In this scenario, one group has a clear linear relationship between x and y. Another group doesn't. However, there is no way to differentiate them in the data. In this case, performing a simple linear regression on the data results in a clear trend in the residual plot. A simple linear regression isn't appropriate in this case and transformations of the variables is meaningless right?
Here's my R code to simulate this scenario, in order to show what I'm talking about:
num_obs = 100 beta_0 = 0 beta_1 = 1 sigma = 2 set.seed(1) epsilon = rnorm(n = num_obs, mean = 0, sd = sigma) x_vals = seq(from = 0, to = 10, length.out = num_obs)#group 1 # (linear relationship between x and y) y_vals = beta_0 + beta_1 * x_vals + epsilon # group 1 x_vals = c(x_vals,rep(c(0:10),10)) #add group 2 (no # relationship between x and y) y_vals = c(y_vals, rep(0:9, each=11)) #add group 2 sim_fit = lm(y_vals ~ x_vals) coef(sim_fit) plot(y_vals ~ x_vals) abline(sim_fit) plot(sim_fit$fitted.values, sim_fit$residuals) #residual plot # shows clear trend
Here's the data and the best fit line: