Question

Use the information given in Exercises \(8-9\) to calculate Spearman's rank correlation coefficient \(r_{s} .\) Do the data present sufficient evidence to indicate an association between variables \(A\) and \(B\) ? Use \(\alpha=.05 .\) $$\begin{array}{l|rrrrrr}\text { A } & 1.2 & .8 & 2.1 & 3.5 & 2.7 & 1.5 \\\\\hline \text { B } & 1.0 & 1.3 & .1 & -.8 & -.2 & .6\end{array}$$

Short answer

Answer: No, there is not enough evidence to indicate a significant association between variables A and B at the 0.05 significance level.

Step-by-step solution

1. Rank the values of A and B

First, we will rank the values in ascending order for both variables A and B. Assign a rank of \(1\) to the smallest value, \(2\) to the second smallest value, and so on. A ranks: \((2, 1, 3, 6, 5, 4)\) B ranks: \((4, 5, 2, 1, 3, 6)\)

2. Calculate the difference between the ranks

Next, we will find the difference between the ranks of each pair of corresponding values for A and B. Let's call the difference \(d_i\) for each pair \((i=1,2,...,6)\). \(d_i = (\text{rank of A}_i - \text{rank of B}_i)\) \(d = (-2, -4, 1, 5, 2, -2)\)

3. Square the differences and sum them up

Now, we will square each of the differences and calculate the sum of these squared differences. \(d^2 = (4, 16, 1, 25, 4, 4)\) \(\sum d^2 = 54\)

4. Calculate \(r_s\)

We can now calculate Spearman's rank correlation coefficient \(r_s\) using the formula: $$r_s = 1 - \frac{6 \sum d^2}{n(n^2-1)}$$ where \(n\) is the number of pairs. In our case, \(n=6\), so the formula becomes: $$r_s = 1 - \frac{6 \cdot 54}{6(6^2-1)}$$ $$r_s = 1 - \frac{324}{210} \approx 0.453$$

5. Test the association

Finally, we will test whether the data presents sufficient evidence to indicate an association between A and B using a \(0.05\) significance level. We can use the critical value for the Spearman's rank correlation. Looking up on a table for Spearman's rank correlation coefficients at \(n=6\) and significance level \(\alpha=0.05\), we find a critical value of approximately \(0.829\). Since the calculated \(r_s\) value of \(0.453\) is less than the critical value of \(0.829\), we conclude that there is not enough evidence to indicate a significant association between variables A and B at the \(.05\) significance level.

Key concepts

Correlation Analysis

Correlation analysis is a statistical method used to determine the strength and direction of the linear relationship between two variables. The goal is to find out how closely the changes in one variable are associated with changes in another.

In the given exercise, the Spearman's rank correlation coefficient (denoted as \( r_s \)) was used as a measure of correlation. This particular form of correlation analysis is useful when the data does not necessarily come from a normal distribution or when we deal with ordinal data—data that can be ranked but not necessarily measured on an interval scale.

It operates by assigning ranks to the data and then assessing how well the ranks between two variables correspond to each other. The Spearman's rank correlation coefficient is a nonparametric measure, meaning it does not assume that the data is normally distributed, making it quite versatile in application.

Nonparametric Statistics

Nonparametric statistics are a branch of statistics that is not reliant on data belonging to any particular distribution. This type of statistics is particularly useful when the data does not meet the assumptions of parametric tests, for instance, if the data is not normally distributed or if the sample size is too small to determine the distribution.

Spearman's rank correlation is a notable nonparametric statistic because it is based on the ranks of the data rather than the actual data points, which circumvents the need for the data to fit a normal distribution. Furthermore, nonparametric methods are often more robust to outliers, as they rely on the relative ordering of values rather than their specific quantities.

In practice, nonparametric methods can be advantageous when dealing with real-world data that may not adhere to the strict assumptions required for parametric testing, ensuring more reliable and valid results under a wider range of circumstances.

Statistical Significance

Statistical significance is a fundamental concept in hypothesis testing that helps researchers determine whether the results observed in their data are likely due to chance or if they reflect actual effects or associations. The significance level, denoted as \( \alpha \), is a threshold set by researchers, below which a p-value suggests that the observed results would be very unlikely under the null hypothesis. A common choice for \( \alpha \) is 0.05, meaning there is a 5% risk of concluding that an effect or an association exists when there is none.

In the context of Spearman's rank correlation coefficient, the statistical significance of the observed value of \( r_s \) is determined by comparing it to a critical value from a reference table based on the chosen \( \alpha \) level and the sample size. If the observed \( r_s \) is greater than the critical value, the association is considered to be statistically significant. Otherwise, as in the exercise, if the calculated \( r_s \) is below the critical value, there is insufficient evidence to conclude a significant association.