Oscar -

Very interesting.

For a sample size for simple random sampling, page 77 in Cochran, W.G.(1977), Sampling Techniques, 3rd ed., John Wiley & Sons, and other survey sampling books, address that. There are certain characteristics for your variable of interest that you must 'guess' first, so a preliminary small sampling may be useful. It is explained there in Cochran and elsewhere. (Other good books would include (1)Lohr, S.L.(2010), Sampling: Design and Analysis, 2nd ed., Brooks/Cole, and (2) Särndal, C.-E., Swensson, B. and Wretman, J. (1992), Model Assisted Survey Sampling, Springer-Verlag.) Note that this applies to each of your variables. You may decide that certain less important variables, that may require large sample sizes to be estimated well, are not worth the bother. Also, textbook sample size selection formulas generally assume no nonsampling error, such as measurement error, and that won't do. (You could search the Internet on "total survey error" to learn more about that.) If you try to obtain too large a sample size for your resources, you could actually obtain less accurate results with a larger sample. If you look at page 77 of Cochran, you can see that a key part in estimating the sample size needs would change.

One can usually do better than simple random sampling. (The following methods are also in Cochran and other books.) One possibility is stratified random sampling. If you expect sections of the road in different areas to behave differently for a given variable, you can make each area a separate stratum. Further, as a matter of practicality you could try cluster sampling in two stages. First you might have a simple random sample of fairly larger sections of road, and then as a second stage, you could have perhaps a simple random sample within each of those sections. It depends on how much of a unit size you can test or measure for whatever variables you are measuring.

I keep mentioning Cochran because it is likely more easily available than many that are more expensive though newer, yet may not cover so much so well - though Lohr is a book that is very well written. Cochran covers design-based (randomized) techniques very well, and has a little (pages 158-160) on prediction (regression). In short, your choices are design-based (randomized) sampling and estimation methods, model-assisted design-based methods, and strictly model-based (regression) methods. If you have regressor data (called auxiliary data in the case of model-assisted design-based methods), that can greatly assist you in compensating for random samples that are 'unfortunately' drawn. Stratification can often be very helpful too, for all three alternatives.

In your case, if you do not have any auxiliary data, you could still use Cochran, and perhaps use a combination of stratified random and cluster sampling.

If you use the Cochran formula for picking a sample size for simple random sampling, for a given variable of interest, stratification tends to make that an overestimate of what you will need. Cluster sampling will tend to need more, but they may be easier to do from a logistics point of view, so a larger sample size may be easier to do that way.

So you can look first at simple random sampling in Cochran to get an idea. (Stratification does better because the idea is to group under each stratum the sections that vary less and have some variance between strata. Leslie Kish, who also wrote a survey statistics book, came up with the idea of a "design effect" measure, to show how much one design improves over another.)

If you do have regressor data, the first paper under my current profile is one that estimates sample size needs for the classical ratio estimator, for a given stratum.

It sounds like an interesting project. Later, perhaps you could post a progress or status report for discussion.

My guess is that you will need a randomized, design-based approach, and if you use a combination of cluster and stratified random sampling, you may do well. If you start by estimating the simple random sample size that would be needed (Cochran, around page 77 and pages leading up to it), then you can work with that. But remember, some variables of interest will be more accurate than others. Perhaps you should start with your key variable, and work from there. And remember that stretching yourself and resources too thin can just add error.

Best wishes - Jim 

You have a lot of options for sampling design and the number of samples depends on the variance and the cost of variable that you want to measure. For sampling design,  assume that you decided to sample 50 point on the road, if you want to sample by simple random sampling design , derive 50 random numbers by a calculator or MS Excel, multiple random numbers and total road length. The result is the locations of the sample along the road length (for example, 0.27 * 200 km= 44km, means travel 44km and take the sample). If you want to measure your samples systematically,  divid the length of sample to the number of samples which results in sampling lag (200km÷50= 4 km, means every 4 km take a sample).

Thanks for your answer!

