Damien McParland

will speak on

Latent Variable Models for Ordinal Data

Time: 1:00PM
Date: Fri 1st April 2011
Location: Statistics Seminar Room- L550 Library building [map]

Abstract: Ordinal data arise in many contexts and item response modelling is a long established method for analysing this type of data.

The ordinal response for individual $i$ on item $j$ is denoted $Y_{ij}$, where $i=1,\ldots,N$ and $j=1,\ldots,J$. Corresponding to each ordinal data point $Y_{ij}$ is a latent Gaussian variable $Z_{ij}$. The value of $Y_{ij}$ is observed to be level $k$ if the latent Gaussian variable $Z_{ij}$ lies within a specified interval. In addition, another latent Gaussian variable $\theta_i$, often called a latent trait, is used to model the underlying attributes of individual $i$. The mean of $Z_{ij}$ depends on $\theta_i$, i.e.
$$ Z_{ij} \sim \mbox{N}(a_{j} \theta_{i} - b_{j}, 1)$$
In the item response literature, $a_{j}$ and $b_{j}$ are typically known as discrimination and difficulty parameters respectively.

The extension to a mixture of two parameter item response models, which provides clustering capabilities in the context of ordinal data is also explored. In this context the mean of $Z_{ij}$ also depends on which group individual $i$ belongs to, i.e.
$$ Z_{ij} \sim \mbox{N}(a_{gj} \theta_{i} - b_{gj}, 1) $$
where $a_{gj}$ and $b_{gj}$ are group specific discrimination and difficulty parameters.

Estimation of both of these models within the Bayesian paradigm is achieved using a Metropolis-within-Gibbs sampler.

(This talk is part of the Working Group on Statistical Learning series.)

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