Diagnostics for Ordinal Data

Justice Nii-Ayitey

doi:10.7273/000007893

The problem we aim to address in this dissertation is to develop a more straightforward method for examining correlation-related diagnostics in ordinal data. Analyzing data that falls between numerical and categorical types can be challenging, particularly when it involves ordinal variables. One major challenge is the limited availability of diagnostics for evaluating the performance of ordinal regression models. Most often, the diagnostics performed for ordinal regression are the proportional odds assumption and the goodness-of-fit test. The proportional odds assumption evaluates whether the relationship between the predictors and the odds of being in a higher versus a lower category is consistent across thresholds, while the goodness-of-fit test assesses whether the model fits the data better than a null model. While these are somewhat easily available in software, the problem of dependence is not. The two main issues related to correlation in ordinal data are: multicollinearity, when the predictors are highly correlated, and autocorrelation or familial dependence, when the responses are correlated with each other. While diagnostics for these are easily available in the General Linear Model (GLM), this is not the case for ordinal data. The approaches that exist for ordinal data are theoretically and computationally cumbersome and time-consuming. For multicollinearity, we can directly examine the structure of the design matrix and predictors, and consider eigenvalues and condition numbers, as in GLM. However, for ordinal data, our estimates are based on maximum likelihood estimates, and thus a direct examination of the design matrix is not necessarily the best approach. Similarly, we could consider generalized estimating equations or Binary Dynamic Logit for Correlated Ordinal (BDLCO) method for modeling the dependence structure, which are computationally intensive. We aim to determine when BDLCO is needed versus using traditional methods for ordinal data, assuming proportional odds. In this dissertation, our primary objective is to develop a “quick and dirty” method to determine whether multicollinearity is indeed an issue and whether methods like ridge or lasso are warranted. In the same way, we try to quickly diagnose "dependence issues". If dependence is detected, we would suggest using a method like BDLCO. Our approach is to utilize our ordinal response and its underlying relationship with predictors through the slope estimates to estimate the underlying latent response as a function of the predictors, and then use readily available methods to perform quick checks for multicollinearity or autocorrelation. Essentially, we show that the slope estimates from the ordinal response can aid in estimating the slope estimates for the underlying latent variable in an Approximate Latent Linear Model (ALLM). The idea is that once we estimate such a continuous response, then we can use the multivariate normal distribution of the approximate latent linear response to understand autocorrelation and multicollinearity. Thus, we can utilize standard diagnostic techniques developed within the GLM framework, such as the Durbin-Watson test and variance inflation factors, to address issues of autocorrelation and multicollinearity. Our simulation results show that this procedure gives us a quick way to understand and diagnose problems that are theoretically and computationally difficult in a much faster way. We use our method on a data example, which helps to make decisions about ordinal data in the presence of dependence.

Diagnostics for Ordinal Data

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