Adjusted Likelihood Ratio Research Articles

Bootstrap likelihood ratio confidence bands for survival functions under random censorship and its semiparametric extension

Simultaneous confidence bands for survival functions, from randomly right censored data, can be computed by inverting likelihood ratio functions based on appropriate thresholds. Sometimes, however, the requisite asymptotic distributions are intractable, or thresholds based on Brownian bridge approximations are not easy to obtain when simultaneous confidence bands over only sub-regions are possible or desired. We obtain the thresholds by bootstrapping (i) a nonparametric likelihood ratio function via censored data bootstrap and (ii) a semiparametric adjusted likelihood ratio function via a two-stage bootstrap that utilizes a model for the second stage. These two scenarios are grounded respectively in standard random censorship and its semiparametric extension introduced by Dikta. The two bootstraps, which are different in the way resampling is done, are shown to have asymptotic validity. The respective confidence bands are neighborhoods of the well-known Kaplan–Meier estimator and the more recently developed Dikta’s semiparametric counterpart. As evidenced by a validation study, both types of confidence bands provide approximately correct coverage. The model-based confidence bands, however, are tighter than the nonparametric ones. Two sensitivity studies reveal that the model-based method performs well when standard binary regression models are fitted, indicating its robustness to misspecification as well as its practical applicability. An illustration is given using real data.

An Adjusted Likelihood Ratio Approach Analysing Distribution of Food Products to Assist the Investigation of Foodborne Outbreaks.

In order to facilitate foodborne outbreak investigations there is a need to improve the methods for identifying the food products that should be sampled for laboratory analysis. The aim of this study was to examine the applicability of a likelihood ratio approach previously developed on simulated data, to real outbreak data. We used human case and food product distribution data from the Norwegian enterohaemorrhagic Escherichia coli outbreak in 2006. The approach was adjusted to include time, space smoothing and to handle missing or misclassified information. The performance of the adjusted likelihood ratio approach on the data originating from the HUS outbreak and control data indicates that the adjusted approach is promising and indicates that the adjusted approach could be a useful tool to assist and facilitate the investigation of food borne outbreaks in the future if good traceability are available and implemented in the distribution chain. However, the approach needs to be further validated on other outbreak data and also including other food products than meat products in order to make a more general conclusion of the applicability of the developed approach.

Modeling crash frequency and severity with spatiotemporal dependence

This study proposes a novel multinomial generalized Poisson model with error components and spatiotemporal dependence (ST-EMGP) to analyze multi-period crash frequency and severity data. The proposed model not only simultaneously models crash frequency and severity, but also accommodates spatial and temporal dependence (spatiotemporal dependence) by specifying a spatiotemporal function. To demonstrate the applicability of the proposed model, a case study is conducted on five consecutive years׳ (2004–2008) crash data of Taiwan׳s Freeway No. 1. Estimation results show that ST-EMGP model performs better than the models without considering spatiotemporal dependence in terms of adjusted likelihood ratio index, consistent Akaike information criterion and log–likelihood test. Additionally, the estimated ST-EMGP model shows that spatial and temporal dependences exist and correlate mutually. Spatial dependence may overstate its impact magnitude, but underestimate its impact range when temporal dependence is ignored. According to the distribution and regression results of spatiotemporal effects, temporal effects are higher in crash frequency and are mainly affected by traffic characteristics; while spatial effects are higher in severe crash severity levels and are mainly affected by geometric configuration. Obviously, the proposed model can successfully elucidate the sources of spatiotemporal dependence as well as their effects on crash frequency and severity.

