Includes equations extracted mostly from Brown and Maydeu-Olivares, 2011. Other references include: Brown & Maydeu-Olivares, 2012; Brown, 2016.
Outcome variables
Number of binary outcome variables:
\[\begin{equation}
\tag{1}
\tilde{n} = n(n-1)/2
\end{equation}\]
Binary outcome variables:
\[\begin{equation}
\tag{2}
y_l = \begin{cases}
1 & \text{if item } i \text{ is preferred over item } k,\\
0 & \text{if item } k \text{ is preferred over item } i
\end{cases}
\end{equation}\]
where \(l\) indicates the pair \({i,k}\).
Outcome variables and item utilities:
binary version:
\[\begin{equation}
\tag{3}
y_l = \begin{cases}
1 & \text{if } t_i \geq t_k,\\
0 & \text{if } t_i < t_k
\end{cases}
\end{equation}\]
where \(t_i\) is the latent utility associated with item \(i\).
or the latent continuous version:
\[\begin{equation}
\tag{4}
y^*_l = t_i - t_k
\end{equation}\]
thus the observed binary outcomes and unobserved difference scores are linked by:
\[\begin{equation}
\tag{5}
y_l = \begin{cases}
1 & \text{if } y_l^* \geq 0\\
0 & \text{if } y_l^* < 0
\end{cases}
\end{equation}\]
Matrix form:
\[\begin{equation}
\tag{6}
\mathbf{y^* = At}
\end{equation}\]
where:
\(\mathbf{y^*}\) is the \(\tilde{n} \times 1\) vector of latent difference responses
\(\mathbf{A}\) is the \(\tilde{n} \times n\) design matrix; columns correspond to \(n\) items and rows correspond to \(\tilde{n}\) pairwise comparisons
\(\mathbf{t}\) is the \(n \times 1\) vector of latent utilities
with multiple blocks:
\(\mathbf{A}\) is the \((p \times \tilde{n}) \times m\) design matrix
\(p\) is the number of blocks
\(n\) is the number of items per block
\(m = p \times n\) is the total number of items
Thurstonian Factor Model
Second-order factor model:
\[\begin{equation}
\tag{7}
\mathbf{t} = \boldsymbol{\mu}_t + \boldsymbol{\Lambda\theta + \epsilon}
\end{equation}\]
where:
\(\boldsymbol{\mu}_t\) is a \(m \times 1\) vector of \(m\) means of the latent utilities \(\mathbf{t}\)
\(\boldsymbol{\Lambda}\) is an \(m \times d\) matrix of factor loadings with \(d\) factors and is an independent clusters solution where every item measures only one trait
\(\boldsymbol{\theta}\) is a \(d\)-dimensional vector of (normally distributed) common factors
\(\boldsymbol{\epsilon}\) is a \(m\)-dimensional vector of (normally distributed) unique factors
\(\boldsymbol{\Phi}\) is the covariance matrix for common factors
\(\boldsymbol{\Psi}^2\) is the diagonal covariance matrix for the uncorrelated unique factors
Thurstonian IRT Model
Reparameterized first-order factor model:
\[\begin{equation}
\tag{8}
\mathbf{y}^* = \boldsymbol{A(\mu_t + \Lambda\theta + \epsilon)}
= \boldsymbol{A\mu_t + A\Lambda\theta + A\epsilon}
\end{equation}\]
\[\begin{equation}
\tag{9}
\mathbf{y}^* = -\boldsymbol{\gamma} + \breve{\Lambda}\theta + \breve{\epsilon}
\end{equation}\]
where:
\(\mathbf{y}^*\) is normally distributed
\(\boldsymbol{\breve{\Lambda} = A\Lambda}\) is a structured \((p \times \tilde{n}) \times d\) matrix of factor loadings
