Fisher Scoring for crossed factor linear mixed models

The analysis of longitudinal, heterogeneous or unbalanced clustered data is of primary importance to a wide range of applications. The linear mixed model (LMM) is a popular and flexible extension of the linear model specifically designed for such purposes. Historically, a large proportion of material published on the LMM concerns the application of popular numerical optimization algorithms, such as Newton–Raphson, Fisher Scoring and expectation maximization to single-factor LMMs (i.e. LMMs that only contain one “factor” by which observations are grouped). However, in recent years, the focus of the LMM literature has moved towards the development of estimation and inference methods for more complex, multi-factored designs. In this paper, we present and derive new expressions for the extension of an algorithm classically used for single-factor LMM parameter estimation, Fisher Scoring, to multiple, crossed-factor designs. Through simulation and real data examples, we compare five variants of the Fisher Scoring algorithm with one another, as well as against a baseline established by the R package lme4, and find evidence of correctness and strong computational efficiency for four of the five proposed approaches. Additionally, we provide a new method for LMM Satterthwaite degrees of freedom estimation based on analytical results, which does not require iterative gradient estimation. Via simulation, we find that this approach produces estimates with both lower bias and lower variance than the existing methods.

Regresi Logistik untuk Pemodelan Indeks Pembangunan Kesehatan Masyarakat Kabupaten/Kota di Pulau Kalimantan

Regresi logistik merupakan model regresi yang paling sering digunakan untuk pemodelan data kategorik. Pada penelitian ini dilakukan pemodelan regresi logistik dan penerapannya pada Indeks Pembangunan Kesehatan Masyarakat (IPKM) kabupaten/kota di Pulau Kalimantan tahun 2013. Metode Maximum Likelihood Estimation (MLE) digunakan untuk penaksiran parameter. Metode Likelihood Ratio Test (LRT) dan uji Wald digunakan untuk pengujian parameter. Hasil penelitian menunjukkan bahwa penaksir parameter dengan metode MLE berbentuk fungsi yang tidak eksplisit. Sehingga digunakan pendekatan numerik dengan metode Fisher Scoring. Berdasarkan metode LRT dan uji Wald, statistik uji untuk pengujian parameter mendekati distribusi chi-square dan distribusi normal standar. Berdasarkan model regresi logistik terbaik, faktor-faktor yang berpengaruh terhadap IPKM kabupaten/kota di Pulau Kalimantan tahun 2013 adalah Indeks Pembangunan Manusia (IPM), tingkat kepadatan penduduk dan persentase penduduk miskin.

Improving Estimation Accuracy in Non\-randomized Response Questioning Methods by Multiple Answers

When private or stigmatizing characteristics are included in sample surveys, direct questions result in low cooperation of the respondents. To increase cooperation, indirect questioning procedures have been established in the literature. Nonrandomized response methods are one group of such procedures and have attracted much attention in recent years. In this article, we consider four popular nonrandomized response schemes and present a possibility to improve the estimation precision of these schemes. The basic idea is to require multiple indirect answers from each respondent. We develop a Fisher scoring algorithm for the maximum likelihood estimation in the presented new schemes and show the better efficiency of the new schemes compared with the original designs.

Improving Estimation Accuracy in Nonrandomized Response Questioning Methods by Multiple Answers

When private or stigmatizing characteristics are included in sample surveys, direct questions result in low cooperation of the respondents. To increase cooperation, indirect questioning procedures have been established in the literature. Nonrandomized response methods are one group of such procedures and have attracted much attention in recent years. In this article, we consider four popular nonrandomized response schemes and present a possibility to improve the estimation precision of these schemes. The basic idea is to require multiple indirect answers from each respondent. We develop a Fisher scoring algorithm for the maximum likelihood estimation in the presented new schemes and show the better efficiency of the new schemes compared with the original designs.