Difference between revisions of "Logistic distribution"
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| − | = Logistic( mean, scale ) = | + | == Logistic(mean, scale) == |
The logistic distribution describes a distribution with a cumulative density given by | The logistic distribution describes a distribution with a cumulative density given by | ||
| − | + | :<math> | |
| + | F(x)=\frac{1}{1+exp \Big(-\frac{(x-mean)}{scale}\Big)} | ||
| + | </math> | ||
| − | The distribution is symmetric and unimodal with tails that are heavier than the [[Normal|normal | + | The distribution is symmetric and unimodal with tails that are heavier than the [[Normal|normal distribution]]. It has a [[mean]] and mode of «mean», [[variance]] of pi<sup>2</sup>, «scale»<sup>2/3</sup>, [[Kurtosis|kurtosis]] of 6/5 and zero [[Skewness|skew]]. The «scale» parameter is optional and defaults to 1. |
| − | distribution]]. It has a mean and mode of «mean», variance of pi | ||
| − | The logistic distribution is particularly convenient for determining dependent probabilities | + | The logistic distribution is particularly convenient for determining dependent probabilities 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, ''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 |
| − | 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, | ||
| − | 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 | ||
| − | + | :<code>Logit(p(x)) = Ln(p(x) / (1-p(x))) = x/s - m/s</code> | |
| − | 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. | ||
| − | = Parameter Estimation = | + | == Parameter Estimation == |
| + | The parameters of the distribution can be estimated using: | ||
| + | :<code>«mean» := Mean(X, I) </code> | ||
| + | :<code>«scale» := Sqrt(3*Variance(X, I))/Pi</code> | ||
| − | + | ==Example== | |
| − | : | + | :[[Image:Logistic Distribution.jpg]] |
| − | |||
| − | |||
| + | == See Also == | ||
* [[Dens_Logistic]] | * [[Dens_Logistic]] | ||
* [[CumLogistic]] | * [[CumLogistic]] | ||
| + | * [[Mean]] | ||
| + | * [[Variance]] | ||
| + | * [[Distribution Densities Library]] | ||
Revision as of 00:23, 27 January 2016
Logistic(mean, scale)
The logistic distribution describes a distribution with a cumulative density given by
- [math]\displaystyle{ F(x)=\frac{1}{1+exp \Big(-\frac{(x-mean)}{scale}\Big)} }[/math]
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 pi2, «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:
«mean» := Mean(X, I)«scale» := Sqrt(3*Variance(X, I))/Pi
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