Principal Points for Mixed Effects Models:  Applications to Applications to Identifying Placebo Responders
(with  Eva Petkova), submitted for publication
Principal points are cluster means for theoretical distributions. For longitudinal data where each observation corresponds to a curve, principal points can be used to
determine a set of representative curve profiles that optimally represent the distribution. This paper presents a method of determining maximum likelihood estimators of principal points for linear mixed models.  The method can incorporate covariates as well.  The results are applied to an anti-depressant study to identify prototypical drug and placebo response profiles.


Model Misspecification: Finite Mixture or Homogeneous?
(with Dong Yun and Eva Petkova,  Statistical Modelling, to appear in volume 8, 2008)
A common problem in statistical modelling is to distinguish between finite mixture distribution and a homogeneous non-mixture distribution.  Finite mixture models
are widely used in practice and often mixtures of normal densities are indistinguishable from homogenous non-normal densities.  This paper illustrates what happens when
the EM algorithm for normal mixtures is applied to a distribution that is a homogeneous non-mixture distribution. In particular, a population-based EM algorithm for finite mixtures
is introduced and applied directly to density functions instead of sample data.  The population-based EM algorithm is used to find finite mixture approximations to common homogeneous distributions.
An example regarding the nature of a placebo response in drug treated depressed subjects is used to illustrate ideas.

Click here for R code that will fit a 2-component univariate normal mixture to a given density function using the Population-Based EM Algorithm.  The R software is freely available at CRAN.
In the program, the user specifies the density function g(y).  The default density is a gamma density.

Linear Transformations and the k-Means Clustering Algorithm:  Applications to Clustering Curves
(2007, The American Statistician, 61, 34-40)
Functional data can be clustered by plugging estimated regression coefficients from individual curves into the k-means algorithm.  Clustering results can differ
depending on how the curves are fit to the data.  Estimating curves using different sets of basis functions corresponds to different linear transformations of the data.
k-means clustering is not invariant to linear transformations of the data.  The optimal linear transformation for clustering will stretch the distribution
so that the primary direction of variability aligns with actual differences in the clusters.  It is shown that clustering the raw data will often give results
similar to clustering regression coefficients obtained using an orthogonal design matrix.  Clustering functional data using an L^2 metric on function
space can be achieved by clustering a suitable linear transformation of the regression coefficients.  An example where depressed individuals are treated
with an antidepressant is used for illustration.

Latent Regression Analysis
(with Eva Petkova, submitted for publication)
An important question in clincial studies if there exist distinct disease classes (e.g. cancer versus no cancer) or if instead there exists a continuous latent variable representing disease severity (e.g. level of depression).  To answer this question, a latent regression model is proposed which represents a generalization of a finite mixture model.  The finite mixture model is cast as a regression with a latent Bernoulli predictor.  The latent regression model generalizes the finite mixture model by allowing the latent predictor to vary continuously distribution on the interval (0,1). An EM algorithm is used to estimate parameters of the latent regression model.  Examples and simulations are given to illustrate the latent regression model.  In particular, the latent regression model is illustrated in a depression treatment study to determine if there exist two distinct classes of subjects (those who experience a placebo effect and those who do not), or if instead everyone experiences a placebo effect over a continous range.


A Parametric k-Means Algorithm
(2007,   Computational Statistics, 22, 71-89)
The k points that optimally represent a distribution (usually in terms of a squared error loss) are called the k principal points.  This paper presents a computationally intensive method that automatically determines the principal points of a parametric distribution.  Cluster means from the k-means algorithm are nonparametric estimators of principal points.  A parametric k-means approach is introduced
for estimating principal points by running the k-means algorithm on a very large simulated data set from a distribution whose parameters are estimated using maximum likelihood.
Theoretical and simulation results are presented comparing the parametric k-means algorithm to the usual k-means algorithm and an example on determining sizes of gas masks is used to
illustrate the parametric k-means algorithm.

