Profile Likelihood Method Research Articles

Modeling Partial Compliance Through Copulas in a Principal Stratification Framework

Within the principal stratification framework for causal inference, modeling partial compliance is challenging because the continuous nature of the principal strata raises subtle specification issues. In this context, we propose an approach based on the assumption that the joint distribution of the degree of compliance to the treatment and the degree of compliance to the control follows a Plackett copula, so that their association is modeled in a flexible way through a single parameter. Moreover, given the two compliances, the distribution of the outcomes is parameterized in a flexible way through a regression model which may include interaction and quadratic terms and may also be heteroscedastic. In order to estimate the parameters of the resulting model, and then the causal effect of the treatment, we adopt a maximum likelihood approach via the EM algorithm. In applying this approach, the marginal distributions of the two compliances are estimated by their empirical distribution functions, so that no constraints are posed on these distributions. Since the two compliances cannot be jointly observed, there is not direct empirical support for the association parameter. We describe a strategy for studying this parameter by a profile likelihood method and discuss an analysis of the sensitivity of the causal inference results to its value. We apply the proposed approach to data from a study about a drug for lowering cholesterol levels previously analyzed by Efron and Feldman and by Jin and Rubin. Estimated causal effects are in line with those of previous analyses, but the pattern of association between the compliances is qualitatively different, apparently due to the flexibility of the copula and to allowing regression equations in the proposed method to include interactions and heteroscedasticity.

Hunting dark matter gamma-ray lines with the Fermi LAT

Monochromatic photons could be produced in the annihilation or decay of dark matter particles. At high energies, the search for such line features in the cosmic gamma-ray spectrum is essentially background free because plausible astrophysical processes are not expected to produce such a signal. The observation of a gamma-ray line would hence be a `smoking-gun' signature for dark matter, making the search for such signals particularly attractive. Among the different dark matter models predicting gamma-ray lines, the local supersymmetric extension of the standard model with small R-parity violation and gravitino LSP is of particular interest because it provides a framework where primordial nucleosynthesis, gravitino dark matter and thermal leptogenesis are naturally consistent. Using the two-years Fermi LAT data, we present a dedicated search for gamma-ray lines coming from dark matter annihilation or decay in the Galactic halo. Taking into account the full detector response, and using a binned profile likelihood method, we search for significant line features in the energy spectrum of the diffuse flux observed in different regions of the sky. No evidence for a line signal at the 5σ level is found for photon energies between 1 and 300 GeV, and conservative bounds on dark matter decay rates and annihilation cross sections are presented. Implications for gravitino dark matter in presence of small R-parity violation are discussed, as well as the impact of our results on the prospect for seeing long-lived neutralinos or staus at the LHC.

Onset and Rate of Cognitive Change Before Dementia Diagnosis: Findings From Two Swedish Population-Based Longitudinal Studies

We used data from two population-based longitudinal studies to estimate time of onset and rate of accelerated decline across cognitive domains before dementia diagnosis. The H70 includes an age-homogeneous sample (127 cases and 255 non-cases) initially assessed at age 70 with 12 follow-ups over 30 years. The Kungsholmen Project (KP) includes an age-heterogeneous sample (279 cases and 562 non-cases), with an average age of 82 years at initial assessment, and 4 follow-ups spanning 13 years. We fit mixed linear models to the data and determined placement of change points by a profile likelihood method. Results demonstrated onset of accelerated decline for fluid (speed, memory) versus crystallized (verbal, clock reading) abilities occurring approximately 10 and 5 years before diagnosis, respectively. Although decline before change points was greater for fluid abilities, acceleration was more pronounced for crystallized abilities after the change points. This suggests that onset and rate of acceleration vary systematically along the fluid-crystallized ability continuum. There is early onset in fluid abilities, but these changes are difficult to detect due to substantial age-related decline. Onset occurred later and acceleration was greater in crystallized abilities, suggesting that those markers may provide more valid identification of cases in later stages of the prodromal phase.

