class: center, middle, inverse, title-slide # IDS 702: Module 1.8 ## Transformations ### Dr. Olanrewaju Michael Akande --- ## Transformations - As we have already seen, sometimes, we have to deal with data that fail linearity and normality. -- - Transforming variables can help with linearity and normality (for the response variable, since we do not need normality of the predictors). -- - The most common transformation is the .hlight[natural logarithm]. For the response variable, that is, `\(\textrm{log}_e(y)\)` or `\(\textrm{ln}(y)\)`. -- - This is often because it is the easiest to interpret. -- - Suppose .block[ .small[ `$$\textrm{ln}(y_i) = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \ldots + \beta_p x_{ip} + \epsilon_i.$$` ] ] -- - Then it is easy to see that .block[ .small[ `$$y_i = e^{(\beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \ldots + \beta_p x_{ip} + \epsilon_i)} = e^\beta_0 \times e^{\beta_1 x_{i1}} \times e^{\beta_2 x_{i2}} \times \ldots \times e^{\beta_p x_{ip}} \times e^{\epsilon_i}.$$` ] ] -- - That is, .hlight[the predictors actually have a multiplicative effect] on `\(y\)`. --- ## Natural log transformation - The estimated `\(\beta_j\)`'s can be interpreted in terms of approximate proportional differences. -- - For example, suppose `\(\beta_1 = 0.10\)`, then `\(e^{\beta_1} = 1.1052\)`. -- - Thus, a difference of 1 unit in `\(x_1\)` corresponds to an expected positive difference of approximately `\(11\%\)` in `\(y\)`. -- - Similarly, `\(\beta_1 = -0.10\)` implies `\(e^{\beta_1} = 0.9048\)`, which means a difference of 1 unit in `\(x_1\)` corresponds to an expected negative difference of approximately `\(10\%\)` in `\(y\)`. -- - .block[When making predictions using the regression of the transformed variable, remember to transform back to the original scale to make your predictions more meaningful.] --- ## Other transformations - While the natural logarithm transformation is the most common, there are several options. -- - For example, logarithm transformations with other bases, taking squares, taking square roots, etc. -- - <div class="question"> Which one should you use? </div> -- - Well, it depends on what you are trying to fix. -- - For linearity, for example, it is possible to need a logarithm transformation on the response variable but a square root transformation on the one of the predictors, to fix violations of linearity and normality. -- - Overall, if you do not know the options to consider, you could try .hlight[Box-Cox power transformations] (to fix non-normality). -- - We will not spend time on those in this course but I am more than happy to provide resources to anyone who is interested. -- - First, see the .hlight[boxcox] function in R's .hlight[MASS] library. --- class: center, middle # What's next? ### Move on to the readings for the next module!