Proportions confidence intervals

The variance about the mean, and hence, the confidence limits on the predicted values, is calculated from all previous values. The variance, at any time, is the variance at the most recent time plus the variance at the current time. But these are equal because the best estimate of the current time is the most recent time. Thus, the predicted value of period t+2 will have a confidence interval proportional to twice the variance about the mean and, in general, the confidence interval will increase with the square root of the time into the future. 

Classic parameter estimation techniques involve using experimental data to estimate all parameters at once. This allows an estimate of central tendency and a confidence interval for each parameter, but it also allows determination of a matrix of covariances between parameters. To determine parameters and confidence intervals at some level, the requirements for data increase more than proportionally with the number of parameters in the model. Above some number of parameters, simultaneous estimation becomes impractical, and the experiments required to generate the data become impossible or unethical. For models at this level of complexity parameters and covariances can be estimated for each subsection of the model. This assumes that the covariance between parameters in different subsections is zero. This is unsatisfactory to some practitioners, and this (and the complexity of such models and the difficulty and cost of building them) has been a criticism of highly parameterized PBPK and PBPD models. An alternate view assumes that decisions will be made that should be informed by as much information about the system as possible, that the assumption of zero covariance between parameters in differ-... 

Fig. 5.5. Best-fit regression lines of A proportion (the difference in the proportion of larvae developing at 25 and 19°C) into free-living males ( ), free-living females ( ) or iL3s that developed by the homogonic route of development (a) through an infection. Error bars are 95% confidence intervals. Some error bars are smaller than the symbol.

The frequency interpretation of the interval estimates on the unknown amounts is given by the following ( 27 ) With at least 1- a confidence, based on the sampling characteristics of the observations on the standards, at least P proportion of the interval estimates made from a particular calibration will contain the true amounts. The Bonferroni inequality insures the 1-a confidence since the confidence interval about the regression line and the upper bound on cr are each performed using a 1- a/2 confidence coefficient. Hence, the frequency interpretation states that at least (1-a) proportion of the standard calibrations are such that at least P proportion of the intervals produced by the method cover the true unknown amounts. For the remaining a proportion of standard calibrations the proportion of intervals which cover the true unknown values may be less than P. 

Any inferences about the difference between the effects of the two treatments that may be made upon such data are the observed rates, or proportions of deteriorations by the intrathecal route. In this example, amongst those treated by the intrathecal route 22/58 = 0.379 of patients deteriorated, and the corresponding control rate is 37/60 = 0.617. The observed rates are estimates of the population incidence rates, jtt for the test treatment and Jtc for the controls. Any representation of differences between the treatments will be based upon these population rates and the estimated measure of the treatment effect will be reported with an associated 95% confidence interval and/or p-value. 

The previous sections in this chapter are applicable when we are dealing with means. As noted earlier these parameters are relevant when we have continuous, count or score data. With binary data we will be looking to construct confidence intervals for rates or proportions plus differences between those rates. 

Note that for binary data and proportions the multiplying constant is 1.96, the value used previously when we first introduced the confidence interval idea. Again this provides an approximation, but in this case the approximation works well except in the case of very small sample sizes. 

In general, the calculation of the confidence interval for any statistic, be it a single mean, the difference between two means, a median, a proportion, the difference between two proportions and so on, always has the same structure ... 

A more recent area of application for meta-analysis is in the choice of the noninferiority margin, A. As mentioned in Section 12.8, A is often chosen as some proportion of the established treatment effect (over placebo) and meta-analysis can be used to obtain an estimate of that treatment effect and an associated confidence interval. 

Ford 1, Norrie J and Ahmedi S (1995) Model inconsistency, illustrated by the Cox Proportional Flazards model Statistics in Medicine, 14, 735-746 Gardner MJ and Altman DG (1989) Estimation rather than hypothesis testing confidence intervals rather than p-values In Statistics with Confidence (eds MJ Gardner and DG Altman), Fondon British Medical Journal, 6-19... 

Figure 8.8 Scaffold (cyclohexene) hierarchy derived from mutagenicity dataP8] The proportion of Ames negative to Ames positive counts is qualitatively indicated below each scaffold. The confidence interval of the proportions is shown on the right of the scaffolds. Data taken from Kho et al.[3 I...

