Allocation
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ALLOCATION. DETERMINING SAMPLE SIZE. Problem: Want to estimate . How choose n to obtain a margin of error not larger than e ?. Solution: Solve the inequality for n !. Ex. Want to estimate . Solve the following inequality for n :.

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ALLOCATION

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Allocation

ALLOCATION


Determining sample size

DETERMINING SAMPLE SIZE

Problem: Want to estimate . How choose n to obtain a margin of error not larger than e?

Solution: Solve the inequality

for n!

Ex. Want to estimate . Solve the following inequality for n:


Determining sample size cont d

DETERMINING SAMPLE SIZE, CONT’D

  • Let where ah is the proportion of the total sample allocated to stratum h.

  • Ignore the fpc:s.

Then, the solution is

where


Allocation of the sample over strata

ALLOCATION OF THE SAMPLE OVER STRATA

Or: determining .

Makes sense to sample more from a stratum if

  • The stratum constitutes a large part of pop

  • The stratum is suspected to be less homogeneous with respect to what you are studying than other strata

  • The stratum is cheap to observe

How formalize this?


Allocation cont d

ALLOCATION, CONT’D

Consider a linear cost function

where c0=overhead cost, ch=the cost of observing one unit in stratum h.

Assume that you want to estimate Remember that the variance of is given by


Allocation cont d1

ALLOCATION, CONT’D

Determine so that

1. (Fix variance) The total cost C is minimized for a given variance V

or

2. (Fix cost) The variance is minimized for a given total cost C


Allocation cont d2

ALLOCATION, CONT’D

Both problems are solved by

- so called optimal allocation!

Depending on the problem, however, n differs:

In case 1 (fix variance): insert ah in the formula for n

In case 2 (fix cost): insert ah in the cost function and solve for n


Simplifications

SIMPLIFICATIONS

Neyman allocation

A special case of optimal allocation if

Only case 1 (fix variance) relevant.

Allocate according to

and insert the result in the formula for n.


Simplifications cont d

SIMPLIFICATIONS, CONT’D

Proportional allocation

A special case of Neyman allocation if

Only case 1 (fix variance) relevant.

Allocate according to

and insert in the formula for n.


Other allocation problems

OTHER ALLOCATION PROBLEMS

Optimal allocation if you have more than one study variable?

Optimal allocation if you have precision demands both for separate strata and for the pop as a whole?

  • Allocate with regard to the most important variable.

  • Solve the full optimization problem.

  • Determine n separately for each stratum.

  • Solve the full optimization problem.


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