Several textbooks give the incorrect formula to calculate the standard error estimates for standardized regression coefficients. As a remedy, two analytic methods have been developed: (1) the delta method and (2) the covariance structure modeling method. However, neither method is applicable to compute the standard error estimates for unstandardized regression coefficients of products of Z-scores. In the literature, a nonparametric bootstrap procedure is advocated to test the significance of unstandardized regression coefficients of products of Z-scores. In this article, we propose a simple analytic approach that can produce the standard error estimates not only for standardized regression coefficients when interaction terms are not included, but also for unstandardized regression coefficients when interaction terms of Z-scores are included. Two numeric examples are used to compare our analytic approach with the existing methods, and simulation studies are conducted to further evaluate the performances of regular regression, our analytic approach, and the nonparametric bootstrap procedure at finite sample sizes. It is found that (1) regular regression performs well only when the variances of predictor variables are small, (2) our analytic approach performs well at the sample size of 200 or larger, and (3) the nonparametric bootstrap procedure performs (almost) perfectly in all conditions.

Four measurement designs are presented for use with correlation coefficients corrected, in one variable, for attenuation due to unreliability—coefficients that we term partially disattenuated correlation coefficients. Asymptotic expressions are derived for the variances and covariances of the estimates accompanying each design. Empirical simulation results that bear on the preceding mathematical developments are then presented. In addition to providing insights into the distributions of the estimates, the empirical results demonstrate satisfactory Type I error control for typical inferential applications. Power is shown to be equal to or greater than that of corresponding product-moment correlations in three of the four designs. Implications for practice are discussed.

The large sample distribution of total indirect effects in covariance structure models in well known. Using Monte Carlo methods, this study examines the applicability of the large sample theory to maximum likelihood estimates oftotal indirect effects in sample sizes of 50, 100, 200, 400, and 800. Two models are studied. Model 1 is a recursive model with observable variables and Model 2 is a nonrecursive model with latent variables. For the large sample theory to apply, the results suggest that sample sizes of 200 or more and 400 or more are required for models such as Model 1 and Model 2, respectively.

A model and computational procedure based on classical test score theory are presented for determination of a correlation coefficient corrected for attenuation due to unreliability. Next, variance-covariance expressions for the sample estimates defined earlier are derived, based on application of the delta method. Results of a Monte Carlo study are presented in which the adequacy of the derived expressions was assessed for a large number of data forms and potential hypotheses encountered in the behavioral sciences. It is shown that, based on the proposed procedures, confidence intervals for single coefficients are reasonably precise. Two-sample hypothesis tests, for both independent and dependent samples, are also accurate. However, for hypothesis tests involving a larger number of coefficients than two—both independent and dependent—the proposed procedures require largens for adequate precision. Results of a preliminary power analysis reveal no serious loss in efficiency resulting from correction for attenuation. Implications for practice are discussed.

In this paper, we prove uniform bounds for $\operatorname {GL}(3)\times \operatorname {GL}(2) \ L$-functions in the $\operatorname {GL}(2)$ spectral aspect and the t aspect by a delta method. More precisely, let $\phi $ be a Hecke–Maass cusp form for $\operatorname {SL}(3,\mathbb {Z})$ and f a Hecke–Maass cusp form for $\operatorname {SL}(2,\mathbb {Z})$ with the spectral parameter $t_f$. Then for $t\in \mathbb {R}$ and any $\varepsilon>0$, we have

$$\begin{align*}L(1/2+it,\phi\times f) \ll_{\phi,\varepsilon} (t_f+|t|)^{27/20+\varepsilon}. \end{align*}$$

$L(1/2+it,\phi \times f)$

$|t|-t_f \gg (|t|+t_f)^{3/5+\varepsilon }$

Presented and illustrated is an easy-to-implement and flexible methodology for the analysis of synergistic and antagonistic effects when the effects are defined as nonlinear functions of means. The methodology augments standard mixed-model analyses with the Delta method for standard errors of nonlinear functions of means. Explained is why standard ANOVA methods that have been adopted in the literature are not recommended. To illustrate the methodology, the joint-action effects of fenoxaprop with companion herbicides in two-component mixtures for weed control in rice were evaluated. The companion herbicides were halosulfuron, bispyribac-sodium, bensulfuron, penoxsulam, carfentrazone, quinclorac, and imazethapyr. Weeds evaluated were barnyardgrass and broadleaf signalgrass. Experiments in Louisiana and Mississippi in 2009 revealed a preponderance of antagonistic effects. The analysis showed that mixtures with bispyribac-sodium, penoxsulam, quinclorac, and imazethapyr were generally the most antagonistic and provided the least control.

A generalization of the delta method is proposed. The asymptotic distribution of g(Y n ) is obtained when g is not necessarily a differentiable function but verifies some weaker regularity conditions. An application to diversity profiles is also presented.