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Chapter 11: Problem 2

Consider the Spearman rank correlation coefficient \(r_{s}\) for data pairs \((x, y) .\) What is the monotone relationship, if any, between \(x\) and \(y\) implied by a value of (a) \(r_{s}=0 ?\) (b) \(r_{s}\) close to \(1 ?\) (c) \(r_{s}\) close to \(-1 ?\)

Short Answer

Expert verified
(a) No monotonic relationship; (b) strong positive; (c) strong negative.

Step by step solution

01

Understanding Spearman's Rank Correlation Coefficient

The Spearman rank correlation coefficient, denoted as \(r_{s}\), measures the strength and direction of a monotonic relationship between two ranked variables. The value of \(r_{s}\) ranges from \(-1\) to \(1\).
02

Evaluating \(r_{s} = 0\)

When \(r_{s} = 0\), it indicates that there is no monotonic relationship between the ranks of \(x\) and \(y\) variables. This means that as \(x\) increases or decreases, \(y\) does not consistently increase or decrease.
03

Evaluating \(r_{s}\) close to \(1\)

When \(r_{s}\) is close to \(1\), it implies a strong positive monotonic relationship between \(x\) and \(y\). This means that as \(x\) increases, \(y\) tends to increase as well, and the ranks of the data points are similar.
04

Evaluating \(r_{s}\) close to \(-1\)

When \(r_{s}\) is close to \(-1\), it implies a strong negative monotonic relationship between \(x\) and \(y\). This means that as \(x\) increases, \(y\) tends to decrease, and the ranks of the data points are inversely related.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Monotonic Relationship
A monotonic relationship in statistics occurs when two variables change in a consistent direction, but not necessarily at a constant rate. This concept is central to understanding the Spearman rank correlation coefficient. In simple terms, if two variables are related in a monotonic fashion, then either increasing one variable consistently leads to increasing or decreasing the other. However, it's important to note that this doesn't mean the changes are predictable or uniform like in a linear relationship. Instead, a monotonic relationship simply means the direction of change is consistent.
  • Types: These relationships can be either increasing, where one variable ascends as the other ascends (positive monotonic), or decreasing, where one variable descends as the other ascends (negative monotonic).
  • Implications: A monotonic relationship helps in predicting the behavior of one variable based on the other, though with less precision compared to linear relationships.
Ranked Variables
When we talk about ranked variables in the context of the Spearman rank correlation, we refer to data that has been ordered from highest to lowest or vice versa. Ranking is used to simplify complex data by organizing it in a way that highlights relationships between the variables.
  • Ranking Process: To rank data, each value of the variables is replaced with its respective rank. If two or more values are identical, they are given an average rank.
  • Why Rank? Spearman's rank correlation coefficient relies on ranks to assess the strength and direction of a monotonic relationship. This makes it ideal for non-linear data.
  • Applications: Rankings can be particularly useful when data does not meet the assumptions required for parametric tests, such as linearity or homoscedasticity.
Positive Correlation
In statistics, a positive correlation indicates that two variables tend to move in the same direction. Within the context of Spearman rank correlation, a positive correlation would be represented by a value of the correlation coefficient, denoted by \(r_{s}\), that approaches 1.
  • Description: As one variable increases, the other also tends to increase. This is exemplified by variables that follow similar ranks in ascending order.
  • Effect on Data: In data sets with a high positive Spearman rank correlation, you can expect the ranks of one variable to mirror those of the other.
  • Example: Consider height and weight; generally, taller individuals tend to weigh more, reflecting a positive correlation.
Negative Correlation
A negative correlation describes a relationship where two variables move in opposite directions. In terms of Spearman rank correlation, a negative relationship would be reflected by an \(r_{s}\) value that nears -1.
  • Characteristics: As one variable increases, the other tends to decrease, resulting in inversely related ranks.
  • Impact on Outcomes: With strong negative correlation, rank increases in one variable will accompany rank decreases in the other.
  • Example: Take, for example, the relationship between elevation and temperature; as elevation increases, temperatures typically decrease, showing a negative correlation.

