MEDD8815 Chap.2 Scatterplots, Correlation and Least-Squares Regression
Scatterplots, Correlation and Least-Squares Regression
Two variables on the same cases
The second session moves from one variable to how two quantitative variables recorded for the same cases move together. It starts with roles, an explanatory variable that may explain or influence and a response variable that measures the outcome, and with the scatterplot, which puts the explanatory variable on the horizontal axis.
A scatterplot is described by direction, form and strength, plus any striking departures, and a categorical variable can be added by marking the points in different colours.
Correlation and its limits
The correlation coefficient r measures the direction and strength of a straight-line relationship.
It lies between minus one and one, has no units, is unchanged when either variable is rescaled, and does not distinguish the two variables.
The chapter spends as long on what r cannot do as on what it can: it needs two quantitative variables, it misses curved patterns, and one outlying point can move it a long way.
Fitting and questioning a line
The least-squares line minimises the sum of squared vertical distances from the points, passes through the point of means, and has slope equal to r times the ratio of the standard deviations.
Its square of correlation is the fraction of the variation in the response that the line accounts for.
The chapter works a line from five summary numbers, then examines residuals, influential points and extrapolation using an AskSia-authored data set in which adding a single case drops r sharply and flattens the slope.
Association is not causation
Three explanations can lie behind an observed association: genuine causation, common response to a lurking variable, and confounding of two influences that cannot be separated.
A well-designed experiment is the most convincing route to a causal claim; without one, the chapter sets out the criteria that together make a causal argument stronger. The SPSS routes from the session handout, for correlation, simple regression and scatterplots with group markers, are shown alongside the hand calculations.
What this chapter covers
- 01
Explanatory and response variables
- 02
Reading direction, form and strength from a scatterplot
- 03
The correlation coefficient and its properties
- 04
The least-squares regression line
- 05
Slope, intercept and r squared
- 06
Residuals and residual plots
- 07
Outliers, influential points and extrapolation
- 08
Causation, common response and confounding
Worked example · free
Which quantities survive a change of units?
- 1Rescale the slope for minutes rather than hours.
- 1Decide whether the intercept moves.
- 2State what happens to r and to r squared, with the reason.
Key terms
- Explanatory Variable
- A variable that may explain or influence changes in another, placed on the horizontal axis of a scatterplot.
- Response Variable
- The variable that measures an outcome of a study, placed on the vertical axis and predicted by a regression line.
- Correlation Coefficient
- A number r between minus one and one summarising how closely, and in which direction, two quantitative variables follow a straight line.
- Least-Squares Line
- The straight line that makes the sum of squared vertical distances from the data points to the line as small as possible.
- Residual
- An observed response minus the response the fitted line predicts for the same case.
- Influential Observation
- A case whose removal would markedly change the fitted line; cases lying far out along the x axis are the usual source.
- Lurking Variable
- A variable not included in a study that nonetheless affects the relationship between the variables studied.
- Extrapolation
- Using a regression line to predict outside the range of explanatory values used to fit it, which is often unreliable.
Scatterplots, Correlation and Least-Squares Regression FAQ
Does a strong correlation mean one variable causes the other?
No. A lurking variable can drive both, which is common response, or two influences can be tangled together, which is confounding. Only a well-designed experiment gives fully convincing evidence of cause; without one, look for strength, consistency, dose-response, time order and a plausible mechanism together.
What does r squared tell me in a regression?
It gives the share of the variability in the response that is explained by the straight-line relationship with the explanatory variable. An r of 0.8 gives 0.64, so about 64 per cent of the variation is explained and the rest is left to other influences.
How can a point lie on the line and still be a problem?
A case far out in the x direction can pull the line towards itself, so its residual looks small while the whole fit depends on it. Refit without that case: if the slope or r changes markedly, the point is influential and both fits should be reported.
How do I get r and the line in SPSS?
Analyze, Correlate, Bivariate gives the Pearson correlation. Analyze, Regression, Linear gives the fitted line: in the Coefficients table, B for the constant is the intercept and B beside the predictor is the slope, and the Model Summary gives R and R Square.
Assessment move
Keep a two-column sheet of the pairs this chapter separates: r and slope, residual and influence, outlier and influential point, common response and confounding. For each pair, write one sentence and one tiny example of your own. The homework and the quiz both reward naming the right member of a pair, and the sheet makes the difference automatic.
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