Interval estimation for response adaptive clinical trials

In this paper we examine a new method for constructing confidence intervals for the difference of success probabilities to analyze dependent data from response adaptive designs with binary responses. Specifically we investigate the feasibility of the Jeffreys–Perks procedure for interval estimation. Simulation results are derived to demonstrate the performance of the Jeffreys–Perks procedure compared with the profile likelihood method. It is found that both asymptotic methods perform well for small sample sizes despite being approximate procedures.

Deriving Prostate Alpha-Beta Ratio Using Carefully Matched Groups, Long Follow-Up and the Phoenix Definition of Biochemical Failure

Prior studies have derived low values of alpha-beta ratio (a/ß) for prostate cancer of approximately 1-2 Gy. These studies used poorly matched groups, differing definitions of biochemical failure, and insufficient follow-up. National Comprehensive Cancer Network low- or low-intermediate risk prostate cancer patients, treated with external beam radiotherapy or permanent prostate brachytherapy, were matched for prostate-specific antigen, Gleason score, T-stage, percentage of positive cores, androgen deprivation therapy, and era, yielding 118 patient pairs. The Phoenix definition of biochemical failure was used. The best-fitting value for a/ß was found for up to 90-month follow-up using maximum likelihood analysis, and the 95% confidence interval using the profile likelihood method. Linear quadratic formalism was applied with the radiobiological parameters of relative biological effectiveness = 1.0, potential doubling time = 45 days, and repair half-time = 1 hour. Bootstrap analysis was performed to estimate uncertainties in outcomes, and hence in a/ß. Sensitivity analysis was performed by varying the values of the radiobiological parameters to extreme values. The value of a/ß best fitting the outcomes data was >30 Gy, with lower 95% confidence limit of 5.2 Gy. This was confirmed on bootstrap analysis. Varying parameters to extreme values still yielded best-fit a/ß of >30 Gy, although the lower 95% confidence interval limit was reduced to 0.6 Gy. Using carefully matched groups, long follow-up, the Phoenix definition of biochemical failure, and well-established statistical methods, the best estimate of a/ß for low and low-tier intermediate-risk prostate cancer is likely to be higher than that of normal tissues, although a low value cannot be excluded.

Confidence intervals for correlated proportion differences from paired data in a two-arm randomised clinical trial

Many key outcome measures for monitoring disease progression are based on summing across several paired domains. The scores from the paired domain in an individual are likely to be correlated and we present here an analysis of treatment effect at each domain that accounts for the correlation. We use the profile likelihood method, asymptotic score method, and three simple asymptotic methods and construct confidence intervals to compare proportions of responders in a two-arm randomised trial. We evaluate the performance of these confidence interval estimators with respect to their mean coverage probabilities, mean left-tail and right-tail non-coverage rates, and mean confidence widths. These methods are then applied to analyse a multi-centre randomised clinical trial for scleroderma patients.

Multiple imputation method for the semiparametric accelerated failure time mixture cure model

There are few discussions on the semiparametric accelerated failure time mixture cure model due to its complexity in estimation. In this paper, we propose a multiple imputation method for the semiparametric accelerated failure time mixture cure model based on the rank estimation method and the profile likelihood method. Both approaches can be easily implemented in R environment. However, the computation time for the rank estimation method is longer than that from the profile likelihood method. Simulation studies demonstrate that the performances of estimated parameters from the proposed methods are comparable to those from the expectation maximization (EM) algorithm, and the estimated variances are comparable to those from the empirical approach. For illustration, we apply the proposed method to a data set of failure times from the bone marrow transplantation.

A profile likelihood method for normal mixture with unequal variance

It is well known that the normal mixture with unequal variance has unbounded likelihood and thus the corresponding global maximum likelihood estimator (MLE) is undefined. One of the commonly used solutions is to put a constraint on the parameter space so that the likelihood is bounded and then one can run the EM algorithm on this constrained parameter space to find the constrained global MLE. However, choosing the constraint parameter is a difficult issue and in many cases different choices may give different constrained global MLE. In this article, we propose a profile log likelihood method and a graphical way to find the maximum interior mode. Based on our proposed method, we can also see how the constraint parameter, used in the constrained EM algorithm, affects the constrained global MLE. Using two simulation examples and a real data application, we demonstrate the success of our new method in solving the unboundness of the mixture likelihood and locating the maximum interior mode.

