Estimation Of Parameters

The entire process of statistical design and analysis can be described briefly as follows. The target of a scientist's investigation is a population with certain characteristic of interest: for example, a man's systolic blood pressure or his cholesterol level, or whether a leukemia patient responds to an investigative drug. A numerical characteristic of a target population is called a parameter: for example, the population mean m (average SBP) or the population proportion p (a drug's response rate). Generally, it would be too time consuming or too costly to obtain the totality of population information in order to learn about the parameter(s) of interest. For example, there are millions of men to survey in a target population, and the value of the information may not justify the high cost. Sometimes the population does not even exist. For example, in the case of an investigative drug for leukemia, we are interested in future patients as well as present patients. To deal with the problem, the researcher may decide to take a sample or to conduct a small phase II clinical trial. Chapters 1 and 2 provide methods by which we can learn about data from the sample or samples. We learned how to organize data, how to summarize data, and how to present them. The topic of probability in Chapter 3 sets the framework for dealing with uncertainties. By this point the researcher is ready to draw inferences about the population of interest based on what he or she learned from his or her sample(s). Depending on the research's objectives, we can classify inferences into two categories: one in which we want to estimate the value of a parameter, for example the response rate of a leukemia investigative drug, and one where we want to compare the parameters for two subpopulations using statistical tests of significance. For example, we want to know whether men have higher cholesterol levels, on average, than women. In this chapter we deal with the first category and the statistical procedure called estimation. It is extremely useful, one of the most useful procedures of statistics. The word estimate actually has a language problem, the opposite of the language problem of statistical "tests" (the topic of Chapter 5). The colloquial meaning of the word test carries the implication that statistical tests are especially objective, no-nonsense procedures that reveal the truth. Conversely, the colloquial meaning of the word estimate is that of guessing, perhaps off the top of the head and uninformed, not to be taken too seriously. It is used by car body repair shops, which "estimate" how much it will cost to fix a car after an accident. The estimate in that case is actually a bid from a for-profit business establishment seeking your trade. In our case, the word estimation is used in the usual sense that provides a "substitute" for an unknown truth, but it isn't that bad a choice of word once you understand how to do it. But it is important to make it clear that statistical estimation is no less objective than any other formal statistical procedure; statistical estimation requires calculations and tables just as statistical testing does. In addition, it is very important to differentiate formal statistical estimation from ordinary guessing. In formal statistical estimation, we can determine the amount of uncertainty (and so the error) in the estimate. How often have you heard of someone making a guess and then giving you a number measuring the "margin of error" of the guess? That's what statistical estimation does. It gives you the best guess and then tells you how "wrong" the guess could be, in quite precise terms. Certain media, sophisticated newspapers in particular, are starting to educate the public about statistical estimation. They do it when they report the results of polls. They say things like, "74% of the voters disagree with the governor's budget proposal," and then go on to say that the margin error is plus or minus 3%. What they are saying is that whoever conducted the poll is claiming to have polled about 1000 people chosen at random and that statistical estimation theory tells us to be 95% certain that if all the voters were polled, their disagreement percentage would be discovered to be within 3% of 74%. In other words, it's very unlikely that the 74% is off the mark by more than 3%; the truth is almost certainly between 71 and 77%. In subsequent sections of this chapter we introduce the strict interpretation of these confidence intervals.

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