| align {scaleAlign} | R Documentation |
Scale Alignment Wrapper Function for 'TAM' Objects
Description
Apply scale alignment methods to models previously fit with tam.mml in the 'TAM' package.
Usage
align(mod, method = "best", refdim = 1)
Arguments
mod |
Fitted model of class tam.mml. Importantly, mod$irtmodel must be either "1PL", "PCM", or "PCM2" |
method |
Either "DDA1", "DDA2", "LRA", or "best", see details |
refdim |
Which is the reference dimension (unchanged during alignment) |
Details
Scales can be said to be aligned if the item sufficient statistics imply the same item parameter estimates, regardless of dimension. Scale alignment is currently defined only for Rasch family models with between-items multidimensionality (i.e., each scored item belongs to exactly one dimension).
MODEL PARAMETERIZATIONS
The partial credit model is a general Rasch family model for polytomous item
responses. Within 'TAM', the partial credit model can be parameterized in two
ways. If a 'TAM' model is fit with the option irtmodel = "PCM", then the
following model is specified for an item with m + 1 response categories:
\log(\frac{P(x | \theta_d)}{P(x-1 | \theta_d)}) = \alpha_d \theta_d - \xi_{i(d)x}
for response category x = 1,...,m, and
P(x = 0 | \theta_d) = \frac{1}{\sum_{j=0}^{m}\exp \sum_{k=0}^j (\alpha_d \theta_d -
\xi_{i(d)k})}
for response category x=0. \alpha_d is a dimension
steepness parameter, typically fixed to 1, \theta_d is a latent
variable on dimension d, and \xi_i(d)x is a step parameter for item
step x on item i belonging to dimension d.
If instead a TAM model is fit with the option irtmodel = "PCM2", the model is specified as
\log(\frac{P(x | \theta_d)}{P(x-1 | \theta_d)}) = \alpha_d \theta_d - \delta_{i(d)} +
\tau_{i(d)x}.
MODEL TRANSFORMATIONS
Under Rasch family models, the latent trait metric can be linearly
transformed. For each dimension d the parameters on the transformed
metric (denoted by the \sim symbol) are found through the transformation
parameters r_d and s_d as described by the following equations:
\tilde{\theta}_d = r_{d} \theta_{d} + s_{d}
\tilde{\alpha}_d = \alpha_{d} / r_{d}
\tilde{\xi}_{i(d)x} = \xi_{i(d)x} + \alpha_d s_d / r_d
\tilde{\delta}_{i(d)} = \delta_{i(d)} + \alpha_d s_d / r_d
\tilde{\tau}_{i(d)x} = \tau_{i(d)x}
SUFFICIENT STATISTICS
Under Rasch family models, the item sufficient statistics are the number of
examinees that score in response category x or higher,
x = 1,...,m. For the purpose of scale alignment, we consider sufficient
statistics to be the proportion of examinees that score in response category
x or higher. This definition allows for scale alignment in the
presence of missing data.
THURSTONE THRESHOLDS
Scales are aligned if the same sufficient statistics imply the same item parameters, regardless of dimension. The success of scale alignment is difficult to assess because the item sufficient statistics typically differ across items and dimensions. Under the Rasch model for binary item responses, the success of scale alignment can be assessed by looking at the rank-order correlation (e.g., Kendall's tau) between item sufficient statistics and item parameter estimates.
However, under the partial credit model, item sufficient statistics need not
be monotonically related to estimated item parameters. Under this model,
we can assess the quality of scale alignment by taking the rank-order
correlation between item sufficient statistics and Thurstone thresholds.
Thurstone thresholds are defined as the \theta value at which the
probability of responding in category x or higher equals .5. Thurstone
thresholds, in most cases, will be monotonically related to item sufficient
statistics (within dimensions). Note that the item difficulty estimates
under the Rasch model for binary items are also Thurstone thresholds.
ALIGNMENT METHODS
Two types of scale alignment methods have been developed.
The first class of methods, historically called delta-dimensional alignment
(DDA), requires fitting both a multidimenisonal model and a model in which
all items belong to a single dimension. With these two sets of parameter
estimates, the transformation parameters r_d and s_d are then
found so that, for each dimension, the means and standard deviation of
parameters from the transformed multidimensional models equal the means and
standard deviations of parameters from the unidimensional model. Under the
ordinary Rasch model, the estimated item difficulties can be used for
transformation (which is done if either method "DDA1" or "DDA2" is selected).
