Estimating confidence intervals for cost-effectiveness ratios: an example from a randomized trial

Cost-effectiveness ratios usually appear as point estimates without confidence intervals, since the numerator and denominator are both stochastic and one cannot estimate the variance of the estimator exactly. The recent literature, however, stresses the importance of presenting confidence intervals for cost-effectiveness ratios in the analysis of health care programmes. This paper compares the use of several methods to obtain confidence intervals for the cost-effectiveness of a randomized intervention to increase the use of Medicaid's Early and Periodic Screening, Diagnosis and Treatment (EPSDT) programme. Comparisons of the intervals show that methods that account for skewness in the distribution of the ratio estimator may be substantially preferable in practice to methods that assume the cost-effectiveness ratio estimator is normally distributed. We show that non-parametric bootstrap methods that are mathematically less complex but computationally more rigorous result in confidence intervals that are similar to the intervals from a parametric method that adjusts for skewness in the distribution of the ratio. The analyses also show that the modest sample sizes needed to detect statistically significant effects in a randomized trial may result in confidence intervals for estimates of cost-effectiveness that are much wider than the boundaries obtained from deterministic sensitivity analyses.     

Divergent effects of different antihypertensive drugs on endothelium-dependent vasodilation in the human forearm

OBJECTIVES: This paper describes 2 statistical methods designed to correct for bias from exposure measurement error in point and interval estimates of relative risk. METHODS: The first method takes the usual point and interval estimates of the log relative risk obtained from logistic regression and corrects them for nondifferential measurement error using an exposure measurement error model estimated from validation data. The second, likelihood-based method fits an arbitrary measurement error model suitable for the data at hand and then derives the model for the outcome of interest. RESULTS: Data from Valanis and colleagues' study of the health effects of antineoplastics exposure among hospital pharmacists were used to estimate the prevalence ratio of fever in the previous 3 months from this exposure. For an interdecile increase in weekly number of drugs mixed, the prevalence ratio, adjusted for confounding, changed from 1.06 to 1.17 (95% confidence interval [CI] = 1.04, 1.26) after correction for exposure measurement error. CONCLUSIONS: Exposure measurement error is often an important source of bias in public health research. Methods are available to correct such biases.     

The cellular substrates of the acute myocardial infarct syndrome

Immunoassay techniques yield estimates of concentrations of analytes based on comparison to known concentrations of a reference solution. The use of the nonlinear logistic model makes the error estimates and confidence levels approximate. When the goal of such a study is estimation of several unknowns, methods in common usage do not account for 'simultaneous' inference, i.e. the repeated use of the standard curve for estimating several concentrations. Alternative methods are described which take multiple use of the reference curve into account. Simulations using normally distributed data with variance proportional to a power of the mean compare different methods of obtaining calibration intervals and illustrate the approximate nature of all such techniques. Calibration intervals based on simple, commonly used methods do not provide the coverage promised, even for one-at-a-time estimation, and are not suited for multiple estimation and comparison.     

Clusters of marijuana use in the United States

In this study, the authors test for and estimate the clustering of marijuana use within United States neighborhoods, making use of data from annual nationally representative household sample surveys conducted during the period 1990-1995. A recently developed statistical method, alternating logistic regression, was used to quantify the clustering of marijuana users in neighborhoods. The resulting estimates of pairwise odds ratios ranged from 1.3 (95% confidence interval 1.22-1.42) for the lifetime history of marijuana use to 2.0 (95% confidence interval 1.6-2.6) for recent sharing of marijuana from one person to another. Exploratory analysis showed a slight decrease of clustering effects after adjustment for individual-level covariates: age, sex, race, education, annual family income, and history of tobacco use. Nevertheless, the main factors that account for clustering remain to be determined. Alternating logistic regression provided useful estimates of marijuana use clustering and can be used to estimate clustering of the other drug-related behavior, including sharing of needle injection equipment and other human immunodeficiency virus risk behaviors. As a form of multilevel analysis, the alternating logistic regression can accommodate shared, community-level characteristics that might influence drug taking (e.g., collective efficacy), as well as individual-level covariates, such as age and sex.     

Correcting for measurement error in the analysis of case-control data with repeated measurements of exposure

The authors present a technique for correcting for exposure measurement error in the analysis of case-control data when subjects have a variable number of repeated measurements, and the average is used as the subject's measure of exposure. The true exposure as well as the measurement error are assumed to be normally distributed. The method transforms each subject's observed average by a factor which is a function of the measurement error parameters, prior to fitting the logistic regression model. The resulting logistic regression coefficient estimate based on the transformed average is corrected for error. A bootstrap method for obtaining confidence intervals for the true regression coefficient, which takes into account the variability due to estimation of the measurement error parameters, is also described. The method is applied to data from a nested case-control study of hormones and breast cancer.     