Limits, discovery and cut optimization for a Poisson process with uncertainty in background and signal efficiency: TRolke 2.0

A C++ class was written for the calculation of frequentist confidence intervals using the profile likelihood method. Seven combinations of Binomial, Gaussian, Poissonian and Binomial uncertainties are implemented. The package provides routines for the calculation of upper and lower limits, sensitivity and related properties. It also supports hypothesis tests which take uncertainties into account. It can be used in compiled C++ code, in Python or interactively via the ROOT analysis framework. Program summary Program title: TRolke version 2.0 Catalogue identifier: AEFT_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEFT_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: MIT license No. of lines in distributed program, including test data, etc.: 3431 No. of bytes in distributed program, including test data, etc.: 21 789 Distribution format: tar.gz Programming language: ISO C++. Computer: Unix, GNU/Linux, Mac. Operating system: Linux 2.6 (Scientific Linux 4 and 5, Ubuntu 8.10), Darwin 9.0 (Mac-OS X 10.5.8). RAM: ∼ 20 MB Classification: 14.13. External routines: ROOT ( http://root.cern.ch/drupal/) Nature of problem: The problem is to calculate a frequentist confidence interval on the parameter of a Poisson process with statistical or systematic uncertainties in signal efficiency or background. Solution method: Profile likelihood method, Analytical Running time: < 10 − 4 seconds per extracted limit.

Estimating gametic introgression rates in a risk assessment context: a case study with Scots pine relicts

The estimation of recent gene immigration is fundamental to a wide range of evolutionary and conservation studies. In a risk assessment context, gene flow estimation procedures are needed that are both accurate and readily amenable to formal evaluation of statistical uncertainty. However, genetic methods for estimating recent migration rates that are specific and have been thoroughly evaluated are scarce. Here we use an original and straightforward maximum-likelihood method to estimate recent uniparental gametic immigration from non-local plantations into an endangered population of the Iberian relict pine variety Pinus sylvestris var. nevadensis D. H. Christ. Our approach is not intended to ascertain population membership of individuals, but rather to obtain accurate immigration rate estimates with reliable confidence limits. We found very high (approximately 40%) pollen introgression at the seed-crop level into the Scots pine relict, and substantial (10-15%) male gametic introgression among naturally regenerated recruits. Using numerical simulation, we show that our method yields uniparental gametic immigration estimates that are expected to be virtually unbiased and usually accurate under our sampling conditions. Among four tested methods to estimate the confidence intervals for immigration estimates, the profile-likelihood method was the best, as it outperformed bootstrapping procedures and yielded coverage close to nominal limits under different sample sizes and migration rates. This study presents a method by which researchers can facilitate decision making within a gene flow risk assessment context.

Maximum likelihood inference for the Cox regression model with applications to missing covariates

In this paper, we carry out an in-depth theoretical investigation for existence of maximum likelihood estimates for the Cox model [D.R. Cox, Regression models and life tables (with discussion), Journal of the Royal Statistical Society, Series B 34 (1972) 187–220; D.R. Cox, Partial likelihood, Biometrika 62 (1975) 269–276] both in the full data setting as well as in the presence of missing covariate data. The main motivation for this work arises from missing data problems, where models can easily become difficult to estimate with certain missing data configurations or large missing data fractions. We establish necessary and sufficient conditions for existence of the maximum partial likelihood estimate (MPLE) for completely observed data (i.e., no missing data) settings as well as sufficient conditions for existence of the maximum likelihood estimate (MLE) for survival data with missing covariates via a profile likelihood method. Several theorems are given to establish these conditions. A real dataset from a cancer clinical trial is presented to further illustrate the proposed methodology.

