Solved – Why does Box-Cox transformation fail in following situation

data transformation

In the book, Linear Models with R by Julian Faraway, it says that if max(y)/min(y) is small, then Box Cox won't do anything because power transformations are well approximated by linear transformations over short intervals.
Why does it say that power transformation is approximated by linear transformation over short intervals? And if so, why when max(y)/min(y) is small, Box Cox fails?

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(mostly copied from the comment by Nick Cox) The Box-Cox transform does not really fail: it is more that it is unnecessary, as there will be no need of transformation if max/min is small. Mostly, with max/min small all the observations are away from zero (relatively), so the power transform will be well approximated linearly over a short interval!

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Solved – Non-normally distributed data – Box-Cox transformation

The data are highly skewed & take just a few discrete values: the within-pair differences must consist of predominantly noughts & ones; no transformation will make them look much like normal variates. This is typical of count data where counts are fairly low.

If you assume that counts for each individual $j$ follow a different Poisson distribution, & that the change from low to high load condition has the same multiplicative effect on the rate parameter of each, you can extend the idea in significance of difference between two counts to a matched-pair design by conditioning on the total count for each pair, $n_j$:

$ \sum_{j=1}^m X_{1j} \sim \mathrm{Bin} (\sum_{j=1}^n n_j, \theta)$

where $m$ is the no. pairs. So the analysis reduces to inference about the Bernoulli parameter in a binomial experiment— 7 "successes" out of 24 trials if I read your graphs right.

Check the homogeneity of proportions across pairs—& note if they're too homogeneous it might indicate underdispersion (relative to a Poisson) of the original count variables.

Note that this approach is equivalent to the generalized linear model suggested for Poisson Repeated Measures ANOVA^†: while it tells you nothing about the nuisance parameters, point & interval estimates for the parameter of interest can be worked out on the back of a fag packet (so you don't need to worry about software requirements).

† Parametrize your model with the log odds $\zeta=\log_\mathrm{e} \frac{\theta}{1-\theta}$: then the maximum-likelihood estimator is $$\hat\zeta=\log_\mathrm{e}\frac{\sum x_{1j}}{\sum n_j - \sum x_{1j}}=\log_\mathrm{e}\frac{7}{24-7}\approx -0.887$$ with standard error $$\sqrt\frac{\sum n_j}{\sum x_{1j}(\sum n_j-\sum x_{1j})}=\sqrt\frac{24}{7\cdot(24-7)}\approx 0.449$$ for Wald tests & confidence intervals. If you want to adjust for over-/under-dispersion (i.e. use "quasi-Poisson" regression) , estimate the dispersion parameter as Pearson's chi-squared statistic (for association) divided by its degrees of freedom (22) & multiply the standard error by its square root.

Solved – Check transformations over variables in model

I am going to broaden the question by pointing out some things that worry me, some enormously, about your project.

Trivially, smoothness_worst is listed twice as a predictor.
Assuming that to be fixed, you are still throwing in 32 predictors into your model! That isn't anything except a recipe for poor statistical science. The predictors are a ragbag of size and shape measures of various kinds. Some have dimensions, but there is no dimensional thinking evident in your choice of predictors. For example, if some response, time in your case, is linear in area, it is most unlikely to be linear in perimeter too.
Without knowing anything specific about your application, except a hint from some names that it might be in oncology, I am prepared to bet that you really need to thin down your predictors because they mean anything, many will be highly correlated with each other. Think in terms of groups of linear size measures, area size measures, shape measures, etc. and look carefully at their correlation.
Most crucially of all, perhaps, I would guess that time is in practice always positive, in which case it's unlikely that linear regression really is the default model of choice. See e.g. this lucid post for an introduction to the argument that generalised linear models with logarithmic link are the starting point in modelling any response of this kind.

Box-Cox, its wonderful name apart, is in my view oversold. Worrying about marginal distributions should take second place in regression to choosing a functional form that makes sense for the science and statistics of your problem. That doesn't rule out transforming some of the predictors, say on dimensional grounds.

Linear regression isn't a washing machine that takes in dirty, messy data and removes the dirt and mess. You have to think your way towards a model that does justice to your data and the underlying science too.

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Solved – Non-normally distributed data – Box-Cox transformation

Solved – Check transformations over variables in model

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