Interpreting confidence intervals

The confidence interval Cl(fi) serves the same purpose as Cl(Xmean) in Section 1.3.2 the quality of these average values is described in a manner that is graphic and allows meaningful comparisons to be made. An example from photometry, see Table 2.2, is used to illustrate the calculations (see also data file UV.dat) further calculations, comments, and interpretations are found in the appropriate Sections. Results in Table 2.3 are tabulated with more significant digits than is warranted, but this allows the reader to check... 

The test for the significance of a slope b is formally the same as a t-test (Section 1.5.2) if the confidence interval CI( ) includes zero, b cannot significantly differ from zero, thus ( = 0. If a horizontal line can be fitted between the plotted CL, the same interpretation applies, cf. Figures 2.6a-c. Note that si, corresponds to fx ean). that is, the standard deviation of a mean. In the above example the confidence interval clearly does not include zero this remains so even if a higher confidence level with t(f = 3, p = 0.001) = 12.92 is used. 

While it is useful to know X(y ), knowing the CL(A ) or, alternatively, whether X is within the preordained limits, given a certain confidence level, is a prerequisite to interpretation, see Figure 2.11. The variance and confidence intervals are calculated according to Eq. (2.18). 

Vogel R, Fennessey NM (1994) Flow duration curves I A new interpretation and confidence intervals. ASCE, J Water Resour Plan Manage 120(4) 485-504... 

All in vivo data, including the human and rat absorption data used by both Egan and Zhao et al., have considerable variability. Zhao et al. comment that measurements of percent absorbed for the same molecule may vary by 30%, and that the 95% confidence interval for a prediction is approximately 30% given a model RMSE of 15%. This is approximately the same as the normal experimental error for absorption values. This means that models predicting percent absorbed have to be carefully interpreted, i.e., a prediction of 30% absorbed really means the molecule is predicted to have absorption from 15 to 45%, and that classification models should work nearly as well as regression models. 

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. 

In some textbooks, a confidence interval is described as the interval within which there is a certain probability of finding the true value of the estimated quantity. Does the term true used in this sense indicate the statistical population value (e.g., p if one is estimating a mean) or the bias-free value (e.g., 6.21% iron in a mineral) Could these two interpretations of true value be a source of misunderstanding in conversations between a statistician and a geologist ... 

As the interpretation of data in some cases is quite definite, while, in other eases, a wider range of assessments seems possible, the qualitative assessment seheme is complemented by providing a subjective confidenee interval for each indicator. This subjective confidence interval is the result of critical discussion among the authors with respect to the possible margins of error of the assessment made. 

Confidence intervals are interpreted differently by frequentists and Bayesians. The 95% confidence interval derived by a frequentist suggests that the true value of some parameter (0) will be contained within the interval 95% of the time in an infinite number of trials. Note that each trial results in a different interval because the data are different. This statement is dependent on the assumed conditions under which the calculations were done, e.g., an infinite number of trials and identical conditions for each trial (O Hagan 2001). Nothing can be said about whether or not the interval contains the true 0. 

The Bayesian approach reverses the role of the sample and model the sample is fixed and unique, and the model itself is uncertain. This viewpoint corresponds more closely to the practical situation facing the individual researcher there is only 1 sample, and there are doubts either what model to use, or, for a specified model, what parameter values to assign. The model uncertainty is addressed by considering that the model parameters are distributed. In other words Bayesian interpretation of a confidence interval is that it indicates the level of belief warranted by the data the... 

Fig. 3. Interpretation of the signals from a multidetector SEC a. direct calculations b. 95% confidence interval included. Low conversion, intermediate styrene content SAN copolymer.

Fig. 4. Interpretation of the plgnal from a multidetector SEC a. direct cal culat 1 one b. 95 confidence Interval Included. High conversion styrene-rloh SAN copolymer.

We will first look at the way we calculate the confidence interval for a single mean p and then talk about its interpretation. Later in this chapter we will extend the methodology to deal with pj — p2 and other parameters of interest. 

Most of the 95 per cent confidence intervals do contain the true mean of 80 mmHg, but not all. Sample number 4 gave a mean value 3c = 81.58 mmHg with a 95 per cent confidence interval (80.33, 82.83), which has missed the true mean at the lower end. Similarly samples 35, 46, 66, 98 and 99 have given confidence intervals that do not contain p = 80 mmHg. So we have a method that seems to work most of the time, but not all of the time. For this simulation we have a 94 per cent (94/100) success rate. If we were to extend the simulation and take many thousands of samples from this population, constructing 95 per cent confidence intervals each time, we would in fact see a success rate of 95 per cent exactly 95 per cent of those intervals would contain the true (population) mean value. This provides us with the interpretation of a 95 per cent confidence interval in... 