Elevated symptom prevalence associated with ventilation type in office buildings

The California Healthy Building Study was designed to assess relations between ventilation system type and office worker symptoms in a set of U.S. buildings selected without regard to worker complaints. Twelve public office buildings in northern California meeting specific eligibility criteria were studied in the summer of 1990: three naturally ventilated, three mechanically ventilated (without air conditioning), and six air-conditioned buildings. Questionnaire data were collected from 880 workers in selected spaces within the study buildings. We adjusted effect estimates for various ventilation types for personal, job, and work place factors using logistic regression, and alternatively, using a mixed effects model (SAS/GLIMMIX) to adjust for correlated responses within study spaces. Higher adjusted prevalences of most symptom outcomes were associated with both mechanical and air-conditioned ventilation, relative to natural. With a conservative adjustment for problem building status, the highest adjusted prevalence odds ratios from logistic regression models were for dry or itchy skin [mechanical: odds ratio (OR) = 6.0, 95% confidence interval (CI) = 1.6-22; air-conditioned: OR = 6.0, 95% CI = 1.7-21] and lower respiratory symptoms (mechanical: OR = 2.9, 95% CI = 0.7-11; air-conditioned: OR = 4.0, 95% CI = 1.1-15). GLIMMIX estimates were similar, with slightly wider confidence intervals. Reporting bias was small. These findings of symptom increases within mechanically ventilated and air-conditioned U.S. buildings support previous findings available only from European buildings.     

Approximate standard errors and confidence intervals for indices of positive and negative agreement

Indices of positive and negative agreement for observer reliability studies, in which neither observer can be regarded as the standard, have been proposed. In this article, it is demonstrated by means of an example and a small simulation study that a recently published method for constructing confidence intervals for these indices leads to intervals that are too wide. Appropriate asymptotic (i.e., large sample) variance estimates and confidence intervals for the positive and negative agreement indices are presented and compared with bootstrap confidence intervals. We also discuss an alternative method of interval estimation motivated from a Bayesian viewpoint. The asymptotic intervals performed adequately for sample sizes of 200 or more. For smaller samples, alternative confidence intervals such as bootstrap intervals or Bayesian intervals should be considered.     

Interpersonal change in brief supportive psychotherapy

The pathogenesis, treatment, and outcomes of type 1 and type 2 diabetes differ. Current surveys derive population-based estimates of diabetes prevalence by type using limited clinical information and applying classification rules developed in white populations. How well these rules perform when deriving similar estimates in African American populations is unknown. For this study, data were collected on a group of African Americans with diabetes who enrolled at the Diabetes Unit of Grady Memorial Hospital in Atlanta, Georgia, from April 16, 1991, to November 1, 1996. The data were used to develop some simple classification rules for African Americans based on a classification tree and a logistic regression model. Sensitivities and specificities, in which fasting C-peptide was used as the gold standard, were determined for these rules and for two current rules developed in mostly white, non-Hispanic populations. Rules that yielded precise (minimum variance unbiased) estimates of the prevalence of type 1 diabetes were preferred. The authors found that a rule based on the logistic regression model was best for estimating type 1 prevalences ranging from 1% to 17%. They concluded that simple classification rules can be used to estimate prevalence of diabetes by type in African American populations and that the optimal rule differs somewhat from the current rules.     

Confidence intervals for differences in correlated binary proportions

An experiment to assess the efficacy of a particular treatment or process often produces dichotomous responses, either favourable or unfavourable. When we administer the treatment on two occasions to the same subjects, we often use McNemar's test to investigate the hypothesis of no difference in the proportions on the two occasions, that is, the hypothesis of marginal homogeneity. A disadvantage in using McNemar's statistic is that we estimate the variance of the sample difference under the restriction that the marginal proportions are equal. A competitor to McNemar's statistic is a Wald statistic that uses an unrestricted estimator of the variance. Because the Wald statistic tends to reject too often in small samples, we investigate an adjusted form that is useful for constructing confidence intervals. Quesenberry and Hurst and Goodman discussed methods of construction that we adapt for constructing confidence intervals for the differences in correlated proportions. We empirically compare the coverage probabilities and average interval lengths for the competing methods through simulation and give recommendations based on the simulation results.     

Confidence intervals for the excess risk in case-control studies

Kaufman et al. compute the 'excess risk' of a disease in the presence of an exposure as the product of the incidence rate of the disease in the source population, the complement of the aetiologic fraction and the relative risk minus one. Methods for calculating confidence intervals for this quantity are derived when (as in case-control studies) the relative risk is estimated by the odds ratio, firstly from multiple logistic regression analysis and secondly without adjustment for covariates. For the latter an innovative approach based on confidence bounds for the two exposure parameters is suggested. The performance of these systems of confidence intervals is assessed by simulation for the former and by exact enumeration of the distributions involved in the latter. Illustrative examples from a study of agranulocytosis and indomethacin are presented.     