Predicting Risk of Spontaneous Preterm Delivery in Women with a Singleton Pregnancy

Prediction of a woman's risk of a spontaneous preterm delivery (PTD) is a core challenge and an unresolved problem in today's obstetric practice. The objective of this study was to develop prediction models for spontaneous PTD (<37 weeks). A population-based register study of women born in Sweden with spontaneous onset of delivery was designed using Swedish Medical Birth Register data for 1992-2008. Predictive variables were identified by multiple logistic regression analysis, and outputs were used to calculate adjusted likelihood ratios in primiparous (n = 199 272) and multiparous (n = 249 580) singleton pregnant women. The predictive ability of each model was validated in a separate test sample for primiparous (n = 190 936) and multiparous (n = 239 203) women, respectively. For multiparous women, the area under the ROC curve (AUC) of 0.74 [95% confidence interval (CI) 0.73, 0.74] indicated a satisfying performance of the model, while for primiparous women, it was rather poor {AUC: 0.58 [95% CI 0.57, 0.58]}. For both primiparous and multiparous women, the prediction models were quite good for pregnancies with comparatively low risk for spontaneous PTD, whereas more limited to predict pregnancies with ≥30% risk of spontaneous PTD. Spontaneous PTD is difficult to predict in multiparous women and nearly impossible in primiparous, by using this statistical method in a large and unselected sample. However, adding clinical data (like cervical length) may in the future further improve its predictive performance.

A Safety Probability Assessment at the Transportation-Planning Level: Should an Ordered or Unordered Response Be Used?

In the context of urban network safety management, this article presents an alternative approach based on a qualitative response model, namely, the multinomial logit model. The main objective of this article is to compare and analyze the use of ordered and unordered mechanisms (such as the multinomial logit model) and consequently to select the appropriate model. To achieve this objective, urban segment data from Porto, Portugal, for a 5-year period were used. The ordered and the multinomial models were applied to estimate the parameters and marginal effects, considering three responses to reflect degrees of hazard (low, medium, and high). To compare and select the best model, statistical tests were conducted, and performance measures were obtained. Although similar results were obtained, the adjusted likelihood ratio test showed that the multinomial model was preferred. Moreover, to illustrate and compare a safety assessment using both models, the probability of each hazard level occurring was computed for two urban segments and hypothetical planned scenarios were defined. It was interesting to note that despite the slight apparent differences between the two models, great differences were found in the category probability model results, proving the importance of model structure selection.

Using latent class analysis to model prescription medications in the measurement of falling among a community elderly population

BackgroundFalls among the elderly are a major public health concern. Therefore, the possibility of a modeling technique which could better estimate fall probability is both timely and needed. Using biomedical, pharmacological and demographic variables as predictors, latent class analysis (LCA) is demonstrated as a tool for the prediction of falls among community dwelling elderly.MethodsUsing a retrospective data-set a two-step LCA modeling approach was employed. First, we looked for the optimal number of latent classes for the seven medical indicators, along with the patients’ prescription medication and three covariates (age, gender, and number of medications). Second, the appropriate latent class structure, with the covariates, were modeled on the distal outcome (fall/no fall). The default estimator was maximum likelihood with robust standard errors. The Pearson chi-square, likelihood ratio chi-square, BIC, Lo-Mendell-Rubin Adjusted Likelihood Ratio test and the bootstrap likelihood ratio test were used for model comparisons.ResultsA review of the model fit indices with covariates shows that a six-class solution was preferred. The predictive probability for latent classes ranged from 84% to 97%. Entropy, a measure of classification accuracy, was good at 90%. Specific prescription medications were found to strongly influence group membership.ConclusionsIn conclusion the LCA method was effective at finding relevant subgroups within a heterogenous at-risk population for falling. This study demonstrated that LCA offers researchers a valuable tool to model medical data.

Data-dependent probability matching priors for likelihood ratio and adjusted likelihood ratio statistics

We consider likelihood ratio statistics based on the usual profile likelihood and the standard adjustments thereof proposed in the literature in the presence of nuisance parameters. The role of data-dependent priors in ensuring approximate frequentist validity of posterior credible regions based on the inversion of these statistics is investigated. Unlike what happens with data-free priors, it is seen that the resulting probability matching conditions readily admit solutions which entail approximate frequentist validity of the highest posterior density region as well.

