\(H_0^C\): is the supported number of classes four in every stratum?#

Definition

\(H_0^C\) (order). Null: the supported number of latent classes is \(K = 4\) within every stratum along both axes, with no class splitting or merging. Alternative: the supported order changes in one or more strata. Estimand: the general finite mixture model order per coarse stratum, read comparatively against the pooled cohort put through the identical procedure.

Status

Not yet run. The stratified order stage has not been executed, so there is no per-stratum supported order to report. Running it will decide whether four components are supported within every stratum of age at diagnosis and diagnostic era, or whether a class splits or merges in some stratum, and it will report the pooled reference order the strata are read against.

Method#

The invariance and prevalence questions ask whether the four reference classes change shape or size along an axis. This question is different: whether the number of classes the data support changes from one stratum to the next. The estimand is the supported order per stratum, and the read is comparative. Each stratum is judged against the pooled cohort put through the identical procedure, not against the number four, because the pooled information criteria over-extract at this sample size (their minimum sits at nine classes, so a raw criterion is not a usable target). Whatever drives that over-extraction, most likely feature misspecification, is shared by the pooled cohort and every stratum, so comparing a stratum’s supported order to the pooled cohort’s cancels it. An order change is a stratum whose supported order differs from the pooled cohort’s, and the pooled order is itself a reported finding.

The confirmatory statistic is a warm-started parametric bootstrap likelihood-ratio test. The number of components is not a nested comparison a chi-square can adjudicate: the null that puts one class’s mixing weight at zero sits on the boundary of the parameter space, so the usual asymptotics fail. The bootstrap test builds the null distribution of the statistic by simulation instead. For one step comparing \(K\) classes against \(K+1\), the observed statistic is twice the log-likelihood gain from the extra class, \(\text{LR} = 2(\ell_{K+1} - \ell_K)\); datasets are simulated from the fitted \(K\)-component model at the stratum’s own sample size and each is put through the same fitting recipe to give a null \(\text{LR}\); and the \(p\)-value is the Phipson-Smyth add-one bootstrap \(p\), so its smallest attainable value is \(1 / (B + 1)\) rather than zero. The fits are measurement-only, on hard subsets of the cohort matrix, so the order question is about the measurement model alone. The test is valid only when the observed and simulated datasets are fitted the same way, so the recipe is fixed and applied identically on both sides: the \(K\)-class fit keeps the best of a handful of random restarts, and the \(K+1\)-class fit is warm-started by splitting each of the \(K\) classes in turn (halving its mixing weight into two children and perturbing its class-conditional parameters symmetrically), giving a good starting point near the class most likely to divide.

The search is anchored at four classes and steps outward. The primary splitting direction tests four against five, and while a step rejects it steps up to a cap of seven; the secondary merging direction tests four against three when the first splitting step does not reject, and steps down to a floor of two while a step fails to reject. The search stops at the first splitting step that does not reject, or the first merging step that does, and the supported order is where it stops. The identical search on the pooled cohort gives the reference order every stratum is read against. Two further reads sit beside the bootstrap test as corroborators, neither deciding on its own: the cross-validated log-likelihood elbow, whose knee is found by an inline Kneedle construction, and the adjusted Lo-Mendell-Rubin test, the proper corrected likelihood-ratio referred to a chi-square. The confirmatory draws are staged: each step is screened at a smaller number of draws and escalated to the full schedule only where the screen \(p\) is close, so the expensive draws are spent only where the decision is tight.

This is the analysis order stage, run per axis and reusing the pooled search across both axes. The full procedure, its knobs, and its cache are documented in the guide, testing the number of classes.

Experimental design decisions#

Comparative to the pooled cohort, not to four. The pooled information criteria over-extract at this sample size, so a raw criterion target is unusable. Reading each stratum against the pooled cohort through the identical recipe cancels the shared over-extraction, and the pooled order is reported in its own right.
The cross-validated elbow, not the raw information criteria. The information criteria fall across the whole grid and reach their minimum at nine classes on the pooled cohort, the known over-extraction of these criteria at a large sample, so they do not point to a usable order. The out-of-sample cross-validated log-likelihood gains little past four classes and is the honest substitute; its knee is read objectively by the Kneedle construction rather than off a plot.
Coarse strata, not the finest bins. The strata are the four equal-frequency quantile bins of the axis. A thousand-proband bin cannot estimate a fifth class, so a null result in a bin that small would be a power artefact rather than evidence of stability; the finer binning is deliberately not used for this question.
Three corroborators must agree for a positive claim. A stratum matching the pooled order is the stability result, and the bootstrap test not rejecting past the pooled order is enough for it. A positive order-change claim asks for more: the bootstrap order must differ from the pooled order, the cross-validated elbow knee must move off it too, and the adjusted Lo-Mendell-Rubin test must agree at the decisive comparison. Requiring the three to agree guards against reading a single close call as a genuine change.

Results#

The stratified order test is not yet run, so there are no results to report. When it runs it will report, per axis, the pooled reference order and each stratum’s supported order anchored at four, together with the per-step bootstrap \(p\)-values, the cross-validated elbow knee, the adjusted Lo-Mendell-Rubin \(p\), whether the three corroborators agree, and whether the order changed against the pooled cohort. The read is comparative, so the finding will be stated as whether every stratum matches the pooled order (stability) or some stratum departs from it (a class splitting or merging). No numbers or figures are shown here until the stage has been executed.

Handling the null#

When the stage runs, the decision rule will be applied in the conditional. A stratum whose supported order matches the pooled order is the stability result, and the bootstrap test failing to reject past the pooled order is enough to keep \(H_0^C\) for that stratum. \(H_0^C\) would be rejected in a stratum only if three conditions hold together: the bootstrap order differs from the pooled order, the cross-validated elbow knee moves off the pooled order in the same stratum, and the adjusted Lo-Mendell-Rubin test agrees at the decisive comparison. A single close call among the three is not enough. Degeneracy is handled by the identical-recipe null itself, since a spurious empty-class fit that inflates the observed statistic inflates the null draws the same way; only numerically broken fits are dropped, and a stratum is flagged if more than one draw in ten is dropped.

Discussion#

\(H_0^C\) is the third of the primary, unconditionally tested questions, alongside the invariance and prevalence reads. It asks whether the four-way partition itself survives stratification, rather than whether its class shapes or sizes drift. Because the pooled criteria over-extract, the answer is framed relative to the pooled cohort rather than to an absolute count of four, and a positive order-change claim is deliberately demanding. The verdict, and the pooled reference order it rests on, await the run.