Confidence intervals for the cross-validated multiple correlation in predictive regression models.

Presents improved procedures to approximate confidence intervals for ρ–2 and ρc–2 in both fixed and random predictor models. These approximations require neither point estimates nor variance estimates and are analytically shown to be precise enough for most practical prediction purposes. An application of confidence intervals in regression model development is also given. (16 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Bcl-2 expression and apoptosis in nephrotoxic nephritis

The aim of this paper is to present a conceptual and practical framework of the case-control design in medical research. To illustrate this method, practical examples directed to clinicians and other health professionals interested in medical research are presented. The case-control method is very versatile and allows for multiple applications. Guidelines for the selection of cases and controls, and some considerations on sample size are presented. In the statistical analysis we use concrete examples of how to estimate odds ratios, confidence intervals, and methods to control for potential confounders, from stratified analysis to logistic regression.     

Semiology and subtyping of panic disorders

OBJECTIVES: To review the appropriateness of the prevalence odds ratio (POR) and the prevalence ratio (PR) as effect measures in the analysis of cross sectional data and to evaluate different models for the multivariate estimation of the PR. METHODS: A system of linear differential equations corresponding to a dynamic model of a cohort with a chronic disease was developed. At any point in time, a cross sectional analysis of the people then in the cohort provided a prevalence based measure of the effect of exposure on disease. This formed the basis for exploring the relations between the POR, the PR, and the incidence rate ratio (IRR). Examples illustrate relations for various IRRs, prevalences, and differential exodus rates. Multivariate point and interval estimation of the PR by logistic regression is illustrated and compared with the results from proportional hazards regression (PH) and generalised linear modelling (GLM). RESULTS: The POR is difficult to interpret without making restrictive assumptions and the POR and PR may lead to different conclusions with regard to confounding and effect modification. The PR is always conservative relative to the IRR and, if PR > 1, the POR is always > PR. In a fixed cohort and with an adverse exposure, the POR is always > or = IRR, but in a dynamic cohort with sufficient underlying follow up the POR may overestimate or underestimate the IRR, depending on the duration of follow up. Logistic regression models provide point and interval estimates of the PR (and POR) but may be intractable in the presence of many covariates. Proportional hazards and generalised linear models provide statistical methods directed specifically at the PR, but the interval estimation in the case of PH is conservative and the GLM procedure may require constrained estimation. CONCLUSIONS: The PR is conservative, consistent, and interpretable relative to the IRR and should be used in preference to the POR. Multivariate estimation of the PR should be executed by means of generalised linear models or, conservatively, by proportional hazards regression.     

Comparing dichotomous screening tests when individuals negative on both tests are not verified

Two dichotomous screening tests are often compared by performing both tests in a sampled population, and submitting positive results on either test to verification by the reference standard. Unbiased estimates of the true positive and false positive rates of each test cannot be estimated directly. However, unbiased estimates of the relative true positive and relative false positive rates may be obtained. When one test has a higher true positive rate at the expense of a higher false positive rate, the trade-off is represented by the ratio of extra false positives detected to extra true positives detected. A 95% confidence interval for this ratio is derived. This ratio is prevalence dependent and only applies to the sampled population. For target populations of different prevalence, estimates of the ratio may be obtained if one of the following applies: (i) the test characteristics of one test are known; (ii) the relative prevalence is known; and (iii) certain assumptions are made.     

Obstetric care and proneness of offspring to suicide as adults: case-control study

OBJECTIVE: To investigate any long term effects of traumatic birth and obstetric procedures in relation to suicide by violent means in offspring as adults. DESIGN: Prospective case-control study. SETTING: Stockholm, Sweden. SUBJECTS: 242 adults who committed suicide by violent means from 1978 to 1995, and who were born in one of seven hospitals in Stockholm during 1945-80, matched with 403 biological siblings born during the same period and at the same group of hospitals. MAIN OUTCOME MEASURES: Adverse and beneficial perinatal factors expressed as relative risks (odds ratios) and 95% confidence intervals, derived from logistic regression of cases matched with their siblings. RESULTS: For multiple birth trauma the estimated relative risks of offspring subsequently committing suicide by violent means were 4.9 (95% confidence interval 1.8 to 13) for men and 1.04 (0.2 to 4.6) for women. In mothers who received multiple opiate treatment during delivery, the estimated relative risk of offspring subsequently committing suicide was equal for both sexes (0.26, 0.09 to 0.69). CONCLUSION: Minimising pain and discomfort to the infant during birth seems to be of importance in reducing the risk of committing suicide by violent means as an adult.     

