• 7.1 Reconciling true and incurred personal taxi costs
• 7.2 Reconciling true and incurred personal taxi costs ignoring the known censoring
• 7.3 Reconciling true and incurred daily living costs
• 7.4 Conclusion

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In this analysis, the fact that many survey respondents said they don't use taxis more than they currently do, because they are "too expensive" is used to provide more meaningful estimates of the costs of blindness. It is possible to adjust the estimates for the average weekly personal cost of taxis by taking the extra information that Gravitas obtained from survey respondents into consideration.

The approach is common in medical research trials when patients are administered certain treatments. Their lifetimes are measured, and hopefully the new improved treatment is the one that in the end leads to longer lifetimes. The problem in this sort of research is that some patients die and we know their lifetimes exactly, but other patients are still alive at the end of the trial. Being alive is a good thing and should be credited to the treatment that patient was given. We know how long the individual has managed to stay alive and we can use this amount of time as a real data point. We must record the fact that this person was still alive and we say that this observations is censored.

Another analogy comes from sport. When we measure a cricketer's batting average, we add up the total number of runs scored, and divide by the number of times the batsman was dismissed. Often a batsman is left not out at the end of play, and this counts in their favour. The not out scores are called censored observations.

Returning to the data for personal costs for taxis, we found that 33 respondents (representing an estimated 1,782 RNZFB members) who gave a personal cost for taxis also gave their main reason for not using them more as "Don't need to". We can be sure that the value these respondents gave is as accurate as possible, and is not censored in any way due to affordability.

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There were 34 respondents (representing an estimated 1,929 RNZFB members) who gave a personal taxi cost value also gave their main reason for not using taxis more, as "Too expensive". These observations are censored. These people are considered to have run out of resources to mitigate their costs of blindness through expenditure on taxis.

The remaining reasons given for not using taxis more are not so clearly categorized into "definitely censored and "definitely not censored". We have investigated the estimated average weekly personal cost for taxis by randomly assigning these people to the groups "censored" or "not censored". Because we have introduced a random element, we try to eliminate the negative effects of this randomness by repeating the assignments over and over again.

In the sections that follow, results that have been averaged over hundreds of simulation runs are presented. In each simulation run, we randomly assign censoring status to the observations; work out what this means for the estimated mean, median, lower and upper quartiles; and estimate the proportion of RNZFB members that the simulation represents. After all the runs have been accumulated, we have constructed figures that illustrate the pattern of the estimated average weekly personal cost for taxis as the level of censoring changes. We do not have a very good idea of the exact level of censoring that exists in the population, but we can be fairly sure that we have a good estimate of the lower and upper bounds on the amount of censoring.

For statistical reasons, the maximum observation was not allowed to become censored in any simulation runs. Whenever this occurred the results from that simulation run were discarded. We have also ensured that no two simulation runs had exactly the same assignment of censoring status. For these reasons, the number of admissible simulations changes from analysis to analysis.

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7.1 Reconciling true and incurred personal taxi costs

In this analysis, a total of 100 observations were able to be used. These were the 98 observations used in Section 5, and two more individuals that had 100% subsidy and therefore no incurred personal costs. One of these people said they did not need to use taxis any more than they currently did, but the other indicated their reason as "other".

A total of 406 different censoring simulations were produced over different implicit levels of right censoring from just over 0.33 (the 'base' simulation) to just under 0.5 (which represents about 50% of the estimated RNZFB membership that have this cost). There were more simulations carried out at lower implicit levels of censoring than for higher levels (the number of simulations produced over the different implicit levels of right censoring is given in Exhibit H1.1).

The 'base' simulation represents the minimal level of censoring to allow for costs being constrained by cost (according to the main reasons given for not using taxi more often). Table 19 provides the estimated measures for the Base simulation and also the same measures without considering any restriction on costs (i.e. without any censored costs at all). As expected, all measures are increased when censoring is used, but this is very minimal for the lower quartile. Having allowed for personal taxi costs to be constrained by their affordability, and not introducing any additional censoring, the mean of personal taxi costs for RNZFB members (who have this cost) has increased by an estimated $9 to over $23. The median also increases from under $7 to over $15.

