Sample size: two stage sample

Theory

In most of cases it is difficult (if not impossible) to take a simple random selection of individuals from a population with the aim of estimating the prevalence of an event. This is due to two reasons: the list of all individuals does not exist and the high cost of sampling.

For example: 300 individuals must be randomly selected from a population of 200,000 cattle distributed in 1,500 farms. In order to take a simple random selection a list of the 200,000 animals would be required. Generally these lists do not exist. In addition, if all 200,000 animals could be identified it would be necessary to visit a large number of farms. The cost of visiting farms is normally high, both in money and time. In the case in which the list of farms and an estimate of the number of individuals on each farm is available it is possible to carry out two stage sampling. Using a two stage approach ProMESA calculates the number of farms (or clusters) that must be selected, dependent on the expected prevalence, the acceptable error, the level of confidence required, the number of individuals to be sampled per farm and the rate of homogeneity. The number of individuals to include in the sample per farm is determined according to operative factors. Initially an approximate value may be estimated and then the programme run to obtain a first result. After doing that, the number may be adjusted and the process repeated until a convenient result is found.

The rate of homogeneity is a parameter difficult to estimate. Its actual value may be calculated from data belonging to a previous sampling of similar characteristics of the one being planned. A table of rho estimates for different disease conditions is provided below. The procedure for the exact calculation is shown as well. The following points must be kept in mind:

• Roh may take negative values, but generally it is between 0 and 1;
• The conditions for the implementation of a two stages sampling are the best when roh is small (< 0.10);
• The greater the value of roh, the greater the estimated sample size;
• When roh exceeds 0.40 it would be convenient to implement other type of sampling (possibly stratified sampling).

The formula for calculating the required number of clusters (n) is as follows:

Where:

n	The number of clusters that have to be selected.
p	The expected prevalence.
D	The design effect.
z	The critical value obtained from a standard normal distribution. For each level of confidence there is a corresponding value of z. The levels of confidence frequently used in biological studies are 90%, 95%, and 99%. The corresponding z values are 1.64, 1.96, and 2.58 respectively.
e	The acceptable absolute error.
b	The number of individuals to select per cluster.

Practice

Expected prevalence	The assumed prevalence of the event in the population under study (usually based on previous studies, field data or the literature). When no information is available a value of 0.50 will yield the maximum sample size. Acceptable values: ≥ 0 and ≤ 1.
Acceptible relative error	A measure of the desired precision. For example, if you assume a prevalence of 0.40 and a relative error of 0.10, the result will have a precision of ± 0.04 (that is, 0.40 × 0.10). In this case 0.04 is the absolute error. In general, the relative error should be ≤ 0.20. Acceptable values: ≥ 0 and ≤ 1.
Level of confidence	The confidence that the user wants to have in the results. Acceptable values: 90%, 95% or 99%.
Number of samples per cluster	The number of samples to be taken per cluster (farm or village) depends on the analysis of operative factors and available resources. It is convenient to be ≥ 5. The number of clusters to be selected will be determined as function of this parameter, among others. There is a kind of compensation between both values: the smaller the number of samples to take per cluster, the greater the number of clusters to be selected. Acceptable values: any integer number.

Disease	Prevalence		Number clusters	Design effect	Rho
	%	n
Enzootic bovine leucosis	1.51	2907	104	3.52	0.09
Enzootic bovine leucosis	11.75	945	81	2.11	0.10
Enzootic bovine leucosis	1.93	466	90	1.34	0.08
Infectious bovine rhinotracheitis	31.97	2852	104	2.76	0.07
Infectious bovine rhinotracheitis	47.88	969	82	1.71	0.07
Infectious bovine rhinotracheitis	28.11	466	90	2.62	0.39
Bovine virus diarrhoea	6.30	2799	108	6.95	0.23
Bovine virus diarrhoea	19.07	970	82	5.74	0.42
Bovine virus diarrhoea	69.74	466	90	2.76	0.42
Newcastle disease	37.89	1470	253	1.89	0.18
Infectious bursal disease	41.56	1470	253	2.56	0.37
Leptospira hardjo	38.55	2861	104	2.54	0.06
Leptospira icterohaemorrhagica	13.60	2861	104	4.24	0.12
Leptospira grippotyphosa	16.57	2861	104	3.91	0.11
Leptospira canicola	5.38	2861	104	3.04	0.08
Brucella abortus	7.74	1512	104	2.18	0.09
Brucella ovis	11.71	1529	40	6.94	0.16
Brucella ovis	10.99	1529	40	9.20	0.22
Anaplasma marginale	3.78	2909	104	2.19	0.04
Anaplasma marginale	4.32	1111	91	2.11	0.10
Trypanosoma vivax	2.75	2909	104	2.56	0.06
Trypanosoma vivax	30.87	1111	91	2.68	0.15
Trypanosoma congolense	23.94	1111	91	2.51	0.13
Trypanosoma brucei	24.39	1111	91	2.39	0.12
Eimeria spp.	27.13	1010	104	2.53	0.18
Eimeria spp.	15.63	1113	91	4.32	0.30
Strongyloides spp.	12.08	1010	104	2.35	0.16
Strongyloides spp.	4.04	1113	91	2.27	0.11
Trychostrongylus spp.	69.01	1010	104	1.70	0.08
Trychostrongylus spp.	48.43	1113	91	2.22	0.10
Moniezia spp.	3.07	1010	104	1.46	0.05
Moniezia spp.	15.90	1113	91	3.21	0.20
Fasciola spp.	6.92	1113	91	4.09	0.27

Table 1: Suumary information used for the calculation of the design effect and the intra-class correlation of selected infections of doemstic animals. Source: Otte J, Gumm I (1997) Intra-cluster correlation coefficients of 20 infections calculated from the results of cluster-sample surveys. Preventive Veterinary Medicine 31, 147 - 150.