BIOL2001 Chap.11 Simple Linear Regression and Model Assumptions
Simple Linear Regression and Model Assumptions
A line as a model
Part III opens with regression: describing how a response changes with a predictor, or more precisely the conditional distribution of Y given X. The chapter defines the linear model, its response, covariate, intercept, slope and error term, and the method of least squares that chooses the line minimising the sum of squared residuals.
A computed figure fits the line to five lizards' bite forces and shows the residuals, the point of means the line always passes through, and residuals that sum to zero.
Judging the fit and the assumptions
A worked example computes the slope, intercept and R-squared by hand.
The chapter then sets out the four assumptions behind inference, normality, homogeneity, independence and fixed x, with the diagnostic for each; a computed pair of residual plots contrasts an even band with the trumpet shape of heteroscedasticity. It closes with influential points, transformations of the covariate or the response, what linear means for a model, and why regression shows association rather than causation.
What this chapter covers
- 01
Regression as a conditional distribution
- 02
Response, covariate, intercept, slope and error
- 03
Least squares and residuals
- 04
Properties of the fitted line
- 05
R-squared and Pearson's r
- 06
Normality, homogeneity, independence and fixed x
- 07
Residual plots and influential points
- 08
Transformations and association versus causation
Worked example · free
Fitting bite force against snout length
- 1The means are 6 cm and 14.4 N; Sxy = 22.0 and Sxx = 10.
- 1Slope = 22.0 / 10 = 2.2 N per cm; intercept = 14.4 − 2.2 × 6 = 1.2 N.
- 1Residuals are 0, 0.8, −1.4, 0.4 and 0.2, which sum to 0; SSerror = 2.8.
- 1SStotal = 51.2, so R2 = 1 − 2.8 / 51.2 = 0.945.
Key terms
- Response Variable
- The variable a regression model explains, plotted on the vertical axis.
- Residual
- The difference between an observed value and the value the fitted model predicts.
- Coefficient of Determination
- R-squared, the share of the variation in the response explained by the model.
- Homoscedasticity
- Equal spread of the errors for every value of the covariate.
- Influential Point
- An observation whose removal would change the fitted model substantially.
- Extrapolation
- Using a fitted model to predict outside the range of the observed covariate values.
Simple Linear Regression and Model Assumptions FAQ
What are the four assumptions of a linear regression model?
The errors are normally distributed for each value of x, they have the same variance for every x, the observations are independent, and the covariate is measured without error, or with error small compared with the response. Each assumption has its own diagnostic plot or check.
How do I check homogeneity of variance in a regression?
Plot the residuals against the fitted values. An even horizontal band supports homogeneity, while a spread that widens as the fitted values grow, the trumpet shape, signals heteroscedasticity. The lecture marks this as the most important check and notes violations are common in ecology.
Why can I not read a regression slope as a cause?
A regression finds associations. Reading a coefficient causally needs every relevant covariate measured and included, every relationship modelled correctly, and prior reasoning about how the variables relate. Without those, a slope describes how the response tends to differ, not what changing x would do.
Is a model with x squared still a linear model?
Yes. What makes a model linear is that its parameters enter linearly, so y equal to alpha plus beta times x squared still counts. A model in which a parameter itself is squared does not. Transforming x keeps the model linear, but the slope then refers to the transformed covariate.
Assessment move
Fit one small regression entirely by hand, then reproduce it with lm() and match every number in summary(). For each workshop dataset, plot the data first, fit the model, then work through the residual plots in order. Practise writing one-sentence interpretations of slopes with units, and say when the intercept cannot be interpreted.
When an assumption fails, try one remedy at a time, such as a log transformation of the covariate, refit, and look at the residual plots again before deciding whether the change helped.
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