Conditions for Regression Inference
State the four conditions for regression inference: Linear (scatterplot shows linear pattern), Independent (independent observations), Normal (residuals are approximately normal), Equal Variance (residuals have constant spread). Check each with appropriate plots.
A lesson of AP Statistics, in Inference for Regression, about 10 minutes.
By the end of this lesson, you will be able to state and verify all conditions for performing inference about the slope of a regression line.
The Regression Model
The true (population) regression line is:
$$\mu_y = \alpha + \beta x$$
where $\alpha$ and $\beta$ are population parameters (unknown), and $\mu_y$ is the mean of y for a given x.
The LSRL $\hat{y} = a + bx$ is an estimate of this true line, using sample data.
We want to make inference about $\beta$ — the true slope. Does the explanatory variable x actually have a linear relationship with y in the population?
Conditions for Regression Inference (LINER)
L — Linear: The true relationship between x and the mean of y is linear. Check: scatterplot should show a linear pattern; residual plot should show random scatter (no curves).
I — Independent: Individual observations are independent of one another. Check: random sampling or random assignment; 10% condition if sampling without replacement.
N — Normal: For each value of x, y is normally distributed around the true mean $\mu_y$. Check: histogram or normal probability plot of residuals should be approximately normal; look for severe skewness or outliers in the residuals.
E — Equal variance: The standard deviation of y is the same for all values of x (homoscedasticity). Check: residual plot should show constant spread — no fan shape.
R — Random: Data were gathered through a random process.
Checking Conditions in Practice
| Condition | What to examine |
|---|---|
| Linearity | Scatterplot; residual plot (no curved pattern) |
| Independence | Study design; 10% condition |
| Normality | Histogram of residuals; normal probability plot of residuals |
| Equal variance | Residual plot (no fan shape) |
| Random | Study design |
The residual plot is the most important diagnostic tool. It simultaneously helps check linearity and equal variance.