How I know if the 50 points are statistically representative? 

Oscar -

Very interesting.

For a sample size for simple random sampling, page 77 in Cochran, W.G.(1977), Sampling Techniques, 3rd ed., John Wiley & Sons, and other survey sampling books, address that. There are certain characteristics for your variable of interest that you must 'guess' first, so a preliminary small sampling may be useful. It is explained there in Cochran and elsewhere. (Other good books would include (1)Lohr, S.L.(2010), Sampling: Design and Analysis, 2nd ed., Brooks/Cole, and (2) Särndal, C.-E., Swensson, B. and Wretman, J. (1992), Model Assisted Survey Sampling, Springer-Verlag.) Note that this applies to each of your variables. You may decide that certain less important variables, that may require large sample sizes to be estimated well, are not worth the bother. Also, textbook sample size selection formulas generally assume no nonsampling error, such as measurement error, and that won't do. (You could search the Internet on "total survey error" to learn more about that.) If you try to obtain too large a sample size for your resources, you could actually obtain less accurate results with a larger sample. If you look at page 77 of Cochran, you can see that a key part in estimating the sample size needs would change.

One can usually do better than simple random sampling. (The following methods are also in Cochran and other books.) One possibility is stratified random sampling. If you expect sections of the road in different areas to behave differently for a given variable, you can make each area a separate stratum. Further, as a matter of practicality you could try cluster sampling in two stages. First you might have a simple random sample of fairly larger sections of road, and then as a second stage, you could have perhaps a simple random sample within each of those sections. It depends on how much of a unit size you can test or measure for whatever variables you are measuring.

I keep mentioning Cochran because it is likely more easily available than many that are more expensive though newer, yet may not cover so much so well - though Lohr is a book that is very well written. Cochran covers design-based (randomized) techniques very well, and has a little (pages 158-160) on prediction (regression). In short, your choices are design-based (randomized) sampling and estimation methods, model-assisted design-based methods, and strictly model-based (regression) methods. If you have regressor data (called auxiliary data in the case of model-assisted design-based methods), that can greatly assist you in compensating for random samples that are 'unfortunately' drawn. Stratification can often be very helpful too, for all three alternatives.

In your case, if you do not have any auxiliary data, you could still use Cochran, and perhaps use a combination of stratified random and cluster sampling.

If you use the Cochran formula for picking a sample size for simple random sampling, for a given variable of interest, stratification tends to make that an overestimate of what you will need. Cluster sampling will tend to need more, but they may be easier to do from a logistics point of view, so a larger sample size may be easier to do that way.

So you can look first at simple random sampling in Cochran to get an idea. (Stratification does better because the idea is to group under each stratum the sections that vary less and have some variance between strata. Leslie Kish, who also wrote a survey statistics book, came up with the idea of a "design effect" measure, to show how much one design improves over another.)

If you do have regressor data, the first paper under my current profile is one that estimates sample size needs for the classical ratio estimator, for a given stratum.

It sounds like an interesting project. Later, perhaps you could post a progress or status report for discussion.

My guess is that you will need a randomized, design-based approach, and if you use a combination of cluster and stratified random sampling, you may do well. If you start by estimating the simple random sample size that would be needed (Cochran, around page 77 and pages leading up to it), then you can work with that. But remember, some variables of interest will be more accurate than others. Perhaps you should start with your key variable, and work from there. And remember that stretching yourself and resources too thin can just add error.

Best wishes - Jim 

PS -  It occurred to me that I assumed you wanted to know how many segments to include in your sample, each segment being one unit.  Not knowing your variables, that was my guess.  -  But if you are just sampling the same section a number of times, then that sounds like a place for simple random sampling for that one section, and you can use that part around page 77 in Cochran or another book.  If this is what you are doing but can break the road into more than one segment, then each one could have a simple random sample, and overall it would be stratified random sampling. 

Many thanks for your answer! Now I have more things for thinking about. In my case, I have many heterogeneous variables that will be measured in different road segments. i.e. The distance of trees from the road segment. 

