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In Discovering Statistics using SPSS 4th Edition by Andy Field, it was recommend to include the interaction term between the independent variable $x$ and its corresponding natural logarithm transform $\ln(x)$ variable to check for violation of the linearity assumption. What is the statistical theory behind this?

This a quote from the book:

This assumption can be tested by looking at whether the interaction term between the predictor and its log transformation is significant (Hosmer & Lemeshow, 1989).

I've also recently found out that this transformation is called Box-Tidwell transformation.

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    $\begingroup$ I recall that book giving dubious advice from another question here: stats.stackexchange.com/questions/157217/…. This include x ln(x) also strikes me as also dubious. $\endgroup$ Jun 6, 2016 at 4:07
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    $\begingroup$ tatami There may well be a good reason for it, but context will probably help locate it more quickly. What basis did Field use to suggest it was a good idea? Did he offer any references? Can you quote what the book says? $\endgroup$
    – Glen_b
    Jun 6, 2016 at 4:48
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    $\begingroup$ A better way (given enough data) might be to use a logistic GAM (generalized additive model) and include a spline fit of x. $\endgroup$ Jun 6, 2016 at 9:36
  • $\begingroup$ @Glen_b I've added a quote from Andy Field's book $\endgroup$
    – tatami
    Jun 6, 2016 at 12:55
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    $\begingroup$ New to me, but seems to make sense for a particular kind of non-linearity: stats.uwo.ca/faculty/braun/ss3859/notes/Chapter6/ch5notes.pdf $\endgroup$ Jun 6, 2016 at 13:04

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Box and Tidwell (1962) [1] presented a somewhat general approach for estimating transformations of the individual predictors (IVs), and work through the specific case of estimating power transformations of the predictor variables (including that power 0, which - with appropriate scaling - corresponds to taking logs as a limiting case).

In that particular case of power transformations, it turns out that there's a connection to regressing on $X_j\log(X_j)$.

So if you have nonlinearity of the kind where the true (conditional) relationship between $Y$ and $X_j$ is linear in $X_j^{\alpha_j}$ then it can be used to check for $\alpha_j\neq 1$, or indeed to estimate $\alpha$ values.

Specifically, when regressing on $X_j$ and $X_j\log(X_j)$ the coefficient of the second term divided by that of the first is an approximate estimate of $\alpha_j-1$. (This estimate can be iterated to convergence.)

If that estimated $\alpha_j$ is close to 1 then there's little indication of a need to transform.

Note that since the two terms in the product $X_j\log(X_j)$ are both functions of $X_j$, this is simply a transformed $X_j$ so I wouldn't call that an interaction; it's just a transformed predictor. (Indeed, even if I were somehow tempted to do so, since $\log(X_j)$ is not included as a predictor I still wouldn't tend to describe that second term as an interaction.)

[1]: Box, G. E. P. and Tidwell, P. W. (1962), "Transformation of the independent variables." Technometrics 4, 531-550.

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    $\begingroup$ Maybe this has some limited value, especially if one has reason to expect nonlinearity of that specific power form, or if one is forced to use software which does not offer more modern alternatives, such as using a GAM (generalized additive model), effectively estimating the nonlinearity directly with splines. Or, if thwe number of observations is to low to permit use of splines. $\endgroup$ Jun 9, 2016 at 7:52
  • $\begingroup$ Just a very small clarification that I've struggled with: Box & Tidewell first fit a regression model only with $X_j$, resulting in the estimate $\hat{b}$. Then, they refit the model with both $X_j$ and $X_j\log(X_j)$, resulting in estimates $\hat{b'}$ and $\hat{c}$. The first order estimate of $\alpha$ is then $\hat{c}/\hat{b} + 1$. I don't think this estimation of $\alpha$ works if both $X_j$ and $X_j\log(X_j)$ are in the model, as Field (based on based on the book of Hosmer & Lemeshow (1989), also mentioned in Pregibon (1984) 10.2307/2530907) recommends. $\endgroup$ May 25, 2022 at 10:35

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