This analysis assumes that the treatment effect is consistent across the centres. For the above data this seems a reasonable assumption, but we will return to a more formal evaluation of this assumption in the next section. The weighted average of the treatment differences that is the basis of the signal provides the best estimate of the overall treatment effect. In the above example this was 6.74 mmHg and we can construct confidence intervals around this value to allow an interpretation of the size of the (assumed common) true treatment effect. 

While thep-value allows us the ability to judge statistical significance, the clinical relevance of the finding is difficult to evaluate from the calculated confidence interval because this is now on the log scale. It is usual to back-transform the lower and upper confidence limits, together with the difference in the means on the log scale, to give us something on the original data scale which is more readily interpretable. The back-transform for the log transformation is the anti-log. 

In practice I would always recommend using confidence intervals for evaluating equivalence and non-inferiority rather than these associated p-values. This is because the associated p-values tend to get mixed up with conventional p-values for detecting differences. The two are not the same and are looking at quite different things. The confidence interval approach avoids this confusion and provides a technique that is easy to present and interpret. 

In the REMARK system, estimated effects (e.g., hazard ratios) with confidence intervals for the marker were recommended for univariant and key multivariable analyses, and a Kaplan-Meier plot was recommended to represent the effect of a tumor marker in a time-to-event outcome. The discussion section should interpret the results in the context of the pre-specified hypothesis and describe limitations of the study as well as implications for future research and clinical value. These guidelines were advocated for reporting of tumor marker studies in breast cancer research and treatment (63). 

The process of providing an answer to a particular analytical problem is presented in Figure 2. The analytical system—which is a defined method protocol, applicable to a specified type of test material and to a defined concentration rate of the analyte —must be fit for a particular analytical purpose [4]. This analytical purpose reflects the achievement of analytical results with an acceptable standard of accuracy. Without a statement of uncertainty, a result cannot be interpreted and, as such, has no value [8]. A result must be expressed with its expanded uncertainty, which in general represents a 95% confidence interval around the result. The probability that the mean measurement value is included in the expanded uncertainty is 95%, provided that it is an unbiased value which is made traceable to an internationally recognized reference or standard. In this way, the establishment of trace-ability and the calculation of MU are linked to each other. Before MU is estimated, it must be demonstrated that the result is traceable to a reference or standard which is assumed to represent the truth [9,10]. 

In the Montreal case-control study carried out by Siemiatycki (1991 see the monograph on dichloromethane in this volume), the investigators estimated the associations between 293 workplace substances and several types of cancer. Isopropanol was one of the substances. About 4% of the study subjects had ever been exposed to isopropanol. Among the main occupations to which isopropanol exposure was attributed in this study were fire fighters, machinists and electricians. For most types of cancer examined (oesophagus, stomach, colon, rectum, pancreas, prostate, bladder, kidney, skin melanoma, lymphoma), there was no indication of an excess risk due to isopropanol. For lung cancer, based on 16 cases exposed at the substantial level, the odds ratio was 1.4 (90% confidence interv al, 0.8-2.7). [The interpretation of the null results has to take into account the small numbers and presumed low levels of exposure.]... 

Problems encountered in HPLC analysis most often stem from a lack of knowledge of the influence of the slight variation of the experimental parameters (pH, temperature, solvent composition, flow rate, etc.). The analyst has to set up the list of parameters and their possible interactions. There are hardware parameters (e.g., flow control, temperature control, lamp current) and software parameters used to interpret and report the results from stored data. The use of factorial designs is of great help. Software such as Validation Manager, from Merck, produces, in a table for each parameter and interaction, its percentage and confidence interval as well as information to help the analyst in concluding the study. 

Suppose we could use b0 and sbo from only one set of experiments to construct a confidence interval about b(l such that there is a given probability that the interval contains the population value of /30 (see Section 3.4). The interpretation of such a confidence interval is this if we find that the interval includes the value zero, then (with our data) we cannot disprove the null hypothesis that fi0 = 0 that is, on the basis of the estimates b0 and sho, it is not improbable that the true value of y30 could be zero. Suppose, however, we find that the confidence interval does not contain the value zero because we know that if /30 were really equal to zero this lack of overlap... 

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