R for Business Guide

R typically recognizes relationships between variables given data in this format:

dependent variable(s) ~ independent variable(s)

Example:

• You're fitting data to this linear model: Y_i = β₀ + β₁X_i + β₂Z_i + ε_i.
• You have a single dependent variable with measurements stored in the object y.
• You have two independent variables with measurements stored in the objects x and z, respectively.
• In R, you can code y ~ x + z.

Functions that accept formula notation often let you reference data objects in either of two ways:

1. Named vectors
   e.g.: y and x are each vectors in your environment, so lm(formula = y ~ x)
2. Names within a data frame
   e.g.: data frame dat has columns dat$y and dat$x, so lm(formula = y ~ x, data = dat)

• One independent variable, or Y_i = β₀ + β₁X_i + ε_i: y ~ x
• Two independent variables, or Y_i = β₀ + β₁X_i + β₂Z_i + ε_i: y ~ x + z
• Two independent variables without intercept, or Y_i = β₁X_i + β₂Z_i + ε_i: y ~ x + z - 1
• All remaining columns in data frame as independent variables: y ~ .
• All remaining columns in data frame (but not z) as independent variables: y ~ . - z

Interactions

• The interaction between x and z: x:z
• x, z, w, and all interactions among them: x * z * w
• x, z, w, and all interactions up to two-way interactions: (x + z + w)^2

Conditional effects

When fitting a mixed effects model (e.g.: lme(...)), fit x grouped by z: x|z

Mathematical transformations

Model the variable resulting from the enclosed mathematical operation: I(...)

• e.g.: The product of x and z: I(x*z)
• e.g.: The sum of x, z, and w: I(x+z+w)