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Analyze Orbits for Leunbach Model

Source: R/orbits.R
analyze_orbits.Rd

Performs orbit analysis for the Leunbach model. For each total score (orbit), computes the expected distribution of (Test1, Test2) score pairs and cumulative probabilities for assessing person fit.

Usage

analyze_orbits(fit, alpha = 0.05, verbose = FALSE)

Arguments

fit

A leunbach_ipf object from leunbach_ipf()

alpha

Significance level for critical values (default 0.05)

verbose

Print detailed output

Value

A list of class "leunbach_orbits" containing:

  • orbits: Matrix of expected percentages within each total score

  • left_right: Cumulative probabilities P(X <= x | S = s)

  • right_left: Cumulative probabilities P(X >= x | S = s)

  • crit_left: Critical values for left tail by total score

  • crit_right: Critical values for right tail by total score

  • crit_values: Combined critical values (two-tailed)

  • expected_critical: Expected number of persons with significant differences

  • variance_expected: Variance of expected critical count

  • n_significant: Number of observed persons with significant differences

  • orbit_df: Degrees of freedom for each orbit

Details

For each total score S = X + Y, the orbit consists of all valid (X, Y) pairs. The expected probability of each cell within an orbit is: P(X = x | S = s) = (gamma_x * delta_y) / sigma_s

Cumulative probabilities are computed in both directions:

  • Left-to-right: P(X <= x | S = s) - tests if Test1 score is unusually low

  • Right-to-left: P(X >= x | S = s) - tests if Test1 score is unusually high

Examples

set.seed(123)
n <- 400
theta <- rnorm(n)
test1 <- pmin(pmax(round(3 + 1.5 * theta + rnorm(n, sd = 0.8)), 0), 6)
test2 <- pmin(pmax(round(2.5 + 1.3 * theta + rnorm(n, sd = 0.7)), 0), 5)
fit <- leunbach_ipf(data.frame(test1, test2),
                    max_score1 = 6, max_score2 = 5)
orb <- analyze_orbits(fit)
print(orb)
#> Leunbach Orbit Analysis
#> =======================
#> 
#> N = 400 observations
#> Significance level:  5.0%
#> 
#> Critical levels for person fit assessment:
#> 
#>  Score  N Crit_Left Crit_Right Crit_Combined DF
#>      0 14     0.000      0.000         0.000  0
#>      1 16     0.000      0.000         0.000  1
#>      2 24     0.000      0.000         0.000  2
#>      3 43     1.887      0.000         1.887  3
#>      4 52     0.060      0.609         0.670  4
#>      5 46     0.746      0.032         0.778  5
#>      6 54     4.124      0.631         4.755  5
#>      7 57     0.877      0.065         0.942  4
#>      8 34     0.000      1.657         1.657  3
#>      9 23     0.000      0.000         0.000  2
#>     10 21     0.000      0.000         0.000  1
#>     11 16     0.000      0.000         0.000  0
#> 
#> 3 (0.8%) persons with significant differences between measurements
#> 5.2 (1.3%) expected
#> 
#> 95% Confidence interval: [0.2%, 2.4%]
#> Chi-square = 0.95, df = 1, p = 0.3298