Iain Pardoe

2. Average Predictive Comparisons for Models with Nonlinearity, Interactions, and Variance Components

In a predictive model, what is the expected difference in the outcome associated with a unit difference in one of the inputs? In a linear regression model without interactions, this average predictive comparison is simply a regression coefficient (with associated uncertainty). In a model with nonlinearity or interactions, however, the average predictive comparison in general depends on the values of the predictors. We consider various definitions based on averages over a population distribution of the predictors, and we compute standard errors based on uncertainty in model parameters. We illustrate with a study of criminal justice data for urban counties in the United States. The outcome of interest measures whether a convicted felon received a prison sentence rather than a jail or non-custodial sentence, with predictors available at both individual and county levels. We fit three models: (1) a hierarchical logistic regression with varying coefficients for the within-county intercepts as well as for each individual predictor; (2) a hierarchical model with varying intercepts only; and (3) a nonhierarchical model that ignores the multilevel nature of the data. The regression coefficients have different interpretations for the different models; in contrast, the models can be compared directly using predictive comparisons. Furthermore, predictive comparisons clarify the interplay between the individual and county predictors for the hierarchical models and also illustrate the relative size of varying county effects.

EXPLAINING SENTENCE SEVERITY IN LARGE URBAN COUNTIES: A MULTILEVEL ANALYSIS OF CONTEXTUAL AND CASE-LEVEL FACTORS

This study used hierarchical logistic modeling to examine the impact of legal, extralegal, and contextual variables on the decision to sentence felons to prison in a sample of large urban counties in 1996. None of the four contextual (county-level) variables—the level of crime, unemployment rate, racial composition, and region—increased the likelihood of a prison sentence, but 10 case-level factors, both legal and extralegal, and several macro-micro interaction terms were influential. These results demonstrate the importance of considering smaller geographic units (i.e., counties instead of states) and controlling for case-level factors in research on interjurisdictional differences in prison use.

Graphical Tools for Quadratic Discriminant Analysis

Sufficient dimension-reduction methods provide effective ways to visualize discriminant analysis problems. For example, Cook and Yin showed that the dimension-reduction method of sliced average variance estimation (SAVE) identifies variates that are equivalent to a quadratic discriminant analysis (QDA) solution. This article makes this connection explicit to motivate the use of SAVE variates in exploratory graphics for discriminant analysis. Classification can then be based on the SAVE variates using a suitable distance measure. If the chosen measure is Mahalanobis distance, then classification is identical to QDA using the original variables. Just as canonical variates provide a useful way to visualize linear discriminant analysis (LDA), so do SAVE variates help visualize QDA. This would appear to be particularly useful given the lack of graphical tools for QDA in current software. Furthermore, whereas LDA and QDA can be sensitive to nonnormality, SAVE is more robust.

A Graphical Method for Assessing the Fit of a Logistic Regression Model

Before a logistic regression model is used to describe the relationship between a binary response variable and predictors, the fit of the model should be assessed. The nature of any model deficiency may indicate that some aspect of the model should be reformulated or that poorly fitting observations need to be considered separately. We propose graphical methodology based on a Bayesian framework to address issues such as this. Publicly available software allows diagnostic plots to be constructed quickly and easily forany model ofinterest. These plots are more intuitive and meaningful than traditional graphical diagnostics such as residual plots.

A Bayesian Sampling Approach to Regression Model Checking

A necessary step in any regression analysis is checking the fit of the model to the data. Graphical methods are often employed to allow visualization of features that the data should exhibit if the model holds. Judging whether such features are present or absent in any particular diagnostic plot can be problematic. In this article I take a Bayesian approach to aid in this task. The “unusualness” of some data with respect to a model can be assessed using the predictive distribution of the data under the model; an alternative is to use the posterior predictive distribution. Both approaches can be given a sampling interpretation that can then be used to enhance regression diagnostic plots such as marginal model plots.

Modeling Home Prices Using Realtor Data

It can be challenging when teaching regression concepts to find interesting real-life datasets that allow analyses that put all the concepts together in one large example. For example, concepts like interaction and predictor transformations are often illustrated through small-scale, unrealistic examples with just one or two predictor variables that make it difficult for students to appreciate how these concepts might be applied in more realistic multi-variable problems. This article addresses this challenge by describing a complete multiple linear regression analysis of home price data that covers many of the usual regression topics, including interaction and predictor transformations. The analysis also contains useful practical advice on model building—another topic that can be hard to illustrate realistically—and novel statistical graphics for interpreting regression model results. The analysis was motivated by the sale of a home by the author. The statistical ideas discussed range from those suitable for a second college statistics course to those typically found in more advanced linear regression courses.

