Penalized optimal scaling for ordinal variables with an application to international classification of functioning core sets

Ordinal data occur frequently in the social sciences. When applying principal component analysis (PCA), however, those data are often treated as numeric, implying linear relationships between the variables at hand; alternatively, non-linear PCA is applied where the obtained quantifications are sometimes hard to interpret. Non-linear PCA for categorical data, also called optimal scoring/scaling, constructs new variables by assigning numerical values to categories such that the proportion of variance in those new variables that is explained by a predefined number of principal components (PCs) is maximized. We propose a penalized version of non-linear PCA for ordinal variables that is a smoothed intermediate between standard PCA on category labels and non-linear PCA as used so far. The new approach is by no means limited to monotonic effects and offers both better interpretability of the non-linear transformation of the category labels and better performance on validation data than unpenalized non-linear PCA and/or standard linear PCA. In particular, an application of penalized optimal scaling to ordinal data as given with the International Classification of Functioning, Disability and Health (ICF) is provided.

Variability in Hourly Activity Levels: Statistical Noise or Insight Into Older Adult Frailty?

BACKGROUND

Frailty is associated with lower mean activity; however, hourly activity is highly variable among older individuals. We aimed to relate frailty to hourly activity variance beyond frailty's association with mean activity.

METHOD

Using the 2010-2011 National Social Life, Health and Aging Project wrist accelerometry data (n = 647), we employed a mixed-effects location scale model to simultaneously determine whether an adapted phenotypic frailty scale (0-4) was associated with the log10-mean hourly counts per minute (cpm) and between-and within-subject hourly activity variability, adjusting for demographics, health characteristics, season, day-of-week, and time-of-day. We tested the significance of a Frailty × Time-of-day interaction and whether adjusting for sleep time altered relationships.

RESULTS

Each additional frailty point was associated with a 7.6% (10-0.0343, β = -0.0343; 95% CI: -0.05, -0.02) lower mean hourly cpm in the morning, mid-day, and late afternoon but not evening. Each frailty point was also associated with a 24.5% (e0.219, β = 0.219; 95% CI: 0.09, 0.34) greater between-subject hourly activity variance across the day; a 7% (e0.07, β = 0.07; 95% CI: 0.01¸ 0.13), 6% (e0.06, β = 0.06; 95% CI: 0, 0.12), and 10% (e0.091, β = 0.091; 95% CI: 0.03, 0.15) greater within-subject hourly activity variance in the morning, mid-day, and late afternoon, respectively; and a 6% (e-0.06, β = -0.06; 95% CI: -0.12, -0.003) lower within-subject hourly activity variance in the evening. Adjusting for sleep time did not alter results.

CONCLUSIONS

Frail adults have more variable hourly activity levels than robust adults, a potential novel marker of vulnerability. These findings suggest a need for more precise activity assessment in older adults.

Modelling monotonic effects of ordinal predictors in Bayesian regression models

Ordinal predictors are commonly used in regression models. They are often incorrectly treated as either nominal or metric, thus under- or overestimating the information contained. Such practices may lead to worse inference and predictions compared to methods which are specifically designed for this purpose. We propose a new method for modelling ordinal predictors that applies in situations in which it is reasonable to assume their effects to be monotonic. The parameterization of such monotonic effects is realized in terms of a scale parameter b representing the direction and size of the effect and a simplex parameter ς modelling the normalized differences between categories. This ensures that predictions increase or decrease monotonically, while changes between adjacent categories may vary across categories. This formulation generalizes to interaction terms as well as multilevel structures. Monotonic effects may be applied not only to ordinal predictors, but also to other discrete variables for which a monotonic relationship is plausible. In simulation studies we show that the model is well calibrated and, if there is monotonicity present, exhibits predictive performance similar to or even better than other approaches designed to handle ordinal predictors. Using Stan, we developed a Bayesian estimation method for monotonic effects which allows us to incorporate prior information and to check the assumption of monotonicity. We have implemented this method in the R package brms, so that fitting monotonic effects in a fully Bayesian framework is now straightforward.

Regularized regression for categorical data

In the last two decades, regularization techniques, in particular penalty-based methods, have become very popular in statistical modelling. Driven by technological developments, most approaches have been designed for high-dimensional problems with metric variables, whereas categorical data has largely been neglected. In recent years, however, it has become clear that regularization is also very promising when modelling categorical data. A specific trait of categorical data is that many parameters are typically needed to model the underlying structure. This results in complex estimation problems that call for structured penalties which are tailored to the categorical nature of the data. This article gives a systematic overview of penalty-based methods for categorical data developed so far and highlights some issues where further research is needed. We deal with categorical predictors as well as models for categorical response variables. The primary interest of this article is to give insight into basic properties of and differences between methods that are important with respect to statistical modelling in practice, without going into technical details or extensive discussion of asymptotic properties.

Testing differentially expressed genes in dose-response studies and with ordinal phenotypes

When testing for differentially expressed genes between more than two groups, the groups are often defined by dose levels in dose-response experiments or ordinal phenotypes, such as disease stages. We discuss the potential of a new approach that uses the levels’ ordering without making any structural assumptions, such as monotonicity, by testing for zero variance components in a mixed models framework. Since the mixed effects model approach borrows strength across doses/levels, the test proposed can also be applied when the number of dose levels/phenotypes is large and/or the number of subjects per group is small. We illustrate the new test in simulation studies and on several publicly available datasets and compare it to alternative testing procedures. All tests considered are implemented in R and are publicly available. The new approach offers a very fast and powerful way to test for differentially expressed genes between ordered groups without making restrictive assumptions with respect to the true relationship between factor levels and response.

Rating scales as predictors--the old question of scale level and some answers

Rating scales as predictors in regression models are typically treated as metrically scaled variables or, alternatively, are coded in dummy variables. The first approach implies a scale level that is not justified, the latter approach results in a large number of parameters to be estimated. Therefore, when rating scales are dummy-coded, applications are often restricted to the use of a few predictors. The penalization approach advocated here takes the scale level serious by using only the ordering of categories but is shown to work in the high dimensional case. We consider the proper modeling of rating scales as predictors and selection procedures by using penalization methods that are tailored to ordinal predictors. In addition to the selection of predictors, the clustering of categories is investigated. Existing methodology is extended to the wider class of generalized linear models. Moreover, higher order differences that allow shrinkage towards a polynomial as well as monotonicity constraints and alternative penalties are introduced. The proposed penalization approaches are illustrated by use of the Motivational States Questionnaire.