A plausible s-factor solution for many types of psychological and educational tests is one that exhibits a general factor and s − 1 group or method related factors. The bi-factor solution results from the constraint that each item has a nonzero loading on the primary dimension and at most one of the s − 1 group factors. This paper derives a bi-factor item-response model for binary response data. In marginal maximum likelihood estimation of item parameters, the bi-factor restriction leads to a major simplification of likelihood equations and (a) permits analysis of models with large numbers of group factors; (b) permits conditional dependence within identified subsets of items; and (c) provides more parsimonious factor solutions than an unrestricted full-information item factor analysis in some cases.

A random effects probit model is developed for the case in which the same units are sampled repeatedly at each level of an independent variable. Because the observed proportions may be correlated under these conditions, estimating their trend with respect to the independent variable is no longer a standard problem for probit, logit or loglinear analysis. Using a qualitative analogue of a random regressions model, we employ instead marginal maximum likelihood to estimate the average latent trend line. Likelihood ratio tests of the hypothesis of no trend in the average line, and the hypothesis of no differences in average trend lines between experimental treatments, are proposed. We illustrate the model both with simulated data and with observed data from a clinical experiment in which psychiatric patients on two drug therapies are rated on five occasions for the presence or absence of symptoms.

We previously showed that folic acid prescriptions for any indication were associated with lower rates of suicidal behaviour. Given that future randomised clinical trials are likely to focus on psychiatric disorders carrying elevated risk for suicide, we now report on the moderating effects of prior suicidal behaviour, psychiatric diagnoses and psychotropic medications on potential antisuicidal effects of folic acid. Data were obtained from the MarketScan Commercial Claims and Encounters databases that cover 164 million insured persons from 2005–2017, from which a cohort of 866 586 patients was derived. Analysis revealed no significant moderation effects on the antisuicidal effect of folic acid. These findings indicate that the potential benefit of folic acid for preventing suicidal behaviour is comparable in psychiatric populations at higher risk of suicide and that it may be additive to any benefit from psychotropic medications.

The issue of intermittency in numerical solutions of the 3D Navier–Stokes equations on a periodic box ${[0, L] }^{3} $ is addressed through four sets of numerical simulations that calculate a new set of variables defined by ${D}_{m} (t)= {({ \varpi }_{0}^{- 1} {\Omega }_{m} )}^{{\alpha }_{m} } $ for $1\leq m\leq \infty $ where ${\alpha }_{m} = 2m/ (4m- 3)$ and ${[{\Omega }_{m} (t)] }^{2m} = {L}^{- 3} \int \nolimits _{\mathscr{V}} {\vert \boldsymbol{\omega} \vert }^{2m} \hspace{0.167em} \mathrm{d} V$ with ${\varpi }_{0} = \nu {L}^{- 2} $ . All four simulations unexpectedly show that the ${D}_{m} $ are ordered for $m= 1, \ldots , 9$ such that ${D}_{m+ 1} \lt {D}_{m} $ . Moreover, the ${D}_{m} $ squeeze together such that ${D}_{m+ 1} / {D}_{m} \nearrow 1$ as $m$ increases. The values of ${D}_{1} $ lie far above the values of the rest of the ${D}_{m} $ , giving rise to a suggestion that a depletion of nonlinearity is occurring which could be the cause of Navier–Stokes regularity. The first simulation is of very anisotropic decaying turbulence; the second and third are of decaying isotropic turbulence from random initial conditions and forced isotropic turbulence at fixed Grashof number respectively; the fourth is of very-high-Reynolds-number forced, stationary, isotropic turbulence at up to resolutions of $409{6}^{3} $ .