Question

A biologist collected data on the height (in inches) and peak expiratory flow (PEF-a measure of how much air a person can expire, measured in \(1 / \mathrm{min}\) ) for 10 women. \({ }^{15}\) Here are the data: $$ \begin{array}{|rcc|} \hline \text { Subject } & \text { Height } & \text { PEF } \\ \hline 1 & 63 & 410 \\ 2 & 63 & 440 \\ 3 & 66 & 450 \\ 4 & 65 & 510 \\ 5 & 64 & 340 \\ 6 & 62 & 360 \\ 7 & 67 & 380 \\ 8 & 64 & 380 \\ 9 & 65 & 360 \\ 10 & 67 & 570 \\ \hline \end{array} $$ Is PEF related to height? Identify the type of statistical method that is appropriate for these data and this question, but do not actually conduct the analysis.

Short answer

Pearson's correlation and linear regression are appropriate to analyze the relationship between height and PEF.

Step-by-step solution

- Determine the type of variables

Identify the type of variables being compared. In this case, the biologist is looking at two quantitative variables: the height of the women (in inches) and their peak expiratory flow (PEF, measured in L/min).

- Choose the appropriate statistical analysis

Given that both the height and PEF are continuous variables, we can analyze the relationship between them using correlation and regression methods. Specifically, a Pearson's correlation coefficient could be used to determine the strength and direction of the linear relationship, and linear regression could help to ascertain whether the height is a good predictor of PEF.

- Conclusion

The suitable statistical method to determine if there is a relationship between height and PEF for the given data is Pearson's correlation and linear regression analysis.

Key concepts

Pearson's correlation coefficient

When it comes to understanding how two quantitative variables are related, one of the most fundamental tools we use is Pearson's correlation coefficient, often symbolized by the letter 'r'. This coefficient measures the strength and direction of a linear relationship between two variables on a scale from -1 to 1. The correlation coefficient equal to 1 indicates a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and a correlation of 0 signifies no linear relationship. It's crucial to remember though, correlation does not imply causation; it only suggests that as one variable changes, the other might change in a specific direction, but it doesn't necessarily tell us why.

In the exercise provided, we would calculate Pearson's correlation coefficient to examine the linear relationship between the height of women and their peak expiratory flow (PEF). If the resulting coefficient is close to 1 or -1, it would suggest a strong relationship, while a value near 0 would suggest a weak or no linear relationship.

Linear regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables. The idea is to find the best-fitting straight line through the data points, which is known as the regression line. In mathematical terms, this line can be represented by the equation of a line, y = mx + b, where 'y' is the dependent variable, 'x' is the independent variable, 'm' is the slope of the line, and 'b' is the y-intercept.

The primary purpose of linear regression is to predict the value of the dependent variable based on the value of the independent variable(s). For the exercise, we would use linear regression to predict PEF based on a given height. A strong linear relationship indicated by Pearson's correlation would suggest that linear regression could be a good method for making these predictions, with height being the independent variable and PEF the dependent variable.

Quantitative variables

Quantitative variables are numerical and allow for mathematical computations. They are key in statistical analysis, because they can represent data that can be measured or counted - like height in inches or peak expiratory flow rates in liters per minute. There are two main types of quantitative variables: discrete and continuous. Discrete variables represent countable data, such as the number of pets people have, while continuous variables represent measurable quantities, such as height and weight, which can have an infinite number of values within a selected range.

In the context of the biologist's data, both height and PEF are examples of continuous quantitative variables because they can take on many different values and measurements can be made as precise as needed.

Peak expiratory flow

Peak Expiratory Flow (PEF) is a critical measure in respiratory health, representing the maximum speed at which a person can exhale. It's measured in liters per minute (L/min) and can be critically important for diagnosing and monitoring conditions such as asthma. The PEF can be affected by numerous factors including age, sex, height, and lung health.

Understanding PEF is vital for the exercise at hand as it's the dependent variable we're examining in relation to the independent variable of height. Health professionals often use PEF as a quantitative variable in studies to gauge how various factors correlate with lung function, making it a perfect example to illustrate how statistical analysis can find relationships between different types of health-related data.

Variable relationship

When discussing variable relationship, we're referring to how one variable changes in relation to another. In statistics, uncovering these relationships can help us make predictions or understand patterns. Relationships can be positive, negative, or non-existent. A positive relationship means that as one variable increases, the other tends to increase as well, whereas in a negative relationship, as one variable increases, the other tends to decrease. If there's no apparent pattern as one variable changes, then there may be no relationship.

Looking at the exercise, we aim to understand the relationship between height and PEF. With Pearson's correlation coefficient and linear regression, we can quantify this relationship. A meaningful relationship here would mean there is a consistent pattern showing that a woman's height could help us predict her PEF, which in turn, could have implications for health assessments and interventions.