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PubMed1995

Confidence limits for the population prevalence rate based on the negative binomial distribution.

This paper shows that the extension of the simple procedure of George and Elston in calculation of confidence limits for the underlying prevalence rate to accommodate any finite number of cases in inverse sampling is straightforward. To appreciate the fact that the length of the confidence interval calculated on the basis of the first single case may be too wide for general utility, I include a quantiative discussion on the effect due to an increase in the number of cases requested in the sample on the expected length of confidence intervals. To facilitate further the application of the results presented in this paper, I present a table that summarizes in a variety of situations the minimum required number of cases for the ratio of the expected length of a confidence interval relative to the underlying prevalence rate to be less than or equal to a given value. I also include a discussion on the relation between Clemans's confidence limits on the expected number of trials before the failure of a given device and those presented here.

Binomial Distribution↗
PubMed1983

A third order iterative procedure for computing exact confidence limits for Poisson expectation values.

The calculation of exact confidence limits for the expectation value of a Poisson distribution when a single observation of K events is given by means of a second order iterative algorithm was recently presented. Here it is shown that the structure of the equations to be solved admits the use of a third order algorithm, thereby significantly reducing computation time, especially for pocket calculators, with practically no additional computational effort.

Computers↗
PubMed1995

Notes on conditional confidence limits under inverse sampling.

When the number of subjects in a two-by-two table is small or moderate, we may commonly use the exact conditional distribution with all marginals fixed to derive the conditional confidence limits on the underlying parameter. Under inverse sampling, in which we continue to sample subjects until we obtain exactly a pre-determined number of subjects failing into a specific category, this paper notes that derivation of a confidence interval, which has the coverage probability equal to or larger than a nominal 1-alpha confidence level, for relative risk and relative difference in cohort studies is straightforward. This paper further finds that, when the underlying disease is rare, we can similarly apply an inverse sampling to produce an approximate 1-alpha conditional confidence limits on attributable risk in case-control studies as well. When the number of subjects is small and the test statistic derived on the basis of large sample theory is not strictly adequate for use, this paper also presents an exact hypothesis testing procedure for the above parameters in the corresponding study designs.

Case-Control Studies↗
PubMed1989

[Confidence factors-method for estimating the confidence limits of standardized population rate of indirect method].

The standardized rate of indirect method is widely used, but no method of interval estimation for its population rate has been reported. The authors have discussed the standard error of standardized rate of indirect method and suggested that a method using this standard error should be used to determine the confidence limits for the population rate. In this paper, the authors put forward another method (Confidence Factors-Method) which can be easily applied to determine the confidence limits. It gives approximately the same result with the method mentioned above.

Mathematics↗
PubMed1994

Estimating upper confidence limits for extra risk in quantal multistage models.

Multistage models are frequently applied in carcinogenic risk assessment. In their simplest form, these models relate the probability of tumor presence to some measure of dose. These models are then used to project the excess risk of tumor occurrence at doses frequently well below the lowest experimental dose. Upper confidence limits on the excess risk associated with exposures at these doses are then determined. A likelihood-based method is commonly used to determine these limits. We compare this method to two computationally intensive "bootstrap" methods for determining the 95% upper confidence limit on extra risk. The coverage probabilities and bias of likelihood-based and bootstrap estimates are examined in a simulation study of carcinogenicity experiments. The coverage probabilities of the nonparametric bootstrap method fell below 95% more frequently and by wider margins than the better-performing parametric bootstrap and likelihood-based methods. The relative bias of all estimators are seen to be affected by the amount of curvature in the true underlying dose-response function. In general, the likelihood-based method has the best coverage probability properties while the parametric bootstrap is less biased and less variable than the likelihood-based method. Ultimately, neither method is entirely satisfactory for highly curved dose-response patterns.

Animals↗
PubMed1993

Confidence limits for hazardous concentrations based on logistically distributed NOEC toxicity data.