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Table 19: Comparison of measures for personal weekly taxi costs between the Base simulation and Without considering any restrictions on costs.

Note: The mean for the Without case is 11 cents smaller than that given in Exhibit F2.2 as these measures are now based on the inclusion of the two $0 costs.

The general trend (using lowess smoother) over the implicit level of right censoring of each of these measures is shown in Exhibit H1.5.

Exhibit H1.5: Lowess smoother fitted to each simulated measures for censored personal taxi costs.

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There is very little change in the estimated lower quartile, median or upper quartile across the range of implicit censoring. The estimated mean is always larger than the estimated median, which reflects that the cost values are right skewed (meaning that the top 50% or larger costs are more spread out than the bottom 50% of costs).

As the implicit level of censoring increases the mean increases and moves further away from the median, and the cost values become more right skewed. These results suggest that when allowing for personal taxi costs to be constrained, about half of RNZFB members with a personal short distance taxi cost will have a cost between $2.50 and $40.00 a week.

Exhibit H1.6 shows that the values of the mean, median lower and upper quartiles all increase as the level of censoring increases, not easily seen in Exhibit H1.5.

Exhibit H1.6: Indicative values of summary statistics for grouped implicit levels of right censoring

(Note: Grouped Implicit Level label refers to the starting level of the group i.e. label '0.33' is the group of all simulations from the level 0.33 up to but not including the level 0.36)

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7.2 Reconciling true and incurred personal taxi costs ignoring the known censoring

In this investigation, the censoring status has been randomly assigned for all survey respondents. It therefore ignores the answers survey respondents gave the question about using taxis more than they currently do.

The aim was to see the pattern of increase in the summary statistics of interest as the level of censoring increases. The level of censoring was allowed to vary over the range from 0 (where no individual survey respondent's personal costs have been censored) to just under 0.5 (50% of the estimated RNZFB members with this cost being allocated a censored cost). Any simulations that resulted in having both cost values censored for individuals '145' and '197' were discarded for methodological reasons.

In total, 1,302 distinct and admissible censoring simulation runs were produced. The simulations are relatively equally spread across the different levels, with slightly smaller numbers of runs at the extremes. The number of simulations produced over the different implicit levels of right censoring is given in Exhibit H2.1.

Scatter plots for each of these estimated measures showing all 1,302 simulation results are given in Appendix H (Exhibits H.2.2 & H.2.4). The general trend (using lowess smoother) over the implicit level of right censoring of each of these measures is shown in Exhibit H2.5.

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Exhibit H2.5: Lowess smoother fitted to each simulated measures for censored personal taxi costs.

There is very little change in the estimated lower quartile, median or upper quartile across the first 20% of implicit censoring level. After this the upper quartile increases more rapidly as the level increases. Similarly so does the median, but not as rapidly.

The estimated mean appears to increase at a constant rate as the implicit level of right censoring increases. The estimated mean is always larger than the estimated median, which shows that the cost values are right skewed.

No individual observation was influential in this analysis, so we can be confident that we have illustrated the impact that censoring has on the estimated cost of personal taxi usage. The methodology used in this section has also been applied to the daily living costs in the next section so it is important to understand the reliability of the method.

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We can see by looking at Exhibit H2.8 that the analyses of this section and of Section 7.1 that there is a difference between the median cost for personal taxi usage across the range of implicit censoring used in Section 7.1.

Exhibit H.2.8: Comparison of simulation measures using Lowess smoothers for censoring simulations using reasons and random allocation.

The reason for this difference is not clear, but is definitely due to the fact that the censoring status of some individuals was kept constant in Section 7.1. The upshot of this part of the investigation is that there is a definite difference between the analysis using known censoring status and the analysis where censoring was entirely random. This second analysis shows the pattern that results from an increased level of censoring across a wider range of implicit censoring, but the analysis of Section 7.1 is to be preferred. Researchers must endeavour to obtain the censoring status from survey respondents if the correct estimates of the true cost of personal taxi usage is to be determined.

