A question: What is the relationship between the parameters (e.g., mean, SD) that you will specify for your extracted sample and the corresponding parameters for the population you will sample from?

I'm not sure if I understood your question correctly. I'm going to sample 100 individuals randomly from the population but their weight, height, and age should be in the desirable ranges that I'm looking for. In other words, weight, age and height of the randomly selected individuals should fall within the distributions of my interest. I do have covariance matrix of the variables in the population data but I'm not interested in generating multivariate random data. Instead, I need actual individual data from the population data set that I have full access to (e.g. 1200th individual in the population data is randomly selected as one of the 100 individuals who has desirable weight, age and height).

I should note that sampling one set of 100 individuals might not be very challenging, but as this process is going to a large number of iterations (m times sampling 100 individuals), I'm looking for a fast and efficient sampling method.

Please let me know if I didn't answer your question.

I think your problem is more or less similar to the Knapsack problem or Simulated Annealing. However, different from Knapsack problem or Simulated Annealing, which aims to find the optimum results, your problem aims to find results obeying your rules. You do not need the optimum results. If the samples can fit for your rules, it can be selected. I think you need a function to describe your rules, and then you can random select some instances, if these instances meet your requirements, then you get the results, otherwise, get rid of some of these instances, and random select some new instances, and test the new dataset by your rules...Such a random selection and data replacement process also can be programmed by Genetic Algorithms.

Hope I have understood your problem, and my answer can help you.

if you have a predefined range you could just use logical sub setting, no?

domething like (in R):

minH

Since you have changed your sample generation criterion from one involving some stated mean and SD, to one based on specified ranges for each attribute's values here is an off-the-cuff thought. Consider two attributes, A and B. Specify a range of values for each -- Bucket A and Bucket B.

1. Generate a subset of records whose attribute A values fall in Bucket A. Create another whose attribute B values fall in bucket B.

2. Take the intersection of the two subsets.

Now whether the result meets your sample size requirement remains to be seen. In fact the intersection may well be empty. If you are arbitrarily picking a range or bucket for one attribute without regard for any other attribute, all you can do is repeat the above procedure (with different bucket choices) in the hopes of finding what you seek. And since you wish to have m replications, this will be an arduous task indeed.

Instead, why not just generate an arbitrary sample of n observations and see if the resulting attribute value ranges are acceptable for your purpose. Discard if not and repeat. Go through this m times.

it appears to be a case of multi (you have k criterion)variable stage QUOTA sampling.

I'd add the following to the observations made by others:

1] My initial question to you was prompted by the concern that Benjamin elaborates on when he notes, "remember that you'll still be bound by the population distribution so forcing a mean will likely get you skewed height, weight and age distributions."

2] Assuming you have some reason for not merely taking m random samples of size n each from the population, it would help for us to know why you wish to proceed as you state. That could cue us in on suggesting other ways of accomplishing what you seek. Subrata notes, what you seek is some kind of Quota Sampling and, generally, Quota Sampling is applied during primary data gathering for reasons of convenience, time or budgetary restrictions, and such usually at the cost of reduced sample reliability (which is what Benjamin is pointing out). However, you are dealing with secondary data and have access to computing power to carry out your sampling. This makes me wonder why would you want to establish quotas in the manner that you seek.

3] The GA approach that Ting advocates may be worth thinking about. However, my initial thoughts would be it may be be overkill for what you seek and perhaps just as demanding of computing resources as enumeration schemes that come to mind.

Given the k variables of interest (x, y, z,…), let’s define m=[mx, my, mz,…] and S the target vector of the means and the target variance-covariance matrix of x,y,z,.., respectively.

The problem of drawing from the population U a sample of size n (say, n=100) whose mean vector and covariance matrix are just the specified targets (vector m and matrix S) is to be addressed in a multivariate approach.

A)

If for the correlated variables the assumption of multinormal distribution holds, then to solve the problem one should consider the (hyper-)ellipse of confidence (in the real space Rk) centered at the point m of coordinates [mx, my, mz,…].

This implies the spectral decomposition of the covariance matrix S, which yields the k eigen-values (Lx, Ly, Lz, …) .

For the chosen significance level (e.g., alpha = 0.05), the length of the jth axis (for j=x,y,z,…) of the (hyper-)ellipse is reported in equation 1, where f(k,n-k,1-α) is the F inverse cumulative distribution function with k and n-k degrees of freedom calculated at the probability 1-α. This ellipse of confidence has probability 1-α of containing the target vector of means m.

Now, in the data set U one can select the sample according to the targets (m and S) just by filtering the data across the k variables in this way: mj  aj/2, that is by implementing a simple routine:

for i= 1 to n

for j= 1 to k

get the U(i,j) such that [U(i,j) ≥ (mj - aj/2) & U(i,j) ≤ (mj + aj/2)]

This method is easy to apply with the filter tool in a Excel spreadsheet or with SQL commands into more complex data sets.

B)

If for the correlated variables the assumption of multinormal distribution doesn’t hold, one can rely on a multivariate formulation of the Čebyšëv inequality.

Equation 2 shows the Chen’s generalization, where the terms inside the square brackets represent the Mahalanobis distance around the target vector m with dispersion S. Thus, for a chosen ε>k, the probability of selecting a points from U arbitrarily close to m is not lower than 1 - (k/ε).

Then, effectively, one can calculate for each record of the data set U the corresponding Mahalanobis distance and take the outcomes that are lower than ε. From this stratified data, one can draw the sample of size n.