Compute the Critical Difference Using SEM and SED

Calculates the critical difference required to determine whether the difference between two test scores is statistically significant. This function computes the critical difference based on the test-retest reliability coefficient, the standard deviation of the test scores, and the desired confidence interval. It also returns the z-score, standard error of measurement (SEM), and standard error of difference (SED).

critical_difference(rtt, sd, ci)

Arguments

rtt

Numeric value between 0 and 1. The test-retest reliability coefficient of the test.

sd

Numeric value. The standard deviation of the test scores.

ci

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

Value

A list containing:

critical_difference

The critical difference value.

z

The z-score corresponding to the specified confidence interval.

standard_error_measurement

The standard error of measurement (SEM).

standard_error_difference

The standard error of difference (SED).

Details

The function calculates the critical difference (CD) using the following steps:

  • Calculate the z-score (\(z\)) corresponding to the desired confidence interval: $$z = \text{qnorm}\left( \frac{1 - \text{ci}}{2}, \text{lower.tail} = FALSE \right)$$

  • Compute the standard error of measurement (SEM): $$\text{SEM} = \text{SD} \times \sqrt{1 - r_{tt}}$$

  • Compute the standard error of difference (SED): $$\text{SED} = \sqrt{2} \times \text{SEM}$$

  • Calculate the critical difference (CD): $$\text{CD} = z \times \text{SED}$$

This approach accounts for the reliability of the test and the desired level of confidence to determine if the difference between two scores is statistically significant.

Examples

# Calculate the critical difference for rtt = 0.9, sd = 15, ci = 0.95
critical_difference(rtt = 0.9, sd = 15, ci = 0.95)
#> $critical_difference
#> [1] 13.14784
#> 
#> $z
#> [1] 1.959964
#> 
#> $standard_error_meassurement
#> [1] 4.743416
#> 
#> $standard_error_difference
#> [1] 6.708204
#>