Background: Immunization averts a large number of children in each year. The burden of vaccine preventable diseases remains high in developing countries compared to developed countries. To overcome from this burden different types of immunization programs have been implemented. For better immunization coverage in developing countries, considerable progress is to be made to improve the knowledge and awareness regarding importance of vaccines. In this study a compara-tive study of immunization coverage under two sampling methods has been performed. Methods: In this study variance and design effect of proportion of children vaccinated against different types of vaccines (BCG, OPV, DPT, Hepatitis B, Hib, Measles and MMR) are estimated under two stage (30 × 30) cluster and systematic sampling for comparison of these two survey sampling methods. Also the homogeneity of clusters has been tested by using chi-square test. Results: It is observed that BCG, OPV and DPT vaccination coverage is more than 90% whereas Hepatitis B, Measles, Hib and MMR vaccination coverage is between 50% - 64% only. Here systematic random sampling is more complicated than two stage (30 × 30) cluster sampling. Also the result shows that the clusters are homogeneous with respect to proportion of children vaccinated. Conclusion: There is no significant difference between the two survey methodologies regarding the point estimation of vaccination coverage but estimation of variances of vaccination coverage is less in two stage (30 × 30) cluster sampling than that of the systematic sampling. Also the clusters are homogeneous. Very less improvement has been observed in case of fully vaccination coverage than the previous study. From the study it can be said that two stage (30 × 30) cluster sampling will be preferred to systematic sampling and simple random sampling method.
World Health Organization (WHO) recommends that all children should receive one dose of Bacillis Calmette- Guerin Vaccine (BCG), three doses of diphtheria-tetanus-pertusis vaccine (DPT), three doses of either oral polio vaccine (OPV) or inactivated polio vaccine (IPV), three doses of hepatitis B vaccine, and one dose of a measles virus-containing vaccine (MVCV), either anti-measles alone or in combination with other antigens. It also recommends three doses of vaccine against infection with Haemophilus influenza type b (Hib). To boost immunity at older ages, additional immunizations are recommended for healthcare workers, travelers, high-risk groups and people in areas where the risk of specific vaccine-preventable diseases is high [1] . The important role played by the WHO’s EPI (Expanded Programme on Immunization) Cluster Survey in the success of national immunization programme efforts in many countries is widely recognized. The programme monitoring capability provided through the conduct of periodic cluster surveys has been especially important in developing country settings, where administrative records are often incomplete [2] . Together with EPI sampling other survey sampling has been compared in different studies [3] -[5] . According to WHO coverage of BCG vaccine is 87%, DPT3 vaccine is 72% and OPV3 vaccine is 70% in 2011 [6] . In a study Phukan et al. reported that the children of Assam in the North-East Region of India have consistently evidenced low rates for routine childhood immunizations. About 62.2% of the children were fully immunized [7] . Children are considered fully immunized if they receive one dose of BCG, three doses of OPV and DPT each and one dose of measles vaccine before reaching one year of age.
In this study estimates of vaccination coverage have been compared using design effect and variance of estimated proportion of children vaccinated against BCG, OPV, DPT, Hepatitis B, Hib, Measles and MMR (measles mumps rubella) vaccines under two stage (30 × 30) cluster sampling and systematic random sampling.
2. Methods
The data that has been used in this study is taken from a survey “Comparison of Two Survey Methodologies to Estimate Total Vaccination Coverage” sponsored by Indian Council of Medical Research (ICMR), New Delhi. It has been collected during the period from January to October, 2011 using following sampling techniques.
Two stage (30 × 30) cluster sampling: In this method the population needs to be divided into a complete set of non-overlapping subpopulations, usually defined by geographic or political boundaries. These subpopulations are called clusters. In the first stage, 30 of these clusters are sampled with probability proportionate to the size (PPS) of the population in the cluster. Sampling with probability proportionate to size allows the larger clusters to have a greater chance of being selected. The clusters are sampled without replacement. In the second stage of sampling, thirty subjects are selected within each cluster. Although the sampling unit is the individual subject, the sampling is conducted on the household level. Cluster sampling is often a practical approach to surveys because it samples by groups (clusters) of elements rather than by individual elements. It simplifies the task of constructing sampling frames, and it reduces the survey costs [8] . The advantages of two stage (30 × 30) cluster sampling over other designs are same as cluster sampling. A sampling frame listing all elements in the population may be impossible or costly to obtain, whereas to obtain a list of all clusters may be easy. Also the cost of obtaining data may be inflated by travel cost if the sampled elements are spread over a large geographic area.