Sounds interesting. You may want to approach this as more than one effort/survey, grouping the variables to be examined in batches that go together better from a logistical point of view.  So, more than one survey. 

I'd be interested in a link to your report later. Others would too, I'm sure. Might you eventually include it on your ResearchGate contributions? 

Has anything like this been done before?  People in other disciplines may have some additional ideas. 

Thanks.   

Actually I haven't seen something like that. Usually, in the papers, authors do not explain the sample size design.  

Sergei -

Survey statisticians, generally working on estimates of totals or means and their standard errors for variables of interest for finite populations, often consider randomization because it has a chance of covering various heterogeneous features analogous to what you mentioned. But simple random sampling is seldom used , because one can usually do better. Stratification is used, not just to estimate a subcategory, which may not pass the laugh test for accuracy, but also to improve overall accuracy. If you can group things that go together with less variance, you are better off. It is far better if you have regressor data, called "auxiliary" data, on the population, to help modify results when randomization works out unfortunately. (That is called model-assisted design-based sampling and estimation.)

In your case, which I don't pretend to completely understand, it sounds like you have features that could help you to design/model your sample in a more optimal manner than to just do a simple random sample. But a simple random sample is likely a good choice for a starting point. After all, you can at least estimate variances for your statistics to get some handle on accuracy.

I am not aware of all of the ramifications of the features you stated, but they may help you to stratify your sampling and/or better estimate from your sample. I suggest that you make a plan that is backed by logic, and run it by those with your subject matter expertise, as well as statisticians, to see how it flies. Then you follow your plan, and include it as an appendix to your report. It is always good to know later what you did, so you can better compare to other reports, and make future improvements.

If you use regression to show relationships in your report, you can estimate the "variance of the prediction error," which can be found in econometrics books. Standard errors for slopes in linear regression are useful.

Many reports contain p- values for statistical tests, but they are almost universally misinterpreted. (See attached link.) You are better off showing cv (relative standard error), the estimated standard error divided by the estimated statistic (parameter) itself, or a confidence interval. That would be far more practically interpretable.

As in Oscar's challenging situation, and most others, having to deal with more than one variable of interest will mean some practical trade offs will likely be necessary.

Perhaps you may want to organize your work into more than one survey. I don't know.

Cheers - Jim 

Article Practical Interpretation of Hypothesis Tests - letter to the...

Sergei -  Hi. - I am a bit confused.  If I understand you to say that some kind of modeling would be helpful, that sounds good - using real, physical relationships and letting them work the problem.  Or, if you are talking about picking the variables to be measured, I assumed Oscar already had that.  - Jim  

Oscar -  By the way, you mentioned the word "representative" early in this thread, and I thought you might find it interesting how many different ways people interpret that word, so I am attaching a thread/question that discussed that.  -  Terminology can get confusing and usage varies, but I thought this enlightening. -  Most of the attached may not relate to your work very much, but might, now or in the future, and I thought it was interesting.  -  Jim  

https://www.researchgate.net/post/What_is_a_representative_sampling

Thanks you James. 

Oscar, the answer to your question depends on many factors, but assuming no such information is available, and you are trying to characterize the road segment in terms of your variable of interest, you could sub-divide the road in N segments and collect m samples at randomly selected locations within each (mxN = total number of samples). The answer to 'how many samples' depends on 'how much variance' and 'the spatial correlation' that most probably is present in your data. Please email me in case you need specific help. 

Oscar -  FYI:  In discussing road safety with someone else - please see first link - I made the mistake of trying to carry over my experience with continuous data without realizing that the discrete data he had recommends some different treatment.  It appears that an expert in traffic safety has now entered an answer for that question, and I thought you might want to check out his ResearchGate page for papers that might be useful for you too. - Just a thought. -  Cheers  - Jim 

https://www.researchgate.net/post/How_can_I_compare_crash_characteristics_between_two_locations

https://www.researchgate.net/profile/Jiri_Ambros