Figure 1 The relative 6-year mortality hazard ratios are shown for reported usual sleep hr from 2-3 hr/night to 10 or more hr/night, relative to 1.0 assigned to the hazard for 7 hr/night as the reference standard. The solid line with 95% confidence interval bars shows results from a 32-covariate Cox proportional hazards survival model, as reported previously (3). The dotted lines show data from models that excluded subjects who were not initially healthy, i.e., who died within the first year or whose questionnaires reported any cancer, heart disease, stroke, chronic bronchitis, emphysema, asthma, or current illness (a yes answer to the question are you sick at the present time ). The dot-dash lines with X symbols show models controlling only for age, insomnia, and use of sleeping pills. Data were from 635,317 women and 478,619 men. The thin solid lines with diamonds show the percent of subjects with each reported sleep duration (right axis).

Describe the production of a 95 per cent confidence interval for a proportion... 

In the example quoted earlier, we found that 42 out of a sample of 50 patients (84 per cent) showed a successful response to treatment, but, what would happen if we were to adopt this treatment and record the outcomes for thousands of patients over the next few years The proportion of successful outcomes would (hopefully) settle down to a figure in the region of 84 per cent, but it would be most surprising if our original sample provided an exact match to the long-term figure. To deal with this, we quote 95 per cent confidence intervals for the proportion in the population based upon a sample proportion. 

Figure 15.2 The 95 percent confidence intervals for the proportion of successful outcomes to treatment with varying sample sizes...

Figure 15.3 refers back to our trial where 42 out of 50 patients showed a successful outcome. Notice that the interval is asymmetrical. The asymmetry arises because possible values are more tightly constrained on one side than the other. The upper limit of the interval could not logically be greater than 100 per cent, so the upper limit cannot be far above the point estimate of 84 per cent. However, the lower limit could be anything down to 0 per cent. Confidence intervals for proportions are always asymmetrical, unless the point estimate happens to be exactly 50 per cent (as in Figure 15.2). 

Estimates of this type are called tolerance intervals. Such intervals are in a form of x Ks(x) similar to the previously described confidence intervals of X ts(x)/-Jn. The value of K (which is a function of n, a, and y) is selected in such a manner that it can be said with a probability of 100-a (corresponding to an error of a percent) that the interval will cover at least a proportion y of the population, ... 

The results of acute toxicity tests are reported as the LC50 and EC50 (concentration that reduces growth 50%) values and their 95% confidence intervals. Probit analysis is the most commonly used statistical method to determine LC50 values. Graphical interpolation can be used to estimate the LC50 value where the proportion of deaths versus the test concentration is plotted for each observation time. 

Conversely, toxicants that do not directly affect the competing species but instead alter the availability of resources also can alter the species composition of the community. In Figure 11.10, the case of the moving resource confidence interval is presented. In this case, the ratio of resource 2 has been increased relative to resource 1. This could be the alteration in microbial cycling of nutrients or the alteration in relative proportions of prey species for a predator, to name two examples. The confidence region is now outside the equilibrium region and species B becomes extinct. 

The following is an example of a mathematical/statistical calculation of a calibration curve to test for true slope, residual standard deviation, confidence interval and correlation coefficient of a curve for a fixed or relative bias. A fixed bias means that all measurements are exhibiting an error of constant value. A relative bias means that the systematic error is proportional to the concentration being measured i.e. a constant proportional increase with increasing concentration. 

Table 5.4 Proportions of agreement between electrocardiographic patterns and contrast-enhanced cardiovascular magnetic resonance for the different myocardial infarction locations and their 95% confidence interval. (Bayes de Luna 2006a)...

Myocardial infarction location Proportions of Agreement 95% Confidence Interval ... 

A measure of variability of the estimate can be gained from the standard error, but it can be seen from Equations 12.4 and 12.5 that the magnitude of the standard error is inversely proportional to n that is, the larger the sample size, the smaller will be the standard error. Therefore, without prior knowledge of the sample size, a reported standard error cannot be evaluated. A standard error value of 0.2 indicates a great deal more variability in the estimate if n = 100 than if n = 3. One way around this shortcoming is to report n for every estimate of mean standard error. Another, and better, method is to report confidence intervals of the mean. 

---