MPH 560 Module 7 Correlation and Regression Example

Measuring Relationships: Correlation and Linear Regression With Age and Blood Pressure

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Master of Public Health Program, Aspen University

MPH 560: Applied Biostatistics for Public Health

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Month Day, Year

Measuring Relationships: Correlation and Linear Regression With Age and Blood Pressure

Public health often asks how two numerical variables are related: does blood pressure rise with age, does air pollution relate to asthma visits, does income relate to life expectancy? Correlation measures the strength of a linear relationship, and regression describes it with an equation that can be used for prediction. This paper explains both through composite data on age and systolic blood pressure.

The Composite Data

Ten adults aged 25 to 70, in five-year steps, had systolic readings of 114, 117, 121, 119, 126, 130, 133, 138, 136 and 145 mm Hg. A scatter plot would show points rising from lower left to upper right, close to a straight line.

Correlation

Pearson's correlation coefficient, r, ranges from -1 to 1. Values near 1 indicate a strong positive linear relationship, near -1 a strong negative one and near 0 little linear relationship. For these data, r equals 0.98, a very strong positive correlation (Bewick et al., 2003). Real population data are usually much noisier; this composite example was built to show the calculations clearly. Correlation is symmetric: the correlation of age with pressure is the same as that of pressure with age. Regression, by contrast, treats one variable as the outcome and the other as the predictor.

Regression

Simple linear regression fits the line that minimizes the squared vertical distances between points and the line. The table summarizes the results.

Interpreting the Slope and Intercept

The slope indicates that, on average, systolic pressure was 0.66 mm Hg higher for each additional year of age in these data, or about 6.6 mm Hg per decade. The intercept, 96.6, is the predicted value at age zero, which lies far outside the data and has no practical meaning; it simply anchors the line. The slope should be reported with its confidence interval so readers can judge its precision, as statisticians have long recommended for all estimates (Gardner & Altman, 1986).

Prediction

The equation predicted pressure = 96.6 + 0.66 times age allows prediction within the range of the data. For a 50-year-old, the prediction is 96.6 plus 33.0, or about 129.6 mm Hg. Predicting outside the observed range, for a 90-year-old or a child, is extrapolation and can be badly wrong, because the relationship may not stay linear.

R-Squared

R-squared, the square of r, is 0.96, meaning 96% of the variation in systolic pressure in these 10 people is accounted for by age. In real data, age typically explains far less, because diet, genetics, medication and other factors also matter.

Correlation Is Not Causation

A strong correlation does not prove that one variable causes another. Both may be driven by a third factor, the relationship may run in the opposite direction or the association may arise by chance or bias. Causal conclusions require study designs and reasoning that address these alternatives.

Assumptions and Diagnostics

A straight-line model needs the relationship to be roughly linear, the observations independent, the scatter around the line about even and, for tests and intervals, residuals that are close to normal. Plotting residuals, the differences between observed and predicted values, against predicted values reveals curves, funnel shapes or outliers that signal problems (Bewick et al., 2003).

Outliers and Influence

A single unusual point can change a regression line substantially, especially in small samples. Analysts should examine outliers, check for data errors and report how results change if influential points are removed. In skewed data, transforming variables or using rank correlation may be more appropriate (Whitley & Ball, 2002). A sensitivity analysis that reruns the regression without a suspect point shows readers whether the conclusion depends on it.

Toward Multiple Regression

Public health questions usually involve several variables. Multiple regression estimates the relationship between an outcome and one variable while holding others constant, which helps address confounding. For example, the association between income and blood pressure can be examined while adjusting for age and sex.

Confidence Intervals and Tests for the Slope

The slope has its own standard error, confidence interval and test of whether it differs from zero. A slope whose interval excludes zero indicates an association unlikely to be due to chance alone. Reporting the interval conveys how precisely the relationship is estimated.

Ecological Correlations

Correlations calculated from group data, such as county averages, can differ from those within individuals, a problem known as the ecological fallacy. A strong correlation between county income and life expectancy does not mean every wealthier person lives longer. Conclusions should match the level at which data were collected.

Using Regression in Public Health Practice

Health departments use regression to adjust rates for age, to estimate trends over time and to identify factors associated with outcomes, such as neighborhood characteristics linked to asthma visits. Clear reporting of the model, variables and assumptions allows others to evaluate the results.

Nonlinear Relationships

Not all relationships are straight lines. Blood pressure may rise faster at older ages, and some exposures have thresholds or plateaus. Plotting the data first shows whether a straight line fits. Curves can be modeled by adding squared terms or using other methods.

Reporting Regression Results

A clear report states the outcome and predictor, the slope with its confidence interval, the intercept if meaningful, R-squared, the sample size and any checks of assumptions. Graphs of the fitted line with the data help readers see the relationship.

Correlation Versus Agreement

A high correlation does not mean two measures agree. Two blood pressure devices might correlate at 0.95 while one reads consistently 10 mm Hg higher. Agreement between methods is assessed with other approaches, such as plotting differences against averages, not with correlation alone.

Conclusion

Correlation and regression describe and quantify relationships between numerical variables. In the composite data, age and systolic pressure had a correlation of 0.98 and a slope of 0.66 mm Hg per year, with 96% of variation explained. Interpreting slopes carefully, avoiding extrapolation, checking assumptions and remembering that correlation is not causation allow these tools to inform public health responsibly.

References

Bewick, V., Cheek, L., & Ball, J. (2003). Statistics review 7: Correlation and regression. Critical Care, 7(6), 451-459. https://doi.org/10.1186/cc2401

Gardner, M. J., & Altman, D. G. (1986). Confidence intervals rather than P values: Estimation rather than hypothesis testing. BMJ, 292(6522), 746-750. https://doi.org/10.1136/bmj.292.6522.746

Whitley, E., & Ball, J. (2002). Statistics review 1: Presenting and summarising data. Critical Care, 6(1), 66-71. https://doi.org/10.1186/cc1455