# How can I interpret a regression model with a reciprocal response?

I fitted a multiple linear regression model as the following $$BMI = \beta_0+\beta_1Age+\beta_2Gender+\beta_3Age*Gender+e$$ But the residuals obtained after fitting the model have violated both normality and homoscedasticity assumptions. I use the reciprocal transformation of $$BMI$$ to solve this and it satisfies the assumptions of normality and homoscedasticity. I tried log transformation, and square root transformation but the problem still exists. So, I used the reciprocal of $$Y$$, and the model becomes $$\frac{1}{BMI}=\beta_0'+\beta_1'Age+\beta_2'Gender+\beta_3'Age*Gender+e$$ Now, how can I interpret the result (main effects, and the interaction) obtained by the model with reciprocal response $$BMI$$? And how can I link the results to the original response itself?

Since the body-mass index is defined as $$\text{BMI} = \text{weight (kg)}/\text{height (m)}^2$$, you can rewrite your reciprocal model (now dropping the units) as:

$$\text{height}^2 = \text{weight} \times (\beta_0'+\beta_1'\text{Age}+\beta_2' \text{Gender} + \beta_3' \text{Age} \cdot \text{Gender} + \text{error}).$$

So, by modelling BMI as a reciprocal linear equation, you get height-squared as the product of weight and that linear predictor. This is certainly one model you could try, but the restriction of holding BMI fixed relative to height means that you are unlikely to get a realistic model.

An alternative: One thing worth noting here is that you are working with a derived quantity that depends on two underlying quantities (height and weight). If you can get the underlying data for these two quantities then it is probably more sensible to try to model one of them in terms of the other (plus the other explanatory variables). In this context, it probably makes most sense to model log-weight in terms of log-height and the other variables, using a model of the form:

$$\log \text{weight} = \alpha_0 \log \text{height} + \alpha_1 \text{Age} + \alpha_2 \text{Gender} + \alpha_3 \text{Age} \cdot \text{Gender} + \text{error}.$$

As a general physical observation, if a three-dimensional object growths proportionately in all dimensions then its log-volume will increase at three times its log-height, and if weight remains proportionate to volume then its log-weight will similarly increase at three times log-height. Now, under this "simplifying theory" we would have $$\alpha_0=3$$ in the above model. However, it is well known that living organisms like humans do not exhibit exactly this growth pattern, in part because they grow relatively thinner when taller. Nevertheless, you can look at the coefficient $$\alpha_0$$ by comparison to this theoretical baseline ---i.e., a value $$\alpha_0 = 3$$ follows this simplifying theory, whereas a value $$\alpha_0<3$$ means that there is some reduction in relative volume as the object gets taller, relative to the simplifying theory.

• As BMI is quite a standard concept and much analyzed, proposing to bypass it have a low probability of success ... at the wikipedia article I linked, there is a proposal for a $\text{BMI}_\text{New}$ that uses height to the power 2.5, which the proposer says fit the data better ... Jun 8 at 2:54
• Fair enough, but I'm not terribly interested in whether it's popular, only whether it's right. The problem with BMI or BMI-New (now with raisins!) is that it fixes the proportionality coefficient rather than just fitting the underlying variables to the data and seeing what happens.
– Ben
Jun 8 at 2:59

BMI (Body Mass Index) is weight (kg) divided by height$$^2$$ (m$$^2$$) so can be seen as a measure of thickness, so its inverse will be a measure of thinness. Seen that way you have a linear model for thinness, and can interpret as usual for linear regression.

For interpretation on the original BMI scale, maybe make plots of the estimated model. For more specific advice tell us more of the context:

• Is this a longitudinal study, so you are following a cohort of people over time, so at different ages?

• Or a population study, so each person observed only once?

But in most cases the model seems strange, published papers on BMI often use quantile regression ... Here is a paper on new regression methods for BMI, Intelligence Test Score and Educational Level in Relation to BMI Changes and Obesity is a longitudinal study of changes in BMI with age,

• Normal residuals are a bit of a hobgoblin or misleading. The goal is usually to identify the pattern of weight in the population, hence the calculation of BMI to standardize for height. Try some graphs of BMI by age for men and women and separate models for men and women. You might find them to be nonlinear or to be extremely weak. Jun 8 at 2:11