The advantage of random sampling is that the sample will represent the target population and sampling bias is eliminated. The disadvantage is that random sampling does not work if the units of the population are heterogeneous or widely dispersed :)

Unequal probability sampling is still random sampling. 

Ratio estimates are biased in general.

[With regard to unequal probability sampling without a ratio: the Horvitz-Thompson estimator is unbiased.]

The issue here is Why use a probability of selection design that is so inefficient for the estimator that will be applied?

Dear James,

Here you are an article that could help you.

The topic here is about a design-based estimator which is most efficient if data are collected with probability proportional to the square root of x, yet historically, simple random sampling has often been used. 

The paper attached above is on a mean of ratios, which represents higher heteroscedasticity than the classical ratio estimator. The most efficient design-based method is then sampling with probability proportional to x, the usual PPS design.  Bias should generally be very tiny compared to the loss of efficiency if simple random sampling is used. 

Something that could be useful for you, Mariano, would be a book such as one of the following:

Särndal, CE, Swensson, B. and Wretman, J. (1992), Model Assisted Survey Sampling, Springer-Verlang.

or

Thompson, S.K.(2012), Sampling, 3rd ed, John Wiley & Sons.

Note that this question is not on (model-assisted) design-based sampling and estimation versus strictly model-based methodology.  It is about which design-based method to use when only considering a probability of selection design re the classical ratio estimator. 

Essentially: Why use the simple random sample (SRS) selection design instead of an unequal probability of selection sample design, specifically probability of selection proportional to the square root of x (PPsqrootx), for the classical ratio estimator?  Uncertainty should be much higher for a given sample size for a given application of the classical ratio estimator, if you use SSR, instead of PPsqrootx, for cases where you would really want to use the classical ratio estimator, ie, R estimated by sum y divided by sum x. 

Not sure why you are asking the question. At first read I thought you were challenging the design-based paradigm but it is the "simple" you are challenging. You clearly know enough to answer your own question.

Well, yes I do understand my question, and tried to make it clear, but no, I don't know under what circumstances one would use SRS with CRE.

I think that using simple random sampling (SRS) with the classical ratio estimator (CRE) is perhaps a default, but I do not know if/when it is actually justified. (However, I don't know all applications for all disciplines everywhere, and perhaps all possible tradeoffs.)

If something is in common use, it should stand questioning.

Thanks.

I suspect that this is not truly a question but rather a request for a debate. I am retired & a little rusty but I believe you are assuming multiple estimation domains with different sub-populations plus I think you are offering the the French Cube-Method (Deville) as the optimal method. I think there has been lots written on this issue.

Sorry, but that is not the topic here.

Please see Särndal, CE, Swensson, B. and Wretman, J. (1992), Model Assisted Survey Sampling, Springer-Verlang, page 254, where they mention the shape of the population.  Perhaps someone has a helpful comment on that.  Thanks.

Have you tried answering the question through simulation?

Thanks Timothy.  No, I haven't.  We have closed form solutions, but perhaps a simulation-like approach might be useful to help check a lot of population forms quickly to see if one is found where it might be reasonable to use SRS with the CRE, but I'm not in a position to do that. 

Perhaps someone has found SRS and CRE together to be useful for some application, in their experience?  

I am having difficulty fully grasping your question. Isn't the ratio estimator design-consistent rather than design-based. I thought it was only design-based in the limit. It uses a model so it is a hybrid of the design & model based Paradigms. I thought a strict following of classical design-based methods implied even design-consistent models were a no-no. If I remember, PPS is valid in a pure design-based paradigm but it is reputed to give unstable estimates. I suspect it might be because the uncertainty associated with the size variable true value is ignored. (I.e. The true PPS weights are not known, rather they are inferred from closely associated variables.) I am sorry if I am misunderstanding your words & confusing things but i am finding the question confusing. isn't Basu's elephant story based with in a pure design-based paradigm.

Sorry.  Don't know what else to say to better clarify the question at hand. 

Don't want to go off on a tangent. 

(But yes, even when talking about probability of selection designs here, this is model-assisted by virtue of the ratio, if that is really your question.)  - 

Note: PPS is not the only unequal probability design.  It actually infers greater heterocedasticity as a selection method, than the CRE does as an estimator for "R." 

...

There are a lot of good survey stat books.  Of the many I own, I think the most relevant one to this question is Särndal, CE, Swensson, B. and Wretman, J. (1992), Model Assisted Survey Sampling, Springer-Verlang.  In fact, anyone who is really interested might try to borrow that book, and see Section 7.3.3, "Optimal Sampling Design for the pi Weighted Ratio Estimator," and other relevant pages needed to understand that section.   - 

Other good sources possibly helpful for considering optimal probability of selection for ratio estimation might include  Thompson, S.K.(2012), Sampling, 3rd ed, John Wiley & Sons. 

Anders Holmberg considered what to do with multiple attribute surveys, which after all, would be the usual case:  

"Using unequal probability sampling in business surveys to limit anticipated variances of regression estimators," ICES III, Anders Holmberg:

https://www.amstat.org/meetings/ices/2007/proceedings/ICES2007-000162.PDF

 ...