Systematic random sampling: Systematic sampling is a random method of sampling in which only the first unit is selected with the help of random numbers and the rest get selected automatically according to some pre-designed pattern. If the population size N = nk, where n is the sample size and k is an integer, and a random number less than or equal to k be selected and every kth unit thereafter. This procedure is linear systematic sampling. When N ≠ nk then every kth unit be included in a circular manner till the whole list is exhausted, it is called circular systematic sampling. Systematic sampling is commonly used as an alternative to simple random sampling (SRS) because of its simplicity. It selects every kth element after a random start (between 1 and k). Its procedural tasks are simple, and the process can easily be checked, whereas it is difficult to verify SRS by examining the results. It is often used in the final stage of multistage sampling when the fieldworker is instructed to select a predetermined proportion of units from the listing of dwellings in a street block. The systematic sampling procedure assigns each element in a population the same probability of being selected [8] .
With the two stage (30 × 30) cluster sampling method in the first stage 30 wards are selected and in the second stage 30 units from each ward are selected. For the selection of second stage units in a selected ward only the first household is randomly selected. After the first household is visited, the surveyor moves to the “next” household, which is defined as the one whose front door is closest to the one just visited. Where there are bylane in a particular lane survey procedure is carried out in that place according to the serial household number in that bylane. This process continues until all 30 eligible subjects are found. The subjects are chosen by selecting a household and for more than one eligible subject (children from 6 months to 5 years of age) in a household all are selected.
After completing the 1st sampling method (that is two stage (30 × 30) cluster sampling) in a ward, 2nd sampling method (systematic random sampling) is carried out in same ward. In this sampling technique a random number is selected from random number table on the basis of the number of household in a lane where the survey was carried out in case of two stage (30 × 30) cluster sampling and this became the first sampling unit (household) of the systematic random sampling. After that each household is selected at an interval of 10 household and continuing the process until the 30 sampling units are not completed. Here the interval of household is taken as 10 so that the interval is neither too small nor too large. If we take the interval too small then we should get so many repetitions of the samples from two stage (30 × 30) cluster sampling which results same sampling unit in the 2nd sampling method (systematic sampling) and if we take the interval too large then there should not be any similarity between the two sampling methodologies as the larger interval will cover larger area and both the sampling techniques would take different places.
3. Statistical Analysis
Analysis has been carried out in the following two sections.
3.1. Section A
Here, variance of proportion of vaccination coverage and design effect of the same has been estimated.
Let, P = proportion of children who are vaccinated
Since same number of children has sampled per cluster, estimate of P is given by
where = the proportion of surveyed children in ith cluster
n = the number of clusters
Then approximate estimated variance of under cluster sampling [4] is given by
Again the estimated variance of under systematic sampling [9] is
An approximate 95% confidence interval on P can be obtained by using
The design effect may be estimated as
where
is the estimated variance under simple random sampling [4] .
Also the design effect for cluster sampling vs systematic sampling is obtained as
3.2. Section B
In this section homogeneity of clusters have been tested by using chi-square test. That is to test equality of proportion of children vaccinated in each clusters. The test procedure is carried out taking Hepatitis B (at birth) vaccine (two stage (30 × 30) cluster sampling).
The null hypothesis is that there are no significant differences among the proportions of children vaccinated against Hepatitis B (at birth) in each clusters.
H0:
Against the alternative that all the proportions are not equal.
H1: Not all Pj’s are equal (where)
The test statistic is
where
fo = observed frequency in a particular cell of a 2 × 30 contingency table
fe = expected frequency in a particular cell if the null hypothesis is true
If the null hypothesis is true the proportions are all equal across the population. And rejecting the null hypothesis only allows to reach the conclusion that all proportions are not equal. But the test statistics does not give any information about proportions that differ. To identify the differences between proportions we will rely on a multiple comparison procedure. The Marascuilo procedure [10] enables us to make comparisons between all pairs of groups. In this procedure the absolute value of the pairwise difference between sample proportions has to be computed. The absolute values of these differences are the test statistics. For each pairwise comparison a critical value is computed as follows:
where α = level of significance, k = number of clusters
To compare each of test statistics with the corresponding critical value a specific pair is significantly different if the absolute difference in the sample proportion is greater than its critical range.
4. Results
Table 1 gives estimated coverage of BCG (at birth), OPV (OPV1 at birth, OPV2 at 6 weeks, OPV3 at 10 weeks, OPV4 at 14 weeks, OPV5 at 15 - 18 months and OPV6 at 5 years), DPT (DPT1 at 6 weeks, DPT2 at 10 weeks, DPT3 at 14 weeks, DPT4 at 15 - 18 months and DPT5 at 5 years), Hepatitis B (HepB1 at birth and HepB2 at 6 weeks), Hib (Hib1 at 6 weeks, Hib2 at 10 weeks and Hib3 at 14 weeks), Measles (at 9 months) and MMR (at 15 - 18 months) vaccine with 95% confidence intervals under two stage cluster and systematic sampling. Coverage of BCG vaccine is 99%, OPV and DPT vaccine coverage is more than 90% except for OPV6 and DPT5. But coverage of Hepatitis B, Hib, Measles and MMR vaccines are only between 50% - 64%. Though the individual vaccination coverage is high for BCG, OPV and DPT vaccine but fully vaccination coverage is only 63.52%. Both the survey methods have given point estimates of vaccination coverage with less difference.
