# Brewer's method for sampling with unequal probabilities with n>2

I'm trying to use Brewer's method to sample 12 units out of a population of 73.

I read on Brewer and Hanif's "Sampling with unequal probabilities" that the probability must be proportional to $$\frac{P_i (1 - P_i)}{(1-rP_i)}$$.

I know that for $n = 2$ the first unit is selected with probability $$\frac{P_i (1 - P_i)}{D (1-2P_i)}$$ where $$D = \sum_{i=1}^N\frac{P_i(1-P_i)}{(1-2P_i)}$$ and the second unit is selected with probability $$\frac{P_j}{1-P_i}$$.

I would like to know if this selection probabilities are the same for $n > 2$ or if I have to change the probabilities for the first unit according to the sample size.

• Your third formula for the second-draw probability is incorrect; The correct formula is on page 52 of support.sas.com/documentation/onlinedoc/stat/131/…. Please make the correction to your question. I will then up-vote and answer it. (Or, you can answer it yourself with information from the linked document.) – Steve Samuels Aug 10 '14 at 21:20
• Thanks for letting me notice my mistake. The probabilities I used are the correct ones, I just wrote them wrong here. – Roberto Aug 11 '14 at 13:09
• Which method from Brewer & Hanif are you referring to? They list 46. And by the way there are new cool methods that have appeared since B&H 83 although that book is still The Bible of unequal probaility sample (Springer should totally retype it in LaTeX and publish it, along with these new methods). – StasK Aug 11 '14 at 18:01
• @Roberto, you have not described it well and fully; you might be better off reproducing the whole algorithm from B&H book. Steve, I am no Ken Brewer, but I think Roberto is right (I checked with the book); SAS manual gives a simplified version only for $n=2$, while the book gives the procedure for arbitrary $n$. The undefined symbol $r$ is the index of the sampled unit. B&H don't say anything about what happens when $1-rP_i$ gets $<0$, which may happen practically. – StasK Aug 11 '14 at 18:09
• I'll just note Roberto's previous question on this topic: stats.stackexchange.com/questions/108635/… – Steve Samuels Aug 12 '14 at 2:55

My reading of Brewer's procedure is as follows.

1. To sample the first unit, set $r=n$ and compute $$D_1 = \sum_{i=1}^N\frac{P_i(1-P_i)}{(1-nP_i)}$$ Then sample the first unit with probability $$P^{(1)}_i = \frac{P_i (1 - P_i)}{D_1 (1-nP_i)}$$Let the index of the first sampled unit be $I_1$.
2. To sample the second unit, set $r=n-1$ and compute $$D_2 = \sum_{i\in\{1,\ldots,N\}, i\notin \{ I_1 \} }\frac{P_i(1-P_i)}{(1-(n-1)P_i)}$$ Then sample the second unit with probability $$P^{(2)}_i = \frac{P_i (1 - P_i)}{D_2 (1-(n-1)P_i)}$$ Let the index of the first sampled unit be $I_2$.
3. etc.

To sample the $k$-th unit, set $r=n-k+1$ and compute $$D_k = \sum_{i\in\{1,\ldots,N\}, i\notin \{ I_1, \ldots, I_{k-1} \} }\frac{P_i(1-P_i)}{(1-(n-k+1)P_i)}$$ Then sample the $k$-th unit with probability $$P^{(k)}_i = \frac{P_i (1 - P_i)}{D_k (1-(n-k+1)P_i)}$$

B&H 83 refers to Brewer (1975) in Australian J of Statistics which I don't see any way of getting.

• Very nice, Stas! I think Sampford's procedure (see my answer) is probably easier to implement. I too have neither Australian Journal of Statistics reference. – Steve Samuels Aug 11 '14 at 22:16
• Sampford's method may be simpler but it is clearly less efficient, getting worse as $n$ increases. – Xi'an Mar 3 '16 at 7:47

Update: August 19 I've now had a look at both Brewer's 1963 and 1975 articles. The 1975 method is indeed the generalization for (n>2) as Stas stated, with the same draw probabilities for n = 1 & 2.

This has been a very informative topic. I wondered why there were two versions of "Brewer's Method" in Roberto's question.}

The first formula Roberto gave is from "Brewer's Procedure" (Procedure 8) in Brewer and Hanif (1982). As Stas points out, it is based on Brewer (1975). However the second version that Roberto shows is that which Cochran (1977, p. 261) and the SAS manual also call "Brewer's Method" for n = 2. This turns out to be from Brewer (1963). It doesn't appear in Brewer & Hanif (1982) It does, as Procedure 8.

Durbin (1967, p.54) created another method for n = 2 (Method I). It is Procedure 9 in B & H. Durbin was unaware of Brewer (1963), and prior to publication, JNK Rao pointed out to Durbin that the pairwise inclusion probabilities are identical to those in Brewer's 1963 method.

Sampford (1967) extended Brewer's 1963 method to n>2. He gave several suggestions for implementation. That used by SAS is what Sampford called "method c" (p. 502). It is Procedure 11 in B & H, who call it the "Rao-Sampford Rejective" method. It is interesting because all draws are made with replacement; if a duplicate appears, the sample is rejected.

Sampford's method is easy to write down. Define

$$\lambda_i = p_i/(1-n p_i)$$

1. Draw the first unit with probability $p_i$.

2. Draw all others with probability $\lambda_j/\sum_k^N \lambda_k$, with replacement

3. If a drawing duplicates any previously selected item, reject the sample and begin again with Step 1.

The joint probabilities $\pi_{ij}$ of inclusion for elements $i$ and $j$ are needed to compute a proper variance estimate for the Horvitz-Thompson ($p_i$ weighted) estimator.. Sampford gives the formula for $\pi_{ij}$ on page 503 of his article and some examples of its calculation in his section 5. See also the SAS SURVEYSELECT manual, p. 8493.

References

Brewer, Kenneth RW 1963. A Model of Systematic Sampling with Unequal Probabilities. Australian Journal of Statistics 5, no. 1: 5-13.

Brewer, Kenneth RW (1975). A Simple Procedure For Sampling $\pi$pswor1. Australian Journal of Statistics, 17(3), 166-172.

Brewer, Kenneth RW, and Muhammad Hanif. 1982. Sampling with unequal probabilities (lecture notes in statistics).

Cochran, William G. 1977. Sampling Techniques. New York: Wiley.

Durbin, J. 1967. Design of multi-stage surveys for the estimation of sampling errors. Applied Statistics 16, no. 2: 152-164.

Sampford, MR. 1967. On sampling without replacement with unequal probabilities of selection. Biometrika 54, no. 3-4: 499-513.

SAS SURVEYSELECT MANUAL http://support.sas.com/documentation/onlinedoc/stat/131/surveyselect.pdf

• This cleared out my confusion about the differences in the Brewer's procedure in the different references. Thanks. – Roberto Aug 20 '14 at 12:55
• Hey @SteveSamuels, how on earth did you get these papers??? – StasK Mar 16 '16 at 16:37
• From university subscriptions for Durbin and Sampford and inter-library loan for the Brewer. – Steve Samuels Mar 18 '16 at 22:24