Specifically, I am using generalized additive models with penalized splines in R - gam() and gamm4(). The smooth term tells me the F-statistic and p-value, but I would like to get the standardized betas so the results can be more comparable to the linear models within my study.
Can someone point me in the right direction? I am coming from a limited mathematical background so resources that go beyond a pure theoretical explanation would be most beneficial for me.
The following relationship is established between them.
Standardized Coefficients i = Coefficients i * (( SD i )/(SD Y ))
Hi Robert,
once you leave the linear realm, you pay with the loss of clear interpretability. In this regard, the only thing you get with a GAM is a statistical test of the wiggliness of a relationship. When you use the mgcv package in R, the EDF parameter can be interpreted as a kind of strength of wiggliness but in the end, you have to go with an illustration of the curve.
And by the way: Even in the linear case, the interpretation of standardized coefficients as effect sizes is a bit spicy :)
Greenland, S., Schlesselman, J. J., & Criqui, M. H. (1986). The fallacy of employing standardized regression coefficients and correlations as measures of effect. American Journal of Epidemiology, 123(2), 203-208.
Kim, J. O. J.-O., & Mueller, C. W. (1976). Standardized and unstandardized coefficients in causal analysis: An expository note. Sociological Methods & Research, 4(4), 423-438. doi:10.1177/004912417600400402
King, G. G. (1986). How not to lie with statistics: Avoiding common mistakes in quantitative political science. American Journal of Political Science, 30(3), 666-687. doi:10.2307/2111095
HTH
--Holger
Holger Steinmetz is right in the sense that a single measure of effect size is no longer possible because the nonlinear association indicates that the effect size is not constant throughout the range of the predictor variable.
That said, you can, and probably should, use Stata's -margins- command to produce predicted values at key values of the predictor variables (for example, if you have age in your model, predicting at 5-yearly or 10-yearly increments). These will at least give a reader some idea of the magnitude of the effects and their degree of nonlinearity.
Somewhere Terry Pratchett observes that all magic comes at a price. Nonlinear models qv.
Really Appreciate the feedback here and couple follow-up questions.
What if I take the Adjusted R-squared of a GAM model and subtract that by the adjusted R-squared of its equivalent linear model. Would that be a reasonable proxy for effect size that is contributable specifically to the non-linear term?
Also, this is my first time hearing about Estimated Degrees of Freedom (EDF). I was looking into the Restricted Likelihood Ratio test (via) the link below and managed to implement that. Both the EDFs and likelihood seem very similar for my study. Is there a preference for one over the other or are they both conveying similar information?
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