Estimate Leunbach Score Parameters using Iterative Proportional Fitting
Source:R/ipf.R
leunbach_ipf.RdEstimates score parameters for Leunbach test equating based on the power series distribution framework. The total score distribution in Rasch models follows a power series distribution where score parameters are defined by generalized symmetric functions of the item parameters.
Usage
leunbach_ipf(
data,
max_score1 = NULL,
max_score2 = NULL,
max_iter = 1000,
tol = 1e-10,
verbose = FALSE
)Arguments
- data
A data. frame or matrix with two columns: Column 1: Sum scores from Test 1 Column 2: Sum scores from Test 2
- max_score1
Maximum possible score for Test 1 (default: max observed)
- max_score2
Maximum possible score for Test 2 (default: max observed)
- max_iter
Maximum number of iterations for IPF
- tol
Convergence tolerance
- verbose
Print iteration progress
Value
A list of class "leunbach_ipf" containing:
gamma: Score parameters for Test1 (generalized symmetric functions)
delta: Score parameters for Test2 (generalized symmetric functions)
sigma: Score parameters for total score (Test1 + Test2)
log_gamma: Log of gamma parameters
log_delta: Log of delta parameters
log_sigma: Log of sigma parameters
fitted: Fitted frequency table
observed: Observed frequency table
iterations: Number of iterations to convergence
converged: Logical indicating convergence
test1_scores: Score values for Test1
test2_scores: Score values for Test2
total_scores: Score values for total score
g_sq: Likelihood ratio test statistic
chi_sq: Pearson chi-square statistic
df: Degrees of freedom
p_value: P-value for likelihood ratio test
Details
The model is based on the power series distribution (PSD) for sum scores in Rasch models. For a test with k polytomous items, the probability of obtaining sum score r is:
$$P(R = r | \theta) = \frac{\gamma_r \theta^r}{\sum_s \gamma_s \theta^s}$$
where \(\gamma_r\) is the generalized symmetric function of order r of the item parameters, and \(\theta = exp(\beta)\) is the person parameter.
The model is over-parameterized, so we apply identification constraints:
\(\gamma_{min}\) = 1 (lowest observed score parameter fixed at 1)
\(\delta_{min}\) = 1 (lowest observed score parameter fixed at 1)
The highest score parameters are constrained so that their product equals the product of all score parameters (Rasch-style constraint)
The sigma parameters for the total score (S = X + Y) are computed as: $$\sigma_s = \sum_{x+y=s} \gamma_x \delta_y$$
References
Leunbach, G. (1976). A probabilistic measurement model for assessing whether two tests measure the same personal factor. Copenhagen: Danish Institute for Educational Research.
Adroher, N. D., Kreiner, S., Young, C., Mills, R., & Tennant, A. (2019). Test equating sleep scales: Applying the Leunbach’s model. BMC Medical Research Methodology, 19(1), 141. https://doi.org/10.1186/s12874-019-0768-y
Kreiner, S. (2007). Validity and objectivity: Reflections on the role and nature of Rasch models. Nordic Psychology, 59(3), 268-298. https://doi.org/10.1027/1901-2276.59.3.268
Examples
# Simulate test score data
set.seed(123)
n <- 500
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)
score_data <- data.frame(test1 = test1, test2 = test2)
# Estimate parameters (specifying max possible scores)
fit <- leunbach_ipf(score_data, max_score1 = 6, max_score2 = 5, verbose = TRUE)
#> Leunbach Test Equating - Score Parameter Estimation
#> ====================================================
#>
#> Power Series Distribution Model with Generalized Symmetric Functions
#>
#> Data summary:
#> N = 500 observations
#> Test1: scores 0 to 6 (observed: 0 to 6)
#> Test2: scores 0 to 5 (observed: 0 to 5)
#> Total: scores 0 to 11 (observed: 0 to 11)
#>
#> Estimating score parameters via IPF...
