Game 13 · Testing and estimation
How strong is this relationship?
Correlation coefficients, regression lines, and correlation without causation.
Read with: Correlation and regression, Logistic regression31 DNB questions
Play
Read the scatter plot
Guess Pearson's r from a cloud of points. Drag a line through the points and compare it with the least squares line. Add one patient to make or break a correlation. Choose between Pearson and Spearman for a curve, and explain a correlation that is not a cause. All data are made up. You have three lives.
Loading the first round…
Correlation
What r and r² tell you
Pearson's correlation coefficient r measures the direction and closeness of a straight-line relation between two measurements. It runs from −1 (a perfect falling line) through 0 (no straight-line relation) to +1 (a perfect rising line), and it has no units. r², the coefficient of determination, is the share of the variation in one variable that the straight line accounts for: r = 0.5 gives r² = 0.25, or 25%.
A p-value for r asks only whether r could be 0 in the population. With r = 0.10, p is 0.49 in 50 patients and 0.0015 in 1,000 patients, and in both the line accounts for 1% of the variation. Report r with its 95% CI and n.
When r misleads:
- The relation is curved: r underestimates it.
- One or two outliers: a single patient can raise or lower r.
- A narrow range of values: r is lower than in the full population.
- Repeated measurements from the same patients counted as separate pairs: the pairs are not independent.
Regression
The least squares line
Simple linear regression fits the straight line that makes the sum of the squared vertical distances from the points to the line (the residuals) as small as possible. The line always passes through the point (mean x, mean y).
y = a + b × x
- b
- the slope (regression coefficient): the change in y for a 1-unit rise in x, in units of y per unit of x
- a
- the intercept: the predicted y when x = 0, often outside the data and without clinical meaning
Teaching example from the chapter: SBP = 103.7 + 0.58 × age in 10 adults, r = 0.82, r² = 0.68. Ten more years of age go with a 5.8 mmHg higher predicted SBP. The adults were 30 to 75 years old, so predicting SBP at 90 is extrapolation and unreliable.
| Feature | Correlation | Regression |
|---|---|---|
| Question | How closely do two variables move together? | How much does y change with x, and what is y for a given x? |
| Result | r, from −1 to +1, with no units | Slope b in units of y per unit of x, and an intercept a |
| Roles | Both variables treated alike | One outcome (y), one or more predictors (x) |
| Types | Pearson for measurements, Spearman for ranks | Linear for measurements, logistic for yes/no, Cox for time to an event |
r and the slope are linked: b = r × SD of y ÷ SD of x. The same r can go with a steep or a shallow line, depending on the units.
Choosing the coefficient
Pearson or Spearman
| Feature | Pearson's r | Spearman's ρ |
|---|---|---|
| Measures | Closeness to a straight line | Consistency of rank order (a monotonic relation) |
| Data | Two measurements on an interval or ratio scale | Ordinal scores, or measurements that are skewed or have outliers |
| Outliers | Strongly affected | Much less affected |
| Curved but one-directional pattern | Underestimates the relation | Close to ±1 if the order is perfect |
In the chapter's teaching data, points on a curve that always rises give ρ = 1.00 but r = 0.85. Eleven points with r = 0.00 plus one outlier give r = 0.75, while ρ = 0.37.
Correlation and cause
A correlation is not a cause until the alternatives are ruled out
- Chance: test enough pairs of variables and some will correlate by chance alone.
- Reverse causation: the outcome can change the exposure. Patients who are unwell go outdoors less and have lower vitamin D.
- Confounding: a third factor affects both variables. Smoking stains fingers and damages lungs, so the two correlate without one causing the other.
- Selection bias: the way patients were chosen can create a correlation that does not exist in the population.
The full treatment, with partial correlation and multiple regression, is in the Correlation and regression chapter. Regression toward the mean and Simpson's paradox are in the Statistical fallacies chapter.
Writing the answer
How to write this in a 10-mark DNB answer
- Draw three small scatter plots first: r near +1, r near 0 and r near −1.
- Define correlation and the correlation coefficient, its range and that it has no units. Define r² with one number (r = 0.5 → 25%).
- Give Pearson for normal measurements and Spearman for ordinal or skewed data, in a two-column table.
- Define regression, write y = a + b × x, explain a and b with a worked example (SBP and age), and state the least squares principle.
- Tabulate correlation against regression (question, result, units, roles of the variables).
- End with the cautions: correlation is not cause (chance, reverse causation, confounding), outliers, curved relations and extrapolation.
The one thing to remember
r tells you how close the points are to a straight line. It does not tell you how steep the line is, or why the points line up.
Past papers
DNB questions this game helps you answer
Questions from NBE theory papers for DNB, DrNB and Diploma courses, most often asked first. The tag after each question is the specialty and the year it was set.
- ×1Explain correlation and regression in statistical methods.
- ×1How is a scatter plot drawn?
- ×1Interpretation of linear correlation coefficient ‘r’.
- ×1Linear regression analysis.
- ×1Linear regression.
- ×1Pearson's r. [2.5]
- ×1Discuss various findings of National Mental Health Survey of India 2016. What is Firth's penalized logistic regression used in this survey?
- ×1Pearson correlation analysis.
- ×1Define linear regression. Discuss giving examples the types and utility of linear regression in research.
- ×1Correlation Coefficient.
- ×1Discuss logistic regression model.
- ×1What is regression analysis? In which situations it is indicated? Discuss giving examples, how multiple regression analysis is conducted.
- ×1Discuss regression analysis
- ×1Write the steps of Linear regression analysis and how is linear regression analysis data presented.
- ×1Write short notes on: Linear Correlation Coefficient.
- ×1a) Write the difference between correlation and regression analysis. b) List the types of observational study design. c) How is sample size calculated in case-control study?
- ×1Multiple Regression Analysis
- ×1What is correlation coefficient?
- ×1What is the theory of regression analysis? Describe one method of regression analysis.
- ×1What is the use of logistic regression analysis in statistics?
- ×1Correlation and Regression.
- ×1a) ANOVA. b) Chi- Square test and t- test. c) Correlation and regression coefficients in statistical analysis.
- ×1Correlation and regression
- ×1Correlation and regression.
- ×1Regression analysis
- ×1Compare and contrast ‘correlation analysis’ with ‘regression analysis’.
- ×1Cox regression analysis.
- ×1Concepts underlying “Odd’s Ratio”, “Multiple Logistic Regression Analysis” and “Intention to treat analysis”.
- ×1Explain confounding. How logistic regression analysis can be used to handle confounders?
- ×1Regression analysis.
- ×1Regression Analysis.
Read next