Avoiding zero between-study variance estimates in random-effects meta-analysis.

Fixed-effects meta-analysis has been criticized because the assumption of homogeneity is often unrealistic and can result in underestimation of parameter uncertainty. Random-effects meta-analysis and meta-regression are therefore typically used to accommodate explained and unexplained between-study variability. However, it is not unusual to obtain a boundary estimate of zero for the (residual) between-study standard deviation, resulting in fixed-effects estimates of the other parameters and their standard errors. To avoid such boundary estimates, we suggest using Bayes modal (BM) estimation with a gamma prior on the between-study standard deviation. When no prior information is available regarding the magnitude of the between-study standard deviation, a weakly informative default prior can be used (with shape parameter 2 and rate parameter close to 0) that produces positive estimates but does not overrule the data, leading to only a small decrease in the log likelihood from its maximum. We review the most commonly used estimation methods for meta-analysis and meta-regression including classical and Bayesian methods and apply these methods, as well as our BM estimator, to real datasets. We then perform simulations to compare BM estimation with the other methods and find that BM estimation performs well by (i) avoiding boundary estimates; (ii) having smaller root mean squared error for the between-study standard deviation; and (iii) better coverage for the overall effects than the other methods when the true model has at least a small or moderate amount of unexplained heterogeneity.

Diverging probability-density functions for flat-top solitary waves.

We investigate the statistics of flat-top solitary wave parameters in the presence of weak multiplicative dissipative disorder. We consider first propagation of solitary waves of the cubic-quintic nonlinear Schrödinger equation (CQNLSE) in the presence of disorder in the cubic nonlinear gain. We show by a perturbative analytic calculation and by Monte Carlo simulations that the probability-density function (PDF) of the amplitude eta exhibits loglognormal divergence near the maximum possible amplitude eta(m), a behavior that is similar to the one observed earlier for disorder in the linear gain [A. Peleg, Phys. Rev. E 72, 027203 (2005)]. We relate the loglognormal divergence of the amplitude PDF to the superexponential approach of eta to eta(m) in the corresponding deterministic model with linear/nonlinear gain. Furthermore, for solitary waves of the derivative CQNLSE with weak disorder in the linear gain both the amplitude and the group velocity beta become random. We therefore study analytically and by Monte Carlo simulations the PDF of the parameter p, where p = eta/(1-epsilon(s)beta/2) and epsilon(s) is the self-steepening coefficient. Our analytic calculations and numerical simulations show that the PDF of p is loglognormally divergent near the maximum p value.

Effects of dissipative disorder on front formation in pattern forming systems.

We study the effects of weak disorder in the linear gain coefficient on front formation in pattern forming systems described by the cubic-quintic nonlinear Schrödinger equation. We calculate the statistics of the front amplitude and position. We show that the distribution of the front amplitude has a loglognormal diverging form at the maximum possible amplitude and that the distribution of the front position has a lognormal tail. The theory is in good agreement with our numerical simulations. We show that these results are valid for other types of dissipative disorder and relate the loglognormal divergence of the amplitude distribution to the form of the emerging front tail.

A nondegenerate penalized likelihood estimator for variance parameters in multilevel models.

Group-level variance estimates of zero often arise when fitting multilevel or hierarchical linear models, especially when the number of groups is small. For situations where zero variances are implausible a priori, we propose a maximum penalized likelihood approach to avoid such boundary estimates. This approach is equivalent to estimating variance parameters by their posterior mode, given a weakly informative prior distribution. By choosing the penalty from the log-gamma family with shape parameter greater than 1, we ensure that the estimated variance will be positive. We suggest a default log-gamma(2,λ) penalty with λ → 0, which ensures that the maximum penalized likelihood estimate is approximately one standard error from zero when the maximum likelihood estimate is zero, thus remaining consistent with the data while being nondegenerate. We also show that the maximum penalized likelihood estimator with this default penalty is a good approximation to the posterior median obtained under a noninformative prior.Our default method provides better estimates of model parameters and standard errors than the maximum likelihood or the restricted maximum likelihood estimators. The log-gamma family can also be used to convey substantive prior information. In either case-pure penalization or prior information-our recommended procedure gives nondegenerate estimates and in the limit coincides with maximum likelihood as the number of groups increases.

Strong collapse turbulence in a quintic nonlinear Schrödinger equation.

We consider the quintic one-dimensional nonlinear Schrödinger equation with forcing and both linear and nonlinear dissipation. Quintic nonlinearity results in multiple collapse events randomly distributed in space and time, forming forced turbulence. Without dissipation each of these collapses produces finite-time singularity, but dissipative terms prevent actual formation of singularity. In statistical steady state of the developed turbulence, the spatial correlation function has a universal form with the correlation length determined by the modulational instability scale. The amplitude fluctuations at that scale are nearly Gaussian while the large-amplitude tail of the probability density function (PDF) is strongly non-Gaussian with powerlike behavior. The small-amplitude nearly Gaussian fluctuations seed formation of large collapse events. The universal spatiotemporal form of these events together with the PDFs for their maximum amplitudes define the powerlike tail of the PDF for large-amplitude fluctuations, i.e., the intermittency of strong turbulence.