To use or not to use the odds ratio in epidemiologic analyses?

This paper argues that the use of the odds ratio parameter in epidemiology needs to be considered with a view to the specific study design and the types of exposure and disease data at hand. Frequently, the odds ratio measure is being used instead of the risk ratio or the incidence-proportion ratio in cohort studies or as an estimate for the incidence-density ratio in case-referent studies. Therefore, the analyses of epidemiologic data have produced biased estimates and the presentation of results has been misleading. However, the odds ratio can be relinquished as an effect measure for these study designs; and, the application of the case-base sampling approach permits the incidence ratio and difference measures to be estimated without any untenable assumptions. For the Poisson regression, the odds ratio is not a parameter of interest; only the risk or rate ratio and difference are relevant. For the conditional logistic regression in matched case-referent studies, the odds ratio remains useful, but only when it is interpreted as an estimate of the incidence-density ratio. Thus the odds ratio should, in general, give way to the incidence ratio and difference as the measures of choice for exposure effect in epidemiology.     

Sample Size for Multiple Regression: Obtaining Regression Coefficients That Are Accurate, Not Simply Significant.

An approach to sample size planning for multiple regression is presented that emphasizes accuracy in parameter estimation (AIPE). The AIPE approach yields precise estimates of population parameters by providing necessary sample sizes in order for the likely widths of confidence intervals to be sufficiently narrow. One AIPE method yields a sample size such that the expected width of the confidence interval around the standardized population regression coefficient is equal to the width specified. An enhanced formulation ensures, with some stipulated probability, that the width of the confidence interval will be no larger than the width specified. Issues involving standardized regression coefficients and random predictors are discussed, as are the philosophical differences between AIPE and the power analytic approaches to sample size planning. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A simulation study of the number of events per variable in logistic regression analysis

We performed a Monte Carlo study to evaluate the effect of the number of events per variable (EPV) analyzed in logistic regression analysis. The simulations were based on data from a cardiac trial of 673 patients in which 252 deaths occurred and seven variables were cogent predictors of mortality; the number of events per predictive variable was (252/7 =) 36 for the full sample. For the simulations, at values of EPV = 2, 5, 10, 15, 20, and 25, we randomly generated 500 samples of the 673 patients, chosen with replacement, according to a logistic model derived from the full sample. Simulation results for the regression coefficients for each variable in each group of 500 samples were compared for bias, precision, and significance testing against the results of the model fitted to the original sample. For EPV values of 10 or greater, no major problems occurred. For EPV values less than 10, however, the regression coefficients were biased in both positive and negative directions; the large sample variance estimates from the logistic model both overestimated and underestimated the sample variance of the regression coefficients; the 90% confidence limits about the estimated values did not have proper coverage; the Wald statistic was conservative under the null hypothesis; and paradoxical associations (significance in the wrong direction) were increased. Although other factors (such as the total number of events, or sample size) may influence the validity of the logistic model, our findings indicate that low EPV can lead to major problems.     

Confidence intervals or surfaces? Uncertainty on the cost-effectiveness plane

Although cost-effectiveness analysis is not new, it is only recently that economic analysis has been conducted alongside clinical trials. Whereas in the past economic analysts most often used sensitivity analysis to examine the implications of uncertainty for their results, the existence of patient-level data on costs and effects opens up the possibility of statistical analysis of uncertainty. Unfortunately, ratio statistics can cause problems for standard statistical methods of confidence interval estimation. The recent health economics literature contains a number of suggestions for estimating confidence limits for ratios. In this paper, we begin by reviewing the different methods of confidence interval estimation with a view to providing guidance concerning the most appropriate method. We go on to argue that the focus on confidence interval estimation for cost-effectiveness ratios in the recent literature has been concerned more with problems of estimation than with problems of decision-making. We argue that decision-makers are most likely to be interested in one-sided tests of hypothesis and that confidence surfaces are better suited to such tests than confidence intervals. This approach is consistent with decision-making on the cost-effectiveness plane and with the cost-effectiveness acceptability curve approach to presenting uncertainty due to sampling variation in stochastic cost-effectiveness analyses.     

Estimation of a common risk ratio in stratified case-cohort studies

The case-cohort study design is a useful modification of the case-control design, which allows direct estimation of the risk ratio without the rare-disease assumption. While several risk ratio estimation procedures have been proposed under large-strata settings, only the Mantel-Haenszel point estimator is available in sparse stratifications. This paper provides simple confidence limits methods, based on the large-sample distribution of the Mantel-Haenszel risk ratio, that apply to both large-strata and sparse-data situations. For the Tarone risk ratio, I give a new large-strata variance estimate.