original_par_2GM {IRTest}

Recovering original parameters of two-component Gaussian mixture distribution from re-parameterized values

Description

Recovering original parameters of two-component Gaussian mixture distribution from re-parameterized values

Usage

original_par_2GM(
  prob = 0.5,
  d = 0,
  sd_ratio = 1,
  overallmean = 0,
  overallsd = 1
)

Arguments

prob

The \pi = \frac{n_1}{N} parameter of two-component Gaussian mixture distribution, where n_1 is the estimated number of examinees belonging to the first Gaussian component and N is the total number of examinees (Li, 2021).

d

The \delta = \frac{\mu_2 - \mu_1}{\bar{\sigma}} parameter of two-component Gaussian mixture distribution, where \mu_1 and \mu_2 are the estimated means of the first and second Gaussian components, respectively. And \bar{\sigma} is the overall standard deviation of the latent distribution (Li, 2021). Without loss of generality, \mu_2 \ge \mu_1 is assumed, thus \delta \ge 0.

sd_ratio

A numeric value of \zeta = \frac{\sigma_2}{\sigma_1} parameter of two-component Gaussian mixture distribution, where \sigma_1 and \sigma_2 are the estimated standard deviations of the first and second Gaussian components, respectively (Li, 2021).

overallmean

A numeric value of \bar{\mu} that determines the overall mean of two-component Gaussian mixture distribution.

overallsd

A numeric value of \bar{\sigma} that determines the overall standard deviation of two-component Gaussian mixture distribution.

Details

Original two-component Gaussian mixture distribution

f(x)=\pi\times \phi(x | \mu_1, \sigma_1)+(1-\pi)\times \phi(x | \mu_2, \sigma_2)

, where \phi is a Gaussian component.

Re-parameterized two-component Gaussian mixture distribution

f(x)=2GM(x|\pi, \delta, \zeta, \bar{\mu}, \bar{\sigma})

, where \bar{\mu} is overall mean and \bar{\sigma} is overall standard deviation of the distribution.

The original parameters retrieved from re-parameterized values

1) Mean of the first Gaussian component (m1).

\mu_1=-(1-\pi)\delta\bar{\sigma}+\bar{\mu}

2) Mean of the second Gaussian component (m2).

\mu_2=\pi\delta\bar{\sigma}+\bar{\mu}

3) Standard deviation of the first Gaussian component (s1).

\sigma_1^2=\bar{\sigma}^2\left(\frac{1-\pi(1-\pi)\delta^2}{\pi+(1-\pi)\zeta^2}\right)

4) Standard deviation of the second Gaussian component (s2).

\sigma_2^2=\bar{\sigma}^2\left(\frac{1-\pi(1-\pi)\delta^2}{\frac{1}{\zeta^2}\pi+(1-\pi)}\right)=\zeta^2\sigma_1^2

Value

This function returns a vector of length 4: c(m1,m2,s1,s2).

m1

The location parameter (mean) of the first Gaussian component.

m2

The location parameter (mean) of the second Gaussian component.

s1

The scale parameter (standard deviation) of the first Gaussian component.

s2

The scale parameter (standard deviation) of the second Gaussian component.

Author(s)

Seewoo Li [e-mail]

References

Li, S. (2021). Using a two-component normal mixture distribution as a latent distribution in estimating parameters of item response models. Journal of Educational Evaluation, 34(4), 759-789.