Difference between revisions of "Logistic distribution"
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[[Category:Distribution Functions]] | [[Category:Distribution Functions]] | ||
[[Category:Doc Status D]] <!-- For Lumina use, do not change --> | [[Category:Doc Status D]] <!-- For Lumina use, do not change --> | ||
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| + | [[Image:Logistic Distribution.jpg]] | ||
= Logistic( mean, scale ) = | = Logistic( mean, scale ) = | ||
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[[Image:Logistic cdf eq.PNG]] | [[Image:Logistic cdf eq.PNG]] | ||
| − | The distribution is symmetric and unimodal with tails that are heavier than the normal | + | The distribution is symmetric and unimodal with tails that are heavier than the [[Normal|normal |
| − | distribution. It has a mean and mode of | + | distribution]]. It has a mean and mode of «mean», variance of pi^2 «scale»^2 / 3, [[Kurtosis|kurtosis]] of 6/5 and zero [[Skewness|skew]]. The «scale» parameter is optional and defaults to 1. |
The logistic distribution is particularly convenient for determining dependent probabilities | The logistic distribution is particularly convenient for determining dependent probabilities | ||
| − | using linear regression techniques, where the probability of a binomial event | + | using [[Regression|linear regression]] techniques, where the probability of a binomial event |
depends monotonically on a continuous variable x. For example, in a toxicology assay, | depends monotonically on a continuous variable x. For example, in a toxicology assay, | ||
x may be the dosage of a toxin, and p(x) the probability of death for an animal exposed | x may be the dosage of a toxin, and p(x) the probability of death for an animal exposed | ||
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has a simple linear form. This linear form lends itself to linear regression techniques for | has a simple linear form. This linear form lends itself to linear regression techniques for | ||
estimating the distribution — for example, from clinical trial data. | estimating the distribution — for example, from clinical trial data. | ||
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| + | = Parameter Estimation = | ||
| + | |||
| + | The parameters of the distribution can be estimated using: | ||
| + | :«mean» := [[Mean]](X,I) | ||
| + | :«scale» := [[Sqrt]](3 * [[Variance]](X,I)) / [[Pi]] | ||
| + | = See Also = | ||
| + | |||
| + | * [[Dens_Logistic]] | ||
| + | * [[CumLogistic]] | ||
Revision as of 16:54, 5 August 2009
Logistic( mean, scale )
The logistic distribution describes a distribution with a cumulative density given by
The distribution is symmetric and unimodal with tails that are heavier than the normal distribution. It has a mean and mode of «mean», variance of pi^2 «scale»^2 / 3, kurtosis of 6/5 and zero skew. The «scale» parameter is optional and defaults to 1.
The logistic distribution is particularly convenient for determining dependent probabilities using linear regression techniques, where the probability of a binomial event depends monotonically on a continuous variable x. For example, in a toxicology assay, x may be the dosage of a toxin, and p(x) the probability of death for an animal exposed to that dosage. Using p(x) = F(x), the logit of p, given by
Logit(p(x)) = Ln( p(x) / (1-p(x)) ) = x/s - m/s
has a simple linear form. This linear form lends itself to linear regression techniques for estimating the distribution — for example, from clinical trial data.
Parameter Estimation
The parameters of the distribution can be estimated using:
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