A Note on the Transformation of Chi-Squared Variables to Normality

Hawkins, Douglas M.; Wixley, R. A. J.

doi:10.2307/2684608

Cited by 33 publications

(27 citation statements)

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“…Transformation can also hurt the imputation of X, because transformation introduces curvature into the relationship between X and Y. In Figure 4, for example, we have an exponentially distributed X, which, as we saw in Section 2.3, can be transformed to a very good approximation of normality by a fourth root transformation √ (Hawkins and Wixley 1986). Yet if we use this transformation in a bivariate setting, the imputation model will be misspecified and incompatible with the analysis model.…”

Section: Transformationmentioning

confidence: 99%

“…To return to our earlier example, if we sample an exponential variable ~ and transform it with a log, the logged variable , is far from normal (Hawkins and Wixley 1986). If we impute a normal variable , ~ , on the log scale, inverting the transformation will yield an imputed , whose mean and variance are far from the mean and variance of the original exponential variable (He and Raghunathan 2006).…”

Section: Transformationmentioning

confidence: 99%

“…Unlike the log, both the square root and the fourth root are defined at 0. The square root transformation is common in data analysis, while the fourth root is less common but comes about as close to normalizing the exponential distribution as any transformation can (Hawkins and Wixley 1986 , has a mean of 1.006 and a variance of 1.135. The mean of has less than 1% bias, while the variance of , is positively biased by 13.5%.…”

Section: Transformationmentioning

confidence: 99%

“…Note that with 2 the chi-square distribution reduces to the standard exponential distribution that we have used in all our examples up to this point. (Hawkins and Wixley 1986). f. The transform all method: linear regression imputation where √ is regressed on .…”

Section: Bivariate Designmentioning

confidence: 99%

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Should a Normal Imputation Model be Modified to Impute Skewed Variables?

Hippel

2012

Sociological Methods & Research

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(169 words)Researchers often impute continuous variables under an assumption of normality, yet many incomplete variables are skewed. We find that imputing skewed continuous variables under a normal model can lead to bias; the bias is usually mild for popular estimands such as means, standard deviations, and linear regression coefficients, but the bias can be severe for more shapedependent estimands such as percentiles or the coefficient of skewness. We test several methods for adapting a normal imputation model to accommodate skewness, including methods that transform, truncate, or censor (round) normally imputed values, as well as methods that impute values from a quadratic or truncated regression. None of these modifications reliably reduces the biases of the normal model, and some modifications can make the biases much worse. We conclude that, if one has to impute a skewed variable under a normal model, it is usually safest to do so without modifications-unless you are more interested in estimating percentiles and shape that in estimated means, variance, and regressions. In the conclusion, we briefly discuss promising developments in the area of continuous imputation models that do not assume normality.

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Section: Transformationmentioning

confidence: 99%

Section: Transformationmentioning

confidence: 99%

Section: Transformationmentioning

confidence: 99%

Section: Bivariate Designmentioning

confidence: 99%

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Should a Normal Imputation Model be Modified to Impute Skewed Variables?

Hippel

2012

Sociological Methods & Research

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“…10, 2017; We then compared the values of ∆lnL for genes with weak evidence of positive selection ( Figure 1). In order to improve the normality of non-zero ∆lnL, we transformed ∆lnL with fourth root (Hawkins and Wixley 1986;Roux et al 2014;Daub et al 2017). …”

Section: Modularity Analysismentioning

confidence: 99%

Adaptive evolution of animal proteins over development: support for the Darwin selection opportunity hypothesis of Evo-Devo

Liu

Robinson‐Rechavi

2017

Preprint

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Animal morphological diversity is shaped both by adaptation and by developmental constraints. Here we have tested Darwin's "selection opportunity" hypothesis, according to which high evolutionary divergence in late development is due to strong positive selection. We contrasted it to a "developmental constraint" hypothesis, according to which late development is under relaxed negative selection. Indeed, the highest divergence, both at the morphological and molecular levels, is observed late there is a consistent signal that positive selection mainly affects genes and pathways expressed in late embryonic development and in adult. Our results imply that the evolution of embryogenesis is mostly conservative, with most adaptive evolution affecting post-embryonic gene expression, and thus post-embryonic phenotypes. This is consistent with the diversity of environmental challenges to which juveniles and adults are exposed.

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Bayesian Target‐Vector Optimization for Efficient Parameter Reconstruction

Plock

Andrle

Burger

et al. 2022

Advcd Theory and Sims

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Parameter reconstructions are indispensable in metrology. Here, the objective is to explain K experimental measurements by fitting to them a parameterized model of the measurement process. The model parameters are regularly determined by least-square methods, that is, by minimizing the sum of the squared residuals between the K model predictions and the K experimental observations, 𝝌 2 . The model functions often involve computationally demanding numerical simulations. Bayesian optimization methods are specifically suited for minimizing expensive model functions. However, in contrast to least-square methods such as the Levenberg-Marquardt algorithm, they only take the value of 𝝌 2 into account, and neglect the K individual model outputs. A Bayesian target-vector optimization scheme with improved performance over previous developments, that considers all K contributions of the model function and that is specifically suited for parameter reconstruction problems which are often based on hundreds of observations is presented. Its performance is compared to established methods for an optical metrology reconstruction problem and two synthetic least-squares problems. The proposed method outperforms established optimization methods. It also enables to determine accurate uncertainty estimates with very few observations of the actual model function by using Markov chain Monte Carlo sampling on a trained surrogate model.

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A Note on the Transformation of Chi-Squared Variables to Normality

Cited by 33 publications

References 0 publications

Should a Normal Imputation Model be Modified to Impute Skewed Variables?

Should a Normal Imputation Model be Modified to Impute Skewed Variables?

Adaptive evolution of animal proteins over development: support for the Darwin selection opportunity hypothesis of Evo-Devo

Bayesian Target‐Vector Optimization for Efficient Parameter Reconstruction

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