Coefficient Estimates and the Müntz Theorem

2005 ◽  
pp. 71-92
Author(s):  
Vladimir I. Gurariy ◽  
Wolfgang Lusky
Keyword(s):  
Coefficient Estimates ◽  
Müntz Theorem

scholarly journals Understanding, Choosing, and Unifying Multilevel and Fixed Effect Approaches

2020 ◽  
pp. 1-20
Author(s):  
Chad Hazlett ◽  
Leonard Wainstein
Keyword(s):  
Random Effects ◽  
Fixed Effects ◽  
Multilevel Models ◽  
Standard Errors ◽  
Point Estimate ◽  
Fe Model ◽  
Grouped Data ◽  
Coefficient Estimates ◽  
Political Science Education ◽  
Robust Standard Errors

Abstract When working with grouped data, investigators may choose between “fixed effects” models (FE) with specialized (e.g., cluster-robust) standard errors, or “multilevel models” (MLMs) employing “random effects.” We review the claims given in published works regarding this choice, then clarify how these approaches work and compare by showing that: (i) random effects employed in MLMs are simply “regularized” fixed effects; (ii) unmodified MLMs are consequently susceptible to bias—but there is a longstanding remedy; and (iii) the “default” MLM standard errors rely on narrow assumptions that can lead to undercoverage in many settings. Our review of over 100 papers using MLM in political science, education, and sociology show that these “known” concerns have been widely ignored in practice. We describe how to debias MLM’s coefficient estimates, and provide an option to more flexibly estimate their standard errors. Most illuminating, once MLMs are adjusted in these two ways the point estimate and standard error for the target coefficient are exactly equal to those of the analogous FE model with cluster-robust standard errors. For investigators working with observational data and who are interested only in inference on the target coefficient, either approach is equally appropriate and preferable to uncorrected MLM.

scholarly journals Comparing the use of aggregate data and various methods of integrating individual patient data to network meta-analysis and its application to first-line ART

2021 ◽  
Vol 21 (1) ◽  
Author(s):  
Steve Kanters ◽  
Mohammad Ehsanul Karim ◽  
Kristian Thorlund ◽  
Aslam H. Anis ◽  
Michael Zoratti ◽  
...  
Keyword(s):  
Meta Analysis ◽  
Evidence Base ◽  
Aggregate Data ◽  
World Health ◽  
Coefficient Estimates ◽  
Two Stage ◽  
Individual Patient Level ◽  
One Stage ◽  
Meta Regression ◽  
Empirical Priors

Abstract Background The 2018 World Health Organization HIV guidelines were based on the results of a network meta-analysis (NMA) of published trials. This study employed individual patient-level data (IPD) and aggregate data (AgD) and meta-regression methods to assess the evidence supporting the WHO recommendations and whether they needed any refinements. Methods Access to IPD from three trials was granted through ClinicalStudyDataRequest.com (CSDR). Seven modelling approaches were applied and compared: 1) Unadjusted AgD network meta-analysis (NMA) – the original analysis; 2) AgD-NMA with meta-regression; 3) Two-stage IPD-AgD NMA; 4) Unadjusted one-stage IPD-AgD NMA; 5) One-stage IPD-AgD NMA with meta-regression (one-stage approach); 6) Two-stage IPD-AgD NMA with empirical-priors (empirical-priors approach); 7) Hierarchical meta-regression IPD-AgD NMA (HMR approach). The first two were the models used previously. Models were compared with respect to effect estimates, changes in the effect estimates, coefficient estimates, DIC and model fit, rankings and between-study heterogeneity. Results IPD were available for 2160 patients, representing 6.5% of the evidence base and 3 of 24 edges. The aspect of the model affected by the choice of modeling appeared to differ across outcomes. HMR consistently generated larger intervals, often with credible intervals (CrI) containing the null value. Discontinuations due to adverse events and viral suppression at 96 weeks were the only two outcomes for which the unadjusted AgD NMA would not be selected. For the first, the selected model shifted the principal comparison of interest from an odds ratio of 0.28 (95% CrI: 10.17, 0.44) to 0.37 (95% CrI: 0.23, 0.58). Throughout all outcomes, the regression estimates differed substantially between AgD and IPD methods, with the latter being more often larger in magnitude and statistically significant. Conclusions Overall, the use of IPD often impacted the coefficient estimates, but not sufficiently as to necessitate altering the final recommendations of the 2018 WHO Guidelines. Future work should examine the features of a network where adjustments will have an impact, such as how much IPD is required in a given size of network.

Coefficient estimates for univalent polynomials

1977 ◽  
Vol 31 (1) ◽  
pp. 112-124 ◽  
Author(s):  
David Horowitz
Keyword(s):  
Coefficient Estimates

scholarly journals Properties of Certain Subclass of Meromorphic Multivalent Functions Associated with q-Difference Operator

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1035
Author(s):  
Cai-Mei Yan ◽  
Rekha Srivastava ◽  
Jin-Lin Liu
Keyword(s):  
Difference Operator ◽  
Necessary Conditions ◽  
Partial Sums ◽  
Coefficient Estimates ◽  
Radius Of Starlikeness ◽  
Closure Theorems ◽  
Sufficient And Necessary Conditions ◽  
Distortion Theorems ◽  
Growth And Distortion Theorems

A new subclass Σp,q(α,A,B) of meromorphic multivalent functions is defined by means of a q-difference operator. Some properties of the functions in this new subclass, such as sufficient and necessary conditions, coefficient estimates, growth and distortion theorems, radius of starlikeness and convexity, partial sums and closure theorems, are investigated.

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