Deletion, replacement and mean-shift for diagnostics in linear mixed models

Shi, Lei; Chen, Gemai

doi:10.1016/j.csda.2011.07.002

Cited by 6 publications

(6 citation statements)

References 21 publications

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“… The ML estimation method: ML estimator involves an iterative computation. Let

Q ((), τ^{2}) = V^{- 1} ((), τ^{2}) - P ((), τ^{2}), P ((), τ^{2}) = V^{- 1} ((), τ^{2}) X {(X^{Τ} V^{- 1} (τ^{2}) X)}^{- 1} X^{Τ} V^{- 1} ((), τ^{2}),

then it is easy to verify that the iterative formula for estimating τ 2 satisfying , 2.5

τ^{2} = \frac{Y^{Τ} Q ((), {normalτ}^{2}) Q^{Τ} ((), {normalτ}^{2}) Y - italictr ((), V^{- 1} ((), {normalτ}^{2}) S V^{- 1} ((), {normalτ}^{2}))}{italictr ((), V^{- 2} ((), {normalτ}^{2}))} .

…”

Section: Meta‐regression Model and Estimation Methodsmentioning

confidence: 99%

“…The ML estimation of τ 2 is obtained iteratively from Equation . For example, firstly, an initial estimator of τ 2 is inserted into the right hand of Equation to find a new estimator, and this new estimator is again substituted into the right hand of Equation to get a renewed estimator . This process continues until the estimator is converged and a converged estimator is denoted by

{\hat{τ}}_{ML}^{2}

.…”

Section: Meta‐regression Model and Estimation Methodsmentioning

confidence: 99%

“…The deletion formula for the estimator

\overset{β}{true} ((), {normalτ}^{2})

of β is

{\overset{truê}{italicβ}}_{[a]} ((), {normalτ}^{2}) = \overset{β}{true} ((), {normalτ}^{2}) - {(X^{normalT} V^{- 1} (τ^{2}) X)}^{- 1} X^{T} V^{- 1} ((), {normalτ}^{2}) D_{a} {(D_{a}^{T} Q (τ^{2}) D_{a})}^{- 1} D_{a}^{T} Q ((), τ^{2}) Y .

…”

Section: Subset Deletion Formulae In Meta‐regressionmentioning

confidence: 99%

“…For ML estimation, it is impossible to find an exact deletion formula since it need iterative computation. However an one‐step approximation can be used in this situation Pregibon, chapter 5, which is done by using ML estimator

{\hat{normalτ}}_{ML}^{2}

under the full data to replace the τ 2 in the right hand of Equation under deletion model. The result is given by

{\overset{truê}{normalτ}}_{italicML ([], a)}^{2} = \frac{Y^{Τ} H_{a}^{2} ((), {\overset{τ}{true}}_{italicML}^{2}) Y - italictr ((), N_{a} ((), {\overset{τ}{true}}_{italicML}^{2}) S N_{a} ((), {\overset{τ}{true}}_{italicML}^{2}))}{italictr ((), N_{a}^{2} ((), {\overset{τ}{true}}_{italicML}^{2}))},

where N a ( τ 2 ) = π( V −1 (τ 2 ), D a ) and H a (τ 2 ) = π( Q (τ 2 ), D a ).…”

Section: Subset Deletion Formulae In Meta‐regressionmentioning

confidence: 99%

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Influence diagnostics in meta‐regression model

Shi

Zuo

et al. 2017

Research Synthesis Methods

Self Cite

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This paper studies the influence diagnostics in meta-regression model including case deletion diagnostic and local influence analysis. We derive the subset deletion formulae for the estimation of regression coefficient and heterogeneity variance and obtain the corresponding influence measures. The DerSimonian and Laird estimation and maximum likelihood estimation methods in meta-regression are considered, respectively, to derive the results. Internal and external residual and leverage measure are defined. The local influence analysis based on case-weights perturbation scheme, responses perturbation scheme, covariate perturbation scheme, and within-variance perturbation scheme are explored. We introduce a method by simultaneous perturbing responses, covariate, and within-variance to obtain the local influence measure, which has an advantage of capable to compare the influence magnitude of influential studies from different perturbations. An example is used to illustrate the proposed methodology.

