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Chapter 3: Problem 20

Describe a situation in which it is inappropriate to use the correlation to measure the association between two quantitative variables.

Short Answer

Expert verified
Correlation is inappropriate for non-linear relationships, datasets with outliers, unequal variance, categorical variables, or when a third variable affects both primary variables.

Step by step solution

01

Understanding Correlation

The correlation coefficient measures the strength and direction of a linear relationship between two quantitative variables. The value ranges from -1 to 1, where 1 indicates a perfect positive linear relationship, -1 a perfect negative linear relationship, and 0 no linear relationship.
02

Identify Non-linear Relationships

In cases where the relationship between two variables is non-linear, meaning the pattern does not resemble a straight line, using the correlation coefficient is inappropriate. For example, if the data forms a U-shape or a curve, the correlation may not accurately reflect the strength or nature of the association.
03

Check for Outliers

Outliers, or extreme values, may significantly distort the correlation. In a dataset where a few data points are much higher or lower than the rest, the correlation may appear stronger or weaker than the true underlying relationship.
04

Unequal Variance

When the variability of one or both variables changes across different levels, correlation might not be suitable. This is often seen when scatter plots show a funnel shape, with the spread of one variable increasing or decreasing across the range of the other variable.
05

Categorical Variables

Correlation is meant for quantitative variables, so it cannot be used to measure associations when one or both variables are categorical, even if they are numerically coded (e.g., assigning numbers to categories like 1 for red, 2 for blue).
06

Consideration of Influence of a Third Variable

The presence of a third variable that may affect both variables being measured can lead to misleading correlation results. This is known as a confounding variable, where the correlation may not truly reflect the direct relationship between the two primary variables.

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Key Concepts

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

Non-linear Relationships
When examining the relationship between two variables, it's crucial to understand the nature of their connection. Not all relationships are linear, where one variable increases or decreases in a straight line as the other variable changes. Non-linear relationships are those where the association between variables bends or curves. Here are some key characteristics:
  • Such relationships might form a U-shape, curve, or even an S-shape.
  • Non-linear relationships are poorly represented by correlation coefficients, as these coefficients capture linear tendencies only.
For example, consider the growth of a bacterial population over time. Initially, growth may be rapid, then level off as resources become limited. Plotting this would show a curve rather than a straight line, making correlation a poor measure of association.
Outliers Effect
Outliers are extreme values in data that differ significantly from other observations. Their presence can significantly impact the measurement of a correlation. Here's how they influence the analysis:
  • Outliers can skew the correlation coefficient, suggesting a stronger or weaker relationship than actually exists.
  • They can create a false impression of a linear relationship where there might be none.
Imagine analyzing the relationship between study time and test scores. If most students score between 70-90 with consistent study time, but a few scores are much lower or higher due to unusual circumstances, these outliers could distort the perceived relationship.
Quantitative vs Categorical Variables
Correlation is a statistical measure designed to assess the association between two quantitative variables. This means both variables should be numeric and not categorical. To understand why this is important:
  • Quantitative variables are measurable and can be ordered or sequenced, like height or temperature.
  • Categorical variables represent groups or categories, like colors or brands, which do not have a natural order.
When one or both variables are categorical, using correlation is inappropriate because the categories do not provide a continuous range of values needed for correlation calculations.
Confounding Variables
A confounding variable is a third factor that can create a misleading association between two other variables. It "confounds" the relationship by being related to both variables being studied. Here's why it's problematic:
  • Confounding variables can make it appear that two variables are directly related when they might not be.
  • This can mislead conclusions drawn from the correlation between the primary variables.
For instance, consider a study on exercise frequency and health levels. Income level could be a confounding variable if it influences both a person's access to fitness facilities and healthcare, thus impacting the perceived relationship between exercise and health.

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Most popular questions from this chapter

Wage bill of Premier League Clubs Data of the Premier League Clubs' wage bills was obtained from www.tsmplug .com. For the response variable \(y=\) wage bill in millions of pounds in 2014 and the explanatory variable \(x=\) wage bill in millions of pounds in \(2013, \hat{y}=-1.537+1.056 x\). a. How much do you predict the value of a club's wage bill to be in 2014 if in 2013 the club had a wage bill of (i) \(拢 100\) million, (ii) \(拢 200\) million? b. Using the results in part a, explain how to interpret the slope. c. Is the correlation between these variables positive or negative? Why? d. A Premier League club had a wage bill of \(拢 100\) million in 2013 and \(拢 105\) million in \(2014 .\) Find the residual and interpret it.

NL baseball Example 9 related \(y=\) team scoring (per game) and \(x=\) team batting average for American League teams. For National League teams in 2010 , \(\hat{y}=-6.25+41.5 x\). (Data available on the book's website in the NL team statistics file.) a. The team batting averages fell between 0.242 and \(0.272 .\) Explain how to interpret the slope in context. b. The standard deviations were 0.00782 for team batting average and 0.3604 for team scoring. The correlation between these variables was 0.900 . Show how the correlation and slope of 41.5 relate in terms of these standard deviations. c. Software reports \(r^{2}=0.81 .\) Explain how to interpret this measure.

Predicting cost of meal from rating Refer to the previous exercise. The correlation with the cost of a dinner is 0.68 for food quality rating, 0.69 for service rating, and 0.56 for d茅cor rating. According to the definition of \(r^{2}\) as a measure for the reduction in the prediction error, which of these three ratings can be used to make the most accurate predictions for the cost of a dinner: quality of food, service, or d茅cor? Why?

Consider the data: $$ \begin{array}{l|lllll} x & 1 & 3 & 5 & 7 & 9 \\ y & 17 & 11 & 10 & -1 & -7 \end{array} $$ a. Sketch a scatterplot. b. If one pair of \((x, y)\) values is removed, the correlation for the remaining four pairs equals \(-1 .\) Which pair has been removed? c. If one \(y\) value is changed, the correlation for the five pairs equals \(-1 .\) Identify the \(y\) value and how it must be changed for this to happen.

How much do seat belts help? In \(2013,\) data was collected from the U.S. Department of Transportation and the Insurance Institute for Highway Safety. According to the collected data, the number of deaths per 100,000 individuals in the U.S would decrease by 24.45 for every 1 percentage point gain in seat belt usage. Let \(\hat{y}=\) predicted number of deaths per 100,000 individuals in 2013 and \(x=\) seat belt use rate in a given state. a. Report the slope \(b\) for the equation \(\hat{y}=a+b x\). b. If the \(y\) intercept equals \(32.42,\) then predict the number of deaths per 100,000 people in a state if (i) no one wears seat belts, (ii) \(74 \%\) of people wear seat belts (the value for Montana), (iii) \(100 \%\) of people wear seat belts.
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