Probability distributions

This page summarizes the distributions available in Mlxtran. In Simulx, they can be used to define random variables in the [INDIVIDUAL] section (individual parameters) or in the [LONGITUDINAL] section (observations).

Normal distribution and the associated transformed

To define the normal distribution and its transformations, the user can either define the mean of the associated normal distribution or the typical value of the distribution; as well as the standard deviation sd or variance var of the associated normal distribution. For the logitnormal distribution, the min and max can also be specified (optional). Covariates and correlations between random effects can also be added, see the [INDIVIDUAL] section.

• Normal distribution, used with keyword normal: 

  CODE

  y_norm = {distribution=normal, typical=0, sd=1}
  y_norm = {distribution=normal, mean=0, sd=1}

• Log-normal distribution, used with keyword lognormal:

  CODE

  y_ln = {distribution=lognormal, typical=1, sd=0.3} 
  y_ln = {distribution=lognormal, mean=0, sd=0.3}

• Logit-normal distribution, used with keyword logitnormal: with a and b the bounds of the logit-normal distribution indicated in the optional arguments min and max. By default, a=0 and b=1.

  CODE

  y_lgn = {distribution=logitnormal, typical=0.5, sd=0.6, min=0, max=1} 
  y_lgn = {distribution=logitnormal, mean=0, sd=0.6, min=0, max=1}

• Probit-normal distribution, used with keyword probitnormal: , where is the cumulative distribution function of the standard normal distribution

  CODE

  y_pbn = {distribution=probitnormal, typical=0.5, sd=0.6} 
  y_pbn = {distribution=probitnormal, mean=0, sd=0.6}

Other distributions

Additional distributions are available for simulation purpose in Simulx. They include continuous and discrete distributions. These distributions can neither be combined with the argument covariate, nor with the definition of correlations of random effects.

Continuous distributions

• Uniform distribution, used with keyword uniform,

  CODE

  y_uni = {distribution=uniform, min=0, max=100}

• Exponential distribution, used with keyword exponential,

  CODE

  y_exp = {distribution=exponential, rate=0.7}

• Gamma distribution, used with keyword gamma,

  CODE

  y_gamma = {distribution=gamma, shape=2, scale=0.5}

• Weibull distribution, used with keyword weibull,

  CODE

  y_weibull = {distribution=weibull, shape=0.75, scale=1.2}

• ExtremeValue distribution, used with keyword extremeValue,

  CODE

  y_ev = {distribution=extremeValue, location=-10, scale=1.8}

• ChiSquared distribution, used with keyword chiSquared,

  CODE

  y_cs = {distribution=chiSquared, df=2}

• Cauchy distribution, used with keyword cauchy,

  CODE

  y_cauchy = {distribution=cauchy, location=-5.2, scale=2.7}

• FisherF distribution, used with keyword fisherF,

  CODE

  y_fisherF = {distribution=fisherF, df1=10, df2=9}

• StudentT distribution, used with keyword studentT,

  CODE

  y_studentT = {distribution=studentT, df=2}

Discrete distributions

• Bernoulli distribution, used with keyword bernoulli,

  CODE

  y_ber = {distribution=bernoulli, prob=0.3}

• Discrete uniform distribution, used with keyword discreteUniform,

  CODE

  y_du = {distribution=discreteUniform, min=-4, max=2}

• Binomial distribution, used with keyword binomial,

  CODE

  y_bi = {distribution=binomial, size=30, prob=0.7}

• Geometric distribution, used with keyword geometric,

  CODE

  y_ber = {distribution=geometric, prob=0.2}

• Negative binomial distribution, used with keyword negativeBinomial,

  CODE

  y_nb = {distribution=negativeBinomial, size=3, prob=0.1}

• Poisson distribution, used with keyword poisson.

  CODE

  y_nb = {distribution=poisson, lambda=0.1}