Question

Generally, stratified random sampling has a design effect of less than 1, while cluster sampling has a design effect of more than 1. Why?

Answer 1

The design effect is the ratio of the actual variance to the variance expected with SRS(Simple Random Sampling). It can more simply be stated as the actual sample size divided by the effective sample size (the effective sample size is what you would expect if you were using SRS).

Mathematically ,

The design effect (DEFF) is given by

DEFF = 1 + δ(n – 1).

Where:

• δ = interclass correlation for the statistic.
• n = average size of the cluster.

The intraclass correlation “represents the likelihood that two elements in the same cluster have the same value, for a given statistic, relative to two elements chosen completely atrandom in the population.

Cluster sampling increases the design effect since the outcome you are looking at is grouped in the population and not evenly spread throughout.

Clearly for a cluster sampling the δ will be a positive fraction and hence the DEFF will be a fraction greater than 1 since something is getting added to 1.

Now look at a statified sampling.stratified sampling (that is, intentionally picking portions of the sample from different subgroups in the population) often leads to a decrease in the design effect.It tends to have negetive correlation.

Thus the DEFF will be always less than 1 since something is getting substracted from 1.

Roughly it can be understood practically too. A stratified sample is a selective samlpling technique wherein we enrich a sample with selection of data points that suit us.

Thus a stratified sample will alsoways be a enriched data set in comparison to the population data (SRS).

In other words, A stratified sample cannot be less efficient than simple random samples.

or in other words,the variations of a stratified sample will have to be lower than that of a simple random SRS data.

or, Var(stratified)<Var(srs)

or Var(stratified)/Var(srs)<1