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 \(\alpha=.05 .\) $$\begin{array}{l|rrrrrr}\text { A } & 2.6 & .8 & 2.1 & 3.5 & 2.6 & 1.5 \\\\\hline \text { B } & 1.0 & 1.3 & 1.0 & -.8 & 1.2 & -.6\end{array}$$

Short answer

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

Step-by-step solution

Rank the values

We need to assign ranks for each value in variables A and B: Variable A ranks: 3.5, 2, 3, 6, 3.5, 1 Variable B ranks: 4, 5, 4, 1, 6, 2

Calculate difference in ranks

Next, we'll calculate the difference in ranks (d) for each pair of values: \(d_i = R(A_i) - R(B_i) = [3.5-4, 2-5, 3-4, 6-1, 3.5-6, 1-2] = [-0.5, -3, -1, 5, -2.5, -1]\)

Square the differences

Now, we'll square each of the differences \(d_i\): \(d_i^{2} = [(-0.5)^2, (-3)^2, (-1)^2, (5)^2, (-2.5)^2, (-1)^2] = [0.25, 9, 1, 25, 6.25, 1]\)

Calculate the sum of squared differences

Adding up the squared differences: \(\sum d_i^{2} = 0.25 + 9 + 1 + 25 + 6.25 + 1 = 42.5\)

Compute \(r_{s}\)

Using the Spearman's rank correlation coefficient formula: \(r_{s} = 1 - \frac{6 \sum d_i^2}{n (n^2 - 1)} = 1 - \frac{6 \times 42.5}{6(6^2 - 1)} = 1 - \frac{255}{210} = -0.2143\)

Perform a hypothesis test

We want to determine if there's sufficient evidence of an association between variables A and B, given \(\alpha = 0.05\). The null hypothesis states that there is no association between variables A and B,(\(H_0: \rho = 0\)), while the alternative hypothesis states that there is an association between variables A and B (\(H_1: \rho \neq 0\)). Using a critical value table, we find the critical value for the Spearman's rank correlation at two-tailed \(\alpha = 0.05\) and \(n = 6\) is \(p_{crit} = 0.886.\) Since our calculated \(r_s=-0.2143\) lies between \(-0.886\) and \(0.886\), we fail to reject the null hypothesis (\(|r_{s}| < |p_{crit}|\)). In conclusion, there is not enough evidence to indicate an association between variables A and B at a significance level of \(\alpha = 0.05\).

Key concepts

Hypothesis Testing

Hypothesis testing is a fundamental procedure in statistics that allows researchers to make inferences about populations based on sample data. It starts with the establishment of a null hypothesis (\( H_0 \)), which is a statement of no effect or no difference, and an alternative hypothesis (\( H_1 \) or \( H_a \)), which is what the researcher aims to support.

In the context of Spearman's rank correlation coefficient, the null hypothesis typically states that there is no monotonic relationship between the two ranked variables. The alternative hypothesis suggests there is a significant monotonic relationship. The critical value from tables, or a p-value calculated from a distribution, determines whether the null hypothesis can be rejected in favor of the alternative hypothesis based on the chosen significance level, often \( \alpha = 0.05 \).

If the test statistic falls into the region defined by the critical value, it suggests that the result is not due to random chance, and the null hypothesis is rejected. If not, like in the exercise solution provided, we fail to reject the null hypothesis, meaning there is not enough evidence to conclude a significant association between the ranked variables.

Statistical Significance

Statistical significance is a determination by an analyst that the results in the data are not explainable by chance alone. It is a critical concept in hypothesis testing, helping to ascertain whether the observed differences or relationships in data are real or simply due to random variation.

A significance level (\( \alpha \)), usually set at 0.05, indicates the threshold for deciding when to reject the null hypothesis. A result is considered statistically significant if the observed test statistic is in the critical region, or if a p-value is less than \( \alpha \). In the given exercise solution, the Spearman's rank correlation coefficient calculated did not meet the threshold required to reject the null hypothesis, meaning the findings are not statistically significant, and any observed association could very well be due to chance.

Ranked Data Analysis

Ranked data analysis involves procedures that apply to data which can be ordered or ranked. It is particularly useful when dealing with non-parametric data or when the assumptions necessary for parametric tests cannot be met. Spearman's rank correlation coefficient, \( r_{s} \), is a non-parametric measure of rank correlation, meaning it assesses the strength and direction of the association between two ranked variables.

To analyze ranked data, each item is replaced by its rank. Differences between the ranks of paired observations are used to compute the test statistic. This approach lessens the impact of outliers and is less sensitive to the exact numerical values. Consequently, it provides a measure of association that is based more on the order of data points rather than their exact numerical value, which was demonstrated in the step-by-step solution, where the ranks of variables A and B were the focal point of the calculation.