A frequentist approach

Throughout this book, the approach taken to hypothesis testing and statistical analysis has been a frequentist approach. The name frequentist reflects its derivation from the definition of probability in terms of frequencies of outcomes. While this approach is likely the majority approach at this time, it should be noted here that it is not the only approach. One alternative method of statistical inference is the Bayesian approach, named for Thomas Bayes work in the area of probability. 

An alternative to the frequentist approach to statistics is based upon the use of probability to quantify the state of knowledge (or ignorance) regarding a quantity. This view is known as the personalist, subjectivist or Bayesian view (Morgan Henrion, 1990). For consistency throughout the text, we will use the term Bayesian . Unlike a frequentist approach, a Bayesian approach does not require assumptions about repeated trials in order to make... 

In what follows we shall assume that the sample size is going to be determined as a function of the other factors. We shall take the example of a two-arm parallel-group trial comparing an active treatment with a placebo for which the outcome measure of interest is continuous and will be assumed to be Normally distributed. It is assumed that analysis will take place using a frequentist approach and via the two independent-samples t-test. A formula for sample size determination will be presented. No attempt will be made to derive it. Instead we shall show that it behaves in an intuitively reasonable manner. 

Even closer to a frequentist approach than the methods of Section 13.2.15 is that of calculating what O Hagan et al. call assurance (O Hagan et al., 2005). This is the... 

Group sequential method. A frequentist approach to running and analysing sequential trials whereby hypothesis tests are carried out at agreed intervals in such a way as to ensure that the probability, for the trial as a whole, of making a type I error, given that the null hypothesis is true, does not exceed some nominal level chosen... 

The classical, frequentist approach in statistics requires the concept of the sampling distribution of an estimator. In classical statistics, a data set is commonly treated as a random sample from a population. Of course, in some situations the data actually have been collected according to a probability-sampling scheme. Whether that is the case or not, processes generating the data will be snbject to stochastic-ity and variation, which is a sonrce of uncertainty in nse of the data. Therefore, sampling concepts may be invoked in order to provide a model that accounts for the random processes, and that will lead to confidence intervals or standard errors. The population may or may not be conceived as a finite set of individnals. In some situations, such as when forecasting a fnture value, a continuous probability distribution plays the role of the popnlation. 

The standard tools of statistical inference, including the concept and approaches of constructing a null hypotheses and associated p values, are based on the frequentist view of probability. From a frequentist perspective, the probability of an event is defined as the fraction of times that the event occurs in a very large number of trials (known as a probability limit). Given a hypothesis and data addressing it, the classical procedure is to calculate from the data an appropriate statistic, which is typically... 

Critics of the frequentist approach consider this disturbing. The actual observations in 12 tosses of a coin, 9 heads and 3 tails were observed should not lead to 2 different conclusions dependent only upon the choice of when to stop the experiment (at 12 tosses or at 3 tails). 

The above approach, which was attacked as being too vague to be the starting point of any theory of probability, led eventually to the frequentist approach, where probability was defined in a manner that assigns a numerical value, albeit a value that cannot ever be measured, since it requires an inhnite number of trials... 

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. 

When a clinical trial has been conducted, the frequentist approach we have discussed in the book leads to certain statistical analyses being conducted. A p-value is calculated which provides information leading to the rejection of the null hypothesis or the failure to reject the null hypothesis. Additionally, the analyses lead to an estimate of the treatment effect and its associated... 

While the Bayesian approach explicitly acknowledges the role of judgement, the frequentist approach also involves judgement regarding the assumption of a random, representative... 

Linear mixed effects (LME) models express the response variable as a linear function of both the fixed effects and the random effects, with an additive within-unit error, see Laird and Wase (1) or Searle et al. (2) for a good review of methodology. The frequentist approach to LME models is generally Ukelihood-based, with restricted maximum likelihood (REML) being the preferred method of estimation (3). 

