Question

Two art critics each ranked 10 paintings by contemporary (but anonymous) artists according to their appeal to the respective critics. The ratings are shown in the table. Do the critics seem to agree on their ratings of contemporary art? That is, do the data provide sufficient evidence to indicate a positive association between critics \(A\) and \(B ?\) Test by using an \(\alpha\) value near. \(05 .\) $$\begin{array}{ccc}\hline \text { Painting } & \text { Critic } \mathrm{A} & \text { Critic B } \\\\\hline 1 & 6 & 5 \\\2 & 4 & 6 \\\3 & 9 & 10 \\\4 & 1 & 2 \\\5 & 2 & 3 \\\6 & 7 & 8 \\\7 & 3 & 1\end{array}$$ $$\begin{array}{ccc}\hline \text { Painting } & \text { Critic A } & \text { Critic B } \\\\\hline 8 & 8 & 7 \\\9 & 5 & 4 \\\10 & 10 & 9 \\\\\hline\end{array}$$

Short answer

Provide the calculated Spearman's rank correlation coefficient and explain your conclusion. Answer: Yes, there is a positive association between the rankings of Critic A and Critic B. The calculated Spearman's rank correlation coefficient is approximately 0.903, which is close to 1, indicating strong evidence of a positive association. Based on the hypothesis test, we reject the null hypothesis in favor of the alternative hypothesis, which states that there is a positive association between the rankings of the two critics.

Step-by-step solution

Compute each critic's ranking of the paintings

First, write down the ranking that each critic assigns to each painting and create a table with paintings, Critics A rankings, and Critics B rankings. $$\begin{array}{cccc}\hline \text { Painting } & \text { Critic } \mathrm{A} & \text { Critic B } & \text { Diff (A-B) } \\\\\hline 1 & 6 & 5 & 1 \\\2 & 4 & 6 & -2 \\\3 & 9 & 10 & -1 \\\4 & 1 & 2 & -1 \\\5 & 2 & 3 & -1 \\\6 & 7 & 8 & -1 \\\7 & 3 & 1 & 2 \\ 8 & 8 & 7 & 1 \\ 9 & 5 & 4 &1 \\ 10 & 10 & 9 &1 \end{array}$$

Calculate the difference in ranks

For each painting, compute the difference between Critic A's rank and Critic B's rank. Add this data to the table in a new column labelled "Diff (A-B)".

Calculate the square of the differences

Square the difference in rankings for each painting. Add these values to the table in a new column labelled "Diff^2". $$\begin{array}{ccccc}\hline \text { Painting } & \text { Critic } \mathrm{A} & \text { Critic B } & \text { Diff (A-B) } & \text{Diff^2}\\\\\hline 1 & 6 & 5 & 1 & 1 \\\2 & 4 & 6 & -2 & 4 \\\3 & 9 & 10 & -1& 1 \\\4 & 1 & 2 & -1 & 1 \\\5 & 2 & 3 & -1 & 1 \\\6 & 7 & 8 & -1 & 1 \\\7 & 3 & 1 & 2 & 4 \\ 8 & 8 & 7 & 1 & 1 \\ 9 & 5 & 4 &1 & 1\\ 10 & 10 & 9 &1 & 1 \end{array}$$

Calculate the sum of squared differences

Add up all the squared differences to get the sum of squared differences. In this case, the sum is \(1 + 4 + 1 + 1 + 1 + 1 + 4 + 1 + 1 + 1 = 16\).

Compute the Spearman's rank correlation coefficient

Calculate the Spearman's rank correlation coefficient using the formula: $$r_s = 1 - \frac{6 \sum d^2}{n(n^2 - 1)}$$ where \(r_s\) is the correlation coefficient, \(d\) is the difference between the two ranks, and \(n\) is the number of data points. $$r_s = 1 - \frac{6(16)}{10(10^2 - 1)} = 1 - \frac{96}{990} \approx 0.903$$

Perform a hypothesis test

We now perform a hypothesis test with a significance level (\(\alpha\)) of 0.05. Null hypothesis (\(H_0\)): There is no association between the rankings of Critic A and Critic B (i.e., \(r_s = 0\)). Alternative hypothesis (\(H_a\)): There is a positive association between the rankings of Critic A and Critic B (i.e., \(r_s > 0\)). Since \(r_s \approx 0.903\), we have strong evidence that there is a positive association between the rankings of Critic A and Critic B. Given that \(\alpha = 0.05\) and our calculated \(r_s\) value is much larger than 0, it is highly unlikely that the null hypothesis is true. We reject the null hypothesis in favor of the alternative hypothesis, which states that there is a positive association between the rankings of Critic A and Critic B.

Key concepts

Hypothesis Test

When attempting to establish a relationship between two ranked datasets, statisticians use the hypothesis test to determine if any observed association is not due to chance. A hypothesis test evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data. In the context of Spearman's rank correlation coefficient, the null hypothesis (\(H_0\)) typically states there is no correlation between the ranks. Conversely, the alternative hypothesis (\(H_a\) or \(H_1\)) suggests there is a statistically significant correlation.

During the hypothesis testing, we also set a significance level (\(\alpha\)), which is the probability of rejecting the null hypothesis when it is true—also known as a Type I error. Commonly, \(\alpha\) is set at 0.05, meaning there is a 5% risk of concluding there is an effect when there is none. If the calculated Spearman's correlation coefficient is extreme enough (e.g., very close to 1 or -1), it can lead us to reject the null hypothesis in favor of the alternative hypothesis, which suggests a correlation exists beyond what we would expect by chance alone.

Ranked Data Analysis

Ranked data analysis involves organizing data points according to their relative standing to each other, rather than their numerical value. This approach is particularly helpful when dealing with ordinal data or when the precise differences between data points are not as relevant as their order. In the exercise provided, the ranked data from the art critics is analyzed to determine the Spearman's rank correlation coefficient, which provides insight into the degree and direction of association between the two critics' rankings.

In order to calculate Spearman's coefficient, differences between the ranks assigned by each critic to each item are squared and aggregated. This sum of squares is then plugged into a formula that accounts for the number of data points. The resulting coefficient (\(r_s\)) ranges between +1 and -1, where +1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no correlation. The method is robust to outliers because it relies on ranks rather than raw data.

Statistical Significance

Statistical significance is a determination about the likelihood that the observed relationship or difference in an analysis is caused by something other than random chance. It provides a quantifiable threshold to help us decide whether to accept or reject the null hypothesis in a hypothesis test. The level of statistical significance is typically represented by the alpha (\(\alpha\)) level, which is chosen by the researcher before conducting the test.

After calculating a test statistic, in this case, Spearman's correlation coefficient, researchers compare it to critical values or a p-value. If the test statistic falls into the critical region or the p-value is less than the pre-determined alpha level, the results are deemed statistically significant. This means that the effect or association being measured is probably not due to random variation and that the null hypothesis can be rejected. Therefore, a statistically significant result in Spearman's rank correlation coefficient analysis strengthens the researcher's confidence in a real association between the ranked variables.