mix_dist: Descriptive statistics for a mixture distribution

In psychmeta: Psychometric Meta-Analysis Toolkit

Descriptive statistics for a mixture distribution

Description

Compute descriptive statistics for a mixture distribution. This function returns the grand mean, the pooled sample variance (mean square within), variance of sample means (mean square between), portions of the total variance that are within and between groups, and mixture (total sample) variance of the mixture sample data.

Usage

mix_dist(mean_vec, var_vec, n_vec, unbiased = TRUE, na.rm = FALSE)

Arguments

mean_vec	Vector of sample means.
var_vec	Vector of sample variances.
n_vec	Vector of sample sizes.
unbiased	Logical scalar determining whether variance should be unbiased (TRUE; default) or maximum-likelihood (FALSE).
na.rm	Logical scalar determining whether to remove missing values prior to computing output (TRUE) or not (FALSE; default)

Details

The grand mean of a mixture distribution is computed as:

\mu=\frac{\Sigma_{i=1}^{k}\bar{x}_{i}n_{i}}{\Sigma_{i=1}^{k}n_{i}}

where \mu is the grand mean, \bar{x}_{i} represents the sample means, and n_{i} represents the sample sizes.

Maximum-likelihood mixture variances are computed as:

var_{pooled_{ML}}=MSW_{ML}=\frac{\Sigma_{i=1}^{k}\left(\bar{x}_{i}-\mu\right)n_{i}}{\Sigma_{i=1}^{k}n_{i}}

var_{means_{ML}}=MSB_{ML}=\frac{\Sigma_{i=1}^{k}\left(\bar{x}_{i}-\mu\right)n_{i}}{k}

var_{BG_{ML}}=\frac{\Sigma_{i=1}^{k}\left(\bar{x}_{i}-\mu\right)n_{i}}{\Sigma_{i=1}^{k}n_{i}}

var_{WG_{ML}}=\frac{\Sigma_{i=1}^{k}v_{i}n_{i}}{\Sigma_{i=1}^{k}n_{i}}

var_{mix_{ML}}=var_{BG_{ML}}+var_{WG_{ML}}

where v_{i} represents the sample variances.

Unbiased mixture variances are computed as:

var_{pooled_{Unbiased}}=MSW_{Unbiased}=\frac{\Sigma_{i=1}^{k}v_{i}\left(n_{i}-1\right)}{\left(\Sigma_{i=1}^{k}n_{i}\right)-k}

var_{means_{Unbiased}}=MSB_{Unbiased}=\frac{\Sigma_{i=1}^{k}\left(\bar{x}_{i}-\mu\right)n_{i}}{k-1}

var_{BG_{Unbiased}}=\frac{\Sigma_{i=1}^{k}\left(\bar{x}_{i}-\mu\right)n_{i}}{\left(\Sigma_{i=1}^{k}n_{i}\right)-1}

var_{WG_{Unbiased}}=\frac{\Sigma_{i=1}^{k}v_{i}\left(n_{i}-1\right)}{\left(\Sigma_{i=1}^{k}n_{i}\right)-1}

var_{mix_{Unbiased}}=var_{BG_{Unbiased}}+var_{WG_{Unbiased}}

Value

The mean, pooled sample (within-sample) variance, variance of sample means (between-groups), and mixture (total sample) variance of the mixture sample data.

Examples

mix_dist(mean_vec = c(-.5, 0, .5), var_vec = c(.9, 1, 1.1), n_vec = c(100, 100, 100))

psychmeta documentation built on June 22, 2024, 6:52 p.m.