That is all I have to say.  I want to hear what others have to say.  But if you are not clear on the question, say after reading the section noted above, please skip this question.  -  I have nothing more to say.  Thank you.

The point I am making is the model-assisted (or design consistent or ratio estimator) approach is not a classical design-based paradigm. Sarndal, et al note this fact. A true design-based estimator only uses the inverse of the selection probabilities to estimate totals or means, In addition, by definition, a design-based estimator is unbiased. In many design-based scenarios, the ratio estimator is not relevant to the estimation process. The usefulness of the ratio estimator comes with multiple estimation domains & a size variables. In theory when one is estimating things like total wealth or sales (i.e. a size variable), a ratio estimator can give lower variance yet still be design-based in the limit & thus unbiased in the limit. Thus under certain circumstances one can have the best of the design-based & model-based worlds. Sarndal et al's book was a real break through because it created a bridge between two apparently conflicting paradigms.

By the way I have read all those references that you mentioned plus many more. I have worked as a practicing design-based statistician for 20 years plus I consider Anders a close friend & I have been to several of Sarndal's talks plus read many of his papers & both of his books.

The design-based paradigm does not dictate that one must use SRS & in fact other selection methods are regularly used. To be design-based one must be estimating totals for finite population, the selection must be random & one must use the inverse of the selection weights to estimate. How that relates to the ratio estimator being a biased estimator, I am not understanding. It seems to me you might be intellectually moving towards abandoning the design-based paradigm completely & embracing a pure model-based paradigm.

Jack.  Please go to page 254 in Särndal, Swensson, and Wretman, where it concludes that probability of selection proportional to the square root of x is most efficient for the ratio estimator shown there.  The question is 'When would you want to use simple random sampling with that estimator anyway?'  (Perhaps it might be to be more robust against misjudging the variance structure in some specific manner?  Some other reason?)  That is the question.  Please address that.  Thank you. 

Most of what you are saying is true & has been exhaustively discussed for over a 100 years from a model-based viewpoint. Remove all references to "design-based" from your question & your question makes sense to me.

As an aside, SRS' popularity has nothing to do with mathematics. It is easy to do, often cost-effective, easy to explain to non-mathematical people and most importantly it is extremely defensible from a moral & ethical point of view. In a way, it is an implementation of Rawls "veil of ignorance". The ratio estimator is also a concept that non-mathematical people intuitively understand. Think of batting averages or completion rates.

The 3 issues you discuss are not part of a unified theory but rather 3 independent tools. The Design-based paradigm removes all "model" assumptions from the design (veil of ignorance), SRS provides a simple to understand and implement random (fair) strategy & the ratio estimator is a mathematical concept that perhaps 70% of humans intuitively grasp.This is why these 3 tools tend to be dominate when statistics are being presented to a general public.In the modern polarized world simple to understand & defensible are extremely important considerations.

After reading the discussion I believe you might be miss-understanding what "design-based" means. The key point for a "design-based" sample is the sampling procedure must be model-free. The HT-theorem & unbiased estimates are necessary but not sufficient. This was the key point in the great model versus design debate in the 1970s. Any model is human derived and is suspected of being biased (in the non-mathematical sense) & thus a no-no. The killer blow to the modelers was an example where a minor mis-specification of a linear model produced much worse results than the model-free design-based estimates. Your whole discussion is about a specific model that was used in this killer discussion. The ratio estimator is not a model-free "design-based" sampling strategy.

Again.  This was just about which probability of selection design to use with a CRE: simple random sampling, or a specific unequal probability of selection design, as noted.  I gave you a Särndal, Swensson, and Wretman page reference, in particular. 

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The x-values give you information for the estimator, so yes, any ratio is at least model-assisted.  Is the SRS easier, yes, but hardly cost-efficient here. 

You mentioned Särndal.  Yes, he is a good reference.

You also seem to be making this about prediction, which it was not.  But if you want to learn more about that, you could read

Finite Population Sampling and Inference: A Prediction Approach,  2000,

Richard Valliant, Alan H. Dorfman, Richard M. Royall,

Wiley Series in Probability and Statistics.  

Or

An Introduction to Model-Based Survey Sampling with Applications, 2012,

Ray Chambers and Robert Clark,  Oxford Statistical Science Series. 

There are rebuttals/examples to show the usefulness of models which refute the Hansen, Madow, Tepping paper and later article you apparently reference, but that is not what this question was about.  (Some might say that paper was rather artificially 'rigged,' just as one might do for the opposite argument.)  But this question ... again ... was not about that.  There has been a lot of discussion pro and con there.  I'm not rehashing it here.  (Ken Brewer liked Combined Survey Sampling Inference  (2002), by the way.)

You have made your opinions clear, but not in the right forum.  Please stop.  This is inappropriate. 

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By itself, I could have accepted "easier" as an answer for using SRS with the CRE, though giving a small ma & pa operation the same chance of selection as a huge corporation seems hardly reasonable, cost-effective or fair.  Unequal probability sampling is not that unusual.  Even a stratified random sample involves unequal probabilities.  