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7.3 Reconciling true and incurred daily living costs

No information relating to possible constraints for personal daily living task costs was collected in the survey. So each simulation for personal daily living costs has been carried out by randomly assigning censoring to each respondent with a personal cost.

Within each simulation each individual cost value had the same chance of being censored. The chance of being censored was varied so that a range of simulations were obtained across different implicit levels of right censoring, from 0 (where no individual survey respondent's personal costs have been censored) to just under 0.5 (50% of the estimated RNZFB members with this cost having censored cost values). Any simulations that resulted in having the largest personal cost value censored for methodological reasons.

The analyses of this section are to be viewed as indicative of the potential for a difference in the true and incurred costs of daily living, because insufficient data exists from survey respondents to make sound judgments.

7.3.1 Home maintenance costs

In total, 870 different useable censoring simulations were produced. The simulations are relatively equally spread across the different levels, with slightly fewer runs at the extremes, as can be seen in Exhibit I1.1.

Scatter plots for each of these estimated measures showing all 870 simulation results are given in Exhibits I1.2 & I1.4. The general trend (using a lowess smoother) over the implicit level of right censoring of each of these measures is shown in Exhibit I1.5.

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Exhibit I1.5 Lowess smoother fitted to each simulated measures for censored Maintenance costs

There is little change in the estimated lower quartile or median in the first 25% of implicit censoring level and after this both estimated measures increase as the level of censoring increases. Both the estimated mean and upper quartile increase as soon as the level of censoring increases. However, the means appears to increase at a fairly constant rate, while the rate of increase in the upper quartile becomes greater at about 25% of implicit censoring. We also found that the contribution of individual 82 from the sample was influential in determining the mean cost of maintenance when they were censored.

7.3.2 Grounds costs

In total, 961 distinct and admissible simulation runs were produced. The simulations are relatively equally spread over the range of censoring levels, as can be seen in Exhibit I2.1.

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Scatter plots for each of these estimated measures showing all 961 simulation results are given in Appendix I (Exhibits I.2.2 & I.2.4). The general trend (using lowess smoother) over the implicit level of right censoring of each of these measures is shown in Exhibit I2.5.

Exhibit I2.5 Lowess smoother fitted to each simulated measures for censored Grounds costs

Individual '192' when censored increase modelled mean by 55 cents

7.3.3 Five combined Daily living tasks

In total, 384 different useable censoring simulations were produced. The simulations are relatively equally spread across the different levels, with slightly fewer at the extremes, as can be seen in Exhibit I3.1.

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Scatter plots for each of these estimated measures showing all 384 simulation results are given in Exhibits I3.2 and I3.4. The general trend (using lowess smoother) over the implicit level of right censoring of each of these measures is shown in Exhibit I3.5.

Exhibit I3.5 Lowess smoother fitted to each simulated measures for censored Five combined costs.

7.4 Conclusion

In this section, we have shown that censoring is a crucial element in determining the true costs of blindness. The average weekly personal cost for taxis has increased as a consequence of the censoring of some people's costs due to the fact they thought taxis were "too expensive".

We have also shown that just assigning the censoring status randomly is not necessarily going to give a good estimate of the actual cost sought, but does

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show the pattern we can expect as the level of censoring increases. Any future research must seek the reasons for not expending more money on costs of blindness so that accurate estimates of the average costs can be obtained.

As was also shown in Section 5.2, this information can also provide better estimates of the proportion of blind and vision impaired people that do or would like to incur a particular expense to mitigate their costs of blindness.

A further point about the need to ask the reasons for not incurring greater expense, is that the level of censoring may in fact be higher than we may believe based on intuition or anecdotal evidence. In Section 7.1, the initial level of censoring from the data was 33%. Would we have gone as far as 33% censoring if we just used random censoring? Only further research will show if this estimate of 33% is accurate. In the meantime it is the best we can do.

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