I know the question is a very elementary one, but I simply cannot understand the difference other than the fact that an SRS is a form of Multi-Stage Sampling. Can anyone provide a simple example(s) to help me understand the critical difference between these two sampling designs?
Solved – Difference between Multistage Sampling and Stratified Random Sampling
experiment-designsamplingstratification
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There isn't going to be one best answer for this kind of sampling. It depends on the observable covariates in your sampling frame, the variables you expect to be important determinants of survey response, and the analysis you want to run once the survey is complete.
With that said, there are a couple of general principles that can help guide your sampling strategy.
For descriptive surveys, you generally want your sample to closely resemble the population of interest in as many ways as possible. This will help keep your weights even, in order to maximize your effective sample size.
If you intend to do multivariate analysis, you may want to stratify on important variables of interest. This will increase variance in your IVs and DVs, and can help increase your statistical power in later analysis. This is why some studies conduct oversamples of minority populations -- because race and ethnicity are important IVs in many analyses. Case-control studies follow a similar logic for stratifying on the dependent variable.
If you intend to do description and analysis, then these are goals will be partly at odds. No matter what, you need to follow the basic principle of sampling and make sure that every individual in the population has a known, non-zero chance of being selected into the sample. Advanced topics worth looking up in this area include propensity scores, and sample weighting via raking.
Closing thought: these are general principles for sampling design and sample weighting. You don't say much about your specific application, but I'm guessing that most of this is overkill. The main reason to stratify a sample is if you have reason to believe a simple random sample will miss out on some important group of interest (geographic, demographic, or otherwise). That is, the sampled population would be too small for useful analysis. If you don't have that problem, then you don't need to worry too much about stratification.
You have not correctly interpreted user697473's claim. He is not talking about failing to include any data from brand C. He was talking about giving a particular vector of assignemnts $0$ probability. He was not saying that you can magically determine the value of some variable while never testing it. He wants to be able to use a balanced random subset, so that each point is included in the random subset with the right probability, but not a uniformly random one.
For example, if the set is $\lbrace x_1,x_2,x_3,x_4 \rbrace$, then the following random subsets of uniform size $2$ all have the property that the probability that $x_i$ is included is $1/2$:
$S_1 = 1/6\lbrace x_1,x_2\rbrace + 1/6\lbrace x_1,x_3\rbrace + 1/6\lbrace x_1,x_4\rbrace + 1/6\lbrace x_2,x_3\rbrace + 1/6\lbrace x_2,x_4\rbrace + 1/6\lbrace x_3,x_4\rbrace $
$S_2 = 1/4\lbrace x_1,x_3\rbrace + 1/4\lbrace x_1,x_4\rbrace + 1/4\lbrace x_2,x_3\rbrace + 1/4\lbrace x_2,x_4\rbrace $
$S_3 = 1/2\lbrace x_1,x_2\rbrace + 1/2\lbrace x_3,x_4\rbrace $
These are all balanced in the sense that if you compute the average value of some function $f$ over the random set, the expected value is $1/4(f(x_1) + f(x_2) + f(x_3)+f(x_4))$. In the third random subset, the probability of the subset $\lbrace x_1,x_3 \rbrace$ is $0$.
That said, the point of the experiment should not be to produce an unbiased estimate. That is just one consideration. Another goal is to provide useful information. If you know that you may want to estimate $f$ on a subset $T$ (say $\lbrace x_1,x_2 \rbrace$) and its complement and to subtract the two, the quality of your estimate depends on $\#(T \cap S)$ and $\#(T^c \cap S)$. Then not all balanced subsets have the same quality. For that task, $S_3$ is worse than random assignment ($S_1$), while $S_2$ is better than random assignment.
Best Answer
Multi-Stage Sampling:
Population: USA elementary school students
First stage sampling: 10 States from total of 50 States.
Second Stage: 20 Counties from total XX counties in selected XXXXX state in the first stage.
Third stage: 50 elementary schools from total yy elementary schools in the selected ZZZ county in the second stage
Fourth stage: 10 students from each selected school.
Finally, I got 10 * 50 * 20 * 10 = 100,000 students.
Stratified sampling:
Population: USA elementary school students
Geographically stratify USA into ten regions
Randomly select 10,000 students from elementary schools from each region. I got a sample with 100,000 students.