psychmeta (version 2.3.4)

mix_dist: 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)

Value

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

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: $$var_{pooled_{Unbiased}}=MSW_{Unbiased}=\frac{\Sigma_{i=1}^{k}v_{i}\left(n_{i}-1\right)}{\left(\Sigma_{i=1}^{k}n_{i}\right)-k}$$ $$var_{means_{Unbiased}}=MSB_{Unbiased}=\frac{\Sigma_{i=1}^{k}\left(\bar{x}_{i}-\mu\right)n_{i}}{k-1}$$ $$var_{BG_{Unbiased}}=\frac{\Sigma_{i=1}^{k}\left(\bar{x}_{i}-\mu\right)n_{i}}{\left(\Sigma_{i=1}^{k}n_{i}\right)-1}$$ $$var_{WG_{Unbiased}}=\frac{\Sigma_{i=1}^{k}v_{i}\left(n_{i}-1\right)}{\left(\Sigma_{i=1}^{k}n_{i}\right)-1}$$ $$var_{mix_{Unbiased}}=var_{BG_{Unbiased}}+var_{WG_{Unbiased}}$$

Examples

# NOT RUN {
mix_dist(mean_vec = c(-.5, 0, .5), var_vec = c(.9, 1, 1.1), n_vec = c(100, 100, 100))
# }