# LoglogisticDistribution

Loglogistic probability distribution object

## Description

A LoglogisticDistribution object consists of parameters, a model description, and sample data for a loglogistic probability distribution.

The loglogistic distribution is closely related to the logistic distribution. If x is distributed loglogistically with parameters μ and σ, then log(x) is distributed logistically with mean and standard deviation. This distribution is often used in survival analysis to model events that experience an initial rate increase, followed by a rate decrease.

The loglogistic distribution uses the following parameters.

ParameterDescriptionSupport
muMean of logarithmic values$\mu >0$
sigmaScale parameter of logarithmic values$\sigma >0$

## Creation

There are several ways to create a LoglogisticDistribution probability distribution object.

• Create a distribution with specified parameter values using makedist.

• Fit a distribution to data using fitdist.

• Interactively fit a distribution to data using the Distribution Fitter app.

## Properties

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### Distribution Parameters

Mean of logarithmic values for the loglogistic distribution, specified as a positive scalar value.

Data Types: single | double

Scale parameter of logarithmic values for the loglogistic distribution, specified as a positive scalar value.

Data Types: single | double

### Distribution Characteristics

Logical flag for truncated distribution, specified as a logical value. If IsTruncated equals 0, the distribution is not truncated. If IsTruncated equals 1, the distribution is truncated.

Data Types: logical

Number of parameters for the probability distribution, specified as a positive integer value.

Data Types: double

Covariance matrix of the parameter estimates, specified as a p-by-p matrix, where p is the number of parameters in the distribution. The (i,j) element is the covariance between the estimates of the ith parameter and the jth parameter. The (i,i) element is the estimated variance of the ith parameter. If parameter i is fixed rather than estimated by fitting the distribution to data, then the (i,i) elements of the covariance matrix are 0.

Data Types: double

Logical flag for fixed parameters, specified as an array of logical values. If 0, the corresponding parameter in the ParameterNames array is not fixed. If 1, the corresponding parameter in the ParameterNames array is fixed.

Data Types: logical

Distribution parameter values, specified as a vector of scalar values.

Data Types: single | double

Truncation interval for the probability distribution, specified as a vector of scalar values containing the lower and upper truncation boundaries.

Data Types: single | double

### Other Object Properties

Probability distribution name, specified as a character vector.

Data Types: char

Data used for distribution fitting, specified as a structure containing the following:

• data: Data vector used for distribution fitting.

• cens: Censoring vector, or empty if none.

• freq: Frequency vector, or empty if none.

Data Types: struct

Distribution parameter descriptions, specified as a cell array of character vectors. Each cell contains a short description of one distribution parameter.

Data Types: char

Distribution parameter names, specified as a cell array of character vectors.

Data Types: char

## Object Functions

 cdf Cumulative distribution function gather Gather properties of Statistics and Machine Learning Toolbox object from GPU icdf Inverse cumulative distribution function iqr Interquartile range of probability distribution mean Mean of probability distribution median Median of probability distribution negloglik Negative loglikelihood of probability distribution paramci Confidence intervals for probability distribution parameters pdf Probability density function plot Plot probability distribution object proflik Profile likelihood function for probability distribution random Random numbers std Standard deviation of probability distribution truncate Truncate probability distribution object var Variance of probability distribution

## Examples

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Create a loglogistic distribution object using the default parameter values.

pd = makedist('Loglogistic')
pd =
LoglogisticDistribution

Log-Logistic distribution
mu = 0
sigma = 1

Create a loglogistic distribution object by specifying the parameter values.

pd = makedist('Loglogistic','mu',5,'sigma',2)
pd =
LoglogisticDistribution

Log-Logistic distribution
mu = 5
sigma = 2

Generate random numbers from the loglogistic distribution and compute their log values.

rng(19) % for reproducibility
x = random(pd,10000,1);
logx = log(x);

Compute the mean of the log values.

m = mean(logx)
m = 4.9828

The mean of the log of x is equal to the mu parameter of x, since x has a loglogistic distribution.

Plot logx.

histogram(logx,50)

The plot shows that the log values of x have a logistic distribution.

## Version History

Introduced in R2013a