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“… The ML estimation method: ML estimator involves an iterative computation. Let

Q ((), τ^{2}) = V^{- 1} ((), τ^{2}) - P ((), τ^{2}), P ((), τ^{2}) = V^{- 1} ((), τ^{2}) X {(X^{Τ} V^{- 1} (τ^{2}) X)}^{- 1} X^{Τ} V^{- 1} ((), τ^{2}),

then it is easy to verify that the iterative formula for estimating τ 2 satisfying , 2.5

τ^{2} = \frac{Y^{Τ} Q ((), {normalτ}^{2}) Q^{Τ} ((), {normalτ}^{2}) Y - italictr ((), V^{- 1} ((), {normalτ}^{2}) S V^{- 1} ((), {normalτ}^{2}))}{italictr ((), V^{- 2} ((), {normalτ}^{2}))} .

…”

Section: Meta‐regression Model and Estimation Methodsmentioning

confidence: 99%

{\hat{τ}}_{ML}^{2}

.…”

Section: Meta‐regression Model and Estimation Methodsmentioning

confidence: 99%

“…The deletion formula for the estimator

\overset{β}{true} ((), {normalτ}^{2})

of β is

{\overset{truê}{italicβ}}_{[a]} ((), {normalτ}^{2}) = \overset{β}{true} ((), {normalτ}^{2}) - {(X^{normalT} V^{- 1} (τ^{2}) X)}^{- 1} X^{T} V^{- 1} ((), {normalτ}^{2}) D_{a} {(D_{a}^{T} Q (τ^{2}) D_{a})}^{- 1} D_{a}^{T} Q ((), τ^{2}) Y .

…”

Section: Subset Deletion Formulae In Meta‐regressionmentioning

confidence: 99%

{\hat{normalτ}}_{ML}^{2}

under the full data to replace the τ 2 in the right hand of Equation under deletion model. The result is given by

{\overset{truê}{normalτ}}_{italicML ([], a)}^{2} = \frac{Y^{Τ} H_{a}^{2} ((), {\overset{τ}{true}}_{italicML}^{2}) Y - italictr ((), N_{a} ((), {\overset{τ}{true}}_{italicML}^{2}) S N_{a} ((), {\overset{τ}{true}}_{italicML}^{2}))}{italictr ((), N_{a}^{2} ((), {\overset{τ}{true}}_{italicML}^{2}))},

where N a ( τ 2 ) = π( V −1 (τ 2 ), D a ) and H a (τ 2 ) = π( Q (τ 2 ), D a ).…”

Section: Subset Deletion Formulae In Meta‐regressionmentioning

confidence: 99%

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Influence diagnostics in meta‐regression model

Shi

Zuo

et al. 2017

Research Synthesis Methods

Self Cite

View full text Add to dashboard Cite

show abstract

“…One‐step estimates can be used to achieve this 34,36,37. An alternative approach is to apply the ‘delete = replace’ identity, which allows for the update of the variance components from quantities available from the original model fit19; however, Shi and Chen38 show that this method is not equivalent to deletion for the linear mixed model (and thus the HLM).…”

Section: Influence Analysismentioning

confidence: 99%

Diagnostic tools for hierarchical linear models

Loy

Hofmann

2012

WIREs Computational Stats

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Hierarchical structures are omnipresent in today's society—this is reflected in the data that we collect on all aspects of this society. Hierarchical linear models allow a representation of structural levels in a statistical modeling framework. Diagnostic tools are used to assess the quality of model estimation and explore features of the data not well described by the model. Residual and influence diagnostics are familiar tools for the classical regression model with independent observations. For hierarchical linear models, these diagnostic tools must be adjusted to reflect the dependence introduced by the nested data structure. Residual analysis now includes the assessment of distributional assumptions at each level of the model. This requires the use of level‐dependent residual quantities. Similarly, the parameter estimates may be influenced at each level of the model, requiring influence diagnostics that can pinpoint specific levels of the model, as well as specific aspects of the model. We present an overview of the diagnostic tools available for hierarchical linear models that are familiar from linear models. Additionally, we discuss the utility of the lineup protocol for residual analysis with complex models. WIREs Comput Stat 2013, 5:48–61. doi: 10.1002/wics.1238 This article is categorized under: Statistical Models > Bayesian Models Statistical and Graphical Methods of Data Analysis > Bayesian Methods and Theory Statistical and Graphical Methods of Data Analysis > Modeling Methods and Algorithms Statistical and Graphical Methods of Data Analysis > Statistical Graphics and Visualization

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An Exploratory Study of Marital and Quality of Life Ratings Among Male Spouses of Military Members

Lufkin

2017

Contemp Fam Ther

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Deletion, replacement and mean-shift for diagnostics in linear mixed models

Cited by 6 publications

References 21 publications

Influence diagnostics in meta‐regression model

Influence diagnostics in meta‐regression model

Diagnostic tools for hierarchical linear models

An Exploratory Study of Marital and Quality of Life Ratings Among Male Spouses of Military Members

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