Second-Order Powers of a Class of Tests in the Presence of a Nuisance Parameter

The second-order local powers of a broad class of asymptotic chi-squared tests are considered in a composite case where both the parameter of interest and the nuisance parameter are possibly multidimensional for which no assumption has been made regarding global parametric orthogonality or curved exponentiality. The main result is that the second-order (point-by-point) local power identity holds if approximate third cumulants of a square-root version of the (modified) test statistic in the class vanish up to the second-order, which is an extension of Kakizawa (2010a) in the absence of the nuisance parameter. It is also shown that in the presence of the nuisance parameter, such a third cumulant condition does not always imply the second-order local unbiasedness of the resulting test. Then, the adjusted likelihood ratio test by Mukerjee (1993b) can be interpreted as the second-order local unbiased modification after applying the third cumulant condition.

Small-sample likelihood inference in extreme-value regression models

We deal with a general class of extreme-value regression models introduced by Barreto-Souza and Vasconcellos [Bias and skewness in a general extreme-value regression model, Comput. Statist. Data Anal. 55 (2011), pp. 1379–1393]. Our goal is to derive an adjusted likelihood ratio statistic that is approximately distributed as χ2 with a high degree of accuracy. Although the adjusted statistic requires more computational effort than its unadjusted counterpart, it is shown that the adjustment term has a simple compact form that can be easily implemented in standard statistical software. Further, we compare the finite-sample performance of the three classical tests (likelihood ratio, Wald, and score), the gradient test that has been recently proposed by Terrell [The gradient statistic, Comput. Sci. Stat. 34 (2002), pp. 206–215], and the adjusted likelihood ratio test obtained in this article. Our simulations favour the latter. Applications of our results are presented.

On testing inference in beta regressions

This article deals with testing inference in the class of beta regression models with varying dispersion. We focus on inference in small samples. We perform a numerical analysis in order to evaluate the sizes and powers of different tests. We consider the likelihood ratio test, two adjusted likelihood ratio tests proposed by Ferrari and Pinheiro [Improved likelihood inference in beta regression, J. Stat. Comput. Simul. 81 (2011), pp. 431–443], the score test, the Wald test and bootstrap versions of the likelihood ratio, score and Wald tests. We perform tests on the parameters that index the mean submodel and also on the parameters in the linear predictor of the precision submodel. Overall, the numerical evidence favours the bootstrap tests. It is also shown that the score test is considerably less size-distorted than the likelihood ratio and Wald tests. An application that uses real (not simulated) data is presented and discussed.

Diagnostic accuracy of clinical symptoms in obstructive airway diseases varied within different health care sectors

ObjectiveTo determine the diagnostic accuracy and diagnostic patterns of clinical symptoms in patients suspected to suffer from obstructive airway diseases (OADs) within different health care sectors. Study Design and SettingTen general practices (219 patients), one practice of pneumologists (259 patients) and one specialist hospital (300 patients). Sensitivities, specificities, positive (LR+), and negative (LR−) likelihood ratios of clinical symptoms were compared with lung function testing. ResultsThirty-one percent had chronic obstructive pulmonary disease (COPD), 21% had asthma. Sensitivities increased and specificities decreased from outpatient to hospital setting. The multivariate model of adjusted likelihood ratios for COPD showed LR+=4.86 (95% confidence interval [CI]=2.09–11.29) and LR−=0.07 (95% CI=0.01–0.43) of the combination “wheezing,” “dyspnea when going upstairs,” “smoking” in general practice. In hospital, the combination “dyspnea when going upstairs,” “dyspnea during minimal exercise,” and “smoking” showed LR+=3.34 (95% CI=2.08–5.31) and LR−=0.02 (95% CI=0.01–0.12). The combination “no coughing,” “dyspnea attacks,” and “no smoking” showed LR+=4.08 (95% CI=1.67–10.4) and LR−=0.24 (95% CI=0.12–0.58) for asthma in general practice. The combination “dyspnea attacks” and “no dyspnea when walking” showed LR+=6.48 (95% CI=1.01–40.94) and LR−=0.28 (95% CI=0.11–0.75) for asthma in hospital. ConclusionClinical decision rules for OAD need to be derived from original studies in their respective settings or assessed on their transferability to other settings.

