Probability, hypothesis testing, and estimation

In the previous sections we discussed probability distributions for the mean and the variance as well as methods for estimating their confidence intervals. In this section we review the principles of hypothesis testing and how these principles can be used for statistical inference. Hypothesis testing requires the supposition of two hypotheses (1) the null hypothesis, denoted with the symbol //, which designates the hypothesis being tested and (2) the alternative hypothesis denoted by Ha. If the tested null hypothesis is rejected, the alternative hypothesis must be accepted. For example, if... 

Equation (3), which is an application of Bayes theorem, is referred to as the Positive Predictive Value. The parameter p is unknown but believed to be very small (<0.01) for large virtual libraries. 1 - p is the power (or 1 - type II error, where ft is the false negative error rate) and a is the type I error, also called the size of a test in the hypothesis testing context, or the false positive error rate. The last equation defines the probability that a molecule is determined to be a hit in a biochemical assay given that the virtual screen predicts the molecule to be a hit. This probability is of great interest because it is valuable to have an estimate of the hit rate one can expect for a subset of molecules that are selected by a virtual screen. 

In designing and analyzing the various aspects of a trial, the statistician uses inferential reasoning and methodology. The formation of a testable hypothesis, the creation of a protocol for hypothesis testing, probability estimation, and decision-making all involve inferential reasoning. 

As applied here, the F test is one-sided, testing the null hypothesis that and (as estimated by and S2 ) are equal, the alternative hypothesis being (Ti > 2 - The Ftest may also be applied as a rwo-j/dedtest, in which the alternative to the null hypothesis is a. This doubles the probability that the null hypothesis is invalid and has the effect of changing the confidence level in the above example from 95 to 90%. ////... 

Another important consideration is the effect size. Because one is not only attempting to estimate the probability, but also the direction and magnitude of relationships, a direct index of the latter is very useful. Unfortunately, the tradition of null-hypothesis testing has tended to divert the focus of research away from the dimension of magnitude. In fact, any significance test represents the confluence of four mathematical components ... 

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. 

A somewhat different computational procedure is often used in practice to carry out the test described in the previous section. The procedure involves two questions What is the minimum calculated interval about bg that will include the value zero and, Is this minimum calculated interval greater than the confidence interval estimated using the tabular critical value of t If the calculated interval is larger than the critical confidence interval (see Figure 6.7), a significant difference between Po and zero probably exists and the null hypothesis is disproved. If the calculated interval is smaller than the critical confidence interval (see Figure 6.8), there is insufficient reason to believe that a significant difference exists and the null hypothesis cannot be rejected. 

Estimate a probit model, and test the hypothesis that X is not influential in determining the probability that Y equals one. 

Table V lists the estimates of the intercept and the coefficients of Equation 3. The entries labelled T FOR HO PARAMETER = 0 are the t-values for testing the null hypothesis that any parameter equals zero. The value of each entry under PR > T answers the question, "If the parameter is really equal to zero, what is the probability of getting a larger value of t " A small value for this probability indicates that it is unlikely that the parameter is actually equal to zero. For example, the probability-significance value for the coefficient of X1X2X3 is 0.0002, consequently, the hypothesis that ai23 = 0 acceptable. The finding that all the parameters of Equation 3 are significant confirms the results obtained from the RSQUARE procedure.

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