PumasEMModel-Error models

using Pumas

The error models in a PumasEMModel implicitly define the dispersion parameters based on the provided distribution. The general syntax is

    @error begin
        dv ~ ProportionalNormal(μ)
    end

Where dv is the observed variable, and μ the expectation, specified in an earlier block. All currently supported error models follow the observed ~ Distribution(expectation) syntax. When specifying inits, add a σ field to the named tuple with one element per observed variable. Each element should itself be a tuple with a Float64 per parameter in the error model. For example, given

    @error begin
        dv1 ~ ProportionalNormal(cp1) # parameterized by 1 dispersion parameter
        dv2 ~ CombinedNormal(cp2) # parameterized by 2 dispersion parameters
    end

one may, specify inits of the form σ = ((2.0,), (0.9, 0.5)), where 2.0 is the standard deviation parameter for dv1 and 0.9, 0.5 are the parameters for dv2, in this case corresponding to the additive and proportional standard deviations of the combined normal error model. It is not necessary to initialize σ when fitting; if unspecified, each element is initialized to 1.0.

Gaussian models

Additive Normal

    @error begin
        Y ~ Normal(μ)
    end

Indicates that Y ~ Normal(μ, σ).

Proportional Normal

    @error begin
        Y ~ ProportionalNormal(μ)
    end

Indicates that Y ~ Normal(μ, abs(μ)*σ).

Combined Normal

    @error begin
        Y ~ CombinedNormal(μ)
    end

Indicates that Y ~ Normal(μ, √(σₐ² + μ²*σₚ²)).

Log Normal

    @error begin
        Y ~ LogNormal(μ)
    end

Indicates that Y ~ LogNormal(μ, σ).

0-Dispersion-Parameter models

Bernoulli

    @error begin
        Y ~ Bernoulli(μ)
    end

Indicates that Y ~ Bernoulli(μ).

Bernoulli Logit

    @error begin
        Y ~ BernoulliLogit(μ)
    end

Indicates that Y ~ Bernoulli( 1 /(1 + exp(-μ)) ).

Exponential

    @error begin
        Y ~ Exponential(μ)
    end

Indicates that Y ~ Exponential(μ).

Poisson

    @error begin
        Y ~ Poisson(μ)
    end

Indicates that Y ~ Poisson(μ).

1-Dispersion-Parameter models

Beta

    @error begin
        Y ~ Beta(μ)
    end

Indicates that Y ~ Beta(μ * 10/σ, (1 - μ) * 10/σ).

Gamma

    @error begin
        Y ~ Gamma(μ)
    end

Indicates that Y ~ Gamma( 1 / σ², μ * σ² ).

Summary of Error Models