This article discusses the use of a symmetric multiplicative interaction effect to capture certain types of third-order dependence patterns often present in social networks and other dyadic datasets. Such an effect, along with standard linear fixed and random effects, is incorporated into a generalized linear model, and a Markov chain Monte Carlo algorithm is provided for Bayesian estimation and inference. In an example analysis of international relations data, accounting for such patterns improves model fit and predictive performance.

Bilinear Mixed Effects Models for Dyadic Data

The stability of yield is an important characteristic to be considered when judging the value of a cropping system relative to others. In the context of agricultural research, the analysis of yield stability has been largely confined to multienvironment trials of crop cultivars. This review emphasizes that methods for comparing the stability of cultivars can also be used for comparing different agronomic treatments in general, of which cultivars are but a special case. Throughout the paper, different agronomic treatments are referred to as cropping systems. Some of the methods useful for stability analysis of cropping systems are discussed and a brief review of applications of these methods is given. The paper puts different stability measures into a unifying mixed model perspective.

Methods for Comparing the Yield Stability of Cropping Systems

SUMMARY We introduce a family of multivariate binary distributions with certain conditional linear property. This family is particularly useful for efficient and easy simulation of correlated binary variables with a given marginal mean vector and correlation matrix.

A family of multivariate binary distributions for simulating correlated binary variables with specified marginal means and correlations

where Yij mean yield of ith genotype in jth environment, , = general mean, gi = effect of ith genotype, ej = effect of jth environment, (ge)ij = interaction of ith genotype and jth environment, Eij random error associated with means Yij, assumed to be distributed as N(O, ac2). This model is appropriate for the analysis of means from equally replicated data with homoscedastic errors. In an analysis of G x E data, it may be useful to fit a more specific model describing the interaction. The most common of such models is the regression model first suggested by Yates and Cochran (1938), which was further elaborated by Finlay and Wilkinson (1963) and Eberhart and Russell (1966). It may be written as

Analyzing genotype-environment data by mixed models with multiplicative terms

[1] We consider the problem of simulating sequences of daily rainfall at a network of sites in such a way as to reproduce a variety of properties realistically over a range of spatial scales. The properties of interest will vary between applications but typically will include some measures of “extreme” rainfall in addition to means, variances, proportions of wet days, and autocorrelation structure. Our approach is to fit a generalized linear model (GLM) to rain gauge data and, with appropriate incorporation of intersite dependence structure, to use the GLM to generate simulated sequences. We illustrate the methodology using a data set from southern England and show that the GLM is able to reproduce many properties at spatial scales ranging from a single site to 2000 km2 (the limit of the available data).

/pdf/spatial-temporal-rainfall-simulation-using-generalized-4g3xs83f9c.pdf

Spatial-temporal rainfall simulation using generalized linear models

SUMMARY Many applications use simple parametric models for the correlation structure of binary responses which are observed in clusters. The usual approach, to use correlation models appropriate for normally distributed responses, suffers from two drawbacks when the marginal probabilities within the clusters differ. First, as it does not explicitly take into account constraints on the second moments which must be satisfied for binary responses, it may fail to model realistically the range of correlations present in the data. Secondly, computer simulation of observations from these models is very difficult. We present an alternative class of correlation models which reflect the binary nature of the responses and allow for simple simulation. Some key wor-ds: Binary variable; Computer simulation; Correlation structure; Generalised estimating equation.

/pdf/modelling-and-generating-correlated-binary-variables-3w5o8cpy9t.pdf

Modelling and generating correlated binary variables

SUMMARY A class of two-way mixed analysis of variance models is proposed, in which the fixed and random effects enter multiplicatively. Equations are developed for iterative computation of maximum likelihood estimates via a scoring algorithm. Parameter estimation and hypothesis testing are illustrated on a set of plant genetics data.

Multiplicative effects in mixed model analysis of variance

SUMMARY Switch-back designs, originally proposed for cow lactation experiments, are repeated measurement designs appropriate for experiments in which the responses of experimental units vary with time according to different rates. A method for constructing such designs to compare an arbitrary number of treatments is presented, together with their analysis of variance. Robustness of the switch-back design when the treatments have residual as well as main effects is considered. Switch-back and balanced residual designs are compared when the treatment has three levels. In repeated measurement designs, also referred to as before-and-after, reversal, crossover, change-over, and multiple time series designs, each experimental unit is assigned more than once to a treatment. We consider a particular such design, the switch-back design, originally proposed by Brandt (1938) for comparing the effects of two different feeds on the milk yields of dairy cattle. This design adjusts for the natural decline at rates which vary widely from cow to cow in milk yield, during successive periods of lactation, by switching the feeds given to the cows in a balanced manner. Lucas (1956) extended this design to cover up to nine treatments, and computed the analysis of variance for treatment, and possibly block, effects. For ease of exposition our discussion is couched in terms of lactation experiments, although clearly our results are applicable in other contexts as well. In ? 2 we construct switch-back designs for an arbitrary number of treatments applied in three consecutive time periods. Section 3 provides the complete analysis of variance for treatment, period, lactation curve slope, cow and block effects. Lucas's (1956) F statistic for treatment effects is identical to ours if no block effects are assumed; however, when block effects are assumed, his error sum of squares has fewer degrees of freedom, since his analysis is based on a singular transformation of the data, so that the sum of squares for blocks must be partitioned from the error sum of squares. Also, he defines the block effect using the curvatures of the lactation curves of the cows in the block, while we, more appropriately we believe, use the cows' average milk yields. In ? 4 we consider the robustness of switch-back designs to residual effects, which occur when each treatment has a main effect on the experimental unit when it is applied, and a residual effect during the immediately following time period. In this case we derive the distribution of the switch-back test statistic for treatment effects, when residual effects are present.

Switch-back designs

SUMMARY We consider the following linear calibration problem. Two scalar quantities X and Y are related by a simple linear regression of Y on X, and at the calibration step repeated measurements on both X and Y are available for a number of sampling units. At the prediction step a Y measurement, possibly together with previously obtained Y measurements, is available for a new sampling unit, and we wish to estimate the corresponding unknown X. Both the intercept and the slope of the regression are allowed to vary between units, resulting in a random regression coefficient model at the calibration step. As a result, at the prediction step the unknown X affects both the mean and covariance structure of Y Point and interval estimates for X are obtained and illustrated on a set of biomedical data.

Calibration with random slopes

SUMMARY In nonlinear calibration and regression a double point is defined with respect to estimation and to confidence sets. Loosely speaking, if a future observation is a double point then multiple point estimates or disconnected confidence regions will occur. We investigate the extent of the set of double points. A diagnostic, the probability of their occurrence, is computed.

Samuel D. Oman

Papers

Modelling and generating correlated binary variables

Multiplicative effects in mixed model analysis of variance

Switch-back designs

Calibration with random slopes

Double points in nonlinear calibration