Compute Confidence Intervals for Test Scores Adjusted for Reliability

Calculates the confidence interval around an observed test score, adjusting for test reliability. Returns the confidence interval in raw score units, z-scores, and percentiles.

ci_score(x, rtt, m, s, ci)

Arguments

x

Numeric value representing the observed test score.

rtt

Numeric value between 0 and 1 indicating the reliability coefficient of the test (e.g., test-retest reliability).

m

Numeric value representing the mean of the test scores in the population.

s

Numeric value representing the standard deviation of the test scores in the population.

ci

Numeric value between 0 and 1 indicating the desired confidence level (e.g., 0.95 for a 95% confidence interval).

Value

A list containing the following components:

scale

A named numeric vector with the confidence interval in raw score units:

  • lower: Lower bound of the confidence interval.

  • x: The observed test score.

  • upper: Upper bound of the confidence interval.

  • range: Width of the confidence interval.

z

A named numeric vector with the confidence interval in z-score units:

  • lower: Lower bound in z-scores.

  • x: Observed score in z-scores.

  • upper: Upper bound in z-scores.

  • range: Width of the confidence interval in z-scores.

percentile

A named numeric vector with the confidence interval in percentiles:

  • lower: Lower percentile.

  • x: Percentile rank of the observed score.

  • upper: Upper percentile.

  • range: Width of the confidence interval in percentiles.

parameters

A named numeric vector with additional parameters:

  • se: Standard error of measurement adjusted for reliability.

  • ci: Confidence level used.

  • z: Z-score multiplier corresponding to the confidence level.

Details

The standard error (se) is calculated using the formula: $$se = s \times \sqrt{1 - rtt}$$ where s is the standard deviation and rtt is the reliability coefficient.

The confidence interval is computed using the normal distribution quantile corresponding to the specified confidence level.

Examples

# Calculate a 95% confidence interval for an observed score of 100
ci_score(x = 100, rtt = 0.9, m = 100, s = 15, ci = 0.95)
#> $scale
#>  lower      x  upper  range 
#>  90.70 100.00 109.30  18.59 
#> 
#> $z
#> lower     x upper range 
#> -0.62  0.00  0.62  1.24 
#> 
#> $percentile
#> lower     x upper range 
#>  26.8  50.0  73.2  46.5 
#> 
#> $parameters
#>    se    ci     z 
#> 4.743 0.950 1.960 
#>