Variables importance: who can do the most pushups?

Question

I don't know enough math to formulate an intelligent question on this so I'll give an example. I'd like an answer to my example but also I'd like to know the jargon I need to be able to research it further.

Suppose you have a large set of data of information about people: their height, weight, age, and number of pushups they can do. How would you go about finding:

1. Which height, weight, and age combinations can do the most pushups?
2. Given a height, weight, and age, what is the probability they can do x number of pushups?

Ultimately I'd like to know, given a set of 10 people's heights, weights, and ages, who would statistically be the most likely to do the most pushups.

Answer 1

It sounds like you want to perform a regression analysis on your data.

Regression analysis is like drawing a 'line of best fit' through your data. The input to the regression analysis is your data points, consisting of the independent values (height, weight, age) and the dependent value (number of pushups.) The output is a formula you can use to estimate the value of the dependent variable, given some arbitrary combination of independent variables.

For example, the regression analysis might tell you that the number of pushups can be estimated as:

$\textrm{Num. of Pushups} \approx 1.01\times\textrm{weight}[kg]+0.11\times\textrm{height}[cm]+0.13\times\textrm{Age}[years]$

Trying to say which of these variables is 'most important' is open to interpretation.

Regression analysis warrants an entire university subject of its own, but hopefully this should give you a start. Some particular pitfalls are:

• The type of regression used above is a linear regression - it assumes a linear relationship between the independent and dependent variables. This is often not the case, and can lead to grossly misleading results.
• Your model is only as good as the data you put into it. The more data you put in, the more accurate the model will be.
• There are a whole bunch of assumptions underlying the regression process (i.e. that the data is normally distributed.) Significant violations of these assumptions lead to inaccurate models.

The Wikipedia article is good (if dry) reading. Alternatively, try and find some university course notes on the subject.

Have fun!

EDIT: the best thing to start reading is probably the Wikipedia article on linear regression, which should be good enough for your purposes

Answer 2

You might be looking for the conditional density function of the number of pushups given the other data (height, weight and age). It is the quotient of the joint density of the number of pushups and the other data with the marginal density of the other data. (Borel's paradox is an indication that this answer is more ambiguous than it looks.) To apply this, you would need to estimate the probability density function from your large set of data of information. The most basic form of density estimation is a rescaled histogram. Hopefully that is enough jargon now.

Let's try to answer the first question. When using a histogram, you have to subdivide height, weight and age into coarse enough classes, which depend a bit on how large your set of data is. Let 1.5 - 1.6 m be one class and 1.6 - 1.7 m be another class for the height. Similarly, let 50 - 60 kg be one class and 60 - 70 kg be another class for the height. Finally let 10 - 20 years be one class and 20 - 30 years be another class for the age. You could also introduce classes for the number of pushups, but it's not strictly required (even so it may be easier in order to get a 'real' probability density function). For a certain (height, weight, age) combination, you can now lookup the individuals from your data set that are in the same classes. This gives you an empirical distribution of the number of pushups for this (height, weight, age) combination (assuming there are matching individuals for this combination in your data set).

The question who can do the most pushups is more tricky, since we have a distribution for the number of pushups, not a fixed number. This question is easier to answer for your second question, where there are just 10 peoples for which the distributions must be compared.

Let's try to answer the second question. One way is to lookup the empirical distribution for each of the 10 people based on the classes of the histogram described above. Then you could compute the probability for each person to do the most pushups by summing up the probability of each number of pushups for that person multiplied by the probabilities of the smaller or equal number of pushups from the each of the other 9 distributions. This gives you 10 probabilities, one for each person. The sum of these probabilities can be larger than one, because if two persons can do the same number of pushups, both would be considered being able to do the most pushups.

However, you could also answer the second question more accurate independent of the first question. To get the empirical distribution of the number of pushups for one person, you define when you consider another person to have the same height/weight/age. To make it simple, lets say they are the same if the difference is smaller than 5. So now the individuals from your data set that match the height/weight/age of that person according to this criteria give you the distribution of the number of pushups for that person. The final step is once again to compare the distributions of the 10 persons as I tried to describe above. You could even generalize this approach, and measure the similarity of the height/weight/age of two persons by a real number between 0 and 1 (instead of just using 0 or 1 as done above).

Note that this is the answer from an mathematician, not a statistician, because this question was initially posted at math.stackexchange.com.