Model-based likelihood ratio confidence intervals for survival functions

We introduce an adjusted likelihood ratio procedure for computing pointwise confidence intervals for survival functions from censored data. The test statistic, scaled by a ratio of two variance quantities, is shown to converge to a chi-squared distribution with one degree of freedom. The confidence intervals are seen to be a neighborhood of a semiparametric survival function estimator and are shown to have correct empirical coverage. Numerical studies also indicate that the proposed intervals have smaller estimated mean lengths in comparison to the ones that are produced as a neighborhood of the Kaplan–Meier estimator. We illustrate our method using a lung cancer data set.

Risk predictions for individual patients from logistic regression were visualized with bar–line charts

Objective The interface of a computerized decision support system is crucial for its acceptance among end users. We demonstrate how combined bar–line charts can be used to visualize predictions for individual patients from logistic regression models. Study Design and Setting Data from a previous diagnostic study aiming at predicting the immediate risk of acute coronary syndrome (ACS) among 634 patients presenting to an emergency department with chest pain were used. Risk predictions from the logistic regression model were presented for four hypothetical patients in bar–line charts with bars representing empirical Bayes adjusted likelihood ratios (LRs) and the line representing the estimated probability of ACS, sequentially updated from left to right after assessment of each risk factor. Results Two patients had similar low risk for ACS but quite different risk profiles according to the bar–line charts. Such differences in risk profiles could not be detected from the estimated ACS risk alone. The bar–line charts also highlighted important but counteracted risk factors in cases where the overall LR was less informative (close to one). Conclusion The proposed graphical technique conveys additional information from the logistic model that can be important for correct diagnosis and classification of patients and appropriate medical management.

P2-205 Heterogeneity in functional recovery after stroke: an exploratory study using longitudinal latent class analysis

IntroductionPrediction models for functional recovery after stroke can be improved by adjusting for the heterogeneity in functional recovery patterns. This study explored the heterogeneity in functional recovery after stroke using...

An adjusted likelihood ratio test for separability in unbalanced multivariate repeated measures data

We propose an adjusted likelihood ratio test of two-factor separability (Kronecker product structure) for unbalanced multivariate repeated measures data. Here we address the particular case where the within subject correlation is believed to decrease exponentially in both dimensions (e.g., temporal and spatial dimensions). However, the test can be easily generalized to factor specific matrices of any structure. A simulation study is conducted to assess the inference accuracy of the proposed test. Longitudinal medical imaging data concerning schizophrenia and caudate morphology illustrate the methodology.

An Algorithm to Optimize Viral Load Testing in HIV-Positive Patients With Suspected First-Line Antiretroviral Therapy Failure in Cambodia

To develop an algorithm for optimal use of viral load testing in patients with suspected first-line antiretroviral treatment (ART) failure. Data from a cohort of patients on first-line ART in Cambodia were analyzed in a cross-sectional way to detect markers for treatment failure. Markers with an adjusted likelihood ratio <0.67 or >1.5 were retained to calculate a predictor score. The accuracy of a 2-step algorithm based on this score followed by targeted viral load testing was compared with World Health Organization criteria for suspected treatment failure. One thousand eight hundred three viral load measurements of 764 patients were available for analysis. Prior ART exposure, CD4 count below baseline, 25% and 50% drop from peak CD4 count, hemoglobin drop of > or =1 g/dL, CD4 count <100 cells per microliter after 12 months of treatment, new onset of papular pruritic eruption, and visual analog scale <95% were included in the predictor score. A score >or=2 had the best combination of sensitivity and specificity and required confirmatory viral load testing for only 9% of patients. World Health Organization criteria had a similar sensitivity but a lower specificity and required viral load testing for 24.9% of patients. An algorithm combining a predictor score with targeted viral load testing in patients with an intermediate probability of failure optimizes the use of scarce resources.

