The Linear Model and Hypothesis

Seber, G. A. F.

doi:10.1007/978-3-319-21930-1

Cited by 17 publications

(15 citation statements)

References 12 publications

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“…Note that the systematic part of z ij , here μ ij , is approximated by a linear form. The model is referred to as a generalized linear model because errors are allowed to be non-Gaussian and the scale factors, σ ij , are not assumed constant ( Seber, 2015 ). In (1) , σ ij is a product of spatial ( σ i ) and temporal ( ϕ j ) factors.…”

Section: Methods: Basic Models and Analysis Techniquesmentioning

confidence: 99%

“…The duration of the scan time-frame is dt j and e −t j ζ , with ζ the decay constant for the radio-tracer isotope, is the standard tracer decay factor. While the estimates of α i based on these simple weights may not be optimal, under general conditions weighted least squares estimates are unbiased and also consistent, as the scale of the error diminishes ( Seber, 2015 ). In addition the Gauss-Markov theorem tells us that if weighting is inversely proportional to the variance of the measurement error, least squares will have minimum variance among all unbiased estimators ( Seber, 2015 ).…”

Section: Methods: Basic Models and Analysis Techniquesmentioning

confidence: 99%

“…While the estimates of α i based on these simple weights may not be optimal, under general conditions weighted least squares estimates are unbiased and also consistent, as the scale of the error diminishes ( Seber, 2015 ). In addition the Gauss-Markov theorem tells us that if weighting is inversely proportional to the variance of the measurement error, least squares will have minimum variance among all unbiased estimators ( Seber, 2015 ). In practice, unless there is a substantial discrepancy between optimal weights and simplistic weights, the latter will typically be highly efficient – see ( Romano and Wolf, 2017 ).…”

Section: Methods: Basic Models and Analysis Techniquesmentioning

confidence: 99%

“…This is to ensure there is adjustment for the bias arising from fitting the α -coefficients at each voxel. Such adjustments are standard in most parametric model fitting settings - see ( Kutner et al, 2013 ; Seber, 2015 ) for its use in linear regression.…”

Section: Methods: Basic Models and Analysis Techniquesmentioning

confidence: 99%

“…Remarkably, our analysis finds that the dependence is very limited. Based on linear regression theory ( Seber, 2015 ), the primary factor impacting the RMSE of uncertainty estimation relative to the scale of the noise, i.e.

, is the dimension of the model basis in relation to the number of data points ( K / T ).…”

Section: Assessment Of Performancementioning

confidence: 99%

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A Generalized Linear modeling approach to bootstrapping multi-frame PET image data

O’Sullivan

et al. 2021

Medical Image Analysis

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PET imaging is an important diagnostic tool for management of patients with cancer and other diseases. Medical decisions based on quantitative PET information could potentially benefit from the availability of tools for evaluation of associated uncertainties. Raw PET data can be viewed as a sample from an inhomogeneous Poisson process so there is the possibility to directly apply bootstrapping to raw projection-domain list-mode data. Unfortunately this is computationally impractical, particularly if data reconstruction is iterative or the acquisition protocol is dynamic. We develop a flexible statistical linear model analysis to be used with multi-frame PET image data to create valid bootstrap samples. The technique is illustrated using data from dynamic PET studies with fluoro-deoxyglucose (FDG) and fluoro-thymidine (FLT) in brain and breast cancer patients. As is often the case with dynamic PET studies, data have been archived without raw list-mode information. Using the bootstrapping technique maps of kinetic parameters and associated uncertainties are obtained. The quantitative performance of the approach is assessed by simulation. The proposed image-domain bootstrap is found to substantially match the projection-domain alternative. Analysis of results points to a close relation between relative uncertainty in voxel-level kinetic parameters and local reconstruction error. This is consistent with statistical theory.

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Section: Methods: Basic Models and Analysis Techniquesmentioning

confidence: 99%

Section: Methods: Basic Models and Analysis Techniquesmentioning

confidence: 99%

Section: Methods: Basic Models and Analysis Techniquesmentioning

confidence: 99%

Section: Methods: Basic Models and Analysis Techniquesmentioning

confidence: 99%

, is the dimension of the model basis in relation to the number of data points ( K / T ).…”

Section: Assessment Of Performancementioning

confidence: 99%

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A Generalized Linear modeling approach to bootstrapping multi-frame PET image data

O’Sullivan

et al. 2021

Medical Image Analysis

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A case for beta regression in the natural sciences

et al. 2022

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Data in the natural sciences are often in the form of percentages or proportions that are continuous and bounded by 0 and 1. Statistical analysis assuming a normal error structure can produce biased and incorrect estimates when data are doubly bounded. Beta regression uses an error structure appropriate for such data. We conducted a literature review of percent and proportion data from 2004 to 2020 to determine the types of analyses used for (0, 1) bounded data. Our literature review showed that before 2012, angular transformations accounted for 93% of analyses of proportion or percent data. After 2012, angular transformation accounted for 52% of analyses and beta regression accounted for 14% of analyses. We compared a linear model with angular transformation with beta regression using data from two fields of the natural sciences that produce continuous, bounded data: biogeochemistry and ecological elemental composition. We found little difference in model diagnostics, likelihood ratios, and p‐values between the two models. However, we found substantially different coefficient estimates from the back‐calculated beta regression and angular transformation models. Beta regression provides reliable parameter estimates in natural science studies where effect sizes are considered as important as hypothesis testing.

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Estimation

Christensen¹

2020

Springer Texts in Statistics

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The Linear Model and Hypothesis

Cited by 17 publications

References 12 publications

A Generalized Linear modeling approach to bootstrapping multi-frame PET image data

A Generalized Linear modeling approach to bootstrapping multi-frame PET image data

A case for beta regression in the natural sciences

Estimation

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