General a multivariate confidence interval for a set of scores

Usage

multivariate_ci(x, r_xx, mu, sigma, ci = 0.95, v_names = names(x))

Arguments

x: a vector of scores
r_xx: a vector reliability coefficients
mu: a vector means
sigma: a covariance matrix
ci: confidence level
v_names: a vector of names

Value

A data frame with the following columns:

variable - Variable names
x - Variable scores
r_xx - Reliability coefficients
mu_univariate - Expected true score estimated from the corresponding observed score
see_univariate - Standard error of the estimate computed from the corresponding reliability coefficient
mu_multivariate - Expected true score estimated from all observed scores
see_multivariate - Standard error of the estimate computed from the corresponding reliability coefficient
upper_univariate - upper bound of univariate confidence interval
lower_univariate - lower bound of univariate confidence interval
upper_multivariate - upper bound of multivariate confidence interval
lower_multivariate - lower bound of multivariate confidence interval

Examples

# Observed Scores
x <- c(
  vci = 130,
  vsi = 130,
  fri = 70,
  wmi = 130,
  psi = 130
)

# Reliability Coefficients
r_xx <- c(
  vci = .92,
  vsi = .92,
  fri = .93,
  wmi = .92,
  psi = .88
  )

# Correlation matrix
R <- ("
  index  vci   vsi   fri   wmi   psi
  vci    1.00  0.59  0.59  0.53  0.30
  vsi    0.59  1.00  0.62  0.50  0.36
  fri    0.59  0.62  1.00  0.53  0.31
  wmi    0.53  0.50  0.53  1.00  0.36
  psi    0.30  0.36  0.31  0.36  1.00") |>
    readr::read_tsv() |>
    tibble::column_to_rownames("index") |>
    as.matrix()
#> Rows: 5 Columns: 6
#> ── Column specification ────────────────────────────────────────────────────────
#> Delimiter: "\t"
#> chr (1): index
#> dbl (5): vci, vsi, fri, wmi, psi
#> 
#> ℹ Use `spec()` to retrieve the full column specification for this data.
#> ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

 # Covariance matrix
 sigma <- R * 15 ^ 2

 # Population means#'
 mu <- rep(100, 5)

 mci <- multivariate_ci(
   x = x,
   r_xx = r_xx,
   mu = mu,
   sigma = sigma
 )

 mci
#>     variable   x r_xx mu_univariate see_univariate mu_multivariate
#> vci      vci 130 0.92         127.6       4.069398       126.69804
#> vsi      vsi 130 0.92         127.6       4.069398       126.11601
#> fri      fri  70 0.93          72.1       3.827205        77.64056
#> wmi      wmi 130 0.92         127.6       4.069398       127.29155
#> psi      psi 130 0.88         126.4       4.874423       127.38539
#>     see_multivariate upper_univariate lower_univariate upper_multivariate
#> vci         3.911922        135.57587        119.62413          134.36526
#> vsi         3.896561        135.57587        119.62413          133.75313
#> fri         3.685405         79.60118         64.59882           84.86382
#> wmi         3.953840        135.57587        119.62413          135.04093
#> psi         4.803025        135.95369        116.84631          136.79915
#>     lower_multivariate
#> vci           119.0308
#> vsi           118.4789
#> fri            70.4173
#> wmi           119.5422
#> psi           117.9716

 # Conditional covariance of true score estimates
 attr(mci, "conditional_covariance")
#>            vci        vsi        fri        wmi        psi
#> vci 15.3031321  0.7940793  0.6610628  0.6064452  0.1059996
#> vsi  0.7940793 15.1831840  0.8677571  0.3369836  0.5206229
#> fri  0.6610628  0.8677571 13.5822068  0.5011386  0.1153453
#> wmi  0.6064452  0.3369836  0.5011386 15.6328484  0.5570710
#> psi  0.1059996  0.5206229  0.1153453  0.5570710 23.0690477