Title: Fitting Probabilistic Index Models on Large Datasets,
Abstract: Recently, Thas et al. (2012) introduced a new statistical model for the
probability index. This index is defined as $P(Y \leq Y^*|X, X^*)$ where Y and
Y* are independent random response variables associated with covariates X and
X* [...] Crucially to estimate the parameters of the model, a set of
pseudo-observations is constructed. For a sample size n, a total of $n(n-1)/2$
pairwise comparisons between observations is considered. Consequently for large
sample sizes, it becomes computationally infeasible or even impossible to fit
the model as the set of pseudo-observations increases nearly quadratically. In
this dissertation, we provide two solutions to fit a probabilistic index model.
The first algorithm consists of splitting the entire data set into unique
partitions. On each of these, we fit the model and then aggregate the
estimates. A second algorithm is a subsampling scheme in which we select $K <<
n$ observations without replacement and after B iterations aggregate the
estimates. In Monte Carlo simulations, we show how the partitioning algorithm
outperforms the latter [...] We illustrate the partitioning algorithm and the
interpretation of the probabilistic index model on a real data set (Przybylski
and Weinstein, 2017) of n = 116,630 where we compare it against the ordinary
least squares method. By modelling the probabilistic index, we give an
intuitive and meaningful quantification of the effect of the time adolescents
spend using digital devices such as smartphones on self-reported mental
well-being. We show how moderate usage is associated with an increased
probability of reporting a higher mental well-being compared to random
adolescents who do not use a smartphone. On the other hand, adolescents who
excessively use their smartphone are associated with a higher probability of
reporting a lower mental well-being than randomly chosen peers who do not use a
smartphone.[...]