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At the moment I am trying to make a list of the different approaches that could be used to verify the linearity of an effect. In a model (Y = b0 + b1.X + etc.), I want to know whether it is acceptable to assume linearity for (X).

What I've been doing so far is to estimate another model (Y = b0 + b1.X + b2.X**2) based on a quadratic specification and (1) look at significance of the quadratic term (b2), and (2) eventually perform a log-likelihood ratio test.

However, I fear that this relatively simple approach would in some circumstances be misleading (especially if pattern of non-linearity is not in line with a quadratic shape). Indeed, this simple approach would fail to reject the assumption of linearity when I simulate data that would be be described by a S-shaped curve.

What approaches (other than polynomial specification + log-likelihood ratio test) would you recommend? Ideally a test - Not a simulation based approach, and something that would work also for non-nested models (unlike the LR test).

I came across the Vuong test (https://en.wikipedia.org/wiki/Vuong%27s_closeness_test), but I am sure there is more to known on this issue. Thanks for your help!

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    $\begingroup$ You can include splines in your larger model instead of a simple quadratic effect and again run a likelihood ratio test against the model that includes only a linear effect. $\endgroup$ – Stephan Kolassa Jun 11 '17 at 14:14
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If you want to see if the relationship between (the conditional expectation of) $y$ and $x_0$ is linear, after adjusting for control variables $x_1, x_2, \dots, x_p$, a simple graphical approach is to create an added-variable plot using the following procedure.

First, regress $y$ on $x_1, x_2, \dots, x_p$ and obtain the residuals from that regression, $\hat{\epsilon}_y$. Then, regress $X_0$ on $x_1, x_2, \dots, x_p$ and obtain the residuals from that regression, $\hat{\epsilon}_{x_0}$.

Then, create a scatter plot of $\hat{\epsilon}_y$ against $\hat{\epsilon}_{x_0}$ and overlay a nonparametric curve (e.g. loess) along with the linear regression line. The linear regression line will have exactly the same slope as the "long" regression that includes all variables $x_0, x_1, \dots, x_p$ by the Frisch-Waugh theorem. The nonparametric curve will give you a sense of how well the relationship between $y$ and $x_0$ can be approximated as linear.

Some simple R code to demonstrate:

data(mtcars)

# full model, with all control variables 
fullmod = lm(mpg ~ wt + vs + gear + am, mtcars)
coef(mod)[2]
>     wt 
> -3.786

# regress y on controls and x on controls, extract residuals
eps_y = lm(mpg ~ vs + gear + am, mtcars)$residuals
eps_x = lm(wt ~ vs + gear + am, mtcars)$residuals

# regress epsilon_y on epsilon_x, see the coef is the same as above
coef(lm(eps_y ~ eps_x))[2]
>  eps_y 
> -3.786

# make added variable plot
library(ggplot2)
qplot(x = eps_x, y = eps_y) + 
  geom_smooth(method = "lm", colour = "black", se= FALSE) + 
  geom_smooth(method = "loess", colour = "red", se = FALSE)

added-variable plot

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As @stephan-kolassa mentioned. An added spline portion can be more beneficial than a quadratic term, since that will not explicitly determine the nonlinearity of the model. A likelihood ratio test or F-test can be performed from there.

Now, there are problems with such a method that I think need to be considered.

  1. The model will test $H_0: Y=X\beta+\epsilon$ vs $H_a:Y=X\beta + f(x) + \epsilon$, where $f(x)$ is a spline model. In such a situation, all you can ever say is that the data does not supply evidence of that nonlinear term, it will never truly verify the assumption of linearity.

  2. Furthermore, there may be the testing for normality issue where the model may never be truly linear. So like testing normality, the only reason a test will ever fail to reject that assumption on real is because a lack of sample size, since no data is actually normal. The same may apply to testing linearity, linearity is a theoretical assumption, and the lack of rejection may be due to the lack of sample size rather than the assumption being actually true.

  3. What may be the best option is to consider the linear model to be the best model via model selection. This can be done with AIC/BIC (which are actually quite good at testing nested models) or cross-validation or some measure of deviance (does the model seem to be around the expected value of the $\chi^2$ it is supposed to represent).

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