Biased coins

If no toxic event is observed in patient j, flip a biased coin with a probability of heads in the range [0, 0.5]. If it lands on the head side, then assign patient ... 

M. Stylianou and D. A. Follmann, The accelerated biased coin up-and-down design in phase I trials. J Biopharm Stat 14 249-260 (2004). 

M. Stylianou and N. Flournoy, Dose finding using the biased coin up-and-down design and isotonic regression. Biometrics 58 171-177 (2002). 

Atkinson AC (1982) Optimum biased coin designs for sequential clinical trials with prognostic factors. Biometrika 69 61-67. 

For frequent hitter analysis, we defined a frequent hitter score that depends on the number of screens in which a compound participated and on the number of screens where this compound was a hit We aimed at identilying a simple, empirical score that allows us to rank compounds with respect to their promiscuity, also in cases where compounds where tested in a different number of assays. A biological assay system is modeled as a biased coin that yields hit or non-hit with certain probabilities and the various assays to which a compound is subjected as a sequence of independent coin flips. Thus, we use a binomial distribution function to estimate the relative probabiUty of identifying a compound as a hit n times in k independent assays by chance. The probabiUties for the events hit and non-hit were estimated empirically from a set of assays. 

Consider the case of flipping a single biased coin where heads will turn up 70% of the time. Answer the following questions for this case ... 

EXAMPLE 6.4 Biased coins The exponential distribution again. Let s determine a coin s bias. A coin is just a die with two sides, t = 2. Score tails fr = 1 and heads in = 2. The average score per toss a) for an unbiased coin would be 1.5. 

The bound of ASC provides reason to reject a uniform noise explanation for this data, but not the biased coin distribution. 

Dembski (Dembski, 1998) has considered the example of ballot rigging where a political party is almost always given the top billing on the ballot hsting candidates. Since the selection is supposed to be chosen on the basis of a fair coin toss, this is suspicious. ASC can quantify this situation. The outcome can be described by giving the numbers of heads and tails, followed by the same representation as for the biased coin distribution. 

Figure 7.6 ASC for varyingly biased coin sequences and 20 coin tosses...

In the case of a biased coin, let p be the probability that heads will occur. The probability that tails will occur is equal lo — p. The probability of heads coming up m times out of n throws is... 

Assume that a certain biased coin has a 51.0% probability of coming up heads when thrown. 

Calculate the mean and the standard deviation of all of the possible cases of ten throws for the biased coin in the previous problem. 

In the absence of any external influence such as a catalyst which is biased in favor of one configuration over the other, we might expect structures [VIII] and [IX] to occur at random with equal probability as if the configuration at each successive addition were determined by the toss of a coin. Such, indeed, is the ordinary case. However, in the early 1950s, stereospecific catalysts were discovered Ziegler and Natta received the Nobel Prize for this discovery in 1963. 

If the coin is biased, conditional probabilities must be introduced Phhh Ph/hhPh/hPh... 

If we were testing whether a coin were biased or not, we would use ideas like these as the basis for a test. We could count, for example, HHH and HH sequences and divide them according to Eq. (7.48). If Ph/hh Ph would be suspicious ... 

This is a statement of the relative plausibility of the 2 hypotheses based on the observations. If one were a betting person, one would offer odds of 19 to 1 against the coin being biased toward tails. ... 

The statistical test procedures that we use unfortunately are not perfect and from time to time we will be fooled by the data and draw incorrect conclusions. For example, we know that 17 heads and 3 tails can (and will) occur with 20 flips of a fair coin (the probability from Chapter 3 is 0.0011) however, that outcome would give a significant p-value and we would conclude incorrectly that the coin was not fair. Conversely we could construct a coin that was biased 60 per cent/40 per cent in favour of heads and in 20 flips see say 13 heads and 7 tails. That outcome would lead to a non-significant p-value (p = 0.224) and we would fail to pick up the bias. These two potential mistakes are termed type I and type II errors. 

