Thanks for your query.

A similar question to yours has been asked and answered in this forum sometime in February of 2015. The link to that discussion is here.

It should be noted that in this previous discussion, the question and the answer were specific to CMAM programming and were related to coverage assessment of CMAM programmes. In your question, you were not really clear what programme you are wanting to monitor programme quality and data quality of. A more specific idea of the type of programme will be useful in providing you with more specific answers and examples. For this response, I assume you are asking about CMAM programmes as this specific forum on coverage assessment was started / setup initially for coverage assessment of CMAM programmes (though a lot of approaches and techniques are applicable to various programmes albeit with some customisation specific to the programme being assessed).

From how you framed your question, you seem to be clear that LQAS is a data analysis procedure rather than a sampling approach which is one of the key points raised in the discussion I referred you to. Is this correct? It is important this is clear because this will give clarity to the answers to the various questions that you are asking. I will try to address these questions below.

1. Appropriateness of LQAS vs. other approach
Whether or not there is another approach other than LQAS which may be more appropriate for your needs, it's really difficult to be specific because it is not clear what your requirements are in terms of the type of monitoring you want to achieve and the type of programme you wanted to monitor and the type of indicators you want to collect data on. But generally, the approach other than LQAS will be to collect enough data from each of your chosen health facilities on the specific indicators you want to monitor that will allow you to calculate an estimate of the indicators with adequate and useful precision. The appropriateness of either approaches will really depend on your needs and your resources.

Now, going deeper into your question and the limited context that you have provided, you should take into consideration at which level you want to apply LQAS data analysis procedure. Another way of thinking of this is to ask yourself, "What will be the lots to which I will apply the LQAS analysis on?". Depending on what you determine to be your lots would determine also how you will go about your sampling and the sample sizes that you will need.

If your lots are the health facilities, then the total number of health facilities will be your universe population and hence will determine how much sample size you will need (i.e, number of health facilities to sample) to be able to perform LQAS. I think in general, you are looking at a sample size of between 20 to 30 heath facilities for a total number of health facilities in the range of 40 to 100. As you will note, the smaller the number of total health facilities requires an almost complete enumeration of all of them to be able to classify them appropriately.

Now, you are still left with the problem of how to determine whether the sampled health facilities are performing well or not performing well. For this, you will then need to decide again on which approach to use to measure your chosen indicators. This can be LQAS again which means within a sampled health facility, you will have to determine again your lots - either cases or case records, health workers if you are testing health worker performance, health volunteers if you are checking their effectiveness, etc. Or it can be an estimation where depending on your indicator you calculate the sample size you need to estimate the metric you are wanting to measure.

So, from these considerations and taking into account the previous discussion on a related topic, it would make sense to just choose 20 health facilities and then in each health facility, apply an LQAS data analysis procedure to determine whether each health facility is failing or passing the indicators or metrics that you have/are using. And then as a way of summarising, you can report how many of the 20 heath facilities that you have sampled are passing or failing on specific metrics.

2. Whether to use SRS or spatial stratification for selecting 20 health facilities
I think either of these approaches will work fine for the selection of health facilities. I would have a preferential bias for the CSAS approach (which is described in the previous discussion and which I assume is what you meant by spatial stratification) as this approach would factor in an element of health facility location that can inform interpretation of results such as the performance of health facilities. An SRS approach will not really give you this kind of location gradient that would give additional insight other than being able to say that x out of 20 health facilities are performing well / not well.

3. 2nd stage sampling for records in heath facilities
I didn't really understand this question very well. I think you are trying to apply a sample size calculation either based on a hypergeometric or binary model based on whether overall sample is finite or not. I think the previous link suggests to collect as much data as possible (between 100 to 200). At a sample of about 96 records, you will already have enough sample to make an estimate with useful precision (±10%). Doubling that to 192 records will allow for any issues such as clustering of data and/or will increase precision. And, within this range of target sample size, you can confidently classify whether the heath facility is failing / passing based on indicators you are using. So, I would use that sample size recommendation.

I hope this is clear and makes sense. Let me know if you have any further questions.