Robust Joint Modeling of Longitudinal Measurements and Competing Risks Failure Time Data

Existing methods for joint modeling of longitudinal measurements and survival data can be highly influenced by outliers in the longitudinal outcome. We propose a joint model for analysis of longitudinal measurements and competing risks failure time data which is robust in the presence of outlying longitudinal observations during follow-up. Our model consists of a linear mixed effects sub-model for the longitudinal outcome and a proportional cause-specific hazards frailty sub-model for the competing risks data, linked together by latent random effects. Instead of the usual normality assumption for measurement errors in the linear mixed effects sub-model, we adopt a t -distribution which has a longer tail and thus is more robust to outliers. We derive an EM algorithm for the maximum likelihood estimates of the parameters and estimate their standard errors using a profile likelihood method. The proposed method is evaluated by simulation studies and is applied to a scleroderma lung study.

A Change-Point Regression Approach for Efficacy Evaluation of Dietary Supplements

In clinical trials for dietary supplements and functional foods, the study population tends to be a mixture of healthy subjects and those who are not so healthy but are not definitely diseased (called “borderline subjects”). For such heterogeneous populations, the t-test and ANCOVA method often fail to provide the desired treatment efficacy. We propose an alternative approach for the efficacy evaluation of dietary supplements and functional foods based on a change-point linear regression model. The model does not require the assumption of a constant treatment effect and provides clinically interpretable results. By employing the AIC-based profile likelihood method, inferences can be made easily using standard statistical software. The proposed method was applied to the Garcinia study data, and the merit of the method was demonstrated by comparing it with traditional methods.

Alignment of protein mass spectrometry data by integrated Markov chain shifting method

Mass spectrometers such as SELDI-TOF (surface enhanced laser desorption/ionization time-of-flight) and MALDI-TOF (matrix assisted laser desorption and ionization time-of-flight) measure the relative abundance of different protein ions or protein fragments (peptides) indexed by the mass-to-charge ratio (m/z). A special characteristic of the MS spectra is its variabilities in both m/z values and intensity magnitudes. We propose modelling the logintensities by a semiparametric model and the m/z by the integrated Markov chain shifting (IMS) model, for which the second-order differences of the random effects are assumed to follow a second-order Markov chain. Alignment of spectra is done through averaging over the random shifts conditional on the observed intensity information. The unknown parameters are estimated by an iterative nonparametric maximum profile likelihood method and a Gaussian kernel approximation. The bandwidths in kernel approximation are taken to be 0.04%–0.08% of the m/z values. Simulation results show that the proposed approach can achieve satisfactory alignment by reducing the intensity variations of the misalignment spectra by a factor of around 75%. Most alignment algorithms align spectra by clustering neighboring peaks and do not incorporate peak height information. Our semiparametric random shifting method builds a model taking into consideration of both the random shift effects of neighboring m/z values and similarity of the intensity magnitudes of common peaks within the ranges of about 50% of the intensity values.

Parameters of a Dose‐Response Model Are on the Boundary: What Happens with BMDL?

It is well known that, under appropriate regularity conditions, the asymptotic distribution for the likelihood ratio statistic is chi(2). This result is used in EPA's benchmark dose software to obtain a lower confidence bound (BMDL) for the benchmark dose (BMD) by the profile likelihood method. Recently, based on work by Self and Liang, it has been demonstrated that the asymptotic distribution of the likelihood ratio remains the same if some of the regularity conditions are violated, that is, when true values of some nuisance parameters are on the boundary. That is often the situation for BMD analysis of cancer bioassay data. In this article, we study by simulation the coverage of one- and two-sided confidence intervals for BMD when some of the model parameters have true values on the boundary of a parameter space. Fortunately, because two-sided confidence intervals (size 1-2alpha) have coverage close to the nominal level when there are 50 animals in each group, the coverage of nominal 1-alpha one-sided intervals is bounded between roughly 1-2alpha and 1. In many of the simulation scenarios with a nominal one-sided confidence level of 95%, that is, alpha= 0.05, coverage of the BMDL was close to 1, but for some scenarios coverage was close to 90%, both for a group size of 50 animals and asymptotically (group size 100,000). Another important observation is that when the true parameter is below the boundary, as with the shape parameter of a log-logistic model, the coverage of BMDL in a constrained model (a case of model misspecification not uncommon in BMDS analyses) may be very small and even approach 0 asymptotically. We also discuss that whenever profile likelihood is used for one-sided tests, the Self and Liang methodology is needed to derive the correct asymptotic distribution.