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Most popular questions from this chapter

To apply the sign test, do you need independent or dependent (matched pair) data?

Is the high school dropout rate higher for males or females? A random sample of population regions gave the following information about percentage of 15 - to 19 -year-olds who are high school dropouts (Reference: Statistical Abstract of the United States, 121 st Edition). $$ \begin{array}{l|cccccccccc} \hline \text { Region } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \text { Male } & 7.3 & 7.5 & 7.7 & 21.8 & 4.2 & 12.2 & 3.5 & 4.2 & 8.0 & 9.7 \\ \hline \text { Female } & 7.5 & 6.4 & 6.0 & 20.0 & 2.6 & 5.2 & 3.1 & 4.9 & 12.1 & 10.8 \\ \hline \end{array} $$ $$ \begin{array}{l|cccccccccc} \hline \text { Region } & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\ \hline \text { Male } & 14.1 & 3.6 & 3.6 & 4.0 & 5.2 & 6.9 & 15.6 & 6.3 & 8.0 & 6.5 \\ \hline \text { Female } & 15.6 & 6.3 & 4.0 & 3.9 & 9.8 & 9.8 & 12.0 & 3.3 & 7.1 & 8.2 \\ \hline \end{array} $$ Does this information indicate that the dropout rates for males and females are different (either way)? Use \(\alpha=0.01\).

Are yields for organic farming different from conventional farming yields? Independent random samples from method A (organic farming) and method B (conventional farming) gave the following information about yield of lima beans (in tons/acre) (Reference: Agricultural Statistics, U.S. Department of Agriculture). $$ \begin{array}{l|llllllllllll} \hline \text { Method A } & 1.83 & 2.34 & 1.61 & 1.99 & 1.78 & 2.01 & 2.12 & 1.15 & 1.41 & 1.95 & 1.25 & \\ \hline \text { Method B } & 2.15 & 2.17 & 2.11 & 1.89 & 1.34 & 1.88 & 1.96 & 1.10 & 1.75 & 1.80 & 1.53 & 2.21 \\ \hline \end{array} $$ Use a \(5 \%\) level of significance to test the hypothesis that there is no difference between the yield distributions.

Is there a relation between incidents of child abuse and number of runaway children? A random sample of 15 cities (over 10,000 population) gave the following information about the number of reported incidents of child abuse and the number of runaway children. $$ \begin{array}{l|rrrrrrrrrrrrrrr} \hline \text { City } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline \text { Abuse cases } & 49 & 74 & 87 & 10 & 26 & 119 & 35 & 13 & 89 & 45 & 53 & 22 & 65 & 38 & 29 \\ \text { Runaways } & 382 & 510 & 581 & 163 & 210 & 791 & 275 & 153 & 491 & 351 & 402 & 209 & 410 & 312 & 210 \\ \hline \end{array} $$ (i) Rank-order abuse using 1 as the largest data value. Also rank-order runaways using 1 as the largest data value. Then construct a table of ranks to be used for a Spearman rank correlation test. (ii) Use a \(1 \%\) level of significance to test the claim that there is a monotone increasing relationship between the ranks of incidents of abuse and number of runaway children.

A data-processing company has a training program for new salespeople. After completing the training program, each trainee is ranked by his or her instructor. After a year of sales, the same class of trainees is again ranked by a company supervisor according to net value of the contracts they have acquired for the company. The results for a random sample of 11 salespeople trained in the previous year follow, where \(x\) is rank in training class and \(y\) is rank in sales after 1 year. Lower ranks mean higher standing in class and higher net sales. $$ \begin{array}{l|rrrrrrrrrrr} \hline \text { Person } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ \hline x \text { rank } & 6 & 8 & 11 & 2 & 5 & 7 & 3 & 9 & 1 & 10 & 4 \\ y \text { rank } & 4 & 9 & 10 & 1 & 6 & 7 & 8 & 11 & 3 & 5 & 2 \\ \hline \end{array} $$ Using a \(0.05\) level of significance, test the claim that the relation between \(x\) and \(y\) is monotone (either increasing or decreasing).
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