Confidence Intervals for Survival Probabilities: A Comparison Study

The confidence interval of the Kaplan–Meier estimate of the survival probability at a fixed time point is often constructed by the Greenwood formula. This normal approximation-based method can be looked as a Wald type confidence interval for a binomial proportion, the survival probability, using the “effective” sample size defined by Cutler and Ederer. Wald-type binomial confidence interval has been shown to perform poorly comparing to other methods. We choose three methods of binomial confidence intervals for the construction of confidence interval for survival probability: Wilson's method, Agresti–Coull's method, and higher-order asymptotic likelihood method. The methods of “effective” sample size proposed by Peto et al. and Dorey and Korn are also considered. The Greenwood formula is far from satisfactory, while confidence intervals based on the three methods of binomial proportion using Cutler and Ederer's “effective” sample size have much better performance.     

Construction of fixed width confidence intervals for a Bernoulli success probability using sequential sampling: a simulation study

This article considers the construction of level 1?α fixed width 2d confidence intervals for a Bernoulli success probability p, assuming no prior knowledge about p and so p can be anywhere in the interval [0, 1]. It is shown that some fixed width 2d confidence intervals that combine sequential sampling of Hall [Asymptotic theory of triple sampling for sequential estimation of a mean, Ann. Stat. 9 (1981), pp. 1229–1238] and fixed-sample-size confidence intervals of Agresti and Coull [Approximate is better than ‘exact’ for interval estimation of binomial proportions, Am. Stat. 52 (1998), pp. 119–126], Wilson [Probable inference, the law of succession, and statistical inference, J. Am. Stat. Assoc. 22 (1927), pp. 209–212] and Brown et al. [Interval estimation for binomial proportion (with discussion), Stat. Sci. 16 (2001), pp. 101–133] have close to 1?α confidence level. These sequential confidence intervals require a much smaller sample size than a fixed-sample-size confidence interval. For the coin jamming example considered, a fixed-sample-size confidence interval requires a sample size of 9457, while a sequential confidence interval requires a sample size that rarely exceeds 2042.     

Some Small-Sample Properties of the Student t Confidence Intervals

For the two-sided Student t confidence interval for the mean of a normal distribution there is, for any sample size, a sufficiently large confidence level that ensures that the interval covers all the observations; there are also sufficiently small confidence levels guaranteeing, respectively, that (a) the interval does not cover all the observations and (b) the interval lies within the extreme observations. Necessary and sufficient conditions are also obtained for the width of the confidence interval to always exceed the sample range, as well as for the reverse inequality. Some implications of the results are discussed.     

Sample size determination for the confidence interval of mean comparison adjusted by multiple covariates

The current method of determining sample size for confidence intervals does not accommodate multiple covariate adjustment. Under the normality assumption, the effect of multiple covariate adjustment on the standard error of the mean comparison is related to a Hotelling T ² statistic. Sample size can be calculated to obtain a desired probability of achieving a predetermined width in the confidence interval of the mean comparison with multiple covariate adjustment, given that the confidence interval includes the population parameter.     

Approximate is Better than “Exact” for Interval Estimation of Binomial Proportions

For interval estimation of a proportion, coverage probabilities tend to be too large for “exact” confidence intervals based on inverting the binomial test and too small for the interval based on inverting the Wald large-sample normal test (i.e., sample proportion ± z-score × estimated standard error). Wilson's suggestion of inverting the related score test with null rather than estimated standard error yields coverage probabilities close to nominal confidence levels, even for very small sample sizes. The 95% score interval has similar behavior as the adjusted Wald interval obtained after adding two “successes” and two “failures” to the sample. In elementary courses, with the score and adjusted Wald methods it is unnecessary to provide students with awkward sample size guidelines.     

Confidence interval estimation for a linear contrast in intraclass correlation coefficients under unequal family sizes for several populations

Confidence intervals [based on F-distribution and (Z) standard normal distribution] for a linear contrast in intraclass correlation coefficients under unequal family sizes for several populations based on several independent multinormal samples have been proposed. It has been found that the confidence interval based on F-distribution consistently and reliably produced better results in terms of shorter average length of the interval than the confidence interval based on standard normal distribution for various combinations of intraclass correlation coefficient values. The coverage probability of the interval based on F-distribution is competitive with the coverage probability of the interval based on standard normal distribution. The interval based on F-distribution can be used for both small sample and large sample situations. An example with real life data has been presented.     

Confidence intervals for the difference in paired Youden indices

Comparison of accuracy between two diagnostic tests can be implemented by investigating the difference in paired Youden indices. However, few literature articles have discussed the inferences for the difference in paired Youden indices. In this paper, we propose an exact confidence interval for the difference in paired Youden indices based on the generalized pivotal quantities. For comparison, the maximum likelihood estimate‐based interval and a bootstrap‐based interval are also included in the study for the difference in paired Youden indices. Abundant simulation studies are conducted to compare the relative performance of these intervals by evaluating the coverage probability and average interval length. Our simulation results demonstrate that the exact confidence interval outperforms the other two intervals even with small sample size when the underlying distributions are normal. A real application is also used to illustrate the proposed intervals. Copyright © 2012 John Wiley & Sons, Ltd.     

