Question

Two methods for controlling traffic, \(A\) and \(B\), were used at each of \(n=12\) intersections for a period of 1 week, and the numbers of accidents that occurred during this time period were recorded. The order of use (which method would be employed for the first week) was randomly selected. You want to know whether the data provide sufficient evidence to indicate a difference in the distributions of accident rates for traffic control methods \(A\) and \(B\) $$ \begin{array}{ccc|ccccc} \hline & {\text { Method }} & & & {\text { Method }} \\ \text { Intersection } & \text { A } & \text { B } & \text { Intersection } & \text { A } & \text { B } \\ \hline 1 & 5 & 4 & 7 & 2 & 3 \\ 2 & 6 & 4 & 8 & 4 & 1 \\ 3 & 8 & 9 & 9 & 7 & 9 \\ 4 & 3 & 2 & 10 & 5 & 2 \\ 5 & 6 & 3 & 11 & 6 & 5 \\ 6 & 1 & 0 & 12 & 1 & 1 \\ \hline \end{array} $$ a. Analyze using a sign test. b. Analyze using the Wilcoxon signed-rank test for a paired experiment.

Short answer

Answer: The main differences between the sign test and the Wilcoxon signed-rank test are (1) in how they calculate the test statistic and (2) which tables are used to find the critical values. In the sign test, the test statistic is the smaller count of positive or negative differences, and a binomial distribution table is used to find the p-value. In contrast, the Wilcoxon signed-rank test calculates the test statistic as the sum of the ranks corresponding to positive differences minus the sum of the ranks corresponding to negative differences, and a Wilcoxon signed-rank test table is used to determine the critical value.

Step-by-step solution

Calculate the differences

For each intersection, calculate the difference in accidents between the two methods: \(D_i = A_i - B_i\).

Count the positive and negative differences

Count the number of positive differences (\(n_+\)) and negative differences (\(n_-\)). Ignore any intersections with a difference of 0, as these do not provide any information about the difference in methods.

Choose a significance level

Choose a significance level (\(\alpha\)), usually 0.05, to determine if there is a significant difference between the methods.

Compute the test statistic

Calculate the test statistic (\(S\)), which is the smaller of the counts from step 2: \(S=\min(n_+, n_-)\).

Determine the p-value

Using a binomial distribution table or calculator, find the p-value corresponding to the test statistic and the total number of intersections (ignoring those with a difference of 0).

Compare the p-value to the significance level

If the p-value is less than or equal to the significance level (\(\alpha\)), reject the null hypothesis and conclude there is a significant difference in the distributions of accident rates for the two methods. Otherwise, we cannot conclude that there is a significant difference. #b. Wilcoxon Signed-Rank Test#

Calculate the differences

For each intersection, calculate the difference in accidents between the two methods: \(D_i = A_i - B_i\).

Rank the absolute differences

Rank the absolute differences (\(|D_i|\)) in ascending order, ignoring any intersections with a difference of 0. Assign tied ranks the average of the ranks they span.

Calculate the test statistic

Calculate the test statistic (\(W\)) as the sum of the ranks corresponding to positive differences (\(W_+\)) minus the sum of the ranks corresponding to negative differences (\(W_-\)).

Choose a significance level

Choose a significance level (\(\alpha\)) for the test, usually 0.05.

Determine the critical value

Refer to a Wilcoxon signed-rank test table or calculator to determine the critical value for the given number of intersections (ignoring those with a difference of 0) and the chosen significance level.

Compare the test statistic to the critical value

If the test statistic (\(W\)) is less than or equal to the critical value, reject the null hypothesis and conclude there is a significant difference in the distributions of accident rates for the two methods. Otherwise, we cannot conclude that there is a significant difference.

Key concepts

Sign Test

The sign test is a simple non-parametric method used to analyze paired experimental data.
It is particularly useful when the data does not conform to the normal distribution assumptions of parametric tests. Instead of comparing means or medians of two groups, the sign test focuses on the direction of changes between paired observations.

In the context of the exercise, where we compare traffic control methods A and B across different intersections, the sign test would look at whether the number of accidents is consistently higher in one method or the other regardless of the magnitude of the differences.

To perform the sign test:

• Calculate the differences in accident rates between methods for each intersection.
• Count the signs of these differences (positive and negative).
• Ignore differences that are zero as they indicate no change.
• Use the total counts to determine the significance of one method over the other.

It's a simple but powerful test when dealing with non-parametric data, helping to identify whether a consistent trend exists between paired observations.

Wilcoxon Signed-Rank Test

The Wilcoxon signed-rank test is another non-parametric statistical method, enhancing the approach of the sign test by considering not only the direction of the difference but also the magnitude of that difference between paired observations.

This method works under the assumption that the differences for the pairs follow a symmetric distribution around a common median. It is more sensitive than the sign test because it factors in the rank of the difference.

For the traffic control methods A and B:

• Determine the differences for each intersection.
• Rank these differences based on their absolute values, ignoring zeros.
• Sum the ranks separately for positive and negative differences to calculate the test statistic.
• Compare the test statistic against the critical value from a Wilcoxon signed-rank test table to draw a conclusion about the significance of the observed differences.

Through this test, a more nuanced picture of the data emerges, considering both how much and how consistently one method outperforms the other.

Paired Experiment Analysis

Paired experiment analysis is a technique in statistics that involves comparing two measurements taken from the same or matched subjects.

This sort of experimental design controls for variability between subjects, isolating the effect of the treatment or condition being tested. It's widely used in before-and-after studies, cross-over trials, and, as in our exercise, comparing two methods at the same location.

Key points in analyzing paired experiments include:

• Calculating the difference between paired measurements.
• Applying appropriate statistical tests, such as the sign test or Wilcoxon signed-rank test.
• Interpreting the direction and magnitude of differences.

Paired experiment analysis is a robust approach to eliminating confounders and drawing accurate conclusions about the impact of an intervention.

Non-Parametric Statistical Methods

Non-parametric statistical methods are a collection of techniques that do not assume a specific probability distribution for the data.

They are particularly advantageous when dealing with skewed distributions, small sample sizes, ordinal data, or when the assumptions of parametric tests (like normality) are not met.

Examples of these methods include:

• The sign test, which ignores the size of the differences and focuses on their direction.
• The Wilcoxon signed-rank test, which takes into account the magnitude of differences using ranks.

Using non-parametric methods allows for flexible and robust analysis, making them vital tools in the statistics toolbox for handling various data types and experimental designs.