Why cluster sampling will normally result in larger variance as compared to simple random sampling? I want to compare two situations.

*

*Consider an experiment with two settings with treatment, one in which I block subjects by factor $A$ and the other one in which I do not block. I would be interested in estimating treatment effect. Under blocking(mixed effect model), the estimated treatment will normally have smaller variance due to correlation.


*Consider a survey with subjects with two settings, one in which I cluster subjects by factor $A$ and the other one in which I do not form clusters. In the former, I perform clustered random sampling(CRS) and in the latter, I perform simple random sampling(SRS). Then CRS has larger CI than that of SRS. Assume in CRS that I can sample 60% of clusters and all units in sampled clusters are sampled.
$Q:$ According to Lohr's Sampling 2nd edition (right before example 5.1) in 2, CRS does not decrease error estimate as compared to SRS. This is in contrast to 1 where blocking should decrease error estimate of treatment effect due to correlation. It seems that I have some sort of contradictive conclusion where clustering/blocking does not help. I am aware that CRS is not blocking as possibly not all levels were used here.(i.e. CRS is not stratified random sampling here. So it is not appropriate to compare the two.) If I increase percentage of sampled clusters, I would expect increased precision in CRS. However, how come 1 and 2 have contradictive conclusion in general?
 A: The difference is that in blocking we're ultimately trying to make a comparison between these groups, while in sampling we're interested in making some inference on the whole population.
Eg, in testing Drug A vs Drug B we might try to make both groups contain the same proportions of males and females. That increases the similarities between the groups, and thus the correlation between the group outcomes. A larger correlation between the groups means that treatment differences between groups becomes easier to detect, since there's less random noise. Imagine if we knew the outcomes from the two groups would be perfectly correlated except for the treatment effect. In that case all we'd have to do is measure the treatment effect, there'd be no need for complex statistics.
In sampling clusters we aren't trying to compare clusters with each other. We're trying to learn something about the population. In that regard having a new observation which is highly correlated with another observation doesn't give us much value. For example, suppose I want to estimate the average length of the legs of adults. I might randomly select either a person's left or right leg and measure the length. I could also measure their other leg, I've already got them in the office so it won't take much more effort. But since I already know the length of their first leg it's highly unlikely that the length of their other leg will be much different. (Although they will probably be slightly different.) Sampling both "units" inside each "cluster" doesn't help as much as if I measured the leg of an entirely different person.
Now as you say, increasing the percentage of sampled clusters does increase the precision of the estimates. If I go sample 10 more people and their 20 legs that will improve the estimate. But it would be more efficient to sample 20 legs from 20 different people.