This paper deals with the calculation of Hazardous Concentrations of toxic substances from small sets of laboratory toxicity data, e.g., NOECs. A procedure due to Van Straalen and Denneman, as adapted from Kooijman (case n = 1), in which one seeks a concentration that protects 95% of the biological species is modified to account for the uncertainty in the estimates. New constants are obtained by simulation. These allow the calculation of the one-sided 95% left confidence limit of the Hazardous Concentration, from the mean and standard deviation of a sample of (laboratory) toxicity data. This 95% confidence limit is always lower than the 95% certainty value calculated with the Kooijman (n = 1)/Van Straalen method. The authors also derive constants to calculate a one-sided 50% confidence value, that overpredicts as often as it underpredicts. This value may be used as a median guess of the Hazardous Concentration. It will always be higher than the 95% certainty value of the Kooijman (n = 1)/Van Straalen method. However, by using the 50% value, one runs the risk of protecting substantially less than 95% of the biological species.

Confidence Intervals↗
PubMed2006

Comparing patients' predicted test scores from a regression equation with their obtained scores: a significance test and point estimate of abnormality with accompanying confidence limits.

In contrast to the standard use of regression, in which an individual's score on the dependent variable is unknown, neuropsychologists are often interested in comparing a predicted score with a known obtained score. Existing inferential methods use the standard error for a new case (s-subN+1) to provide confidence limits on a predicted score and hence are tailored to the standard usage. However, s-subN+1 can be used to test whether the discrepancy between a patient's predicted and obtained scores was drawn from the distribution of discrepancies in a control population. This method simultaneously provides a point estimate of the percentage of the control population that would exhibit a larger discrepancy. A method for obtaining confidence limits on this percentage is also developed. These methods can be used with existing regression equations and are particularly useful when the sample used to generate a regression equation is modest in size. Monte Carlo simulations confirm the validity of the methods, and computer programs that implement them are described and made available.

Confidence Intervals↗
PubMed1987

Confidence limits to the risk ratio.

For the problem of obtaining confidence limits to the risk ratio, or the ratio of two binomial probabilities, Katz et al. (1978, Biometrics 34, 469-474) proposed a method based on the logarithm of the observed ratio and using Fieller's result. We propose a method based on a power of the observed ratio, the power being chosen to minimize the skewness of the pivotal random variable. The method is simple to use and more stable than that of Katz et al. A continuity correction is suggested if a conservative interval is desired.

Abortion, Spontaneous↗
PubMed2003

Estimation of parameters of dose-volume models and their confidence limits.

Predictions of the normal-tissue complication probability (NTCP) for the ranking of treatment plans are based on fits of dose-volume models to clinical and/or experimental data. In the literature several different fit methods are used. In this work frequently used methods and techniques to fit NTCP models to dose response data for establishing dose-volume effects, are discussed. The techniques are tested for their usability with dose-volume data and NTCP models. Different methods to estimate the confidence intervals of the model parameters are part of this study. From a critical-volume (CV) model with biologically realistic parameters a primary dataset was generated, serving as the reference for this study and describable by the NTCP model. The CV model was fitted to this dataset. From the resulting parameters and the CV model, 1000 secondary datasets were generated by Monte Carlo simulation. All secondary datasets were fitted to obtain 1000 parameter sets of the CV model. Thus the 'real' spread in fit results due to statistical spreading in the data is obtained and has been compared with estimates of the confidence intervals obtained by different methods applied to the primary dataset. The confidence limits of the parameters of one dataset were estimated using the methods, employing the covariance matrix, the jackknife method and directly from the likelihood landscape. These results were compared with the spread of the parameters, obtained from the secondary parameter sets. For the estimation of confidence intervals on NTCP predictions, three methods were tested. Firstly, propagation of errors using the covariance matrix was used. Secondly, the meaning of the width of a bundle of curves that resulted from parameters that were within the one standard deviation region in the likelihood space was investigated. Thirdly, many parameter sets and their likelihood were used to create a likelihood-weighted probability distribution of the NTCP. It is concluded that for the type of dose response data used here, only a full likelihood analysis will produce reliable results. The often-used approximations, such as the usage of the covariance matrix, produce inconsistent confidence limits on both the parameter sets and the resulting NTCP values.

Dose-Response Relationship, Radiation↗
PubMed1982

The performance of three approximate confidence limit methods for the odds ratio.

