# Scale Alignment Wrapper Function for 'TAM' Objects Apply scale alignment methods to models previously fit with tam.mml in the 'TAM' package. ```r 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) ## Returns 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 ## 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: $$ \log(\frac{P(x | \theta_d)}{P(x-1 | \theta_d)}) = \alpha_d \theta_d - \xi_{i(d)x}log(P(x | \theta_d)/(P(x - 1 | \theta_d))) = \alpha_d \theta_d - \xi_i(d)x $$ for response category $x = 1,...,m$, and $$ P(x = 0 | \theta_d) = \frac{1}{\sum_{j=0}^{m}\exp \sum_{k=0}^j (\alpha_d \theta_d -\xi_{i(d)k})}P(x = 0 | \theta_d) = 1 / (\sum_{j=0}^{m}\exp \sum_{k=0}^j (\alpha_d \theta_d -\xi_i(d)k)) $$ 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 $$ \log(\frac{P(x | \theta_d)}{P(x-1 | \theta_d)}) = \alpha_d \theta_d - \delta_{i(d)} +\tau_{i(d)x}.log(P(x | \theta_d)/(P(x - 1 | \theta_d))) = \alpha_d \theta_d - \delta_i(d) +\tau_i(d)x. $$ 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 $'$ symbol) are found through the transformation parameters $r_d$ and $s_d$ as described by the following equations: $$ \tilde{\theta}_d = r_{d} \theta_{d} + s_{d}\theta'_d = r_{d}\theta_{d} + s_{d} $$ $$ \tilde{\alpha}_d = \alpha_{d} / r_{d}\alpha'_d = \alpha_d / r_d $$ $$ \tilde{\xi}_{i(d)x} = \xi_{i(d)x} + \alpha_d s_d / r_d\xi'_i(d)x = \xi_i(d)x + \alpha_d * s_d / r_d $$ $$ \tilde{\delta}_{i(d)} = \delta_{i(d)} + \alpha_d s_d / r_d\delta'_i(d) = \delta_i(d) + \alpha_d * s_d / r_d $$ $$ \tilde{\tau}_{i(d)x} = \tau_{i(d)x}\tau'_i(d)x = \tau_{i(d)x} $$ 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. ## Examples ```r ## 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 ``` ## References Feuerstahler, L. M., & Wilson, M. (2019). Scale alignment in between-item multidimensional Rasch models. Journal of Educational Measurement, 56(2), 280--301. Feuerstahler, L. M., & Wilson, M. (under review). Scale alignment in the between-items multidimensional partial credit model.

align function