Thus, for 82 infants, both tests make the correct...

Thus, for 82 infants, both tests make the correct classification, for 3 infants both tests make the wrong classification, for 5 infants the standard test makes the correct diagnosis while the new test does not, and for 10 infants the new test makes the correct diagnosis while the standard test does not. Suppose the investigators plan a larger study comparing the two tests. How many infants need to be tested to have an $80 \%$ probability of finding a significant difference using a two-sided test with $\alpha=0.05$ if the proportion of test results in the four groups in Table 10.44 are considered to be the true proportions?

Thus, for 82 infants, both tests make the correct classification, for 3 infants both tests make the wrong classification, for 5 infants the standard test makes the correct diagnosis while the new test does not, and for 10 infants the new test makes the correct diagnosis while the standard test does not.
Suppose the investigators plan a larger study comparing the two tests. How many infants need to be tested to have an $80 \%$ probability of finding a significant difference using a two-sided test with $\alpha=0.05$ if the proportion of test results in the four groups in Table 10.44 are considered to be the true proportions?

Key Concepts

Sample Size Calculation

Determining the required sample size in a paired diagnostic study involves using the observed proportions in the discordant cells along with the desired power (80%) and significance level (0.05). The calculation ensures that the study is adequately powered to detect a statistically significant difference between the diagnostic tests based on the expected effect size derived from the proportion of discordant results.

Discordant Pairs

In the context of comparing two diagnostic tests registered on the same subjects, discordant pairs are those for which the test outcomes differ. The number of discordant pairs is critical in testing the null hypothesis and influences the power of the study, because only these pairs provide information on the difference between the tests.

Statistical Power and Type I Error

Statistical power is the probability that the test will correctly reject a false null hypothesis (detect a true difference) and is typically set at 80% or higher, balancing the risk of Type II error. The significance level (alpha), such as 0.05 in a two-sided test, defines the acceptable probability of committing a Type I error, i.e., falsely declaring a difference when there is none.

Paired Data Analysis

When comparing diagnostic tests in the same subjects, observations are paired, meaning that each subject receives both tests. This design controls for between-subject variability and requires statistical methods that handle dependent data rather than treating the observations as independent, which ultimately increases the study’s efficiency.

McNemar's Test

McNemar's test is used to analyze paired dichotomous data, particularly for assessing differences in proportions where the interest lies in the discordant pairs—cases where one test gives a correct diagnosis and the other does not. This test specifically focuses on the change or disagreement between paired observations.

Transcript

00:01 So we had a 2002 poll that said 47 % of parents with children in school, k through 12, were satisfied with the education that the student was receiving.

00:10 And then a recent gallup poll found that 437 out of 1013, which is 43%, had said that they had quality of education.

00:21 And we want to know, does this suggest that the proportion has decreased? so we would assume that the proportion is still 0 .47 and alternately that the proportion is less than 0 .47.

00:35 And a asks, what does it mean to have a type 2 error? a type 2 error means we would fail to reject the null.

00:45 So we conclude that we don't have evidence lower, but the proportion is lower than 47%.

01:03 Part b says if a researcher decides to test the hypothesis at a 0 .01 significance level, we want to find the probability of making a type 2 error if the true proportion is 0 .42.

01:20 So the first thing we need to do is find where would we reject the null or fail to reject the null.

01:27 And this is a left tail test and we put all 0 .1 in that left tail.

01:33 And that corresponds with a z value of negative 1 .282.

01:39 So the z value or the proportion cutoff point, we take that 0 .47 and subtract away 1 .282 standard deviations.

01:52 And a standard deviation for this setting is that 0 .47 times 0 .53.

01:58 And then over that sample size that they use, which is that 1 ,013.

02:06 And so let's get that value, 0 .47 minus 1 .282 times the square root of 0 .47 times 0 .53.

02:19 And then divided by that 1 ,013.

02:22 And i have that cutoff point is 0 .4499.

02:28 So basically 0 .45.

02:31 So we need to find for the type 2 error to answer that question.

02:36 What is the probability of getting a p hat that is greater than 0 .4499 if the true proportion is 0 .42? and so we find that probability by converting it to a z value...

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