Model Assessment Plots for Multilevel Logistic Regression

Abstract This paper extends the Bayes marginal model plot (BMMP) model assessment technique from a traditional logistic regression setting to a multilevel application in the area of criminal justice. Convicted felons in the United States receive either a prison sentence or a less severe jail or non-custodial sentence. Researchers have identified many determinants of sentencing variation across the country, some individual such as type of crime and race, and some based on geographical units such as county crime rate. Multilevel rather than conventional regression should be used to quantify any interplay between such individual- and county-level effects since the covariates have a hierarchical structure. Questions arise, however, as to whether a multilevel model provides an adequate fit to the data, and whether the computational burden of a multilevel model over a conventional model is justified. Residual plots, traditionally used to assess regression models, are difficult to interpret with a binary response variable and multilevel covariates, as in this case. BMMPs, an alternative graphical technique, can be used to visualize goodness of fit in such settings. The plots clearly demonstrate the need to use multilevel modeling when analyzing data such as these.

Stochastic shipyard simulation with SimYard

SimYard is a stochastic shipyard simulation tool designed to evaluate the labor costs of executing different schedules in a shipyard production environment. SimYard simulates common production problems such as task delays and labor shortages. A simulated floor manager reacts to problems as they arise. Repeatedly simulating multiple schedules allows the user to compare the schedules on many different metrics, such as expected labor costs and the probability of missing the deadline. A SimYard simulation is driven by many inputs that describe the shipyard being simulated. Determining the correct values for these inputs can be framed as a multivariate calibration problem, which can be solved using inverse regression methods. Predictive sampling from the resulting model provides an appropriate adjustment for statistical uncertainty

Just How Predictable Are the Oscars

Each year, hundreds of millions of people worldwide watch the television broadcast of the Academy Awards ceremony, at which the Academy of Motion Picture Arts and Sciences (AMPAS) honors film-making from the previous year. Almost 6000 members of AMPAS vote for the nominees and final winners of Academy Awards, more commonly known as Oscars, in a wide range of categories for directing, acting, writing, editing, etc. Oscars have been presented for outstanding achievement in film every year since 1928, and are generally recognized to be the premier awards of their kind since AMPAS voting members are themselves the foremost workers in the motion picture industry. In a comparison with other movie awards and movie guide ratings, Simonton (2004) finds substantial validity for the Oscars, and notes that “Those who take an Oscar home can have a strong likelihood of having exhibited superlative cinematic creativity or achievement.” As well as honoring film-makers, Oscars can boost the box-office performance of nominated and winning films. It has even been shown that winning a Best Actor or Best Actress Oscar is associated with a gain in life expectancy, perhaps four extra years of life (Redelmeier and Singh, 2001). However, while studies into the factors that impact a movie’s economic success show that awards can boost revenues, there is little overall association between budget and box office variables and the most important movie awards, such as the Oscars. This article does not consider the economic and aesthetic aspects of movies in relation to the Oscars, but rather focuses purely on the goal of predicting the winners of the four major awards— picture, director, actor in a leading role, actress in a leading role—from those nominated each year. Although many in the media (as well as movie-loving members of the public) make their own annual predictions, it appears that very few researchers have conducted a formal statistical analysis

Picking Apples: Can Multi-Attribute Ecolabels Compete?

Abstract Global food markets in Europe, the U.S. and elsewhere, are experiencing a rapid growth in the number of private party and government environmental labeling programs. Most current ecolabels are defined by standards related to multiple environmental practices. This study presents an analysis of consumers’ choice of food products, in this case apples with or without ecolabels, where the ecolabels present varying combinations of farm practices with implications for environmental quality. These practices include: whether or not standards are met specific to on-farm pest management; presence of stream or groundwater quality protection; presence of on-farm wildlife habitat provision; and which certifier provides the guarantee. Factors influencing consumer preferences for ecolabel attributes are evaluated as a choice-based conjoint analysis. To empirically test the effect of heterogeneity of consumers on preferences for ecolabel attributes, surveys were conducted in a stratified sample in three regions (Portland, Oregon; Minneapolis, Minnesota; Rhode Island) with a focus on sampling across shoppers at different types of markets including conventional supermarkets, farmers markets, natural food stores and food co-ops. Results show that preferences for ecolabels are most strongly driven by type of pesticide usage, in particular for non-synthetic pesticides which were identified with organic production. With an appropriate price premium, ecolabels with an alternative pest management practice and other environmental practices were preferred to conventionally produced apples. These results varied according to age and gender of respondents, and type of store at which respondents shopped.