#>
#> Iteration 1: max param change = 3.28e+00 (gamma: 3.23e+00, delta: 3.28e+00)
#> Iteration 2: max param change = 4.95e+00 (gamma: 4.95e+00, delta: 3.42e+00)
#> Iteration 3: max param change = 7.06e+00 (gamma: 7.06e+00, delta: 3.66e+00)
#> Iteration 4: max param change = 9.25e+00 (gamma: 9.25e+00, delta: 3.91e+00)
#> Iteration 5: max param change = 1.14e+01 (gamma: 1.14e+01, delta: 4.17e+00)
#> Iteration 6: max param change = 1.36e+01 (gamma: 1.36e+01, delta: 4.41e+00)
#> Iteration 7: max param change = 1.57e+01 (gamma: 1.57e+01, delta: 4.61e+00)
#> Iteration 8: max param change = 1.77e+01 (gamma: 1.77e+01, delta: 4.79e+00)
#> Iteration 9: max param change = 1.96e+01 (gamma: 1.96e+01, delta: 4.94e+00)
#> Iteration 10: max param change = 2.13e+01 (gamma: 2.13e+01, delta: 5.06e+00)
#> Iteration 11: max param change = 2.29e+01 (gamma: 2.29e+01, delta: 5.15e+00)
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#>
#> Converged after 584 iterations
print(fit)
#> Leunbach Score Parameter Estimation
#> ====================================
#> Power Series Distribution with Generalized Symmetric Functions
#>
#> N = 500 observations
#>
#> Test 1 score parameters (gamma):
#>
#> Score Frequency Gamma Log_Gamma
#> 0 32 1.00000 0.000000
#> 1 49 47.50220 3.860776
#> 2 99 607.69584 6.409675
#> 3 135 1480.82436 7.300354
#> 4 96 529.56519 6.272056
#> 5 56 50.08795 3.913781
#> 6 33 0.80244 -0.220098
#>
#> Test 2 score parameters (delta):
#>
#> Score Frequency Delta Log_Delta
#> 0 39 1.000000 0.000000
#> 1 84 42.348644 3.745936
#> 2 120 221.281483 5.399436
#> 3 134 234.333759 5.456746
#> 4 73 34.813941 3.550018
#> 5 50 1.246199 0.220098
#>
#> Total score parameters (sigma = Test1 + Test2):
#>
#> Score Frequency Sigma Log_Sigma
#> 0 19 1.00000 0.000000
#> 1 22 89.85085 4.498151
#> 2 30 2840.63124 7.951782
#> 3 46 37961.61108 10.544331
#> 4 59 208878.48986 12.249508
#> 5 70 494214.10151 13.110724
#> 6 69 487527.55199 13.097102
#> 7 62 187523.16197 12.141658
#> 8 51 32196.51658 10.379614
#> 9 23 2591.74144 7.860085
#> 10 26 90.35566 4.503754
#> 11 23 1.00000 0.000000
#>
#> Converged: TRUE (after 584 iterations)
#>
#> --- Goodness of Fit ---
#>
#> 1. Likelihood Ratio Test:
#> LR = 18.30 DF = 21 p = 0.6299
#>
#> 2. Goodman-Kruskal Gamma Test (one-sided):
#> Gamma (observed) = 0.7649
#> Gamma (expected) = 0.7630
#> SE = 0.0219
#> Z = -0.09 p = 0.5339
#>
#> 3. Orbit Analysis:
#> Use analyze_orbits() to assess person fit within total score strata
summary(fit)
#> Leunbach Score Parameter Estimation - Summary
#> ==============================================
#> Power Series Distribution with Generalized Symmetric Functions
#>
#> N = 500 observations
#> Test 1: scores 0 to 6 (observed: 0 to 6)
#> Test 2: scores 0 to 5 (observed: 0 to 5)
#> Total: scores 0 to 11 (observed: 0 to 11)
#>
#> === Goodness of Fit ===
#>
#> 1. Likelihood Ratio Test (observed vs expected counts):
#> Likelihood ratio G² = 18.30 (df = 21, p = 0.6299)
#> Pearson chi-square = 17.31
#>
#> 2. Goodman-Kruskal Gamma Test (one-sided):
#> Tests if observed correlation exceeds expected under the model.
#> Gamma (observed) = 0.7649
#> Gamma (expected) = 0.7630
#> Standard error = 0.0219
#> Z statistic = -0.09 (p = 0.5339, one-sided)
#>
#> 3. Orbit Analysis (person fit):
#> Run analyze_orbits() separately to assess the number of cases
#> outside 95% confidence regions of orbit distributions.
#>
#> Converged: TRUE (after 584 iterations)