PerformEquating: Equating parameter estimates of a Rasch model across...

In Nwosu/RM.weights: Weighted Rasch modeling and extensions using Conditional Maximum Likelihood

Description

This functions allows to equate parameter estimates across different cultural, linguistic or environmental contexts, providing coefficients to report item and raw score parameters on the same metric for cross-context comparisons.

Usage

PerformEquating(listcountry, tol, tol2, mincommon, step)

Arguments

listcountry	A list of RM.w objects.
tol	Tolerance from the second iteration step on. After the first iteration, a difference of discr = 1+tol is considered to define common and unique items. discr is decreased iteration by iteration.
tol2	Initial tolerance. At the first iteration step, items are considered common if their item severity differ of tol2 from the standard, hence, it is advisable that tol2 is defined sufficieltly large (1 or 2 logits). At each iteration, tol2 = min(tol2 - step, discr), where discr depends on the maximum observed parameter difference.
mincommon	Number of minimum common items. In a specific country, if the number of common items is less than mincommon, the country is not used to compute the standard.
step	Speed of decreasing of the algorithm tolerance. After the first iteration, tol2 = min(tol2 - step, discr), where discr depends on the maximum observed parameter difference.

Details

This function allows to define a common reference scale across different context (depending on, for instance, the different environment, country, language the scale has been administered). In this document, such differences are refered to as country effects, but they have in fact a broader, context-related, meaning. In each country, common and unique item are defined based on the logit-difference with the standard, and such standard is defined based only on the common items. Because of the link between the definition of common items and the standard reference scale, the algorithm needs to start with a very permissive tolerance, and gradually decrease this tolerance during the subsequent iterations.

Let us indicate with n the number of countries, k the number of items and b_{ij} the estimated item parameter for country i = 1, …, n and item j = 1, …, k. The vector b_i includes all the estimated item parameters for country i, while the vector b_j contains the estimated parameter for item j in all countries. By construction, b_i are zero-centered (we will indicate this property as μ_i = 0), but they can have arbitrary standard deviation (σ_i). σ_i is related to the scale discrimination power and reliability, but it also reflects a country-specific, and therefore not comparable, metric.

The equating algortithm is breafly described below.

EQUATING ALGORITHM

• Set the iteration counter it = 1

• Report the country-specific item parameters on the same metric, i.e. having the same mean and standard deviation. Standardized, country-specific, item parameters (indicated, from here on, as country std.sev.) are obtained by dividing them by their standard deviation σ_i.

• Assuming that all items are common, define the global reference standard (from here on, ref.sev.) as median of severity parameters across countries.

• For each country, calculate the absolute difference between the standardized parameter, and the reference standard (d_i = (d_i1, ..., d_ik)), and define the set of unique and common items for country i as U_i = {j: d_ij > tol2}, and C_i = {j: d_ij < tol2}.

• Define discr = 1+tol

• WHILE discr > tol

  1. The new metric is defined based on common items in each country, leading to the computation of expansion coefficients α_i and β_i:

     β_i = sd(comm.ref.sev.)/sd(comm.country.std.sev.),

     and

     α_i = mean(comm.ref.sev.)- β_i * mean(comm.country.std.sev.)

     Here, comm. stands for common that reference and country severities are considered only for items defined as common. Therefore, for each country, the new metric is defined in terms of mean and standard deviation of the set of common items.

  2. The country-specific parameters are re-scaled and re-centered based on the expansion coefficients as follows:

     new country std.sev. = α_i + β_i * old country std.sev.

  3. Calculate the new reference standard (new ref.sev.) based on the median of the new country std. sev. (based only on common items). Based on the new reference standard, the new parameter difference (d_i), and the new set of unique (U_i) and common (C_i) items is defined.

  4. Re-define the tolerance as discr = max(tol, max(d_ij')), where j' in C^_i.

  5. If C_i has cardinality less than mincommon, country i is eliminated from the set of countries that contributes to the definition of the reference standard.

  6. Set it = it + 1

• end WHILE

• Define the expansion coefficients on the actual contry-specific severities. This final coefficients will be used to equate the country-specific metric to a common metric, allowing comparable results.

  β_i final = sd(comm.ref.sev.)/sd(comm.country.sev.),

  and

  α_i final = mean(comm.ref.sev.)- β_i final * mean(comm.country.sev.)

  It is important to note that β_i final and α_i final consider the actual country severities, and not the standardized ones.

Value

A list with the following elements:

aa	Vector of the α_i, one for each country, at convergence.
bb	Vector of the β_i, one for each country, at convergence.
aaa	Vector of the α_i final, one for each country, at convergence.
bbb	Vector of the β_i final, one for each country, at convergence.
id.comm	TRUE/FALSE matrix with n rows and k columns. TRUE indicates that that item in that country is common.
b	Matrix with n rows and k columns containing the estimated item severities for each country.
b.scaled	Matrix with n rows and k columns containing the scaled item severities for each country at convergence.
b.tot	Reference standard severity vector at convergence.
diff.mat	Matrix with n rows and k columns containing the differences between the country-specific severities (b.scaled) and the reference standard severities (b.tot).
items	Names of the items.
names	Names of the countries.
diff	The tolerance (discr) at convergence.
eqset	Set of countries that contribute to the reference standard definition, at convergence.
number	Number of common items per country, at convergence.

Author(s)

Carlo Cafiero [e-mail]

See Also

RM.w, prob.assign

Examples

## Not run: 
data(data.FAO_country1)
data(data.FAO_country2)
data(data.FAO_country3)
data(data.FAO_country4)
# Questionnaire data and weights for countries 1, 2, 3 and 4
XX.country1 = data.FAO_country1[,1:8]
XX.country2 = data.FAO_country2[,1:8]
XX.country3 = data.FAO_country3[,1:8]
XX.country4 = data.FAO_country4[,1:8]

wt.country1 = data.FAO_country1$wt
wt.country2 = data.FAO_country2$wt
wt.country3 = data.FAO_country3$wt
wt.country4 = data.FAO_country4$wt

# Fit weighted Rasch for each country separately
rr.country1 = RM.w(XX.country1, wt.country1, country = "Country 1")
rr.country2 = RM.w(XX.country2, wt.country2, country = "Country 2")
rr.country3 = RM.w(XX.country3, wt.country3, country = "Country 3")
rr.country4 = RM.w(XX.country4, wt.country4, country = "Country 4")

# Prepare function input
listcountry = list(rr.country1, rr.country2, rr.country3, rr.country4)

# Perform equating
result = PerformEquating(listcountry, tol = 0.35, tol2 = 2, mincommon = 5, step = 1e-9)

# Expansion factors
alphai = result$aaa   
betai = result$bbb

# Compare prevalence curves on a common metric 
s.levels = seq(-5, 5, length.out = 200)
s.levels.eq = sapply(1:length(listcountry), function(i) alphai[i] + betai[i] * s.levels)

plot(s.levels.eq[,1], prob.assign(listcountry[[1]], sthres = s.levels)$sprob, 
type = "l", col = 1, ylim = c(0, 1), xlab = "Latent trait", ylab = "Cumulative prevalence profile")  

for(i in 2:length(listcountry)) {
	lines(s.levels.eq[,i], prob.assign(listcountry[[i]], sthres = s.levels)$sprob, 
  type = "l", col = i) 
}     
legend("topright", lty = 1, col = 1:length(listcountry), legend = result$names, 
bty = "n", x.intersp = 0.5, cex = 0.8)
text(result$b.tot, 0.2, result$items, srt = 90, cex = 0.6)

## End(Not run)

Nwosu/RM.weights documentation built on May 7, 2019, 8:20 p.m.