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TECHNICAL NOTE
4 Since the estimates in this publication are based on information obtained from occupants of a sample of dwellings, they are subject to sampling variability. That is, they may differ from those estimates that would have been produced if all occupants of all dwellings had been included in the survey. One measure of the likely difference is given by the standard error (SE), which indicates the extent to which an estimate might have varied by chance due to only a sample of dwellings being included. There are about two chances in three (67%) that a sample estimate will differ by less than one SE from the number that would have been obtained if all dwellings had been included, and about 19 chances in 20 (95%) that the difference will be less than two SEs. Relative Standard Error (RSE) 5 Another measure of the likely difference is the relative standard error (RSE), which is obtained by expressing the SE as a percentage of the estimate. 6 RSEs for patient experience estimates (numbers or percentages) have been calculated using the Jackknife method of variance estimation. 7 RSEs were calculated for each separate estimate and are available to download free of charge as data cubes (Excel spreadsheets) from the ABS website <www.abs.gov.au> as an attachment to this publication. 8 In the tables in this publication, only estimates with RSEs less than 25% are considered sufficiently reliable for most purposes. However, estimates with larger RSEs have been included and are preceded by an asterisk (e.g. *2.2) to indicate they are subject to high SEs relative to their estimate and should be used with caution. Estimates with RSEs greater than 50% are preceded by a double asterisk (e.g. **1.5) to indicate that they are considered too unreliable for general use. Replicate Weights Technique 9 A class of techniques called 'replication methods' provide a general method of estimating variances for the types of complex sample designs and weighting procedures employed in ABS household surveys. 10 The basic idea behind the replication approach is to select subsamples repeatedly from the whole sample, for each of which the statistic of interest is calculated. The variance of the full sample statistic is then estimated using the variability among the replicate statistics calculated from these subsamples. The subsamples are called 'replicate groups', and the statistics calculated from these replicates are called 'replicate estimates'. 11 There are various ways of creating replicate subsamples from the full sample. The replicate weights produced for the survey were created under the deleteagroup Jackknife method of replication. There are numerous advantages to using the replicate weighting approach, including the fact that:
Derivation of replicate weights 12 Under the deleteagroup Jackknife method of replicate weighting, weights were derived as follows:
Application of replicate weights 13 As noted above, replicate weights enable variances of estimates to be calculated relatively simply. They also enable unit record analyses such as chisquare and logistic regression to be conducted, which take into account the sample design. 14 Replicate weights for any variable of interest can be calculated from the 30 replicate groups, giving 30 replicate estimates. The distribution of this set of replicate estimates, in conjunction with the full sample estimate, is then used to approximate the variance of the full sample. 15 This method can also be used when modelling relationships from unit record data, regardless of the modelling technique used. In modelling, the full sample would be used to estimate the parameter being studied (such as a regression coefficient); i.e, the 30 replicate groups would be used to provide 30 replicate estimates of the survey parameter. The variance of the estimate of the parameter from the full sample is then approximated, as above, by the variability of the replicate estimates. CALCULATION OF STANDARD ERROR Standard error of an estimate 16 Standard errors can be calculated using the estimate and the corresponding RSEs. For example, Table T1 at the end of this Technical Note shows the estimated number of persons aged 15 years and over who rated their health as excellent, very good or good in the last 12 months is 14,438,500. The RSE table corresponding to the estimate in Table T2 (also below) shows the RSE for this estimate is 0.6%. The SE is calculated by: 17 Therefore, there are about two chances in three that the value that would have been produced if all dwellings had been included in the survey will fall within the range 14,351,900 to 14,525,100 and about 19 chances in 20 that the value will fall within the range 14,265,300 to 14,611,700. This example is illustrated in the diagram below: RELATIVE STANDARD ERRORS OF COMPARITIVE ESTIMATES Proportions 18 Proportions formed from the ratio of two estimates are also subject to sampling error. The size of the error depends on the accuracy of both the numerator and the denominator. A formula to approximate the RSE of a proportion is given below. This formula is only valid when x is a subset of y. 19 As an example, using estimates from Table T1, 11,357,800 persons aged 15 years and over who rated their health as excellent, very good or good saw a general practitioner in the last 12 months, representing 78.7% of the 14,438,500 persons aged 15 years and over who rated their health as excellent, very good or good. As can be seen in Table T2, the RSE of the estimated number of persons aged 15 years and over who rated their health as excellent, very good or good and saw a general practitioner in the last 12 months is 1.2%, and the RSE of the estimated number of persons aged 15 years and over who rated their health as excellent, very good or good is 0.6%. Applying the above formula, the RSE of the proportion is: 20 Therefore, the SE for persons aged 15 years and over who rated their health as excellent, very good or good and saw a general practitioner in the last 12 months as a proportion of persons aged 15 years and over who rated their health as excellent, very good or good is 0.8 percentage points (=78.7×(1.0/100)). Therefore, there are about two chances in three that the proportion is between 77.9% and 79.5% and 19 chances in 20 that the proportion is within the range 77.1% to 80.3%. 21 The exact RSEs of various proportions for patient experience can be found in the data cubes in the download tab of this publication. Sum of or Differences between estimates 22 Published estimates may also be used to calculate the sum of or difference between two survey estimates. Such estimates are also subject to sampling error. The sampling error of the difference between two estimates depends on their SEs and the correlation between them. An approximate SE of the difference between two estimates (xy) may be calculated by the following formula: 23 The sampling error of the sum of two estimates is calculated in a similar way. An approximate SE of the sum of two estimates (x+y) may be calculated by the following formula: 24 As an example, from paragraph 16, the estimated number of persons aged 15 years and over who rated their health as excellent, very good or good in the last 12 months is 14,438,500 and the SE is 86,600. Performing the appropriate calculations, the estimated number of persons aged 15 years and over who rated their health as fair or poor is 2,308,600 and the SE is 73,900 (rounded to nearest 100). The estimated number of persons aged 15 years and over who provided a selfassessed health status rating is: 25 The SE of the estimated number of persons aged 15 years and over who provided a selfassessed health status rating is: 26 Therefore, there are about two chances in three that the value that would have been produced if all dwellings had been included in the survey will fall within the range 16,633,300 to 16,860,900 and about 19 chances in 20 that the value will fall within the range 16,519,500 to 16,974,700. 27 While these formulae will only be exact for differences between separate and uncorrelated characteristics or subpopulations, it is expected to provide a good approximation for all differences likely to be of interest in this publication. STATISTICAL SIGNIFICANCE TESTING 28 To determine whether, with a certain level of confidence, there was evidence of a true difference between corresponding population characteristics, a statistical significance test for comparisons between estimates was performed. The standard error of the difference between two corresponding estimates (x and y) can be calculated using the formula in paragraph 22. This standard error is then used to calculate the following test statistic: 29 If the value of the test statistic is greater than 1.96, there is statistical evidence (with 95% confidence) of a difference in the two populations with respect to that characteristic. Otherwise, it cannot be stated with confidence that there is a real difference between the populations. SELECTED ESTIMATES AND RSES
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