Transformations and non-linear - Statistics AP Study Notes

Transformations in two-variable data involve mathematically modifying one or both variables to linearize non-linear relationships, making them easier to analyze using linear regression techniques.

Non-linear relationships occur when the association between variables follows patterns other than straight lines—common forms include exponential, power (polynomial), and logarithmic functions.

Key Transformation Types:

Exponential Model: y = ab^x transforms to log(y) = log(a) + x·log(b). Plot x against log(y) to linearize.

Power Model: y = ax^b transforms to log(y) = log(a) + b·log(x). Plot log(x) against log(y) to linearize.

Logarithmic Model: y = a + b·log(x). Plot log(x) against y to linearize.

Critical Formula: After transformation, apply linear regression to find the least-squares regression line: ŷ = a + bx, where b = r(s_y/s_x) and a = ȳ - bx̄.

Re-transformation reverses the process. If you found log(y) = 2 + 0.5x, then y = 10^(2+0.5x) = 100 × 10^(0.5x).

Residual plots after transformation should show random scatter around zero—this confirms the transformation successfully linearized the relationship. Patterns in residuals indicate poor model fit.

Memory Aid (PEEL): Power needs both logs, Exponential needs y-log, Logarithmic needs x-log.

Coefficient of determination (r²) measures proportion of variance explained—higher r² after transformation indicates better model fit. Always check r² values to compare model effectiveness.

Why Transform Data? Linear models are simpler to interpret and predict from, but nature rarely follows straight lines. Transformations act as "mathematical translators," converting curved patterns into straight ones.

Exponential Growth Example: Bacterial colonies double every hour. If you plot time vs. population, you'll see a steep curve. Bacteria don't grow linearly—they multiply exponentially (y = ab^x where b > 1). Taking log(population) straightens this curve because logarithms convert multiplication into addition.

Real Application: Epidemiologists use exponential transformations to model disease spread during pandemic early stages. When log(cases) vs. time is linear, doubling time remains constant—crucial for resource planning.

Power Relationship Example: The allometric scaling in biology follows power laws. An elephant's metabolic rate doesn't scale linearly with mass; it follows y = ax^(3/4). Plotting log(metabolic rate) against log(mass) for different animals produces a straight line with slope 0.75.

Think of it this way: Imagine photographing a spiral staircase. From ground level (original data), it curves dramatically. But photograph it from directly above (after transformation), and it appears as straight radiating lines—same staircase, different perspective reveals underlying structure.

Logarithmic Decay Example: Earthquake intensity decreases logarithmically with distance. Doubling the distance doesn't halve intensity—it reduces it by a logarithmic factor. Seismologists use y = a + b·log(distance) to predict ground shaking at various locations.

Analogy: Transformations are like using different lenses on a camera—wide-angle, telephoto, fisheye—each reveals different aspects of the same scene.