A logistic approximation to the cumulative normal distribution

Bowling, Shannon R.; Khasawneh, Mohammad T.; Kaewkuekool, Sittichai; Cho, Byung Rae

doi:10.3926/jiem.2009.v2n1.p114-127

Cited by 99 publications

(83 citation statements)

References 2 publications

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“…The approximate equivalence of the CDF for the standard normal distribution and the logistic distribution has long been noted [7-10]; an example of this can be seen in Figure 1. Having demonstrated that the conditional probability of disease from a model on the logit scale can be written as a logistic CDF, and similarly for the liability scale as a standard normal CDF, we use this approximate equivalence to equate the disease risks from both models.…”

Section: Methodsmentioning

confidence: 99%

Transforming Summary Statistics from Logistic Regression to the Liability Scale: Application to Genetic and Environmental Risk Scores

2018

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Objective: Stratified medicine requires models of disease risk incorporating genetic and environmental factors. These may combine estimates from different studies, and the models must be easily updatable when new estimates become available. The logit scale is often used in genetic and environmental association studies; however, the liability scale is used for polygenic risk scores and measures of heritability, but combining parameters across studies requires a common scale for the estimates. Methods: We present equations to approximate the relationship between univariate effect size estimates on the logit scale and the liability scale, allowing model parameters to be translated between scales. Results: These equations are used to build a risk score on the liability scale, using effect size estimates originally estimated on the logit scale. Such a score can then be used in a joint effects model to estimate the risk of disease, and this is demonstrated for schizophrenia using a polygenic risk score and environmental risk factors. Conclusion: This straightforward method allows the conversion of model parameters between the logit and liability scales and may be a key tool to integrate risk estimates into a comprehensive risk model, particularly for joint models with environmental and genetic risk factors.

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Section: Methodsmentioning

confidence: 99%

Transforming Summary Statistics from Logistic Regression to the Liability Scale: Application to Genetic and Environmental Risk Scores

2018

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“…Therefore, we use a generalized logistic (GL) function to approximate the ECDF. It has been proven that a logistic function can be used to accuractely approximate the CDF of a normal distribution [11]. In this algorithm, we do not make any assumption on the distribution of the data; therefore, we use a more general form of the logistic function, called the generalized logistic (GL) function …”

Section: Methodsmentioning

confidence: 99%

A robust data scaling algorithm to improve classification accuracies in biomedical data

2016

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BackgroundMachine learning models have been adapted in biomedical research and practice for knowledge discovery and decision support. While mainstream biomedical informatics research focuses on developing more accurate models, the importance of data preprocessing draws less attention. We propose the Generalized Logistic (GL) algorithm that scales data uniformly to an appropriate interval by learning a generalized logistic function to fit the empirical cumulative distribution function of the data. The GL algorithm is simple yet effective; it is intrinsically robust to outliers, so it is particularly suitable for diagnostic/classification models in clinical/medical applications where the number of samples is usually small; it scales the data in a nonlinear fashion, which leads to potential improvement in accuracy.ResultsTo evaluate the effectiveness of the proposed algorithm, we conducted experiments on 16 binary classification tasks with different variable types and cover a wide range of applications. The resultant performance in terms of area under the receiver operation characteristic curve (AUROC) and percentage of correct classification showed that models learned using data scaled by the GL algorithm outperform the ones using data scaled by the Min-max and the Z-score algorithm, which are the most commonly used data scaling algorithms.ConclusionThe proposed GL algorithm is simple and effective. It is robust to outliers, so no additional denoising or outlier detection step is needed in data preprocessing. Empirical results also show models learned from data scaled by the GL algorithm have higher accuracy compared to the commonly used data scaling algorithms.

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“…Compared to the Logistic Function used in [10], this general form of Logistic Function provides us with the flexibility to approximate a more variety of distributions. One of the notable properties of (7) is that it maps the values in the interval (∞,−∞) to the interval the interval (0,1 In order to approximate the ecdf, we need to learn the parameters Q, B, M , and ν from the data, so that the GLF could best fit the ecdf .…”

Section: B Cumulative Density Function Approximationmentioning

confidence: 99%

“…Therefore, we propose to use a generalized logistic function (GLF) to approximate the ecdf. Using a Logistic Function to approximate the cdf of a normal distribution was proven viable and accurate [10]. In this algorithm, we do not make any assumption on the distribution of the data; therefore, we use a more general form of the Logistic function…”

Section: B Cumulative Density Function Approximationmentioning

confidence: 99%

A robust data scaling algorithm for gene expression classification

Cao

Obradović

2015

2015 IEEE 15th International Conference on Bioinformatics and Bioengineering (BIBE)

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Gene expression data are widely used in classification tasks for medical diagnosis. Data scaling is recommended and helpful for learning the classification models. In this study, we propose a data scaling algorithm to transform the data uniformly to an appropriate interval by learning a generalized logistic function to fit the empirical cumulative density function of the data. The proposed algorithm is robust to outliers, and experimental results show that models learned using data scaled by the proposed algorithm generally outperform the ones using min-max mapping and z-score which are currently the most commonly used data scaling algorithms.

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A logistic approximation to the cumulative normal distribution

Cited by 99 publications

References 2 publications

Transforming Summary Statistics from Logistic Regression to the Liability Scale: Application to Genetic and Environmental Risk Scores

Transforming Summary Statistics from Logistic Regression to the Liability Scale: Application to Genetic and Environmental Risk Scores

A robust data scaling algorithm to improve classification accuracies in biomedical data

A robust data scaling algorithm for gene expression classification

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