4228.0 - Programme for the International Assessment of Adult Competencies, Australia, 2011-12 Quality Declaration 
Latest ISSUE Released at 11:30 AM (CANBERRA TIME) 09/10/2013   
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TECHNICAL NOTE DATA QUALITY


RELIABILITY OF THE ESTIMATES

1 Two types of error are possible in an estimate based on a sample survey: sampling error and non-sampling error. Since the estimates in this publication are based on information obtained from a sample, they are subject to sampling variability. That is, due to randomness in the composition of the sample, the estimates may differ from those population values that would have been produced if all dwellings had been included in the survey. One measure of the likely difference is given by the standard error (SE). 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.

2 In contrast to most other Australian Bureau of Statistics (ABS) surveys, PIAAC estimates also include significant imputation variability, due to the use of multiple possible assessment tasks and the complex scaling procedures. The effect of this on the estimation can be reliably estimated and is included in the calculated SEs. An accepted procedure for estimating the imputation variance using plausible values is to measure the variance of the plausible values (with an appropriate scaling factor) as follows:

Equation: the imputation variance of theta mean equals the summation over i from one to M of theta i minus theta mean all squared all then divided by M minus 1 all then multiplied by the factor one plus the fraction one divided by M

where:
Equation: the mean equals the mean estimate of the plausible values

Equation: i equals one to ten respectively for the plausible values theta one to theta ten

Equation: M equals the total number of plausible values used M equals ten for the Programme for the International Assessment of Adult Competencies

3 Together, the sampling variance and imputation variance can be added to provide a suitable measure of the total variance, and total SE. This SE indicates the extent to which an estimate might have varied by chance because only a sample of persons was included, and/or because of the significant imputation used in the literacy scaling procedures.

4 There are a number of more convenient ways of expressing the sampling variability than the SE. One is called the relative standard error (RSE), which is obtained by expressing the SE as a percentage of the estimate:

Equation: Relative standard error in percent is equal to the standard error divided by the estimate all multiplied by one hundred

5 Another way of expressing the sampling variability is called the margin of error (MOE) which may be more useful for proportion estimates, in particular where the estimated proportion is large or small. MOEs are provided for all proportion estimates at the 95% confidence level. At this confidence level the MOE indicates that there are about 19 chances in 20 that the estimate will differ by less than the specified MOE from the population value. The 95% margin of error is obtained by multiplying the SE by 1.96:

MOE = SE 1.96

6 The estimate combined with the MOE defines a range, known as a confidence interval, which is expected to include the true population value with a given level of confidence, known as a confidence interval. The confidence interval can easily be constructed from the MOE of the same level of confidence by taking the estimate plus or minus the MOE of the estimate. This range should be considered when using the estimates to make assertions about the population or to inform decisions.

7 Whilst the MOEs in this publication are calculated at the 95% confidence level, they can easily be converted to a 90% confidence level by multiplying the MOE by 1.654/1.96 or to a 99% confidence level by multiplying by a factor of 2.576/1.96.

8 The 95% MOE can also be calculated from the RSE by:

Equation: Margin of error is equal to relative standard error in percent multiplied by the estimate multiplied by 1.96 divided by 100

9 Sampling error for estimates from PIAAC 2011-2012 have been calculated using the Jackknife method of variance estimation. This involves the calculation of 60 'replicate' estimates based on 60 different subsamples of the obtained sample. The variability of estimates obtained from these subsamples is used to estimate the sample variability surrounding the estimate.

10 A data cube (spreadsheet) containing tables produced for this publication and the calculated RSEs for each of the estimates, and MOEs for each proportion estimate, is available from the Downloads tab of the publication.

11 Estimates with RSEs less than 25% are considered sufficiently reliable for most purposes. Estimates with RSEs between 25% to 50% have been included and are annotated to indicate they are subject to high sample variability relative to the size of the estimate and should be used with caution. In addition, estimates with RSEs greater than 50% have also been included and annotated to indicate they are usually considered unreliable for most purposes. All cells in the data cube with RSEs greater than 25% contain a comment indicating the size of the RSE. These cells can be identified by a red indicator in the corner of the cell. The comment appears when the mouse pointer hovers over the cell.


CALCULATION OF STANDARD ERROR

12 SEs can be calculated using the estimates (counts or proportions) and the corresponding RSEs. For example, the estimated number of persons in Australia aged 15 to 74 years that have scores at Level 3 on the literacy scale is 6,379,600 and the RSE for this estimate is 1.8%. The SE is calculated by:

Equation: Standard error of estimate equals relative standard error over 100 times estimate example

13 Therefore, there are about two chances in three that the actual number of persons that have scores at Level 3 on the literacy scale is in the range of 6,264,800 to 6,494,400 and about 19 chances in 20 that the value was in the range 6,150,000 to 6,609,200. This example is illustrated in the diagram below.

Image: Calculation of standard error example


PROPORTION AND PERCENTAGES

14 Proportions and percentages formed from the ratio of two estimates are also subject to sampling errors. 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. The formula is only valid when the numerator is a subset of the denominator:

Equation: RSE of Proportion equals square root of RSE x squared minus RSE y squared

15 The proportion of Australians aged 15 to 74 years who have scores at Level 3 on the literacy scale is 39% and the associated RSE is 2.4% and associated MOE is +/- 1.8 percentage points. Hence there are about two chances in three that the true proportion of Australians aged 15 to 74 years who have scores at Level 3 on the literacy scale is between 38.1% and 39.9%, and 19 chances in 20 that the true proportion is within 1.8 percentage points from 39%, that is between the interval 37.2% and 40.8%.

16 The RSEs of proportions within the data cubes have been provided. Calculations of RSEs for other proportions using the above formula should be seen as only indicative of the true RSE.


DIFFERENCES

17 Published estimates may also be used to calculate the difference between two survey estimates (numbers or proportions). Such an estimate is also subject to sampling error. The sampling error of the difference between two estimates depends on their SEs and the relationship (correlation) between them. An approximate SE of the difference between two estimates (x-y) may be calculated by the following formula:

Equation: Standard error x minus y equals square root of standard error x squared plus standard error y squared

18 An approximate MOE of the difference between two estimates (x-y) may be calculated by the following formula:

Equation: Margin of error x minus y equals square root of margin of error x squared plus margin of error y squared

19 These formula will only be exact for differences between separate and uncorrelated characteristics or sub populations and only provides an indication for the differences likely to be of interest in this publication.


SIGNIFICANCE TESTING

20 A statistical significance test for any comparisons between estimates can be performed to determine whether it is likely that there is a difference between two corresponding population characteristics. The approximate standard error of the difference between two corresponding estimates (x and y) can be calculated using the formula in paragraph 17. The standard error is then used to create the following test statistic:

Equation: Statistical significance test statistic equals x minus y over standard error of x minus y

21 If the value of this test statistic is greater than 1.96 then there is evidence, with a 95% level of confidence, of a statistically significant 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 with respect to that characteristic. Any calculations using the above formula should be seen as only indicative of a statistically significant difference.