Dear Purna Bahadur Subba ,

Using purposive sampling in quantitative research is problematic due to the risk of bias as you have no guarantee that you sample is representative of the wider population. If you are only interested in descriptive statistics then this is acceptable but I would be very cautious at attempting and inferential statistics with such a sampling approach.

Thank you Peter for the answer.

Appreciated.

Can I have views from more people? Seeking for a help.

Statistical analysis is generally based on random sampling, but if your purposive sampling procedures are simply intended to select a target population, then you can still use random sampling within that targeted population.

Thank you Atira for coming up with a similar problem as of mine.

Thank you David for sharing the idea of randomization within a select target population if the sample is a purposive

If you don't use probability-of-selection-based (randomized) sampling, you can use model-based (prediction-based) sampling.  That requires auxiliary/predictor data on the population or subpopulation to which you wish to infer.  This is very useful for repeated sample surveys of establishments, where there is a less frequent census which can be used for predictor data on the same items.  See this invited presentation for mathematical statisticians at the US Energy Information Administration:   

https://www.researchgate.net/publication/319914742_Quasi-Cutoff_Sampling_and_the_Classical_Ratio_Estimator_-_Application_to_Establishment_Surveys_for_Official_Statistics_at_the_US_Energy_Information_Administration_-_Historical_Development, 2017, using prediction. 

Further, "balanced sampling" in a model-based approach will generally be more consistently 'representative' than randomized sampling.

Note Royall, R.M.(1992), "The model based (prediction) approach to finite population sampling theory," Institute of Mathematical Statistics Lecture Notes - Monograph Series, Volume 17, pp. 225-240. 

The paper is available under Project Euclid, open access:

 https://projecteuclid.org/euclid.lnms/1215458849.

This is pointed out in Richard Royall's ResearchGate contributions.

Here is a book:

Chambers, R, and Clark, R(2012), An Introduction to Model-Based Survey Sampling with Applications, Oxford Statistical Science Series. 

You might find this of interest: 

Brewer(2014), "Three controversies in the history of survey sampling," Survey Methodology, Dec 2013 -  article Ken Brewer wrote due to receiving the Waksberg Award: 

https://www150.statcan.gc.ca/n1/pub/12-001-x/2013002/article/11883-eng.htm 

However, data collected without available predictor data and a plan, or randomized sampling, or both, may be very unreliable for inference to a population. 

Perhaps if you can explain why you picked the cases that you did, then there would be some basis for evaluating the results.  However, as I indicated, this is generally unreliable.  Also, I don't know what you mean by a "critical" purposive sample.   Also the term "quasi-experimental" is a bit vague.  But if you want to infer to a population, then without a model-based or a probability-of-selection-based design, you will generally not be on 'solid ground.'