Estimates of Sampling Error
Estimates from sample surveys are affected by two types of errors: non-sampling errors and sampling errors. Non-sampling errors result from mistakes made during data collection, e.g., misinterpretation of an HIV test result and data management errors such as transcription errors during data entry. While NAIIS implemented numerous quality assurance and control measures to minimize non-sampling errors, these were impossible to avoid and difficult to evaluate statistically. In contrast, sampling errors can be evaluated statistically. Sampling errors are a measure of the variability between all possible samples. The
sample of respondents selected for NAIIS was only one of many samples that could have been selected from the same population, using the same design and expected size. Each of these samples could yield
results that differed somewhat from the results of the actual sample selected. Although the degree of variability cannot be known exactly, it can be estimated from the survey results.
The standard error, which is the square root of the variance, is the usual measurement of sampling error for a statistic (e.g., proportion, mean, rate, count). In turn, the standard error can be used to calculate
confidence intervals within which the true value for the population can reasonably be assumed to fall. For example, for any given statistic calculated from a sample survey, the value of that statistic will fall
within a range of approximately plus or minus two times the standard error of that statistic in 95% of all possible samples of identical size and design.
NAIIS utilized a multi-stage stratified sample design, which required complex calculations to obtainsampling errors. The Taylor linearization method of variance estimation was used for survey estimates that are proportions, e.g., HIV prevalence. The Jackknife repeated replication method was used for variance estimation of more complex statistics such as rates, e.g., annual HIV incidence and counts such as the number of people living with HIV.
The Taylor linearization method treats any percentage or average as a ratio estimate, , where y represents the total sample value for variable y and x represents the total number of cases in the group or subgroup under consideration. The variance of r is computed using the formula given below, with the standard error being the square root of the variance: in which Where represents the stratum, which varies from 1 to H, is the total number of clusters selected in the hth stratum, is the sum of the weighted values of variable y in the ith cluster in the hth stratum, is the sum of the weighted number of cases in the ith cluster in the hth stratum and, f is the overall sampling fraction, which is so small that it is ignored.
In addition to the standard error, the design effect for each estimate is also calculated. The design effect is defined as the ratio of the standard error using the given sample design to the standard error that would result if a simple random sample had been used. A design effect of 1.0 indicates that the sample design is as efficient as a simple random sample, while a value greater than 1.0 indicates the increase in the sampling error due to the use of a more complex and less statistically efficient design. Confidence limits for the estimates, which are calculated as
where t(0.975, K) is the 97.5th percentile of a t-distribution with K degrees of freedom, are also computed.
Sampling errors for selected variables from NAIIS are presented in Tables C.1 through C.9. For most variables, sampling error tables include the weighted estimate, unweighted denominator, standard error or
design effect and lower- and upper-95% confidence limits.
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