There are many other distributions used in statistics besides the normal distribution. Common ones are the yl and the F-distributions (see later) and the binomial distribution. The binomial distribution involves binomial events, i.e. events for which there are only two possible outcomes (yes/no, success/failure). The binomial distribution is skewed to the right, and is characterised by two parameters n, the number of individuals in the sample (or repetitions of a trial), and n, the true probability of success for each individual or trial. The mean is n n and the variance is nn(l-n). The binomial test, based on the binomial distribution, can be used to make inferences about probabilities. If we toss a true coin a iarge number of times we expect the coin to faii heads up on 50% of the tosses. Suppose we toss the coin 10 times and get 7 heads, does this mean that the coin is biased. From a binomiai tabie we can find that P(x=7)=0.117 for n=10 and n=0.5. Since 0.117>0.05 (P=0.05 is the commoniy... 

Sometimes we know a great deal about the expected statistics of a measurement. Suppose we actually flip the same coin 10,000 times, and get 5500 heads we showed that the chance of getting this many heads or more is less than 10-23. We could conclude that we were just extremely lucky. However, it is more reasonable to conclude that something is biased about the coin itself or the way we tossed it, so that heads and tails do not really have equal probability. Would we draw the same conclusion if we got 5200 heads (the chance of getting this many heads or more is 3.91 x 10-5) Would we draw the same conclusion if we got 5050 heads ... 

The rule that you intuitively arrived at was that if you observed as few as 0 or 1 or as many as 9 or 10 heads out of 10 coin flips, you would conclude that the coin was not fair. How likely is it that such a result would happen In other words, suppose you repeated this experiment a number of times with a truly fair coin. What proportion of experiments conducted in the same manner would result in an erroneous conclusion on your part because you followed the evidence in this way This is the point where the rules of probability come into play. You can find the probability of making the wrong conclusion (calling the fair coin biased) by... 

You will recall that we might have chosen other results before we concluded that the coin was biased, but we chose results that would rarely be expected by chance alone, in fact, the decision rule is based on our chosen probability of rejecting the null hypothesis when it is really true. For the coin example, this is the probability of claiming that the coin is biased when it is really fair. When asked to take part in this experiment the fairness of the coin remains unknown to us, but we choose a decision rule that is consistent with results that would not be expected by chance very often. 

In the rest of this chapter, we illustrate the principles that we need by concocting a class of problems involving die rolls and coin flips instead of molecules. How would you know if a die is biased You could roll it N times and count the numbers of I s, 2 s,. .., 6 s. If the probability distribution were perfectly flat, the die would not be biased. You could use the same test for the orientations of pencils or to determine whether atoms or molecules have biased spatial orientations or bond angle distributions. However the options available to molecules are usually so numerous that you could not possibly measure each one. In statistical mechanics you seldom have the luxury of knowing the full distribution, corresponding to all six numbers p, for i = 1,2,3.6 on die rolls. 

In summary, the Ukelihood L is a function returning the probability of observed outcomes (e.g. HHT), given a parameter value (i.e. pn)- We now ask ourselves how the likelihood L = 1 (1-pn) can be maximised. Mathematically this is easy calculus teUs us that dL/dpn = d/dpn [Ph(I-Ph)] = 2pn-3 Pn, which vanishes when Ph = 2/3. A plot, or a quick calculation of the second derivative, tells us that pn = 2/3 is indeed a maximum, at which point L = 4/27. The result that Ph, max l = 2/3 can be intuitively understood by stating that the coin is biased towards heads up, by a factor 2 over tail up. Indeed, with such a bias, the probability of the observed outcomes HHT, given pn = 2/3, is maximal. How does all this help understanding a key aspect behind Kriging ... 

We started this review with a brief discussion of some apparent dilemmas that we face in chemistry qualitative versus quantitative approaches, observables versus non-observables, structural criteria versus properties as criteria for characterization of aromaticity, chemical graph theory versus quantum chemistry, Clar 6/j rule versus Huckel An+ 2 rule, and hydrocarbons versus heteroatomic systems. As we have seen, most of the mentioned dilemmas are man-made and reflect inbred biases of different circles of chemists. It is not uncommon to come across critics with strong opinions and weak arguments, and it would be a waste of time to try to point out to them the other side of the coin . Max Planck apparently experienced difficult times before his quantum constant was accepted, as is reflected in the following quotations ... 

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