Onset of terminal decline in cognitive abilities in individuals without dementia

To identify time of onset and rate of mortality-related change (terminal decline) in cognitive abilities in later life. The sample consisted of 288 individuals without dementia (born 1901-1902) drawn from the population of Göteborg, Sweden. Participants were followed from age 70 until death, with up to 12 measurement occasions on three cognitive abilities. Change-point analysis was performed using an automated piecewise linear mixed modeling approach to identify the inflection point indicating accelerated within-person change related to mortality. A profile likelihood method was used to identify the change point that best fit the data for each of three cognitive abilities. Onset of terminal decline was identified 6.6 years prior to death for verbal ability, 7.8 years for spatial ability, and 14.8 years for perceptual speed. There is substantial acceleration in cognitive decline many years prior to death among individuals without dementia. Time of onset and rate of terminal decline vary considerably across cognitive abilities.

Evaluation of three methods for calculating statistical significance when incorporating a systematic uncertainty into a test of the background-only hypothesis for a Poisson process

Hypothesis tests for the presence of new sources of Poisson counts amidst background processes are frequently performed in high energy physics (HEP), gamma ray astronomy (GRA), and other branches of science. While there are conceptual issues already when the mean rate of background is precisely known, the issues are even more difficult when the mean background rate has non-negligible uncertainty. After describing a variety of methods to be found in the HEP and GRA literature, we consider in detail three classes of algorithms and evaluate them over a wide range of parameter space, by the criterion of how close the ensemble-average Type I error rate (rejection of the background-only hypothesis when it is true) compares with the nominal significance level given by the algorithm. We recommend wider use of an algorithm firmly grounded in frequentist tests of the ratio of Poisson means, although for very low counts the overcoverage can be severe due to the effect of discreteness. We extend the studies of Cranmer, who found that a popular Bayesian-frequentist hybrid can undercover severely when taken to high Z-values. We also examine the profile likelihood method, which has long been used in GRA and HEP; it provides an excellent approximation in much of the parameter space, as previously studied by Rolke and collaborators.

Estimation of benchmark doses as threshold levels of urinary cadmium, based on excretion of β 2-microglobulin in cadmium-polluted and non-polluted regions in Japan

The threshold level of urinary cadmium (Cd) for β 2-microglobulin (MG)-uria was calculated using a benchmark dose ( BMD) approach. Total number of subjects was 3103 for Cd-polluted areas and 2929 for non-polluted areas. Multiple logistic regression analysis was employed to fit the dose–response model taking into consideration an age effect. Cut-off values for urinary β 2-MG were defined as those corresponding to the 84th and 97.5th percentile of β 2-MG levels in the controls, and 1000 μg/g creatinine (cr). The BMD low (BMDL) was calculated using the profile likelihood method. When the benchmark response was 5%, the BMD/BMDL of Cd for the 84th percentile of β 2-MG for mean age, 55, 65, and 75 years was 3.0/2.7, 4.6/4.2, 2.8/2.6, and 1.8/1.6 μg/g cr in men and 3.4/3.2, 5.8/5.5, 3.2/3.1, and 1.8/1.7 μg/g cr in women, respectively. The value for the 97.5th percentile for each age was 4.9/4.5, 7.6/7.0, 4.6/4.3, and 2.6/2.4 μg/g cr in men and 5.9/5.6, 9.7/9.2, 5.6/5.3, and 2.8/2.6 μg/g cr in women. Namely it became clear that the margin between the threshold level and average excretion level of urinary Cd was small in the older population in Japan. To prevent the adverse health effects caused by exposure to Cd, it is important to establish the threshold level of Cd exposure at each age.