Estimated variance of proportion of vaccination coverage is given in Table 2. It is seen that variances are less in case of two stage cluster sampling than the systematic sampling for all the vaccines namely BCG, OPV, DPT, Hepatitis B, Hib, Measles and MMR that are considered in the study. So the interval estimation of vaccination coverage has given better estimate in case of two stage (30 × 30) cluster sampling than the systematic sampling with less standard error (SE).
Estimated coverage of vaccines under two stage cluster (30 × 30) and systematic sampling
Vaccine
Two stage cluster (30 × 30)
Systematic sampling
Coverage estimate
95% CI
Coverage estimate
95% CI
BCG
0.99
(0.98, 0.99)
0.99
(0.98, 0.99)
OPV
OPV1
0.99
(0.98, 0.99)
0.99
(0.98,0.99)
OPV2
0.98
(0.97, 0.98)
0.99
(0.98,0.99)
OPV3
0.98
(0.97, 0.98)
0.99
(0.98,0.99)
OPV4
0.97
(0.96, 0.97)
0.99
(0.98,0.99)
OPV5
0.90
(0.89, 0.90)
0.89
(0.86,0.91)
OPV6
0.54
(0.53, 0.54)
0.54
(0.50,0.57)
DPT
DPT1
0.98
(0.97, 0.98)
0.99
(0.98,0.99)
DPT2
0.98
(0.97, 0.98)
0.99
(0.98,0.99)
DPT3
0.97
(0.96, 0.97)
0.98
(0.98,0.99)
DPT4
0.90
(0.89, 0.90)
0.90
(0.88,0.91)
DPT5
0.52
(0.51, 0.52)
0.51
(0.47,0.54)
Hepatitis B
HepB1
0.58
(0.57, 0.58)
0.56
(0.52,0.59)
HepB2
0.59
(0.58, 0.59)
0.56
(0.52,0.59)
Hib
Hib1
0.57
(0.56, 0.57)
0.55
(0.51,0.58)
Hib2
0.57
(0.56, 0.57)
0.55
(0.51,0.58)
Hib3
0.57
(0.50, 0.64)
0.55
(0.51,0.58)
Measles
0.64
(0.63, 0.64)
0.64
(0.60, 0.67)
MMR
0.52
(0.51, 0.52)
0.50
(0.46, 0.53)
Estimated variance of proportion of vaccination coverage<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/18-8203489x23.png" xlink:type="simple"/></inline-formula>
Vaccines
Methodology
Two stage cluster (30 × 30)
Systematic sampling
BCG
9.2009 × 10−09
2.44173 × 10−06
OPV
OPV1
1.1947 × 10−08
4.87257 × 10−06
OPV2
3.2958 × 10−08
9.70164 × 10−06
OPV3
3.2134 × 10−08
1.32949 × 10−05
OPV4
1.2785 × 10−07
1.56768 × 10−05
OPV5
5.4684 × 10−06
2.29435 × 10−04
OPV6
6.0007 × 10−07
1.06516 × 10−04
OPV7
9.5874 × 10−07
2.73425 × 10−04
DPT
DPT1
3.2958 × 10−08
9.70164 × 10−06
DPT2
6.4818 × 10−08
1.20999 × 10−05
DPT3
9.4344 × 10−08
1.32949 × 10−05
DPT4
6.4134 × 10−07
1.03118 × 10−04
DPT5
9.8328 × 10−07
2.75069 × 10−04
Hepatitis B
HepB1
1.1741 × 10−06
2.7177 × 10−04
HepB2
1.1741 × 10−06
2.71907 × 10−04
Hib
Hib1
1.2841 × 10−06
2.72553 × 10−04
Hib2
1.2814 × 10−06
2.72303 × 10−04
Hib3
1.305 × 10−06
2.72429 × 10−04
Measles
1.6381 × 10−06
2.53068 × 10−04
MMR
1.702 × 10−06
2.75305 × 10−04
Table 3 represents estimates of design effect of proportion of children vaccinated against different types of vaccines. Design effect estimates are calculated for two stage cluster sampling vs simple random sampling, systematic sampling vs simple random sampling and cluster sampling vs systematic sampling. It is seen that design effect estimates are high in systematic sampling vs simple random sampling rather than the two stage cluster sampling vs simple random sampling and cluster sampling vs systematic sampling for all the vaccines considered here.