The performance of three approximate confidence limit methods for the odds ratio, R, is studied at the 95% level in the unconditional sample space. There are: the method proposed by Cornfield (Proceedings of the 3rd Berkeley Symposium 1956;4:135-48), the logit method with 1/2 corrections first considered by Woolf (Ann Hum Genet 1955;19:251-3), and the test-based method proposed by Miettinen (Am J Epidemiol 1976;103:226-35). Cornfield's method comes closest to attaining the normal confidence coefficient. The logit method typically has actual confidence coefficients somewhat too large with disparate tail areas. The latter is ascribed in part to the enhanced skewness induced by the logit transformation itself. The test-based method has actual coefficients uniformly less than nominal when R not equal to 1. This underestimation is worse in finite samples than Halperin found it to be asymptotically. Although the Cornfield and test-based methods have the same confidence coefficients for R = 1, the test-based method is more likely to cover distant values of R not equal to 1 when in fact R = 1. It is concluded that Cornfield's method without the continuity correction is the preferred approximate method in the unconditional space as, with the continuity correction, it was previously found to be in the conditional space.

Abnormalities, Drug-Induced↗
PubMed1995

Confidence limits for maximum word-recognition scores.

Clinical judgments are often made regarding whether maximum word-recognition scores (PBmax) are appropriate in relation to degree of sensorineural hearing loss. In order to determine if word recognition is significantly poorer than expected, it is necessary to consider the lower boundary of PBmax associated with a particular degree of hearing loss for speech materials commonly used to measure word recognition. The purpose of this experiment was to define a confidence limit for PBmax from Northwestern University Test #6 (NU-6) word-recognition scores obtained from a large group of young and aged subjects with confirmed cochlear hearing loss. Word-recognition scores at several speech levels were obtained from 407 ears with a wide range of pure-tone averages. Because the characteristics of the distribution of maximum scores are not known, a procedure was developed using computer simulations to approximate the distribution of word-recognition scores corresponding to PBmax and determine the 95% confidence limit (CL). Results of the simulation were confirmed by comparing means and standard deviations of PBmax derived from experimental and simulation data. Percentages of young and aged subjects with scores outside the 95% CL are equal to their proportions in the entire subject sample. If PBmax determined from a score-level psychometric function is poorer than the 95% CL, PBmax may be considered "disproportionately" poor in relation to the degree of hearing loss. One score measured at a single arbitrary suprathreshold level that is poorer than the 95% CL suggests that the score may underestimate PBmax and that word recognition should be measured at additional levels to obtain a more reasonable estimate of the listener's maximum word-recognition score.

Adult↗
PubMed2006

Comparison of fixed percentage method and lower confidence limits for defining limits of normality for interpretation of spirometry.

BACKGROUND: The use of the lower 90% confidence limit of the lower limit of normal (LLN(CI)), rather than a fixed percentage of the predicted value (LLN(%)), appears to be statistically more appropriate for interpretation of spirometry results. There has been no comparative assessment of these 2 definitions of the LLN in routine clinical practice. METHODS: We studied results of spirometry interpretations made with these 2 approaches, and assessed various factors that influence discordant classification of spirometry results. Spirometry records from 18,112 consecutive adult patients referred for spirometry were interpreted as normal, obstructive, or restrictive, based on both LLN(CI) and LLN(%). Discordant results were analyzed using multiple logistic regression techniques to identify variables that significantly affected discordant classification of results. RESULTS: Overall, 11.7% of the results were discordant between the 2 methods. Agreement between the 2 methods, calculated using the kappa estimate, was poorer with spirometry values from women and from patients at the extremes of height and age. Age, sex, and height independently influenced discordant classification. Limits of agreement between LLN(CI) and LLN(%) were wide for all the spirometric variables studied--more so in women and in shorter and older patients. CONCLUSIONS: LLN(CI) and LLN(%) yielded different interpretations of spirometry data in several instances, and the 2 methods cannot be used interchangeably. When interpreting spirometry data in routine clinical practice, LLN(CI) should be preferred over LLN(%).

Adolescent↗
PubMed1989

A note on confidence limits for quartiles with right censored data.

Two new methods are proposed for constructing the confidence limits for quartiles. The small sample behaviour of these two methods is compared with the Jennison and Turnbull modified Brookmeyer-Crowley method at three quartiles. The simulation study indicates that one of the new methods, the smoothed modified reflected method, is preferred over the other methods when the censoring rate is less than 40 per cent while the Jennison and Turnbull method is preferred for higher censoring. The results for the upper quartile are similar to the median. For the lower quartile with small sample and high censoring, semi-infinite intervals may often occur. The correct practice is to permit the semi-infinite intervals without modifying them since it is more informative and gives closer to nominal level coverage.

