Transformations to Achieve Linearity

Why Transform?

When the relationship between $x$ and $y$ is not linear, we can transform one or both variables to make the relationship linear. This allows us to use linear regression methods.

Common Nonlinear Patterns

Exponential Growth: $y = ab^x$

Take the logarithm of y: $\ln(y) = \ln(a) + x \cdot \ln(b)$

This is linear in $x$ vs. $\ln(y)$.

Power Model: $y = ax^b$

Take the logarithm of both x and y: $\ln(y) = \ln(a) + b \cdot \ln(x)$

This is linear in $\ln(x)$ vs. $\ln(y)$.

Steps for Transformation

1. Examine the scatterplot: Identify the type of curve
2. Apply the appropriate transformation
3. Check the residual plot: Should show random scatter
4. Check $r^2$: Should be high
5. Write the model: In transformed and original form

Logarithmic Transformation Summary

| Original Model | Transform | Linear Form | |----------------|-----------|-------------| | $y = ab^x$ (exponential) | $\ln(y)$ | $\ln(y) = \ln(a) + (\ln b)x$ | | $y = ax^b$ (power) | $\ln(x)$ and $\ln(y)$ | $\ln(y) = \ln(a) + b \cdot \ln(x)$ |

Making Predictions with Transformed Models

After fitting a model to $\ln(y)$:

1. Find $\widehat{\ln(y)}$ using the regression equation
2. Back-transform: $\hat{y} = e^{\widehat{\ln(y)}}$

Example: Exponential Model

Data suggests exponential growth. After plotting $x$ vs. $\ln(y)$:

Regression: $\widehat{\ln(y)} = 1.2 + 0.5x$

To predict $y$ when $x = 3$:

1. $\widehat{\ln(y)} = 1.2 + 0.5(3) = 2.7$
2. $\hat{y} = e^{2.7} \approx 14.88$

Evaluating the Transformation

A good transformation produces:

• A linear scatterplot (in the transformed variables)
• A random residual plot
• A high $r^2$ value
• A reasonable model in context

Square Root Transformation

Sometimes $\sqrt{y}$ vs. $x$ works well for count data or data where variability increases with the mean.

AP Tip: On the AP exam, you may be given transformed data and asked to interpret the regression. Be comfortable going back and forth between $\ln(y)$ and $y$ using the exponential function.