# How many cases are needed for an extrapolation?

In social science it is comon to use sample weight to post adjust the sample so that it fits to a given basci population. There are algorithms to calculate how many cases you need to have a representive sample of the population. But often we would like to analyse a bit deeper than just the sample as a whole e.g. male vs. female or age groups. Theoretically - if I'm not wrong - we need for each field a representive number of cases. So if I need 360 cases in the whole sample, I need for the analysis of male vs. female 360 * 2 (one for each field)

If we use weights, the question is on who many cases we the extrapolation is based. But how many cases I need at least to extrapolate my sample? I have the following table with observed cases per field:

                       Male                 ||            Female
Education                   ||      Education
age   |  low  | middle| high  |  sum     ||  low  | middle|  high |  sum  |  SUM
---------+-------+-------+-------+----------++-------+-------+-------+-------+------
18 - 24  |   26  |   60  |   23  |  109     ||   33  |   45  |   20  |   98  |  207
25 - 39  |    6  |   78  |   94  |  178     ||    8  |  114  |  147  |  269  |  447
40 - 49  |    1  |  105  |  134  |  240     ||    6  |  124  |  172  |  302  |  542
50 - 59  |    6  |   63  |  117  |  186     ||    8  |  117  |  146  |  271  |  457
60 - 69  |    3  |   48  |  110  |  161     ||   17  |   99  |  122  |  238  |  399
70+      |    7  |   30  |   75  |  112     ||   41  |   89  |   58  |  188  |  300
---------+-------+-------+-------+----------++-------+-------+-------+-------+-------
SUM      |   49  |  384  |  553  |  986     ||  113  |  588  |  665  | 1366  | 2352


If we take a look at the males with a low education in the different agegroups, I have doubt, that we can extrapolate them. But I can't argue that right now. Even if we take just the low educated males at all, I don't think that the amount of cases are enough to extrapolate them.

What I have to do is to analyse injuries in sex, education leavel and age groups. But because of this distribution table I don't know, if the reslaults are representive for the basic population wich is over all about 2 million people.

So here my questions:

• How I can figure out if a number of cases is big enough to get extrapolated
• Do I really need, as I mentioned above, the amount of samples per field e.g. 360 males in the age of 18-24 with a low education level to analyse this group?
• On which categories (e.g. age groups, sex or combined) I can run representive calculations?

Maybe there is also something with which I can do the extrapolation with R and something like a test how good my extrapolation is...

Thanks for y'all help. Dominik

• lejohn has a very good answer, you should vote it up (and potentially accept it if you don't get one you like better) – John Aug 17 '11 at 20:29

There is no absolute threshold for this. Neither is there a test. It really depends on you, on how far you want to go, and on how much confidence you have in your data.

You could have a look at the precision of the estimates by computing confidence intervals or coefficients of variation. This could give you some guidance, or could help you to make an assessment.

As far as R is concerned, you might have a look at the Official Statistics task view. It contains several packages to handle survey data.

You might also be interested in what is called "small area estimation". These are techniques to estimate parameters from small (sub)populations. Small area estimation is covered in Sharon Lohr's textbook. A more complete reference on the topic is Rao (2003).

Follow-up

Regarding the threshold question, I would like to quote from Rao's 2003 book mentioned above:

A domain (area) is regarded as large if the domain specific sample is large enough to support direct estimates of adequate precision.

If you have a look at texts on small area estimation, they will hardly get more precise on the size issue. This isn't really a surprise. If you have a rather homogenous (sub)population, you can get away with rather small samples. In the extreme case, where a (sub)population is perfectly homogenous, a sample size of 1 will be enough to obtain perfectly representative estimates. If the (sub)populations become more heterogeneous, you will need larger sample sizes, ceteris paribus. That's why I said in the beginning that the answer depends on the situation at hand and that coefficients of variation and confidence intervals can provide some guidance.

• Thanks for your answer, especially for the both book links. But I can't imagin, that there is no rule, not even a rule of thumb. It seems obvious for me, that one respondend can't be taken as a basis for a whole population. But what might be a respectable amount of cases? In the described case, I would have no problem to analyse over age classes, sex or a combination of both, but the education level give me a headache, but I can't explain why exactly there are not enough cases to do an analysis over the educational levels... – Dominik Aug 17 '11 at 13:07
• I have edited my answer. – user5644 Aug 17 '11 at 15:05
• Thank you for the update. But what are CV's and CI's? – Dominik Aug 22 '11 at 12:48
• Coefficients of variation (CV) and confidence intervals (CI) – user5644 Aug 22 '11 at 12:58

lejohn gives a great answer. I might further add that, even though it's basically impossible to say anything about specific groups of low education males in your study, it may well be possible to say something about trends. I don't know how you're doing your analysis but let's use regression as a simple example.

It could well be that you have a regression predicting injuries from education, age, and sex. The fact that one age group is under represented shouldn't impact the slope of the regression much. In that case you'd then be "extrapolating" about the general impact of age and education as opposed to extrapolating about just 40-49 year olds with low education.

Understand that extrapolation is essentially curve fitting. Even when you use a different technique it's conceptually the same as taking a bunch of data points, fitting a curve to the points, and then extending the curve outward beyond those points. If the points are not too noisy, and if you pick the right polynomial or other curve equation to fit to the points, and if you use the right curve-fitting algorithm, your resulting curve may resemble "reality" fairly closely.

Lots of ifs, though.

• I understand that extrapolation is curve fitting. But the problem with that is, which points you have... Take a sinus curve for example. If you just have the points around the zero point, than you would expect a complete different curve than if you would have a wide spreaded points... The question is therefore, if there is an estimation if we have engough points and if the points are representive engough to do a meaningfull curve fiting... – Dominik Aug 22 '11 at 9:31
• I think the main problem is that you never know (without some domain knowledge, at least) what sort of curve fits your data. A second order polynomial, an exponential, and 1/x all look similar for small parts of their curves, but have radically different extremes. Pick the wrong one and you can be off by a mile, only millimeters from where they seemed to match. – Daniel R Hicks Aug 22 '11 at 20:41
• (I ran into this situation in college, where we fit a curve (laboriously, this was ca 1970) to some experimental data and were all set to publish a paper containing the equation when we decided to plot it out first. Turns out there was an asymptote right smack dab in the middle of our data points. And this wasn't even extrapolation.) – Daniel R Hicks Aug 22 '11 at 20:47
• (And, IIRC (it's been a few years) the curve we were fitting to the data was the "right" curve, according to theory. (This was back when I used to test rubbers for the Air Force.)) – Daniel R Hicks Aug 22 '11 at 21:32
• You're right. So I'm looking for an estimation if my data is good engouh... CV's and CI's are a good starting point, but there should be somehing more objective... – Dominik Aug 23 '11 at 11:20