Are the data in this figure normally distributed?

Question

What kind of distribution is figure 2 in this study? If it's a Gaussian distribution for the LH hormone, I can't make any sense of the data.

The article:

• Aksglaede, L. et al. (2009): Recent decline in age at breast development: the Copenhagen Puberty Study.

Link to the article: http://doi.org/10.1542/peds.2008-2491

https://ai2-s2-public.s3.amazonaws.com/figures/2017-08-08/75566aa0e00b43349dba01d4c24dac219bf84952/6-Figure2-1.png

Answer 1

A short answer would be: We don't know, but it certainly isn't normally distributed. There are two variables in the plot, so if anything, these data come from some bivariate distribution. However, perhaps you confuse model assumptions with the actual data. So I'll try to clarify those below:

First off, data is almost never normally distributed. Just think about what that would mean in this figure: The concentration of these hormones in blood would have to range from $-\infty$ to $+\infty$. At best, it could be come from a truncated normal distribution, but there is no way to prove that either.

Second, data need not be be normally distributed for regression models in the first place. Instead, we assume that if we subtract all linear fixed effects specified in the model, that the remaining differences between individuals should be random noise. Specifically, normally distributed random noise:

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(Disclaimer: I don't actually know what methods they used, the article is behind a paywall.)

In figure 2 of the article, scatter plots of hormone concentrations against age are displayed. Some trend line and what looks like confidence intervals are also drawn. I will instead take the BMI plot as an example, since it is the easiest to explain, but the following applies to any regression model assuming normal residuals.

If this were a linear regression model, then the picture below shows you what would be assumed to be approximately normal:

More specifically, in case of the linear regression:

$$\text{hormone concentration} = \beta_0 + \beta_1 \cdot \text{age} + \epsilon,$$

Imagine we subtract from each individual's hormone concentration (1) the intercept ($\beta_0$) and (2) the slope ($\beta_1$) of this model times the age of that individual, then the remainder should be normally distributed noise with mean zero and some unknown variance:

$$\epsilon = \text{hormone concentration} - \beta_0 - \beta_1 \cdot \text{age};$$

$$\epsilon \sim \mathcal{N}(0, \sigma^2).$$

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So what is the distribution of the data? We cannot know for certain, but we can use models to approximate the data. The simple linear regression I gave as an example assumes that the 'data' come from the distribution:

$$\mathcal{N}(\beta_0 + \beta_1 \cdot \text{age}, \sigma^2)$$

Although I hope to have convinced you that 'the distribution of the data' is perhaps not what you intended to ask about.

This model may not truly represent the data generating process,$^*$ but it helps us with inference nevertheless, such as estimating the effect of age on hormone concentration in this case.

$^*$: Remember, all models are wrong, but some are useful.