Inference about Confidence Intervals

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 

The classical or frequentist approach to probability is the one most taught in university conrses. That may change, however, becanse the Bayesian approach is the more easily nnderstood statistical philosophy, both conceptually as well as numerically. Many scientists have difficnlty in articnlating correctly the meaning of a confidence interval within the classical frequentist framework. The common misinterpretation the probability that a parameter lies between certain limits is exactly the correct one from the Bayesian standpoint. 

Apart from this pedagogical aspect (cf Lee 1989, preface), there is a more technical reason to prefer the Bayesian approach to the confidence approach. The Bayesian approach is the more powerfnl one eventnally, for extending a model into directions necessary to deal with its weaknesses. These are various relaxations of distribntional assnmptions. The conceptnal device of an infinite repetition of samples, as in the freqnentist viewpoint, does not yield enongh power to accomplish these extensions. 

Confidence intervals nsing freqnentist and Bayesian approaches have been compared for the normal distribntion with mean p and standard deviation o (Aldenberg and Jaworska 2000). In particnlar, data on species sensitivity to a toxicant was fitted to a normal distribntion to form the species sensitivity distribution (SSD). Fraction affected (FA) and the hazardons concentration (HC), i.e., percentiles and their confidence intervals, were analyzed. Lower and npper confidence limits were developed from t statistics to form 90% 2-sided classical confidence intervals. Bayesian treatment of the uncertainty of p and a of a presupposed normal distribution followed the approach of Box and Tiao (1973, chapter 2, section 2.4). Noninformative prior distributions for the parameters p and o specify the initial state of knowledge. These were constant c and l/o, respectively. Bayes theorem transforms the prior into the posterior distribution by the multiplication of the classic likelihood fnnction of the data and the joint prior distribution of the parameters, in this case p and o (Fignre 5.4). 

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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. 

Automation of inference sometimes produces fundamental tensions. For example, some automated procedures have no guarantees of reliability—no theorems about their error rates, no confidence intervals—but in practice work much better than procedures that do have such guarantees. Which are to be preferred, and why In some cases, theoretically well-founded procedures produce results that are inferior to stupid procedirres. For example, in determining protein homologue families, a simple matching procedure appears to work as well or better than procedures using sophisticated Hidden Markov models. Which sort of procedure is to be preferred, and why ... 

Inference is the act of drawing conclusions from a model, be it making a prediction about a concentration at a particular time, such as the maximal concentration at the end of an infusion, or the average of some pharmacokinetic parameter, like clearance. These inferences are referred to as point estimates because they are estimates of the true value. Since these estimates are not known with certainty they have some error associated with them. For this reason confidence intervals, prediction intervals, or simply the error of the point estimate are included to show what the degree of precision was in the estimation. With models that are developed iteratively until some final optimal model is developed, the estimate of the error associated with inference is conditional on the final model. When inferences from a model are drawn, modelers typically act as though this were the true model. However, because the final model is uncertain (there may be other equally valid models, just this particular one was chosen) all point estimates error predicated on the final model will be underestimated (Chatfield, 1995). As such, the confidence interval or prediction interval around some estimate will be overly optimistic, as will the standard error of all parameters in a model. 

If instead we consider that we wish to make probabilistic statements about patients in general, including those from centres we did not include, we then have to move to regarding the centres themselves as some realization of a random process. The true difference in treatment effects from centre to centre now becomes a further source of random variation because, although, if we restrict inference to these centres only, this variability is frozen (the differences are what they are and that is an end of it), if we talk about future centres then these might also vary and there is no reason why these other differences should be exactly the same which now apply. If we take account of this further source of variation, not only in forming estimators but also in calculating confidence intervals, then we have a random-effects model. 

This Is an issue where consensus now seems to have been achieved. An ingenious theorem due to Fieller (Fieller, 1940, 1944) enables one to calculate a confidence interval for the ratio of two means. The approach does not require transformation of the original data. (Edgar C. Fieller, 1907—1960, is an early example of a statistician employed in the pharmaceutical industry. He worked for the Boots company in the late 1930s and 1940s.) For many years this was a common approach to making inferences about the ratio of the two mean AUCs in the standard bioequivalence experiment (Locke, 1984). 

Additionally, it is assumed that, for any set of predictor values, the corresponding y,s are normally distributed about the regression plane. This is a requirement for general inference making, e.g., confidence intervals, prediction of y, etc. The predictor variables, x,s, are also considered independent of each other, or additive. Therefore, the value of Xj does not, in any way, affect or depend on X2, if they are independent. This is often not the case, so the researcher must check and account for the presence of interaction between the predictor x, variables. 

Statistical inferences such as point estimation, confidence intervals, and hypothesis testing developed under the frequentist framework use the sampling distribution of the statistic given the unknown parameter. They answer questions about where we are in the parameter dimension using a probability distribution in the observation dimension. 

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