\(\boldsymbol{\breve{\epsilon} = A\epsilon}\) is the unique pairwise errors with the covariance matrix cov(\(\boldsymbol{\breve{\epsilon} = \breve{\Psi}}^2 = \boldsymbol{A\Psi^2A'}\))
\(\boldsymbol{\gamma = -A\mu}_t\) is the unrestricted \((p \times \tilde{n}) \times 1\) vector of thresholds, one threshold \(\gamma_l = -(\mu_i - \mu_k)\) is estimated for each binary outcome
with restrictions:
\(\Phi\) is a correlation matrix, with variances of common factors set to 1
For \(n > 2\), fix the uniqueness \(\psi\) of the 1st item per block
\(\lambda_i\) loadings on a trait are of the same magnitude for pairs that involve the same item
\(\boldsymbol{\Psi}^2\) is a block diagonal matrix where unique errors of pairs related to the same item within a block are correlated
residual error variance of a binary outcome equals sum of residual error variances of utilities of the 2 items in that pair
Item Characteristic Function
Item characteristic function for binary outcome variable \(y_l\)
\[\begin{equation}
\tag{10}
\text{Pr}(y_l = 1|\theta_a,\theta_b) = \boldsymbol{\Phi}
\Big(\frac{-\gamma_l + \lambda_i\theta_a - \lambda_k\theta_b}{\sqrt{\psi^2_i+\psi^2_k}}\Big)
\end{equation}\]
where:
\(y_l = 1\) denotes preferring item \(i\) measuring trait \(\theta_a\) over item \(k\) measuring trait \(\theta_b\)
\(\boldsymbol{\Phi}(x)\) is the cumulative standard normal distribution function evaluated at \(x\)
\(\psi^2_l = \psi^2_i + \psi^2_k\) is the uniqueness of the latent response variable \(y_l^*\)
Item characteristic function in intercept/slope form
\[\begin{equation}
\tag{11}
\text{Pr}(y_l = 1|\theta_a, \theta_b) = \boldsymbol{\Phi}(\alpha_l + \beta_i\theta_a - \beta_k\theta_b)
\end{equation}\]
where:
\(\alpha_l = \frac{-\gamma_l}{\sqrt{\psi^2_i + \psi^2_k}}\)
\(\beta_i = \frac{\lambda_i}{\sqrt{\psi^2_i + \psi^2_k}}\)
\(\beta_k = \frac{\lambda_k}{\sqrt{\psi^2_i + \psi^2_k}}\)
\(\boldsymbol{\alpha}\) and \(\boldsymbol{\beta}\) are not mathematically independent except for when \(n = 2\) items per block
Redundancy and adjustment to degree of freedom
\[\begin{equation}
\tag{12}
r = n(n-1)(n-2)/6
\end{equation}\]
where:
\(r\) is the number of redundancies among the thresholds and tetrachoric correlations estimated from binary outcome variables
\(r \times p\) is the number of redundancies for \(p\) ranking blocks
\(n\) is the number of items per block
\[\begin{equation}
\tag{13}
\text{df-adjusted} = \text{df} - (p \times r) \hspace{30pt} n > 2
\end{equation}\]
Information Functions
Directional derivative in direction \(\boldsymbol{\alpha}\)
\[\begin{equation}
\tag{14}
\nabla_\alpha P_l(\boldsymbol{\theta}) = \frac{\partial P_l(\boldsymbol{\theta})}{\partial \theta_1} \cos \alpha_1 +
... + \frac{\partial P_l(\boldsymbol{\theta})}{\partial \theta_d} \cos \alpha_d
\end{equation}\]
where:
\(\alpha\) is a vector of angles to all \(d\) axes that defines the direction from point \(\boldsymbol{\theta}\)
