Is using percentile data for linear regression a valid approach? With this article from the CDC, I'm creating a log-log plot of all percentiles for the weight of men aged 25-34 (page 32) against all percentiles for the height of men aged 25-34 (page 33).
So the data looks something like...
percentile  | log(height) | log(weight)
99         |  4.3307    |   5.5134
95          | 4.301 |       5.407
90      |     4.286   |     5.338
80       |    4.268 |       5.273
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My linear regression equation ended up with a slope of 4.0527, but I know the slope should be somewhere close to 2 from previous analyses for other datasets.
Is the misconception with this approach that the xth percentile observation in height is NOT necessarily related to the xth percentile observation in weight. Therefore, the ordered pairs on the log-log plot are incorrect?
I found a similar question here, but the author is only plotting percentiles for the predictor variable and not the response variable.
 A: No, this doesn't make sense. The simplest reason is that the set of individuals in the top percentile (or any percentile) for height are not the same as the set of individuals in the top percentile for weight. So these quantities do not have any meaningful relationship to each other.
Height is associated with weight in humans, so let's take a different example to make this easier to think about.
Suppose you take a bag of bananas and measure their weights and calculate the average weight per weight decile. Separately, you measure their colour (on a continuous green to yellow scale) and calculate the average colour per colour decile. Then you plot the deciles of weight vs. the deciles of colour. You will by definition get a positive relationship whether there is no relationship between weight and colour, or even a negative relationship, simply because you have sorted them into deciles first. It doesn't matter if the yellowest bananas are the smallest, because you have chosen to make yellow the 'high' part of the colour scale and the yellowest bananas are therefore in the highest colour percentile.
So, what you need to do to understand this relationship (in fruit, or in humans) is ideally use measurements on individuals. You can extract some information from average values of groups - but the groups must be the same for measuring both the quantities you are interested in (whether human height/weight or banana weight/colour). The problem here is that you are using different groups for the height calculations (height deciles) and the weight calculations (weight deciles).
