Question

A political scientist wished to examine the relationship between the voter image of a conservative political candidate and the distance (in miles) between the residences of the voter and the candidate. Each of 12 voters rated the candidate on a scale of 1 to \(20 .\) a. Calculate Spearman's rank correlation coefficient \(r_{s}\) b. Do these data provide sufficient evidence to indicate a negative correlation between rating and distance?

Short answer

Answer: After calculating and interpreting the Spearman's rank correlation coefficient (rs), we can determine if there is a negative correlation between voters' rating of a candidate and the distance between their residences. If rs is close to -1, it indicates a strong negative correlation, suggesting that as the distance between residences increases, the voters' rating of a candidate decreases, and vice versa.

Step-by-step solution

Calculate the rank of each variable

First, we need to rank the data for both variables. Assign a rank to each data point within each variable, with the lowest value receiving a rank of 1 and the highest value receiving the highest rank.

Calculate the difference between the ranks \(d_{i}\)

Now, for each pair of ranks \((R_{i}, D_{i})\), calculate the difference \(d_{i} = R_{i} - D_{i}\).

Calculate the squared difference \(d_{i}^2\)

Next, square each difference \(d_{i}^2\).

Calculate Spearman's rank correlation coefficient (\(r_{s}\))

Now, we can calculate Spearman's rank correlation coefficient using the following formula: \(r_{s} = 1 - \frac{6\sum d_{i}^2}{n(n^2 - 1)}\) Where: \(n\) = number of pairs of ranks (in this case, \(n=12\)) \(\sum d_{i}^2\) = sum of the squared differences After calculating \(r_{s}\), we'll have a value between -1 and 1, where a value close to -1 or 1 indicates a strong negative or positive correlation, respectively.

Interpret the result and determine if there is a negative correlation

Using the value of \(r_{s}\), we can interpret if there is sufficient evidence to indicate a negative correlation between rating and distance. a. If \(r_{s}\) is close to -1, we can say that there is a strong negative correlation between rating and distance. b. If \(r_{s}\) is close to 0, we can say that there is no significant relationship between rating and distance. c. If \(r_{s}\) is close to 1, we can say that there is a strong positive correlation between rating and distance. After interpreting the result, we can answer the question of whether the data provides sufficient evidence for a negative correlation between rating and distance.

Key concepts

Rank Correlation

Rank correlation is a fundamental concept used in statistics to evaluate the strength and direction of a relationship between two ranked variables. Unlike traditional correlation measures, rank correlation does not rely on the actual values of data points, but rather on their rankings. This method is particularly useful when the data do not meet the assumptions of linearity and homoscedasticity required for Pearson’s correlation.

One common measure of rank correlation is Spearman’s rank correlation coefficient, denoted as \( r_s \). This coefficient considers the difference between ranks assigned to the data points of two variables, where each variable is sorted independently. The formula for Spearman's \( r_s \) is:

• Calculate the rank of each item in the datasets.
• Find the difference between the ranks for paired items.
• Square these differences to avoid negative values causing cancellation.
• Substitute the just-calculated squared differences into the rank correlation formula.

A value of \( r_s \) near 1 or -1 suggests a high positive or negative rank correlation, respectively, while a value close to 0 indicates weak or no correlation.

Negative Correlation

Negative correlation refers to a relationship where an increase in one variable is associated with a decrease in another. In the context of rank correlation, a negative correlation is suggested if the rank of one variable consistently decreases as the rank of another variable increases.

In statistical analysis, a negative Spearman's rank correlation coefficient (\( r_s \)) close to -1 indicates a strong inverse relationship between two variables. This means that as one variable goes up, the other tends to go down.

For example, in the exercise about the voter image of a candidate, if the results show a negative \( r_s \), it suggests that the further away a voter resides, the lower their rating of the candidate might be. Observing negative correlation helps researchers and analysts predict and understand inverse behavioral trends, encouraging more informed decision-making.

Statistical Analysis

Statistical analysis is a crucial step in interpreting data in research studies. It involves the use of statistical tools and techniques to identify patterns, relationships, and trends within a data set. In the exercise involving Spearman's rank correlation coefficient, the statistical analysis aims to determine the strength and direction of the relationship between two variables: the voter's rating and their proximity to the candidate.

Such analysis includes several steps:

• Data preparation: Assign ranks to the data points to align with the assumptions of rank-based analysis.
• Calculations: Carry out the calculations for differences and squared differences between ranks to use in the \( r_s \) formula.
• Interpretation: Assess and interpret the resulting coefficient to determine whether there is an evidence-backed correlation.

Statistical analysis through rank correlation methods can be crucial in fields such as political science, where understanding complex human behaviors and preferences is necessary. Proper interpretation of results guides educators, policymakers, and researchers in making predictions and drawing conclusions from their data assessments.