R-code for the Parametric k-means algorithm:

On Estimation in Compartment Modeling with an Input Function
(2006, with Ogden, Biostatistics,  7, 115-129)
Abstract:  In some nonlinear regression situations, one or more of the parameters in the expression for the regression function is estimated from a separate data source.  In such a case, the typical estimation procedure is to estimate the appropriate parameters from the separate data, then plug these estimated values into the expression for the regression function for the estimation of the rest of the parameters. This situation arises frequently in compartment modeling when there is
an external ``input function'' to the system.  This paper addresses the general question of the estimation of parameters and their standard errors in nonparametric regression when some parameters are estimated separately.  One important application of this method is for estimation of rate parameters and their standard errors in a compartmental system when parameters from an input function are estimated from separate data.  An example and a simulation study are provided to illustrate the results and to study the performance of the proposed methodologies.
 

Allometric Extension for Multivariate Regression Models
(2006, with Ivey,  Journal of Data Science, 4, 479-495)
Abstract:  In multivariate regression, interest lies on how the response vector depends on the covariates. A multivariate regression model is proposed where covariates explain variation in the response only in the direction of the first principal component axis. This model provides allows a clear interpretation in situations where the first principal component has a meaningful interpretation. In particular, in allometric studies where the first principal component is considered a size variable, the model stipulates that the covariates effect only the size and not the shape of an organism. We show that the model naturally generalizes the two--group allometric extension model to the framework of multivariate regression where groups differ conditionally on a set of covariates. An example which motivated this model is illustrated.

Linear Conditional Expectation for Discretized Distributions
(2004, with Sanders, Journal of Applied Statistics, 31, 361-372)
Abstract:  Many statistical methods for continuous distributions assume a linear conditional expectation. Components of multivariate distributions are often measured on a discrete ordinal scale based on a discretization of an underlying continuous
latent variable. The results in this paper show that common examples of discretized bivariate and trivariate distributions will have a linear conditional expectation. Examples and simulations are provided to illustrate the results.

Clustering Functional Data
(2003, with Kimberly K. J. Kinateder, The Journal of Classification, 20, 93-114)
Abstract:  The problem of clustering functional data is addressed. Results on principal points (cluster means for probability distributions) are given for functional Gaussian distributions. Examples and simulations are provided to illustrate results.
 

Estimating the Average Slope
(2003, Journal of Applied Statistics,  30, 389-396)
Abstract : In the regression setup Y = f(X) + e, an example is illustrated  where the average slope E[f ' (X)] is to be estimated. A simple solution is to use the slope obtained from fitting a least squares line as an estimator of
E[f ' (x)]. If f(X) is not linear, then the simple linear regression model is the wrong model.  However, in certain circumstances, the slope from the simple linear regression model is a correct estimator for the average slope of the response.  This paper investigates when the slope of a least squares line is a suitable estimator of the average slope of the reponse.
 

Identifying Placebo Responders Among Drug Treated Subjects
(2003, with Petkova and Ogden, Journal of the American Statistical Association, 98, 850-858)
Abstract:  Identification of placebo responders among subjects treated with active drug has significant clinical and research implications.  In clinical practice when a patient treated with medication improves, this improvement may be attributed to the chemical component of the drug itself, a ``placebo effect'', or some combination of these. Determining the proper subsequent treatment and maintenance of the patient may be aided greatly by understanding the mechanism of a patient's improvement.
In a research context, classification of patient response has bearing on the way efficacy and effectiveness clinical trials are designed and conducted. This paper presents a framework for studying placebo response in diverse areas of medicine.
In order to identify placebo responders among drug treated patients, a profile of the clinical status over time (outcome profile) is estimated for each subject.  Self-consistent partitioning techniques are used to group subjects based on the amount of curvature in the profile  as well as the overall trend in the profile.  The resulting partitions determine representative profiles for subjects in the drug group which can subsequently be used to classify patients.  The proposed method is applied to data from a clinical trial for treatment of depression involving placebo and the active drug phenelzine.  Data from the placebo arm of the study is used to help validate the procedure since the drug-treated and placebo-treated subjects should share common profiles.


 
 

Last updated March 20, 2008