Christopher H. Morrell

Linear Transformations of Linear Mixed-Effects Models

Abstract A number of articles have discussed the way lower order polynomial and interaction terms should be handled in linear regression models. Only if all lower order terms are included in the model will the regression model be invariant with respect to coding transformations of the variables. If lower order terms are omitted, the regression model will not be well formulated. In this paper, we extend this work to examine the implications of the ordering of variables in the linear mixed-effects model. We demonstrate how linear transformations of the variables affect the model and tests of significance of fixed effects in the model. We show how the transformations modify the random effects in the model, as well as their covariance matrix and the value of the restricted log-likelihood. We suggest a variable selection strategy for the linear mixed-effects model.

Likelihood ratio testing of variance components in the linear mixed-effects model using restricted maximum likelihood.

This paper reports the results of an extensive Monte Carlo study of the distribution of the likelihood ratio test statistic using the value of the restricted likelihood for testing random components in the linear mixed-effects model when the number of fixed components remains constant. The distribution of this test statistic is considered when one additional random component is added. The distribution of the likelihood ratio test statistic computed using restricted maximum likelihood is compared to the likelihood ratio test statistic computed from the usual maximum likelihood. The rejection proportion is computed under the null hypothesis using a mixture of chi-square distributions. The restricted likelihood ratio statistic has a reasonable agreement with the maximum likelihood test statistic. For the parameter combinations considered, the rejection proportions are, in most cases, less than the nominal 5% level for both test statistics, though, on average, the rejection proportions for REML are closer to the nominal level than for ML.

Estimating Unknown Transition Times Using a Piecewise Nonlinear Mixed-Effects Model in Men with Prostate Cancer

Abstract It may be clinically useful to know when prostate-specific antigen (PSA) levels first begin to rise rapidly and to determine if the natural history of PSA progression is different in men with locally confined prostate cancers compared to men with metastatic tumors. This article uses a nonlinear mixed-effects model to describe longitudinal changes in PSA in men before their prostate cancers were detected clinically. Repeated measurements of PSA are available for 18 subjects with a diagnosis of prostate cancer based on prostate biopsy. PSA measurements were determined on repeated frozen serum samples collected from subjects with at least 10.0 years and up to 25.6 years of observation before the cancer was detected. A piecewise model is used to describe this data. The model is linear long before the cancer was detected and exponential nearer the time the cancer was detected. The time at which the PSA levels change from linear to exponential PSA progression is unknown but can be estimated by includin...

Model Choice Can Obscure Results in Longitudinal Studies

BACKGROUNDnThis article examines how different parameterizations of age and time in modeling observational longitudinal data can affect results.nnnMETHODSnWhen individuals of different ages at study entry are considered, it becomes necessary to distinguish between longitudinal and cross-sectional differences to overcome possible selection biases.nnnRESULTSnVarious models were fitted using data from longitudinal studies with participants with different ages and different follow-up lengths. Decomposing age into two components-age at entry into the study (first age) and the longitudinal follow-up (time) compared with considering age alone-leads to different conclusions.nnnCONCLUSIONSnIn general, models using both first age and time terms performed better, and these terms are usually necessary to correctly analyze longitudinal data.

Screening for prostate cancer by using random‐effects models

Random-effects models are used to screen male participants in a long-term longitudinal study for prostate cancer. By using posterior probabilities, each male can be classified into one of four diagnostic states for prostate disease: normal, benign prostatic hyperplasia, local cancer and metastatic cancer. Repeated measurements of prostate-specific antigen, collected when there was no clinical evidence of prostate disease, are used in the classification process. An individuals screening data are considered one repeated measurement at a time as his data are collected longitudinally over time. Posterior probabilities are calculated on the basis of data from other individuals with confirmed diagnoses of each of the four diagnostic states. Copyright 2003 Royal Statistical Society.

Modeling Disease Progression with Longitudinal Markers.

