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2.9 Analyzing Departures from Linearity

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Recognizing Non-Linear Relationships

Not all relationships are linear. When a scatterplot shows a curved pattern, a linear model is inappropriate.

Signs of non-linearity:

  1. Scatterplot shows a curved (e.g., parabolic, exponential) pattern
  2. Residual plot shows a systematic curved pattern
  3. High-leverage points create an artificially high r{r}

Common non-linear forms:

  • Exponential: y=a⋅bx{y = a \cdot b^x} — growth/decay curves (population growth, radioactive decay)
  • Power: y=a⋅xb{y = a \cdot x^b} — many natural phenomena
  • Logarithmic: y=a+bln⁡(x){y = a + b \ln(x)} — diminishing returns

Example: Bacterial population growth over time — if you try to fit a linear model, the residuals will show a clear pattern (all positive at the ends, all negative in the middle).

In this unit

9 lessons

  1. 2.1Introducing Statistics: Are Variables Related?Preview lesson
  2. 2.2Representing Two Categorical VariablesPreview lesson
  3. 2.3Statistics for Two Categorical VariablesPreview lesson
  4. 2.4Representing the Relationship Between Two Quantitative VariablesPreview lesson
  5. 2.5CorrelationPreview lesson
  6. 2.6Linear Regression ModelsPreview lesson
  7. 2.7ResidualsPreview lesson
  8. 2.8Least Squares RegressionPreview lesson
  9. 2.9Analyzing Departures from LinearityThis lesson

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