If you were testing people for the presence of COVID-19 antibodies, which error would you be most concerned with and why?

Responses

2 pages discussion

After this write 2 reply on others discussion

about 100 words for each reply.

First Part Write 1 page

Prompt:

Find some examples of the misuses of p-values that have been reported in the popular literature.

If you were testing people for the presence of COVID-19 antibodies, which error would you be most concerned with and why?

Second Part Write 1 page

Urban Transportation options

Urban travelers usually have several options for taking trips, especially when commuting to work.

(You are a student live in Los Angeles or San Francisco)

Lesson for first part

If there is any one single area that statistics gets a bad rep it’s hypothesis testing. The issue isn’t the underlying theory, it is the use to which it is put and how sometimes more is made of it than the theory permits. The issues surrounding hypothesis testing and statistical significance include: 1) statistical significance is not the same as practical importance; 2) dichotomizing into significant and nonsignificant results encourages the dismissal of observed differences in favour of the usually less interesting null hypothesis of no difference; 3) and any particular threshold for setting significance is arbitrary. The problem is so large that the American Statistical Association, the US’s society for professional statisticians, found it necessary to issue a statement about the use of p-values. I’ve included in this Module’s Assets folder the ASA’s statement and a relevant paper. I urge you to read them. (I’ve also included three cartoons by Randall Munroe whose website https://xkcd.com/ is almost always amusing and insightful about an amazing variety of topics.)

We begin by setting up the hypothesis testing framework. Hypothesis testing requires a null and an alternative hypothesis. The null hypothesis is often treated as if it is the maintained hypothesis, that is, it is the presumed state of the world prior to having data. The data will inform us whether it is consistent or not with that state of the world. The alternative hypothesis is often taken to be that the state of the world is not as implied by the null hypothesis. The question immediately arises: In what way is that so? The alternative hypothesis presumably embodies the view of the world that rejecting the null hypothesis implies. So, the null and alternative hypotheses need to be viewed as a disjoint pair of sets, with the alternative hypothesis the complement of the parameter space of the null hypothesis. There are a variety of ways to characterize those disjoint sets – H0 >= value, Ha<value, H0=value, Ha n.e. value. (BTW, I generally prefer naming the alternative hypothesis as Ha rather than H1 as a kind of mnemonic for its meaning.) We need to be careful in our choice to make sure that they truly embody the hypotheses in as precise a way as we can. Corresponding to these two sets is a critical region, that is, a set, which, if the random sample should fall in that region, you would reject the null hypothesis in favour of the alternative. Another set that corresponds to the hypotheses is the rejection region. Usually, we have some test statistic which is a function of the data. The test procedure is to determine whether this test statistic is an element of the rejection region, then we reject the null hypothesis in favour of the alternative. When this is is the test procedure, then the critical region and the rejection region coincide.

We then discuss the power function which gives the conditional probability of the sample being an element of the critical region or, equivalently, the test statistic an element of the rejection region given the parameter of interest. In essence, the power function gives the probability you will reject the null hypothesis for given values of the parameter of interest.

This leads us to the kinds of errors we can make in testing hypotheses. Rejecting a true null hypothesis erroneously is called a Type I error. Failing to reject a false null hypothesis is a Type II error. Generally, researchers would like to make the Type I error small, corresponding to the power function having values near zero when the true parameter values are in the region associated with the null hypothesis. Conversely, researchers would like the Type II error small also, corresponding to the power function having values near 1 when the true parameters are in the region associated with the alternative hypothesis. Unfortunately, these two work against one another, that is, there is a trade between the two types of error. Usually, researchers will choose a small value of Type I error and will compromise setting the Type II error somewhat higher. A typical Type II error value is 0.20.

Next, we discuss p-values. Again, I strongly recommend you read the papers in the Assets folder. The p-value is the smallest value at which we reject the null hypothesis given the observed data. We briefly discuss the likelihood ratio test due to Wald. It is a commonly used test in economics but requires an appeal to large-sample properties.

We then discuss simple and two-sided hypotheses, with our concentration on the latter since they are most common in economics. This is followed by the application of the t-test, a test for means, both for a single sample and for two samples.

Finally, we discuss the F-test and its use in testing variances. I’ve also included in our readings for this chapter, section 9.9 because it discusses some of the foundational issues of hypothesis testing.

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