PYB350 Chap.8 Correlation and Multiple Regression
Correlation and Multiple Regression
Regression handles continuous predictors. This chapter starts with Pearson's r, the coefficient of determination and the unexplained variance, then builds the bivariate least-squares equation, residuals and the standard error of estimate.
It moves to multiple regression: the model F test that must be read first, R squared and adjusted R squared, unstandardised B and standardised beta weights, and the squared semi-partial correlation as each predictor's unique variance.
It closes with normality, linearity and homoscedasticity checks, and why causal claims depend on the design rather than the analysis.
The worked examples build a bivariate equation from summary statistics and read a two-predictor output, and the practice cards check variance explained and unique variance from a semi-partial correlation.
What this chapter covers
- 01
Categorical and continuous variables
- 02
Pearson's r, r squared and unexplained variance
- 03
The least-squares line, residuals and standard error of estimate
- 04
The model F test, R squared and adjusted R squared
- 05
B, beta and squared semi-partial correlations
- 06
Residual plots and the limits of causal claims
- 07
Reading model fit, B, beta and unique variance in a two-predictor output
Worked example · free
A prediction equation for sleep and next-day focus
- 1Intercept: a = 34.68 minus 3.369 times 7.0125 = 11.05.
- 1Equation: predicted focus = 11.05 + 3.37 times hours of sleep, with the slope rounded.
- 1Prediction: 11.05 + 3.37 times 8 = 38.01, about 38 points.
- 2Variance: r squared = .62 times .62, about .39, so sleep accounts for about 39% of the variance in focus and 61% is unexplained.
Key terms
- Coefficient of determination
- The square of a correlation, read as the share of the variability in Y that X explains.
- Residual
- An actual score minus the score predicted by the regression equation, the part the equation cannot explain.
- Adjusted R squared
- An estimate of the population proportion of variance explained, lower than R squared in the sample.
- Beta weight
- A regression slope expressed in standard-deviation units, allowing predictors on different scales to be compared.
- Homoscedasticity
- The assumption that residuals are evenly spread at every level of the predicted values rather than fanning out.
Correlation and Multiple Regression FAQ
What should be reported first in a multiple regression?
The test of the whole model: the F test of R squared. If it is not significant, the predictors are not a viable set for predicting the outcome and nothing else is interpreted. Once it is significant, report R squared and adjusted R squared, then each predictor's B with its confidence interval, its beta weight and its unique contribution.
Does a significant regression show that X causes Y?
No. The lecture is emphatic that correlation is not causation, and that the limit comes from the research design rather than the statistic. Regression is usually applied to measured, pre-existing differences, so confounds cannot be ruled out. Experimental data can be analysed by regression too, and ANOVA is itself a special case of regression with categorical predictors.
How do I check the regression assumptions?
Use plots. Normality of residuals is checked on a normal probability plot, where points close to the diagonal indicate normal residuals. Linearity is checked with a scatterplot of each predictor against the outcome. Homoscedasticity is checked by plotting standardised predicted values against standardised residuals: a random band is good, a funnel shape signals a breach.
What is the difference between B and beta in regression output?
B is the unstandardised slope: the change in the outcome, in its own units, for a one-unit change in the predictor, holding the other predictors constant. Beta is the same slope expressed in standard-deviation units, which lets you compare predictors measured on different scales. Report confidence intervals for B; the output does not give intervals for beta.
Exam move
Practise reading regression output in a fixed order: model F, R squared and adjusted R squared, then each predictor's B, beta and squared semi-partial, ending with one sentence about what the strongest predictor means in the scenario. Finish every regression answer with a sentence about design, because the unit treats a causal claim from measured variables as an error in interpretation.
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