We present a brief overview of the methods for making statistical inference (testing statistical hypotheses, construction of confidence and/or prediction intervals and regions) about linear functions of the fixed effects and/or about the fixed and random effects simultaneously, in conventional simple linear mixed model. The presented approach is based on solutions from the Henderson’s mixed model equations.
Exact Likelihood Ratio Test for the Parameters of the Linear Regression Model with Normal Errors
In this paper we present an exact likelihood ratio test (LRT) for testing the simple null hypothesis on all parameters of the linear regression model with normally distributed errors. In particular, we consider the simultaneous test for the regression parameters, β, and the error standard deviation, σ. The critical values of the LRT are presented for small sample sizes and a small number of explanatory variables for usual significance levels, α = 0.1, 0.05, and 0.01. The test is directly applicable for construction of the (1 - α)-confidence region for the parameters (β,σ) and the simultaneous tolerance intervals for future observations in linear regression models. For comparison, the suggested method for construction of the tolerance factors of the symmetric (1 - γ)-content simultaneous (1 - α)-tolerance intervals is illustrated by a simple numerical example.
Omnia in mensura et numero et pondere disposuisti is a famous Latin phrase from Solomon’s Book of Wisdom, dated to the mid first century BC, meaning that all things were ordered in measure, number, and weight. Naturally, the wisdom is appearing in its relation to man. The Wisdom of Solomon is understood as the perfection of knowledge of the righteous as a gift from God showing itself in action. Consequently, a natural and obvious conjecture is that measurement science is the science of sciences. In fact, it is a basis of all experimental and theoretical research activities. Each measuring process assumes an object of measurement. Some science disciplines, such as quantum physics, are still incomprehensible despite complex mathematical interpretations. No phenomenon is a real phenomenon unless it is observable in space and time, that is, unless it is a subject to measurement. The science of measurement is an indispensable ingredient in all scientific fields. Mathematical foundations and interpretation of the measurement science were accepted and further developed in most of the scientific fields, including physics, cosmology, geology, environment, quantum mechanics, statistics, and metrology. In this year, 2020, Measurement Science Review celebrates its 20th anniversary and we are using this special opportunity to highlight the importance of measurement science and to express our faith that the journal will continue to be an excellent place for exchanging bright ideas in the field of measurement science. As an illustration and motivation for usage and further development of mathematical methods in measurement science, we briefly present the simple least squares method, frequently used for measurement evaluation, and its possible modification. The modified least squares estimation method was applied and experimentally tested for magnetic field homogeneity adjustment.
Linear models with variance-covariance components are used in a wide variety of applications. In most situations it is possible to partition the response vector into a set of independent subvectors, such as in longitudinal models where the response is observed repeatedly on a set of sampling units (see, e.g., Laird & Ware 1982). Often the objective of inference is either a test of linear hypotheses about the mean or both, the mean and the variance components. Confidence intervals for parameters of interest can be constructed as an alter- native to a test. These questions have kept many statisticians busy for several decades. Even under the assumption that the response can be modeled by a multivariate normal distribution, it is not clear what test to recommend except in a few settings such as balanced or orthogonal designs. Here we investigate statistical properties, such as accuracy of p-values and powers of exact (Crainiceanu & Ruppert 2004) tests and compare with properties of approximate asymptotic tests. Simultaneous exact confidence regions for variance components and mean parameters are constructed as well.
Model Based Determination of Detection Limits for Proton Transfer Reaction Mass Spectrometer
Proton Transfer Reaction Mass Spectrometry (PTR-MS) is a chemical ionization mass spectrometric technique which allows to measure trace gases as, for example, in exhaled human breath. The quantification of compounds at low concentrations is desirable for medical diagnostics. Typically, an increase of measuring accuracy can be achieved if the duration of the measuring process is extended. For real time measurements the time windows for measurement are relatively short, in order to get a good time resolution (e.g. with breath-to-breath resolution during exercise on a stationary bicycle). Determination of statistical detection limits is typically based on calibration measurements, but this approach is limited, especially for very low concentrations. To overcome this problem, a calculation of limit of quantification (LOQ) and limit of detection (LOD), respectively, based on a theoretical model of the measurement process is outlined.