To study the homogeneity of clusters chi-square test has been performed. Here calculated value of is 116.68 with 29 d.f. and p value is 0.00 that is the test statistic is significant and we reject the null hypothesis and concluded that the proportions of children vaccinated against Hepatitis B (at birth) are not equal. Let us start with computing all the proportions of children vaccinated against Hepatitis B (at birth) (given in Table 4).
Estimates of design effect of proportion of children vaccinated
Vaccine
Design effect
Cluster vs SRS
Systematic vs SRS
Cluster vs systematic
BCG
0.000835516
0.221728395
0.003768
OPV
OPV1
0.001084923
0.442469136
0.002452
OPV2
0.001511716
0.880987654
0.003397
OPV3
0.0014739
1.207284
0.002417
OPV4
0.003949769
1.423580247
0.008155
OPV5
0.022256
0.933037
0.023834
OPV6
0.005094
0.831239
0.005634
OPV7
0.001308
0.411678
0.003506
DPT
DPT1
0.001511716
0.880987654
0.003397
DPT2
0.002973041
1.098765432
0.005357
DPT3
0.002914598
1.207283951
0.007096
DPT4
0.0054443
0.87535464
0.00622
DPT5
0.00133547
0.41166791
0.003575
Hepatitis B
HepB1
0.004333
0.991563
0.00432
HepB2
0.004364
0.992063
0.004318
Hib
Hib1
0.004710096
0.99
0.004712
Hib2
0.0047
0.989091
0.004706
Hib3
0.0047867
0.9895506
0.00479
Measles
0.0059581
0.9380207
0.006473
MMR
0.0052098
0.8413334
0.006182
Estimated proportions of children vaccinated against Hepatitis B (at birth)
Sl. No.
Ward No.
Estimated proportions
1
2
0.17
2
4
0.70
3
5
0.23
4
11
0.93
5
12
0.53
6
15
0.60
7
17
0.47
8
18
0.70
9
24
0.17
10
25
0.67
11
26
0.63
12
33
0.73
13
35
0.40
14
36
0.63
15
37
0.63
16
38
0.43
17
40
0.53
18
42
0.73
19
43
0.67
20
46
0.53
21
47
0.57
22
48
0.60
23
50
0.80
24
51
0.63
25
53
0.67
26
54
0.63
27
55
0.37
28
57
0.70
29
59
0.57
30
60
0.83
It is seen that Hepatitis B (at birth) vaccine coverage is higher for ward number 11 (p4 = 0.93) than all other wards. After that and are computed and compared each of test statistics with the corresponding critical value (given in Table 5).
Results are significant only for proportion of Hepatitis B (at birth) vaccine coverage for ward number 1 vs ward number 4 (vs), ward number 1 vs ward number 30 (vs), ward number 3 vs ward number 4 (vs), ward number 4 vs ward number 9 (vs) and ward number 9 vs ward number 30 (vs). That is these proportions are not equal. Out of 435 pairs of proportions of vaccination coverage only 5 pairs of proportions are unequal.
5. Discussion
Estimates of variances and design effect have been used by Milligan et al. [4] to compare two cluster sampling methods for health surveys in developing countries. Both the methods gave very similar point estimates of vaccination coverage. The estimates of the proportion fully vaccinated were 0.56 (EPI) and 0.54 (segmented method) and suggest that EPI method can give accurate and precise results. On the basis of this previous study the current study tries to estimate the design effect of vaccination coverage of the considered study population. In a study of comparison of survey methodologies relative feasibility of the sampling methodologies was assessed by Luman et al. [3] . Coverage with routine vaccinations among children aged 12 - 23 months was much lower than coverage achieved through the measles SIA (supplemental immunization activities). Also Katz et al. studied bias estimate and design effects associated with the EPI sampling design [11] . Brogan et al. suggested techniques for improving the accuracy of the EPI cluster survey method [12] . In Bangladesh overall only 64.1% of children received the measles vaccine, polio1 has the highest coverage rate in both urban and rural areas. The study also reported that percentage of receiving DPT and polio vaccine decreases when higher doses are given [13] . Chhabra et al. studied the factors affecting the vaccination coverage in two urbanized villages of East Delhi. The coverage levels were highest for BCG (82.7%) and DPT/OPV1 (81.5%) and lowest for HBV3 (24.3%). About 65.3% had received primary immunization while only 41.6% of children had received MMR vaccine [14] .
Pairwise Comparison of test statistics (|p<sub>i</sub> - p<sub>j</sub>|) and critical values (CV<sub>ij</sub>)
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