Calculating the p-value

We will start with a very simple situation to see how we actually calculate p-values. Suppose we want to know whether a coin is a fair coin by that we mean that when we flip the coin, it has an equal chance of coming down heads (H) or tails (T). 

Let pr(H) denote the probability of the coin coming down heads. We can then formulate null and alternative hypotheses as follows  

We now need some data on which to evaluate the hypotheses. Suppose we flip the coin 20 times and end up with 15 heads and 5 tails. Without thinking too much about probabilities and p-values what would your intuition lead you to conclude Would you say that the data provide evidence that the coin is not fair or are the data consistent with the coin being fair  

We will now be a little more structured about this. Because this is such a simple situation we can write down everything that could have happened in this 

Note that we have included a column H —T this is the number of heads minus the number of tails. This is done in order to link with what we do when we are comparing treatments where we use differences to measure treatment effects. 

---

Because age is not normally distributed here, the Wilcoxon signed rank test is used to calculate the p-value and is placed into a data set called pvalue. (Inferential statistics are discussed further in Chapter 7.)... 

Our data were 15 heads and 5 tails, so how do we calculate the p-value Well, remember the earlier definition and translate that into the current setting the probability of getting the observed data or more extreme data in either direction with a fair coin. To get the p-value we add up the probabilities (calculated when the null hypothesis is true - coin fair) associated with our data (15 heads, 5 tails) and more extreme data (a bigger dilference between the number of heads and the number of tails) in either direction ... 

Using the plot we calculate the p-value, firstly by identifying the outcome we saw in our data, and secondly by adding up all those probabilities associated with... 

In part this is because of the way we set up the hypotheses in our earlier discussion we asked is the coin fair or is the coin not fair We could have asked a different set of questions is the coin fair or are heads more likely than tails in which case we could have been justified in calculating the p-value only in the tail corresponding to heads more likely than tails . This would have given us a one-tailed or a one-sided test. Under these circumstances, had we seen 17 tails and 3 heads then this would not have led to a significant p-value, we would have discounted that outcome as a chance finding, it is not in the direction that we are looking for. 

Having calculated the p-value we would also calculate the 95 per cent confidence interval for the difference — P2 to give us information about the magnitude of the treatment effect. For the data in the example in Section 3.3.3 this confidence interval is given by ... 

We observed the 6/1 split in terms of successes in our data and we can calculate the p-value by adding up the probabilities associated with those outcomes which are as extreme, or more extreme, than what we have observed, when the null hypothesis is true (equal treatments). This gives p = 0.097. The corresponding chi-square test applied (inappropriately) to these data would have givenp = 0.041, so the conclusions are potentially impacted by this. 

When two samples are veiy similar, t approaches zero when they are different, t approaches infinity. The value of f is used to calculate the P value using Student s f-test tables, given in the appendix of this book. The P value is tte probability that the two distribution means are the same that is, Aj = Ag. When the P value is greater than a critical accepted value (typically 5% [21] or the experimental error due to both sampling and size determination if it is lai ger) then the null hypothesis (Ho Aj = A2) is accepted (i.e., the two populations are considered to be the same). Ceramic powder size distributions are often represented by log-normal distributions and not by normal distributions. For this reason the t statistic must be augmented for use with lognormal distributions. Equation (2.59) can be modified for this purpose to... 

Calculate the p-value for each of the indicated ions in the following ... 

Calculate the p value as the sum of the observed probability (from the first step) and all probabilities for other permutations that are less than the probability for the observed table. 

It is almost never the case that we run a single n-of-1 trial. Usually we run a series using a number of patients. It has been suggested that such a series may be adequately analysed by calculating the P-value for the treatment effect for each patient and noting in which cases the treatment was significant and in which it was not. 

Method of Continuous Variations (Job s Method). In this method, a series of solutions are prepared in which, to take metal eomplexation formation as an example, the sum of the concentrations of the metal and ligand is held constant, as the mole ratio of M to L is varied from 0.0 to 1.0 in steps of 0.1 (or smaller). As with the mole ratio method, the data is seen to form two linear segments which intersect at a point that reveals the composition of the complex. For ML, this point is at mole ratio 0.5 for ML2, the lines intersect at 0.33[M] [L], etc. The differences in absorbance between the values on the hypothetical extrapolated lines and the experimental values provide the basis of calculating the P value. A comparison of Figures 19.3 and 19.4 show that Job s method gives clearer definition of complex composition when p values are small. 

Nornicotine analogues have been explored as green catalysts for aqueous Aldol reactions, such as the one shown below. To clarify the proposed mechanism, a Hammett study was conducted and the data below obtained. Make a Hammett plot of the data and calculate the p value. What does the value of p tell you about the mechanism Propose a mechanism that is consistent with this information [40]. 

---