Computer Simulation↗
PubMed1991

An efficient program for computing conditional maximum likelihood estimates and exact confidence limits for a common odds ratio.

This paper describes a method and associated computer program for calculating exact confidence limits and P values, along with the conditional maximum likelihood estimate of the common odds ratio for a series of 2 x 2 tables. The program can be used to calculate exact estimates for matched data and is considerably faster than others currently available.

Algorithms↗
PubMed2003

Confidence limits of arteriovenous fistula flow rate measured by the "on-line" thermodilution technique.

BACKGROUND: A method is presented for estimating the confidence limits (CLs), or accuracy, of the arteriovenous fistula flow rate measured at haemodialysis by the "on-line" thermodilution technique. METHODS: This was by derivation of an expression to estimate what variance a set of repeated measures of flow would yield, using values pertaining to a single measure of flow. (Laws of variance were applied to the formula used to calculate flow, to account for its variables' values and measurement errors.) This enabled CLs of a single measure to be estimated. RESULTS: The variance estimated from a single measure was compared with that actually observed upon immediately taking a second measurement; differences in 189 pairs were not significantly different from zero (P=0.56). Applying the results demonstrated that measured flow values of 430-570 ml/min typically had associated 95% CLs that included 500 ml/min; therefore, true flow could not be said to be either side of 500 ml/min. The same was the case for 500-700 ml/min with regard to 600 ml/min. CLs widened considerably with the magnitude of flow rate, limiting the accurate measurement of higher flows and the detection of falls in flow. CONCLUSION: A method to estimate CLs of flow rate measured by the thermodilution technique is presented and validated. Application demonstrates an accurate measurement of low flow, but limitations at higher flow and in detecting falls in flow. Appreciating the magnitude of such is critical to informed clinical decision making when using flow rate in an access surveillance programme.

Arteriovenous Shunt, Surgical↗
PubMed1993

A longitudinal study describing confidence limits of normal fetal cardiac, thoracic, and pulmonary dimensions from 20 to 40 weeks' gestation.

The current prospective, longitudinal study was designed to define the confidence limits and characteristics of growth for fetal cardiac, thoracic, and pulmonary dimensions from 20 to 40 weeks' gestation. We tested the hypotheses that (1) CC/TC, (2) CC/AC, (3) TC/HC, and (4) TC/AC ratios remain constant throughout this period of gestation. The four-chamber view of the fetal heart in diastole, in the absence of fetal breathing, was used as a standard reference for sonographic measurement of cardiac, thoracic, and left and right pulmonary dimensions in a longitudinal study of 45 uncomplicated pregnancies. Cardiac growth is best fit by a quadratic function: -8.147 + 0.900 (GA) - 0.0078 (GA)2. Similarly, thoracic growth is best fit by a quadratic function: -14.072 + 1.757 (GA) - 0.018 (GA)2. The characteristics of growth of the left and right lungs are best described by linear growth equations: -1.71 + 0.426 (GA) and -1.62 + 0.539 (GA), respectively. Each of the CC/TC, CC/AC, TC/HC, and TC/AC ratios changed significantly over time.

Adult↗
PubMed1977

A method for the determination of confidence limits for the P/2e- ratio for chosen values of ymaxatp form the results of continuous culture experiments.

A model is described, which allows the determination of 95% confidence limits for the maintenance coefficient and the efficiency of oxidative phosphorylation for chosen values of the growth yield for ATP corrected for energy maintenace (YmaxATP). As experimental data the specific rates of substrate consumption, product formation and oxygen uptake in chemostat cultures at various growth rates are use.

Bacteria↗
PubMed1994

MC-Fit: using Monte-Carlo methods to get accurate confidence limits on enzyme parameters.

A program is described for estimating enzymatic parameters from experimental data using Apple Macintosh computers. MC-Fit uses iterative least-square fitting and Monte-Carlo sampling to get accurate estimates of the confidence limits. This approach is more robust than the conventional covariance matrix estimation, especially in cases where experimental data is partially lacking or when the standard error on individual measurements is large. This happens quite often when analysing the properties of variant enzymes obtained by mutagenesis, as these can have severely impaired activities and reduced affinities for their substrates.

Confidence Intervals↗