Matching pseudocounts for interval estimation of binomial and Poisson parameters

ABSTRACT

For interval estimation of a binomial proportion and a Poisson mean, matching pseudocounts are derived, which give the one-sided Wald confidence intervals with second-order accuracy. The confidence intervals remove the bias of coverage probabilities given by the score confidence intervals. Partial poor behavior of the confidence intervals by the matching pseudocounts is corrected by hybrid methods using the score confidence interval depending on sample values.     

New confidence interval estimator of the signal-to-noise ratio based on asymptotic sampling distribution

In this paper, the asymptotic distribution of the signal-to-noise ratio (SNR) is derived and a new confidence interval for the SNR is introduced. An evaluation of the performance of the new interval compared to Sharma and Krishna (S–K) (1994) confidence interval for the SNR using Monte Carlo simulations is conducted. Data were randomly generated from normal, log-normal, χ², Gamma, and Weibull distributions. Simulations revealed that the performance of S–K interval is totally dependent on the amount of noise introduced and that it has a constant width for a given sample size. The S–K interval performs poorly in four of the distributions unless the SNR is around one. It is recommended against using the S–K interval for data from log-normal distribution even with SNR = 1. Unlike the S–K interval which does not account for skewness and kurtosis of the distribution, the new confidence interval for the SNR outperforms S–K for all five distributions discussed, especially when SNR?? 2. The proposed ranked set sampling (RSS) instead of simple random sampling (SRS) has improved the performance of both intervals as measured by coverage probability.     

Estimating Desired Sample Size for Simple Random Sampling of a Skewed Population

A simulation study was conducted to assess how well the necessary sample size to achieve a stipulated margin of error can be estimated prior to sampling. Our concern was particularly focused on performance when sampling from a very skewed distribution, which is a common feature of many biological, economic, and other populations. We examined two approaches for estimating sample size—one being the commonly used strategy aimed at regulating the average magnitude of the stipulated margin of error and the second being a previously proposed strategy to control the tolerance probability with which the stipulated margin of error is exceeded. Results of the simulation revealed that (1) skewness does not much affect the average estimated sample size but can greatly extend the range of estimated sample sizes; and (2) skewness does reduce the effectiveness of Kupper and Hafner's sample size estimator, yet its effectiveness is negatively impacted less by skewness directly, and to a much greater degree by the common practice of estimating the population variance via a pilot sampling from the skewed population. Nonetheless, the simulations suggest that estimating sample size to control the probability with which the desired margin of error is achieved is a worthwhile alternative to the usual sample size formula that controls the average width of the confidence interval only.     

Exact confidence limits for the probability of response in two-stage designs

In addition to point estimate for the probability of response in a two-stage design (e.g. Simon's two-stage design for binary endpoints), confidence limits should be computed and reported. The current method of inverting the p-value function to compute the confidence interval does not guarantee coverage probability in a two-stage setting. The existing exact approach to calculate one-sided limits is based on the overall number of responses to order the sample space. This approach could be conservative because many sample points have the same limits. We propose a new exact one-sided interval based on p-value for the sample space ordering. Exact intervals are computed by using binomial distributions directly, instead of a normal approximation. Both exact intervals preserve the nominal confidence level. The proposed exact interval based on the p-value generally performs better than the other exact interval with regard to expected length and simple average length of confidence intervals.     

Robust Confidence Intervals for the Bernoulli Parameter

Despite the simplicity of the Bernoulli process, developing good confidence interval procedures for its parameter—the probability of success p—is deceptively difficult. The binary data yield a discrete number of successes from a discrete number of trials, n. This discreteness results in actual coverage probabilities that oscillate with the n for fixed values of p (and with p for fixed n). Moreover, this oscillation necessitates a large sample size to guarantee a good coverage probability when p is close to 0 or 1.

It is well known that the Wilson procedure is superior to many existing procedures because it is less sensitive to p than any other procedures, therefore it is less costly. The procedures proposed in this article work as well as the Wilson procedure when 0.1 ≤p ≤ 0.9, and are even less sensitive (i.e., more robust) than the Wilson procedure when p is close to 0 or 1. Specifically, when the nominal coverage probability is 0.95, the Wilson procedure requires a sample size 1, 021 to guarantee that the coverage probabilities stay above 0.92 for any 0.001 ≤ min {p, 1 ?p} <0.01. By contrast, our procedures guarantee the same coverage probabilities but only need a sample size 177 without increasing either the expected interval width or the standard deviation of the interval width.     

