Allocation of Sample Size in Different Strata PDF

## 3.7 Allocation of Sample Size in Different Strata In stratified sampling, the allocation of the sample to different strata is done by the consideration of three factors, viz., * The total number of units in the stratum, i.e. stratum size * The variability within the stratum, and * The cost...

## 3.7 Allocation of Sample Size in Different Strata In stratified sampling, the allocation of the sample to different strata is done by the consideration of three factors, viz., * The total number of units in the stratum, i.e. stratum size * The variability within the stratum, and * The cost in taking observations per sampling unit in the stratum A good allocation is one where maximum precision is obtained with minimum resources, or in other words, the criterion for allocation is to minimize the budget for a given variance or minimize the variance for a fixed budget, thus making the most effective use of the available resources. There are four methods of allocation of sample sizes to different strata in a stratified sampling procedure. These are : * Equal allocation * Proportional allocation * Neyman allocation * Optimum allocation ### 3.7.1 Equal Samples From Each Stratum This is a situation of considerable practical interest for reasons of administrative or field work convenience. In this method, the total sample size n is divided equally among all the strata, i.e. for the ith stratum $ n_i=n/k $ (3.7.1) ### 3.7.2 Proportional Allocation This allocation, generally known as proportional allocation, was originally proposed by Bowley (1926). This procedure of allocation is very common in practice because of its simplicity. When no other information except N_i, the total number of units in the ith stratum, is available, the allocation of a given sample of size n to different strata is done in proportion to their sizes, i.e. in the ith stratum, $ n_i = nN_i/N (or f_i = f) $ (3.7.2) This means that the sampling fraction is the same in all strata. It gives a self-weighting sample by which numerous estimates can be made with greater speed and a higher degree of precision. ### 3.7.3 Neyman Allocation This allocation of the total sample size to strata is called minimum-variance allocation and is due to Neyman (1934). This result appears to have been first discovered by Tschuprow (1923) but remained unknown until it was rediscovered independently by Neyman. The allocation of samples among different strata is based on a joint consideration of the stratum size and the stratum variation. In this allocation, it is assumed that the sampling cost per unit among different strata is the same and the size of the sample is fixed. Sample sizes are allocated by $ n_i = W_i S_i / \sum W_i S_i = n (W_i S_i) / \sum (W_i S_i) $ (3.7.3). A formula for the minimum variance with fixed n is obtained by substituting the value of n_i in Eq. (3.5.3) when we get $ V_{min}(Pst) = (W/S_i^2/n - \sum W_i S_i^2 / N $ (3.7.4) There may be difficulty in using this method as the value of S_i will usually be unknown. However, the stratum variances may be obtained from previous surveys or from a specially planned pilot survey. The other alternative is to conduct the main survey in a phased manner and utilize the data collected in the first phase for ensuring better allocation in the second phase. ### 3.7.4 Optimum Allocation In this method of allocation the sample sizes n_i in the respective strata are determined with a view to minimize V(̄st) for a specified cost of conducting the sample survey or to minimize the cost for a specified value of V(̄st). The simplest cost function in stratified sampling that can be taken is $ C = a + \sum n_i c_i $ (3.7.5) where the overhead cost a is constant and c_i is the average cost of surveying one unit in the ith stratum, which may depend upon the nature and size of the units in the stratum. To determine the optimum value of n_i, we consider the function $ \psi = V(\overline{st}) + \lambda C $ (3.7.6) where \lambda is some unknown constant. Using the calculus method of Lagrange multipliers, we select n_i and the constant \lambda to minimise \(\psi\). Differentiating with respect to n_i, we have $ -W_i^2 S_i^2 / n_i^2 + \lambda c_i = 0 (i = 1, 2,... , k) $ or $ n_i = W_i S_i / \sqrt{\lambda c_i} $ (3.7.7) Summing over all strata, we get $ n= \sum(W_i S_i / \sqrt{\lambda c_i}) $ (3.7.8) From relations (3.7.7) and (3.7.8) we can obtain $ n_i = \frac{(W_i S_i/ \sqrt{\lambda c_i})}{\sum(W_i S_i/ \sqrt{\lambda c_i})} = \frac{n (W_i S_i / \sqrt{\lambda c_i})}{\sum (W_i S_i / \sqrt{\lambda c_i})} = n \frac{N_i S_i / \sqrt{c_i}}{\sum (N_i S_i / \sqrt{c_i})} $ (3.7.9) Thus the relation (3.7.9) leads to the following important conclusions that, in a given stratum, we have to take a larger sample if * the stratum size is larger; * the stratum has larger variability; and * the cost per unit is cheaper in the stratum. If c_i’s are the same from stratum to stratum, relation (3.7.9) will lead to the Neyman allocation. Similarly, if c_i’s and S_i’s do not vary from stratum to stratum, relation (3.7.9) will lead to proportional allocation. The total sample size n required for estimating the population with a specified cost C is given by $ n = \frac{(C-a)}{\sum (W_i S_i / \sqrt{c_i})} $ (3.7.10) If V is fixed, we find $ n = \frac{V + \sum W_i S_i^2 / N}{\sum (W_i S_i / \sqrt{c_i})} $ (3.7.11) The values of the stratum variances can be obtained from earlier surveys or from a knowledge of the measurements within each stratum. An alternative proof of the allocation results due to Stuart (1954) may be summarized as follows: Let V’ and C’ be the parts of V and C that depend on n_i or in a simple form, V’ = \(\sum A_i /n_i\) and C’ = \(\sum c_in_i\) Minimizing V’ for fixed C’ or vice versa are both equivalent to minimizing the product $ V’ C’ = (\sum A_i /n_i) (\sum c_in_i) $ (3.7.12) Now, by Cauchy-Schwartz’s inequality, $ (\sum x_i^2)(\sum y_i^2) \geq (\sum x_i y_i)^2 $ we have $ (\sum A_i / n_i) (\sum c_i n_i) \geq (\sum\sqrt{A_i c_i})^2 $ The equality being if and only if y_i is proportional to x_i, i.e. $ \frac{c_i n_i} {A_i/n_i} $ = constant which yields $ n_i \propto \sqrt{\frac {A_i c_i}{}} $ (3.7.13) * In case A_i = W_iS_i², we get from relation (3.7.13) the optimum sample sizes within strata as given in relation (3.7.9). * In case c_i = c and A_i = W_i²S_i², the optimum sample sizes for Neyman allocation are given by the above relation. * In case A_i = W_i and c_i = c, proportional allocation is also obtained by the above relation.