Refer to Example 20.2. Suppose that the responses were coded...

Refer to Example 20.2. Suppose that the responses were coded as follows: $100=$ The drug was extremely effective. $60=$ The drug was quite effective. $40=$ The drug was somewhat effective. $35=$ The drug was slightly effective. $10=$ The drug was not at all effective. Determine whether we can infer that the new painkiller is more effective than aspirin. b Why are the results of Example 20.2 and part (a) identical? Sample statistics: $$ T_1=276.5, n_1=15, T_2=188.5, n_2=15 . $$

Refer to Example 20.2. Suppose that the responses were coded as follows: $100=$ The drug was extremely effective.
$60=$ The drug was quite effective.
$40=$ The drug was somewhat effective.
$35=$ The drug was slightly effective.
$10=$ The drug was not at all effective.
Determine whether we can infer that the new painkiller is more effective than aspirin.
b Why are the results of Example 20.2 and part (a) identical?

Sample statistics:
$$
T_1=276.5, n_1=15, T_2=188.5, n_2=15 .
$$

Key Concepts

Monotonic Transformation Invariance

Monotonic transformation invariance implies that if the data are transformed by any monotonic function (one that preserves order), the relative rankings among data points remain unchanged. Consequently, nonparametric tests that rely solely on data ranks yield identical results regardless of such coding or scaling changes, which is why the results in the two parts of the example are identical.

Rank Sum Test

The rank sum test (or Mann-Whitney U test) is a nonparametric test used to compare two independent groups. It works by ranking all data points together and then comparing the sum of the ranks between groups. This test is particularly appropriate when sample data are ordinal or when the underlying assumptions of parametric tests are not met.

Hypothesis Testing

Hypothesis testing in the context of comparing treatments involves formulating a null hypothesis (usually that there is no difference between groups) and an alternative hypothesis (indicating, for example, that one treatment is more effective than the other). Nonparametric tests such as the rank sum test are used to statistically assess the likelihood of the observed differences in the context of these hypotheses.

Nonparametric Methods

Nonparametric methods allow us to perform statistical inference without relying on strict distributional assumptions, such as normality. These methods typically use the order or rank of the data rather than the actual numerical values, making them particularly useful when dealing with ordinal data or data measured on scales where intervals are not equally spaced.

Ordinal Data

Ordinal data refers to measurements that indicate a ranking or order among observations but do not necessarily imply equal intervals between the ranks. When responses are coded with numerical scores in ordinal data, the numbers merely represent relative positions rather than true magnitudes, meaning that changes or transformations that preserve the order do not alter the underlying information.

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