\(P_l(\boldsymbol{\theta}) = \text{Pr}(y_l = 1|\theta_a, \theta_b)\)
\(\frac{\partial P_l(\theta_a, \theta_b)}{\partial \theta_a} = \beta_i \phi(\alpha_l + \beta_i\theta_a - \beta_k\theta_b)\)
\(\frac{\partial P_l(\theta_a, \theta_b)}{\partial \theta_b} = -\beta_k \phi(\alpha_l + \beta_i\theta_a - \beta_k\theta_b)\)
\(\phi(z)\) is the standard normal density function evaluated at \(z\)
Information information surfaces in direction of trait \(a\) and \(b\)
\[\begin{equation}
\tag{15}
\mathbf{I}^a_l(\theta_a, \theta_b) = \frac{[\beta_i - \beta_k \text{corr}(\theta_a, \theta_b)]^2[\phi(\alpha_l + \beta_i\theta_a - \beta_k\theta_b)]^2}
{P_l(\theta_a, \theta_b)[1 - P_l(\theta_a, \theta_b)]}
\end{equation}\]
\[\begin{equation}
\tag{16}
\mathbf{I}^b_l(\theta_a, \theta_b) = \frac{[- \beta_k + \beta_i \text{corr}(\theta_a, \theta_b)]^2[\phi(\alpha_l + \beta_i\theta_a - \beta_k\theta_b)]^2}
{P_l(\theta_a, \theta_b)[1 - P_l(\theta_a, \theta_b)]}
\end{equation}\]
Total information about trait \(\theta_a\)
\[\begin{equation}
\tag{17}
\mathbf{I}^a(\boldsymbol{\theta}) = \sum_l\mathbf{I}^a_l(\boldsymbol{\theta})
\end{equation}\]
Posterior test information for trait \(\theta_a\) - Bayes MAP estimation
\[\begin{equation}
\tag{18}
\mathbf{I}^a_P(\boldsymbol{\theta}) = \mathbf{I}^a(\boldsymbol{\theta}) -
\frac{\partial^2\ln(\phi(\boldsymbol{\theta}))}{\partial^2\theta_a}
= \mathbf{I}^a(\boldsymbol{\theta}) + \omega^a_a
\end{equation}\]
where:
\(\omega^a_a\) is the diagonal element of the inverted latent trait covariance matrix \(\Phi^-1\)
\(SE(\hat{\boldsymbol{\theta}_a}) = \frac{1}{\sqrt{\mathbf{I}^a_P(\boldsymbol{\theta})}}\) is the standard error of MAP-estimated score
Error variance
\[\begin{equation}
\tag{19}
\bar{\sigma}^2_{\text{error}}(\hat{\boldsymbol{\theta}}) = \frac{1}{N} \sum^N_{j=1}
\frac{1}{\mathbf{I}^a_P(\hat{\boldsymbol{\theta}}_j)}
\end{equation}\]
where \(j\) denotes each respondent in a sample of size \(N\)
Empirical reliability
\[\begin{equation}
\tag{20}
\rho = \frac{\sigma^2 - \bar{\sigma}^2_{\text{error}}}{\sigma^2}
\end{equation}\]
\[\begin{equation}
\tag{21}
\text{corr}(\theta_a, \hat{\theta}_a) = \sqrt{\rho}
\end{equation}\]
where \(\sigma^2\) is estimated using sample variance of estimated MAP scores
Estimation
Estimating item parameters
Mean- and variance-corrected unweighted least squares (ULSMV)
\[\begin{equation}
\tag{22}
\text{RMSEA} = \sqrt{\frac{\chi^2-\text{df-adj}}{\text{df-adj}\times (N - 1)}}
\end{equation}\]
Goodness-of-fit indices and p-values need to be recalculated using adjusted degrees of freedom from
(13) for
\(n > 2\) items per block.
Estimating person parameters
Maximum a posteriori (MAP) Bayesian modal procedures
\[\begin{equation}
\tag{23}
F(\theta) = \frac{1}{2} \theta'\Phi^{-1}\theta -
\sum_l \ln \{[Pr(y_l = 1)|\theta]^{y_l}[1 - Pr(y_l = 1|\theta)]^{1-y_l}\}
\end{equation}\]
this ignores local dependency within-block, but does not hugely affect accuracy