When is n large enough? Looking for the right sample size to estimate proportions

The central limit theorem indicates that when the sample size goes to infinite, the sampling distribution of means tends to follow a normal distribution; it is the basis for the most usual confidence interval and sample size formulas. This study analyzes what sample size is large enough to assume that the distribution of the estimator of a proportion follows a Normal distribution. Also, we propose the use of a correction factor in sample size formulas to ensure a confidence level even when the central limit theorem does not apply for these distributions.     

Confidence intervals for estimating the population signal-to-noise ratio: a simulation study

This paper considered several confidence intervals for estimating the population signal-to-noise ratio based on parametric, non-parametric and modified methods. A simulation study has been conducted to compare the performance of the interval estimators under both symmetric and skewed distributions. We reported coverage probability and average width of the interval estimators. Based on the simulation study, we observed that some of our proposed interval estimators are performing better in the sense of smaller width and coverage probability and have been recommended for the researchers.     

Confidence Intervals for a Binomial Parameter after Observing No Successes

The easily computed, one-sided confidence interval for the binomial parameter provides the basis for an interesting classroom example of scientific thinking and its relationship to confidence intervals. The upper limit can be represented as the sample proportion from a number of “successes” in a future experiment of the same sample size. The upper limit reported by most people corresponds closely to that producing a 95 percent classical confidence interval and has a Bayesian interpretation.     

Conservative confidence intervals for the intraclass correlation coefficient for clustered binary data

Asymptotic approaches are traditionally used to calculate confidence intervals for intraclass correlation coefficient in a clustered binary study. When sample size is small to medium, or correlation or response rate is near the boundary, asymptotic intervals often do not have satisfactory performance with regard to coverage. We propose using the importance sampling method to construct the profile confidence limits for the intraclass correlation coefficient. Importance sampling is a simulation based approach to reduce the variance of the estimated parameter. Four existing asymptotic limits are used as statistical quantities for sample space ordering in the importance sampling method. Simulation studies are performed to evaluate the performance of the proposed accurate intervals with regard to coverage and interval width. Simulation results indicate that the accurate intervals based on the asymptotic limits by Fleiss and Cuzick generally have shorter width than others in many cases, while the accurate intervals based on Zou and Donner asymptotic limits outperform others when correlation and response rate are close to their boundaries.     

The large sample coverage probability of confidence intervals in general regression models after a preliminary hypothesis test

We derive a computationally convenient formula for the large sample coverage probability of a confidence interval for a scalar parameter of interest following a preliminary hypothesis test that a specified vector parameter takes a given value in a general regression model. Previously, this large sample coverage probability could only be estimated by simulation. Our formula only requires the evaluation, by numerical integration, of either a double or a triple integral, irrespective of the dimension of this specified vector parameter. We illustrate the application of this formula to a confidence interval for the odds ratio of myocardial infarction when the exposure is recent oral contraceptive use, following a preliminary test where two specified interactions in a logistic regression model are zero. For this real‐life data, we compare this large sample coverage probability with the actual coverage probability of this confidence interval, obtained by simulation.     

Fixed width confidence intervals for the location parameter of an exponential distribution

We study confidence intervals of prescribed width for the lo-cation parameter of an exponential distribution. Asymptotic expan-sions up to terms tending to zero are obtained for the coverage probability and expected sample size. The limiting distribution of the sample size is given from which an asymptotic expression for the variance of the sample size is deduced. Sequential procedures with non-asymptotic coverage probability are also investigated     

Corrected empirical Bayes confidence intervals in nested error regression models

In the small area estimation, the empirical best linear unbiased predictor (EBLUP) or the empirical Bayes estimator (EB) in the linear mixed model is recognized to be useful because it gives a stable and reliable estimate for a mean of a small area. In practical situations where EBLUP is applied to real data, it is important to evaluate how much EBLUP is reliable. One method for the purpose is to construct a confidence interval based on EBLUP. In this paper, we obtain an asymptotically corrected empirical Bayes confidence interval in a nested error regression model with unbalanced sample sizes and unknown components of variance. The coverage probability is shown to satisfy the confidence level in the second-order asymptotics. It is numerically revealed that the corrected confidence interval is superior to the conventional confidence interval based on the sample mean in terms of the coverage probability and the expected width of the interval. Finally, it is applied to the posted land price data in Tokyo and the neighboring prefecture.     

Gompertz model with time-dependent covariate in the presence of interval-, right- and left-censored data

In this paper, the Gompertz model is extended to incorporate time-dependent covariates in the presence of interval-, right-, left-censored and uncensored data. Then, its performance at different sample sizes, study periods and attendance probabilities are studied. Following that, the model is compared to a fixed covariate model. Finally, two confidence interval estimation methods, Wald and likelihood ratio (LR), are explored and conclusions are drawn based on the results of the coverage probability study. The results indicate that bias, standard error and root mean square error values of the parameter estimates decrease with the increase in study period, attendance probability and sample size. Also, LR was found to work slightly better than the Wald for parameters of the model.