Again, Jack.  This was just about which probability of selection design to use with a CRE: simple random sampling, or a specific unequal probability of selection design, as noted.  I gave you a Särndal, Swensson, and Wretman page reference, in particular.  Since you said you enjoyed Särndal, I thought you might relate. 

intersting question, I Follow

I suggest to go through the book

"Sampling Technique"

by W. G. Cochran.

Cochran(3rd ed, 1977) is a very good book.  But better here to see Thompson(2012), page 102, for the generalized ratio estimator, for example, which uses unequal probabilities of selection. 

Thompson, S.K.(2012), Sampling, 3rd ed, John Wiley & Sons.

Cochran, W.G(1977), Sampling Techniques, 3rd ed., John Wiley & Sons. 

Also please see https://www.researchgate.net/publication/319434665_Comparison_of_Model-Based_to_Design-Based_Ratio_Estimators

To clarify, this question is about equal versus unequal probabilities of selection, used with ratio estimation.

You want to assume a model a priori (i.e. the model underlying the ratio estimator) & build your sample selection & estimation strategy around this model. This is the antithesis of the design-based strategy.which must be model free a priori. Sarndal et al introduce the ratio model not because they believe it is the true model but rather because in the limit it is design-based & model free. In finite population cases however, Hansen et al showed that assuming the ratio-model a priori & building both your sampling & estimation strategy around this model can be extremely dangerous. Sarndal et al's proposal may be schizophrenic but they showed it is model free in the limit & furthermore, practice has supported their assumptions. Practice seems to show that in certain types of estimation problems it works better than the design-based & the model-based approach.

I have no difficulty with what you are proposing but what you are proposing is neither design-based nor design-consistent: you are proposing a pure model-based approach. You want a pure & consistent "ideology" but the design-consistent strategy's strength is its hybrid approach based upon melding the two conflicting ideologies. Your answer to rounding the square created by Sarndal's kludge is to make it a pure model-based approach while my answer to rounding the square is to abandon the kludge & use a pure design-based approach. My answer is much easier to defended than your fix.

What you propose has been suggested & tried many times & it usually fails badly because the ancillary variable (x) is only partially correlated with y & the weights themselves become a source of instability. Random errors associated with big x's get magnified. (Note that all real life random shocks in social or economic data come from heavy-tailed & skewed distributions.) In practice, Sarndal's approach tends to be a more stable estimator. That is also one of the reasons why SRS is preferred, it tends to have better stability & convergence properties. Sarndal's approach with SRS is a kludge but it has good theoretical properties & in practice it seems to work.

Plus as Fred Smith & other authors have pointed out design-consistent or design-based approaches using SRS are easily explained to lay-persons & highly defensible using simple heuristic arguments. Trying to explain what you are proposing to groups of people with opposing & hostile viewpoints can be challenging. The information that central statistical agencies publish has real life consequences & must appear to be seen to be reasonable by lay-persons, When you talk to lay-persons, you cannot use any mathematics in your discussions.

You do realize that you will be essentially telling the doubting Thomases that God spoke to & told you the model underlying the universe is the ratio-model. As soon as the critics start to grasp the narrowness of your world-view you will be in trouble. Trying to argue that your chosen model is the true unbiased one will be challenging. On the other hand, if we eliminate Sarndal's kludge & use an ideology pure design-based approach I can argue that I am assuming nothing & I am not injecting any personal biases into the results. My methodology will then be perceived to be "fair" while yours will be suspect.

You want pureness of ideology. When we deal with humans & real-life issues that is rarely possible.

You should read Brewer. He is a modeler by inclination but he gets the philosophy underlying the design-based paradigm.

As I tried to explain before Jack, I think you are going off on a few tangents.  Here I'm just saying that sum y over sum x, as I have noted was pointed out in  Särndal, CE, Swensson, B. and Wretman, J. (1992), Model Assisted Survey Sampling (Springer-Verlang), is just not consistent with simple random sampling.  I'm not sure what you have against unequal probability sampling when you have highly skewed populations.  I'm sorry if this is not clear.  Perhaps I worded things in a way that confused the description of the topic.

Regarding my friend and mentor, Ken Brewer, he did think enough of my work that I was invited to publish an article in the Pakistan Journal of Statistics, in an issue in honor of Ken's 80th birthday, in 2011.  (He was on the board for that journal.)  Yes, I know his work.  I have an autographed copy of a pre-print of his book (with a pretty funny story behind that).

Many may find the following interesting and entertaining:

Brewer, K.R.W. (2014), “Three controversies in the history of survey sampling,” Survey Methodology,

(December 2013/January 2014), Vol 39, No 2, pp. 249-262. Statistics Canada, Catalogue No. 12-001-X.

http://www.statcan.gc.ca/pub/12-001-x/2013002/article/11883-eng.htm

I think that every application should be approached from a practical point of view.  One size does not fit all. 

I think, Jack, that in general we will just have to agree to disagree.  Bye.