In this article we propose a Bayesian natural history model for disease progression based on the joint modeling of longitudinal biomarker levels, age at clinical detection of disease, and disease status at diagnosis. We establish a link between the longitudinal responses and the natural history of the disease by using an underlying latent disease process that describes the onset of the disease and models the transition to an advanced stage of the disease as dependent on the biomarker levels. We apply our model to data from the Baltimore Longitudinal Study of Aging on prostate-specific antigen to investigate the natural history of prostate cancer.

Lines in Random Effects Plots from the Linear Mixed-Effects Model

Abstract After fitting a linear mixed-effects model to a set of repeated-measures or longitudinal data, it is common practice to plot the estimated random effects. On occasion it may be observed that in these plots a straight line may appear. How did this line arise? What influence does it have on the interpretation of the results from the model? This article demonstrates an artifact that can occur in the plots of random effects. If a cluster has exactly one observation, the plot of any estimated random effect against any other estimated random effect will fall on a straight line. The line in the plot of a pair of estimated random effects may even lie in the opposite direction of the direction suggested by the correlation from the random effects covariance matrix. For clusters with two observations at the same design points of the variables for the random effects, the estimated random effects will lie on a plane. Using an example, we demonstrate the patterns in the plots of the estimated random effects. This illustrates that care must be taken when using plots of the estimated random effects.

Construction of hearing percentiles in women with non-constant variance from the linear mixed-effects model

Current age-specific reference standards for adult hearing thresholds are primarily cross-sectional in nature and vary in the degree of screening of the reference sample for noise-induced hearing loss and other hearing problems. We develop methods to construct age-specific percentiles for longitudinal data that have been modelled using the linear mixed-effects model. We apply these methods to construct percentiles of hearing level using data from a carefully screened sample of women from the Baltimore Longitudinal Study of Aging. However, the variation in the residuals and random effects from the linear mixed-effects model does not remain constant with age and frequency of the stimulus tone. In addition, the distribution of the hearing levels is not symmetric about the mean. We develop a number of methods to use the output from the linear mixed-effects model to construct percentiles that do not have constant variance. We use a transformation of the hearing levels to provide for skewness in the final percentile curves. The change in the variation of the residuals and random effects is modelled as a function of beginning age and frequency and we use this variance function to construct the hearing percentiles. We present a number of approaches. First, we use the absolute values of the population residuals to model the total deviation about the mean as a function of beginning age and frequency. Second, we model the standard deviation in the person-specific (cluster) residuals as well as the standard deviation in the estimated random effects. Finally, we use weighted least squares with the regressions on the absolute cluster residuals and absolute estimated random effects where the weights are the reciprocal of the standard deviations of their estimates.

Screening for prostate cancer using multivariate mixed-effects models

Using several variables known to be related to prostate cancer, a multivariate classification method is developed to predict the onset of clinical prostate cancer. A multivariate mixed-effects model is used to describe longitudinal changes in prostate-specific antigen (PSA), a free testosterone index (FTI), and body mass index (BMI) before any clinical evidence of prostate cancer. The patterns of change in these three variables are allowed to vary depending on whether the subject develops prostate cancer or not and the severity of the prostate cancer at diagnosis. An application of Bayes’ theorem provides posterior probabilities that we use to predict whether an individual will develop prostate cancer and, if so, whether it is a high-risk or a low-risk cancer. The classification rule is applied sequentially one multivariate observation at a time until the subject is classified as a cancer case or until the last observation has been used. We perform the analyses using each of the three variables individually, combined together in pairs, and all three variables together in one analysis. We compare the classification results among the various analyses and a simulation study demonstrates how the sensitivity of prediction changes with respect to the number and type of variables used in the prediction process.

An Investigation of the Median-Median Method of Linear Regression.

Least squares regression is the most common method of fitting a straight line to a set of bivariate data. Another less known method that is available on Texas Instruments graphing calculators is median-median regression. This method is proposed as a simple method that may be used with middle and high school students to motivate the idea of fitting a straight line to data. The median-median line may also be viewed as a method that is not greatly affected by outliers (robust to outliers). Our paper briefly reviews the median-median regression method, considers various examples to compare the median-median line to the least squares line, and investigates the properties of the median-median line versus the least squares line using a simulation study.