Some restriction tests in a new class of regression models for proportions

The main purpose of this work is to study the behaviour of Skovgaard’s [Skovgaard, I.M., 2001. Likelihood asymptotics. Scandinavian Journal of Statistics 28, 3–32] adjusted likelihood ratio statistic in testing simple hypothesis in a new class of regression models proposed here. The proposed class of regression models considers Dirichlet distributed observations, and the parameters that index the Dirichlet distributions are related to covariates and unknown regression coefficients. This class is useful for modelling data consisting of multivariate positive observations summing to one and generalizes the beta regression model described in Vasconcellos and Cribari-Neto [Vasconcellos, K.L.P., Cribari-Neto, F., 2005. Improved maximum likelihood estimation in a new class of beta regression models. Brazilian Journal of Probability and Statistics 19, 13–31]. We show that, for our model, Skovgaard’s adjusted likelihood ratio statistics have a simple compact form that can be easily implemented in standard statistical software. The adjusted statistic is approximately chi-squared distributed with a high degree of accuracy. Some numerical simulations show that the modified test is more reliable in finite samples than the usual likelihood ratio procedure. An empirical application is also presented and discussed.

Comparative Assessment of Weighting Comorbidities in the Hematopoietic Cell Transplantation (HCT) Specific Comorbidity Index (HCT-CI)

An increasing number of HCT recipients have pre-existing comorbidities at the time of transplantation. Sorror et al (Blood 2005) have proposed the HCT-CI, which evaluates various comorbidities as a combined categorical weighted score that independently predicts non-relapse mortality (NRM) and overall survival (OS) after HCT. In a recent study that included 373 adult allogeneic HCT recipients at our center, we have shown that the sensitivity and generalizability of the HCT-CI needs to be improved further prior to its routine universal use (Majhail et al, Biology of Blood and Marrow Transplantation 2008). The median age of the original cohort was 47 (range, 18–69) years and included 150 myeloablative and 223 non-myeloablative HCT recipients. The HCT-CI score was zero (low-risk) in 16%, 1–2 (intermediate-risk) in 32%, and ≥3 (high-risk) in 52% patients. We hypothesized that the HCT-CI loses some power in the early and unnecessary step of converting the adjusted hazard ratios (HR) of the comorbidities for NRM to the categorical weights of 0 to 3 prior to summation of the final score. We propose a revised HCT-CI based on a pure multiplicative model that assigns weights to comorbidities more efficiently and increases the discriminating and predictive power of the original HCT-CI for NRM and OS. In our analysis, which included the same study cohort (N=373), we calculated the HR for each comorbidity from a regression analysis on NRM after adjusting for all other comorbidities as well as age, donor type, disease risk and conditioning regimen intensity. Instead of converting the adjusted HR to categorical weights and then summing these weights, we directly calculated a risk index (multiplicative HCT-CI [MHCT-CI]) by exponentiating the sum of all parameter coefficients from the regression analysis. The revised index score is: MHCT-CI = exponent [0.82*(binary indicator (bi) for cardiac disorders) + 0.20*(bi for peptic ulcer) + 0.60*(bi for diabetes) – 0.43*(bi for obesity) + 0.30*(bi for psychiatric disturbance) + 0.34*(bi for Moderate/Severe hepatic function) + 0.63*(bi for infection) + 0.73*(bi for renal insufficiency) + 0.63*(bi for moderate pulmonary abnormalities) + 0.90*(bi for severe pulmonary abnormalities) + 0.22*(bi for prior solid tumor)]. Comorbidities not appearing did not have influence on NRM. The distribution of the MHCT-CI score was ≤1 (low-risk) in 20%, 1–2.5 (intermediate-risk) in 41% and ≥2.5 (high-risk) in 40% patients. MHCT-CI was more predictive for both NRM and OS compared to the original HCT-CI. The HR for intermediate and high risk categories increased (>43% for NRM and >19% for OS). The adjusted likelihood ratio, showing model fit, increased from 13.2 to 22.1 for NRM and increased from 18.8 to 34.1 for OS after substituting MHCT-CI for HCT-CI. There are no statistical tests for this statistic but an increase shows better prediction of the endpoint. The c statistic, which is the proportion of all pairs of patients in the study in which the patient with the higher index score has a worse outcome, increased from 0.641 to 0.752 for NRM (P=0.03) and increased from 0.611 to 0.713 for OS (P=0.01). In conclusion, the MHCT-CI showed higher discriminating and predictive power for post-HCT NRM and OS among our study population. Given that an increasing number of HCT recipients are being transplanted with pre-existing comorbidities, the greater discrimination of assigning patient comorbidity will better inform future studies among HCT recipients by better adjusting for these important risk factors.

