How to deal with non random samples? I have a specific question about random selection, representativeness and inference. It is well known that it's necessary to use random selection to get representative samples from the population of interest. But what happens with non-random samples? 
I am working with an intentional sample. I compared the means of some of the main variables of my database with official data. And I can conclude that there aren't important differences between them. So I have a quite representative sample. However, I am confused about how to deal with this sample. When I estimate the means of some variables or the correlations between them, do I need to compute CI or p-values for this estimations? 
I guess that it's not necessary, since I got the sample without random selection. Therefore, I have no sampling error, and I can't know how my estimations differ from the population. However, I have read some papers where the authors work with non-random samples and they make estimations (they use CI's and p-values). Moreover, it's difficult to use multivariate techniques (like ANOVA or regression) without the help of statistical significance.
Can anybody help me? I am very confused with this matter.
 A: You certainly do have sampling error, even if you don't know what population your sample came from. 
As with random samples, if all you want to do is make statements about the sample itself, then you do not need p values or any form of inferential statistics. Indeed, you don't need any specific sample size either. I can measure myself and my wife and say "I am taller than she is" (N of 2). I can just measure myself and say "I am 5 foot 8" (N = 1).
However, even with non-random samples you are usually interested in inference. You therefore have to assume either a) that the non-randomness in your sample isn't affecting things (a dangerous assumption!) or b) That there is some population from which your sample is random, and that that population is interesting.
In real life, this often gets blurred. In the many cases where there is no way to take a random sample (too expensive, too impractical, unethical, illegal, impossible) people often write as if they are inferring to something sort of in between a and b. 
A: You cannot make statistical inferences (p values, CIs) from non-random samples.  Oh, you can compute them, but they will be essentially uninterpretable as they are based on the assumption that your data are from a random sample (or as with a trial, there was random assignment--though this inference is derived, at root, from randomization tests).  You can attempt to make a logical case that the results in your sample are similar in all relevant ways to a unmeasured population of interest.  The use of p values with non-random samples in research drives me crazy, because it conveys a meaning that is not valid--it is a kind of fraud.
A: Although the concept of random sampling is central to much of statistical theory, in practice it is rare. For example, in surveys involving humans, it is usually not practical to contact most people, let alone to compel them to participate if randomly selected.
Consequently, many alternatives exist to random sampling. A sample that is not a random sample is known as a non-random or non-probability sample. Specific types of non-random sampling include quota sampling, convenience sampling, volunteer sampling, purposive sampling, and snowball sampling. It is common practice to use as much randomization as possible when employing these techniques, in the hope that the resulting sample approximates the qualities of a random sampling.
Nevertheless, when significant randomization does occur within non-random samples, they can have properties that are closer to random samples. For example, with s quota sampling, if randomly contacting people and continuing to do so until all quotas are met, this approach can lead to a sample that is approximately random (Seymour Sudman, Applied Sampling, Academic Press, 1976).
In practice, nearly all samples are non-probability samples. For example, in the case of political polling, some people are not contactable, and others refuse to participate. As a result, all political polls are convenience samples. This is true of just about all samples of living organisms.
You may read more on this at this link