A comparison of the active surveillance of scrapie in the European Union

The abattoir and the fallen stock surveys constitute the active surveillance component aimed at improving the detection of scrapie across the European Union. Previous studies have suggested the occurrence of significant differences in the operation of the surveys across the EU. In the present study we assessed the standardisation of the surveys throughout time across the EU and identified clusters of countries with similar underlying characteristics allowing comparisons between them. In the absence of sufficient covariate information to explain the observed variability across countries, we modelled the unobserved heterogeneity by means of non-parametric distributions on the risk ratios of the fallen stock over the abattoir survey. More specifically, we used the profile likelihood method on 2003, 2004 and 2005 active surveillance data for 18 European countries on classical scrapie, and on 2004 and 2005 data for atypical scrapie separately. We extended our analyses to include the limited covariate information available, more specifically, the proportion of the adult sheep population sampled by the fallen stock survey every year. Our results show that the between-country heterogeneity dropped in 2004 and 2005 relative to that of 2003 for classical scrapie. As a consequence, the number of clusters in the last two years was also reduced indicating the gradual standardisation of the surveillance efforts across the EU. The crude analyses of the atypical data grouped all the countries in one cluster and showed non-significant gain in the detection of this type of scrapie by any of the two sources. The proportion of the population sampled by the fallen stock appeared significantly associated with our risk ratio for both types of scrapie, although in opposite directions: negative for classical and positive for atypical. The initial justification for the fallen stock, targeting a high-risk population to increase the likelihood of case finding, appears compromised for both types of scrapie in some countries.

Lyman–Kutcher–Burman NTCP model parameters for radiation pneumonitis and xerostomia based on combined analysis of published clinical data

Knowledge of accurate parameter estimates is essential for incorporating normal tissue complication probability (NTCP) models into biologically based treatment planning. The purpose of this work is to derive parameter estimates for the Lyman–Kutcher–Burman (LKB) NTCP model using a combined analysis of multi-institutional toxicity data for the lung (radiation pneumonitis) and parotid gland (xerostomia). A series of published clinical datasets describing dose response for radiation pneumonitis (RP) and xerostomia were identified for this analysis. The data support the notion of large volume effect for the lung and parotid gland with the estimates of the n parameter being close to unity. Assuming that n = 1, the m and TD50 parameters of the LKB model were estimated by the maximum likelihood method from plots of complication rate as a function of mean organ dose. Ninety five percent confidence intervals for parameter estimates were obtained by the profile likelihood method. If daily fractions other than 2 Gy had been used in a published report, mean organ doses were converted to 2 Gy/fraction-equivalent doses using the linear-quadratic (LQ) formula with α/β = 3 Gy. The following parameter estimates were obtained for the endpoint of symptomatic RP when the lung is considered a paired organ: m = 0.41 (95% CI 0.38, 0.45) and TD50 = 29.9 Gy (95% CI 28.2, 31.8). When RP incidence was evaluated as a function of dose to the ipsilateral lung rather than total lung, estimates were m = 0.35 (95% CI 0.29, 0.43) and TD50 = 37.6 Gy (95% CI 34.6, 41.4). For xerostomia expressed as reduction in stimulated salivary flow below 25% within six months after radiotherapy, the following values were obtained: m = 0.53 (95% CI 0.45, 0.65) and TD50 = 31.4 Gy (95% CI 29.1, 34.0). Although a large number of parameter estimates for different NTCP models and critical structures exist and continue to appear in the literature, it is hard to justify the use of any single parameter set obtained at a selected institution for the purposes of biologically based treatment planning. Our expectation is that the proposed model parameters based on cumulative experience at various institutions are more representative of the overall practice of radiation therapy than any single-institution data, and could be more readily incorporated into clinical use.