Skovgaard’s adjustment to likelihood ratio tests in exponential family nonlinear models

Likelihood ratio tests can be substantially size distorted in small- and moderate-sized samples. In this paper, we apply Skovgaard’s [Skovgaard, I.M., 2001. Likelihood asymptotics. Scandinavian Journal of Statistics 28, 3–32] adjusted likelihood ratio statistic to exponential family nonlinear models. We show that the adjustment term has a simple compact form that can be easily implemented from standard statistical software. The adjusted statistic is approximately distributed as χ 2 with high degree of accuracy. It is applicable in wide generality since it allows both the parameter of interest and the nuisance parameter to be vector-valued. Unlike the modified profile likelihood ratio statistic obtained from Cox and Reid [Cox, D.R., Reid, N., 1987. Parameter orthogonality and approximate conditional inference. Journal of the Royal Statistical Society B 49, 1–39], the adjusted statistic proposed here does not require an orthogonal parameterization. Numerical comparison of likelihood-based tests of varying dispersion favors the test we propose and a Bartlett-corrected version of the modified profile likelihood ratio test recently obtained by Cysneiros and Ferrari [Cysneiros, A.H.M.A., Ferrari, S.L.P., 2006. An improved likelihood ratio test for varying dispersion in exponential family nonlinear models. Statistics and Probability Letters 76 (3), 255–265].

Cervical length and obstetric history predict spontaneous preterm birth: development and validation of a model to provide individualized risk assessment

To evaluate the ability of combinations of cervical length and maternal history to assess the risk of spontaneous preterm birth, and to provide a simple procedure for the optimal estimation of risk. This prospective observational study was carried out between January 1998 and May 2006. Transvaginal sonographic measurement of cervical length at 20 + 0 to 24 + 6 weeks of gestation was carried out in 58 807 singleton pregnancies as part of routine antenatal care. The outcome measure was spontaneous extreme (< 28 weeks), early (28-30 weeks), moderate (31-33 weeks) and mild (34-36 weeks) preterm birth. Logistic regression analysis was used to derive models for the prediction of spontaneous preterm birth from the maternal obstetric history, demographic characteristics and cervical length. The rates of extreme, early, moderate and mild spontaneous preterm birth were 0.23%, 0.24%, 0.57% and 2.93%, respectively. The best prediction of spontaneous preterm birth was provided by cervical length (area under the receiver-operating characteristics curve (AUC), extreme 0.903, early 0.816, moderate 0.784 and mild 0.617) and this was improved by adding obstetric history (AUC, extreme 0.919, early 0.836, moderate 0.819 and mild 0.650). Addition of other parameters was without material effect. For a 10% screen-positive rate, models using cervical length and obstetric history had a sensitivity of 80.6%, 58.5%, 53.0% and 28.6% for extreme, early, moderate and mild spontaneous preterm birth, respectively. These models were expressed as tables of adjusted likelihood ratios to allow simple estimation of the risk of spontaneous preterm birth. A model combining cervical length and obstetric history provides a better prediction of spontaneous preterm birth than either factor alone, and the sensitivity of screening improves for increasing degrees of prematurity.