Under the partial credit model, either the \delta parameters or the
Thurstone thresholds from the two models may be used within the DDA (note
that DDA using item \xi parameters tends to be unsuccessful). Method
"DDA1" uses the item \delta parameters, and method "DDA2" uses the
Thurstone thresholds. If all items are binary, "DDA1" and "DDA2" are identical.
The second class of methods, called logistic regression alignment (LRA),
requires fitting a logistic regression between item sufficient statistics
and Thurstone thresholds for each dimension. The fitted logistic regression
coefficients can then be used to estimate r_d and s_d so that the
same logistic regression curve expresses the relationship between sufficient
statistics and Thurstone thresholds for all dimensions.
For either the DDA or LRA method, a reference dimension (by default, the
first dimension) is specified such that r_d = 1 and s_d = 0 for
the reference dimension.
Value
Aligned tam.mml object with the following added list items:
method |
Alignment method: "DDA1", "DDA2", or "LRA" |
rhat |
Vector of estimated scaling parameters r, see details |
shat |
Vector of estimated shift parameters s, see details |
cor_before |
Kendall's rank-order correlation between sufficient statistics and Thurstone thresholds before alignment |
cor_after |
Kendall's rank-order correlation between sufficient statistics and Thurstone thresholds after alignment |
References
Feuerstahler, L. M., & Wilson, M. (2019). Scale alignment in between-item multidimensional Rasch models. Journal of Educational Measurement, 56(2), 280–301. <doi: 10.1111/jedm.12209>
Feuerstahler, L. M., & Wilson, M. (under review). Scale alignment in the between-items multidimensional partial credit model.
Examples
## Example 1: binary item response data
## generate data for a 2-dimensional model with 10 items on each dimension
if(require(TAM)){
set.seed(2524)
diff_1 <- rnorm(10)
diff_2 <- rnorm(10)
N <- 500
th <- MASS::mvrnorm(N, mu = c(0, -1),
Sigma = matrix(c(1, .5 * 2, .5 * 2, 4), nrow = 2))
probs_1 <- 1 / (1 + exp(-outer(th[, 1], diff_1, "-")))
probs_2 <- 1 / (1 + exp(-outer(th[, 2], diff_2, "-")))
probs <- cbind(probs_1, probs_2)
dat <- apply(probs, 2, function(p) as.numeric(p > runif(N)))
Q <- cbind(c(rep(1, 10), rep(0, 10)),
c(rep(0, 10), rep(1, 10)))
# fit the model
mod <- TAM::tam.mml(resp = dat, irtmodel = "1PL", Q = Q)
# align the model
mod_aligned <- align(mod)
## check alignment success
mod_aligned$cor_before
mod_aligned$cor_after
## view "best" alignment method
mod_aligned$method
## view alignment parameters
mod_aligned$rhat
mod_aligned$shat
}
## Example 2: Partial Credit Model
# generate 3-category data for a 2-dimensional model with 5 items on each dimension
set.seed(8491)
N <- 500
th <- MASS::mvrnorm(N, mu = c(0, 0),
Sigma = matrix(c(1, .5 * 2, .5 * 2, 4), nrow = 2))
xi_1 <- rnorm(5)
xi_1 <- cbind(xi_1, xi_1 + rnorm(5, mean = 1, sd = .5))
xi_2 <- rnorm(5)
xi_2 <- cbind(xi_2, xi_2 + rnorm(5, mean = 1, sd = .5))
dat1 <- catR::genPattern(th[, 1], it = xi_1, model = "PCM")
dat2 <- catR::genPattern(th[, 2], it = xi_2, model = "PCM")
dat <- cbind(dat1, dat2)
Q <- cbind(c(rep(1, 5), rep(0, 5)),
c(rep(0, 5), rep(1, 5)))
## fit the model using both parameterizations
mod1 <- TAM::tam.mml(resp = dat, irtmodel = "PCM", Q = Q)
mod2 <- TAM::tam.mml(resp = dat, irtmodel = "PCM2", Q = Q)
## align the models
mod1_aligned <- align(mod1)
mod2_aligned <- align(mod2)
## check alignment success
mod1_aligned$cor_before
mod1_aligned$cor_after
mod2_aligned$cor_before
mod2_aligned$cor_after
## view "best" alignment method
mod1_aligned$method
mod2_aligned$method
## view alignment parameters
mod1_aligned$rhat
mod1_aligned$shat
mod2_aligned$rhat
mod2_aligned$shat