dist2 {IRTest}

Re-parameterized two-component normal mixture distribution

Description

Probability density for the re-parameterized two-component normal mixture distribution.

Usage

dist2(x, prob = 0.5, d = 0, sd_ratio = 1, overallmean = 0, overallsd = 1)

Arguments

x

A numeric vector. The location to evaluate the density function.

prob

A numeric value of \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

A numeric value of \delta = \frac{\mu_2 - \mu_1}{\bar{\sigma}} parameter of two-component Gaussian mixture distribution, where \mu_1 and \mu_2 are the estimated mean of the first and second Gaussian component, 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 deviation of the first and second Gaussian component, 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

The overall mean and overall standard deviation obtained from original parameters;

1) Overall mean (\bar{\mu})

\bar{\mu}=\pi\mu_1 + (1-\pi)\mu_2

2) Overall standard deviation (\bar{\sigma})

\bar{\sigma}=\sqrt{\pi\sigma_{1}^{2}+(1-\pi)\sigma_{2}^{2}+\pi(1-\pi)(\mu_2-\mu_1)^2}

Value

The evaluated probability density value(s).

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.

Examples

# Evaluated density
dnst <- dist2(seq(-6,6,.1), prob = 0.3, d = 1, sd_ratio=0.5)

# Plot of the density
plot(seq(-6,6,.1), dnst)