When applied to real-life problems, however, there is little experience reported using the frequentist prior approach in the literature. Gastonguay et al. (1999) used prior information from adults to estimate the model parameters in a PopPK analysis in children. Simonsen et al. (2000) used prior information to estimate the pharmacokinetics and pharmacodynamics of epirubicin in rats. So, while it appears that using prior information may be useful in certain circumstances, but caveat emp-tor applies at the present time—let the buyer beware. Gisleskog, Karlsson, and Beal conclude that considerable care must be taken with the use of a frequentist prior. That would seem good advice. 

The frequentist approach to this sort of problem is closely related to the following philosophical position. If I have a theory which states all swans are white , then it doesn t matter how many white swans are observed, I cannot prove that it is true. However, a single swan of a different colour will succeed in proving that it is false. Now, if we return to the problem of the CD player, we can see that if the first track played is not W then the theory that I pressed play is wrong, whatever the prior probability of its being true. (It is worth pointing out that the Bayesian will also reach this conclusion.)... 

However, at 1/12, the size of the test is rather larger than is usually considered conventionally acceptable a 5% size is more usual. We now leave this specific example to discuss the frequentist approach to hypothesis testing in clinical trials. 

R.A. Fisher s own views on inference are outlined in Fisher (1956) For a modern text covering frequentist approaches see Cox (2006) and for a mathematically elementary but otherwise profound outline of the Bayesian approach see Bindley (2006). Likelihood approaches are covered in Lindsey (1996), Royall (1999), Pawitan (2001) and the classic by Edwards (1992). Classic expositions of the fully subjective Bayesian view of probability are give by Savage (1954) and de Finetti (1974, 1975) and the case for its use in science is made by the philosophers Howson and Urbach (1993). A comparative overview of the various statistical systems is given by Barnett (1999). 

An increasingly employed approach to conducting meta-analyses is to perform a Bayesian analysis. In fact, the most commonly employed analysis is not fully Bayesian but a hybrid. That is because, although prior distributions are employed for the treatment parameters, the nuisance parameters themselves are treated as fixed (Senn, 2007). That is to say that the variances of the treatment contrasts from various trials are treated as if they were known. This means that such analyses do not avoid many of the problems associated with frequentist approaches outlined in section 16.2.14. 

A fifth frequentist approach, that of flexible designs (Bauer and Kohne, 1994 Gallo et al, 2006 Proschan and Hunsberger, 1995), has a rather different flavour and will be considered after looking at Bayesian approaches. 

This is certainly the Bayesian point of view, although, effectively, it is not so much a specific criticism of frequentist approaches to sequential analysis but a general criticism of frequentist methods, which reveal their weaknesses most fully in the sequential... 

In practice, neither Lindley s approach nor any of the other Bayesian suggestions is used. However, some standard frequentist approaches can be given a fiducial interpretation, which is very similar to a Bayesian one involving uninformative priors (O Quigley and Baudoin, 1988). For example, if one requires that the probability that the true unknown bioavailability ratio lies within the limits of equivalence should be at least 95%, then this is approximately equivalent to Westlake s symmetrical confidence limit approach. Alternatively, if one requires that the probability of inferior bioavailability should be less than 5% and also that the probability of superior bioavailability should be less than 5%, then this is approximately equivalent to requiring that conventional 90% limits should lie within the ranges of equivalence. 

Alpha-spending approach. One of a number of frequentist approaches to sequential trials. It controls the type I error rate by proposing a time path for spending the type I error rate allowed for the trial. This avoids having to specify in advance the exact time points at which analysis must take place. 

The general pharmaceutical physician cannot be expected to be able to generate Bayesian statistical plans for him/herself. These require an experienced statistician and, it may be added, a statistician who is not, him/herself, philosophically opposed to Bayesian rather than frequentist thinking. The decision to employ a Bayesian design for a clinical trial will be viewed as courageous in most companies, and there will be many clinical trials for which an orthodox, frequentist approach will be selected, for several good reasons. Overall, the gen-... 

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