Linear regression effect sizes when using transformed variables

Question

When performing linear regression, it is often useful to do a transformation such as log-transformation for the dependent variable to achieve better normal distribution conformation. Often it is also useful to inspect beta's from the regression to better assess the effect size/real relevance of the results.

This raises the problem that when using e.g. log transformation, the effect sizes will be in log scale, and I've been told that because of non-linearity of the used scale, back-transforming these beta's will result in non-meaningful values that do not have any real world usage.

This far we have usually performed linear regression with transformed variables to inspect the significance and then linear regression with the original non-transformed variables to determine the effect size.

Is there a right/better way of doing this? For the most part we work with clinical data, so a real life example would be to determine how a certain exposure affects continues variables such as height, weight or some laboratory measurement, and we would like to conclude something like "exposure A had the effect of increasing weight by 2 kg".

Answer 1

I would suggest that transformations aren't important to get a normal distribution for your errors. Normality isn't a necessary assumption. If you have "enough" data, the central limit theorem kicks in and your standard estimates become asymptotically normal. Alternatively, you can use bootstrapping as a non-parametric means to estimate the standard errors. (Homoskedasticity, a common variance for the observations across units, is required for your standard errors to be right; robust options permit heteroskedasticity).

Instead, transformations help to ensure that a linear model is appropriate. To give a sense of this, let's consider how we can interpret the coefficients in transformed models:

• outcome is units, predictors is units: A one unit change in the predictor leads to a beta unit change in the outcome.
• outcome in units, predictor in log units: A one percent change in the predictor leads to a beta/100 unit change in the outcome.
• outcome in log units, predictor in units: A one unit change in the predictor leads to a beta x 100% change in the outcome.
• outcome in log units, predictor in log units: A one percent change in the predictor leads to a beta percent change in the outcome.

If transformations are necessary to have your model make sense (i.e., for linearity to hold), then the estimate from this model should be used for inference. An estimate from a model that you don't believe isn't very helpful. The interpretations above can be quite useful in understanding the estimates from a transformed model and can often be more relevant to the question at hand. For example, economists like the log-log formulation because the interpretation of beta is an elasticity, an important measure in economics.

I'd add that the back transformation doesn't work because the expectation of a function is not the function of the expectation; the log of the expected value of beta is not the expected value of the log of beta. Hence, your estimator is not unbiased. This throws off standard errors, too.

Answer 2

The question is about marginal effects (of X on Y), I think, not so much about interpreting individual coefficients. As folk have usefully noted, these are only sometimes identifiable with an effect size, e.g. when there are linear and additive relationships.

If that's the focus then the (conceptually, if not practically) simplest way to think about the problem would seem to be this:

To get the marginal effect of X on Y in a linear normal regression model with no interactions, you can just look at the coefficient on X. But that's not quite enough since it is estimated not known. In any case, what one really wants for marginal effects is some kind of plot or summary that provides a prediction about Y for a range of values of X, and a measure of uncertainty. Typically one might want the predicted mean Y and a confidence interval, but one might also want predictions for the complete conditional distribution of Y for an X. That distribution is wider than the fitted model's sigma estimate because it takes into account uncertainty about the model coefficients.

There are various closed form solutions for simple models like this one. For current purposes we can ignore them and think instead more generally about how to get that marginal effects graph by simulation, in a way that deals with arbitrarily complex models.

Assume you want the effects of varying X on the mean of Y, and you're happy to fix all the other variables at some meaningful values. For each new value of X, take a size B sample from the distribution of model coefficients. An easy way to do so in R is to assume that it is Normal with mean coef(model) and covariance matrix vcov(model). Compute a new expected Y for each set of coefficients and summarize the lot with an interval. Then move on to the next value of X.

It seems to me that this method should be unaffected by any fancy transformations applied to any of the variables, provided you also apply them (or their inverses) in each sampling step. So, if the fitted model has log(X) as a predictor then log your new X before multiplying it by the sampled coefficient. If the fitted model has sqrt(Y) as a dependent variable then square each predicted mean in the sample before summarizing them as an interval.

In short, more programming but less probability calculation, and clinically comprehensible marginal effects as a result. This 'method' is sometimes referred to CLARIFY in the political science literature, but is quite general.

Answer 3

SHORT ANSWER: Absolutely correct, the back transformation of the beta value is meaningless. However, you can report the non-linearity as something like. "If you weigh 100kg then eating two pieces of cake a day will increase your weight by approximately 2kg in one week. However, if you weigh 200kg your weight would increase 2.5kg. See figure 1 for a depiction of this non-linear relationship (figure 1 being a fit of the curve over the raw data)."

LONG ANSWER:

The meaningfulness of the back transformed value varies but when properly done it usually has some meaning.

If you have a regression of natural log values on two x predictors with a beta of 0.13, and an intercept of 7.0, then the back transformation of 0.13 (1.14) is pretty much meaningless. That is correct. However, the back transformation of 7.13 is going to be a value that can be interpreted with some meaning. You could then subtract out the back transformation of 7.0 and be left with a remainder value that is your effect in a meaningful scale (152.2). If you want to look at any predicted value you would need to first calculate it all out in log values and then back-transform. This would have to be done separately for every predicted value and result in a curve if graphed.

This is often reasonable to do if your transformation has a relatively small effect on your data. Log transformation of reaction times are one kind of value that can be back transformed. When it's done correctly you'll find that the values seem close to median values doing simple calculations on the raw data.

Even then though one must be careful with interactions and non-interactions. The relative values vary across the scale. The analysis was sensitive to the log value while the back transformed values may show different patterns that make interactions seem like they shouldn't be there or vice versa. In other words, you can back transform things that make small changes to the data as long as you're careful.

Some changes, like logistic transform of probability, can have quite massive impacts, especially near the end of the scale. An example of a place you should never back transform is interaction plots near the high or low end of probability.