see them:

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

Note. Odds ratios (OR) of "case = 1" vs. "case = 0" within the matched sets, with 95% confidence intervals. Factors: OR against the reference level (OR = 1); on the reference row of a factor with three or more levels, p of the likelihood ratio test of the whole factor. Covariates: OR per one-unit increase, or per the unit chosen in Covariate Scaling (SD of the rows of the model). Multivariable ORs are adjusted for the other variables in the model.

Warnings are shown whatever this option says: predictors that cannot be estimated (Not estimable: …), fitting problems, the number of uninformative matched sets, and reference levels not found.

Collinearity Statistics

Collinearity Statistics
	VIF	Tolerance
induced	2.150	0.465
spontaneous_f	2.150	0.465

Variance inflation factors from the correlation matrix of the estimated coefficients. For a term with more than one coefficient, such as a factor with three levels, the table gives the generalised VIF of the term as a whole (Fox and Monette 1992), with tolerance = 1/VIF. This is the formula of car::vif, except that the strata are not counted as a term.

What changed from jmv’s logistic regression

jmv Binomial Logistic Regression	condLogReg
`glm(family = binomial)`	`coxph(... + strata(set), method = "exact")`, as `clogit`
—	Matching Variable, Likelihood option, Matched Sets table
—	Descriptive table by case/control status, with missing values
estimates, SE, Z first; odds ratios and their intervals on request	odds ratios, CI and p by default, one row per level with the reference level at OR = 1 (as the prognosis module); log OR, SE and Z on request
—	univariable beside multivariable odds ratios, trend test, forest plot
notes always shown	explanatory notes on request (Show notes); warnings always
intercept	none (absorbed by the matched sets)
Tjur’s R²	removed (needs fitted probabilities)
omnibus tests from `car::Anova`	likelihood ratio tests computed by refitting
VIF from `car::vif`	the same GVIF, with the strata excluded
estimated marginal means	removed (need absolute probabilities)
prediction: classification, ROC, AUC, cut-off	removed (need absolute probabilities)
saved predicted values, residuals, Cook’s distance	removed
data weights	ignored: a weight would break the matched sets

References

Fox, John, and Georges Monette. 1992. “Generalized Collinearity Diagnostics.” Journal of the American Statistical Association 87 (417): 178–83. https://doi.org/10.1080/01621459.1992.10475190.