Download Some initial plots comparing different scenarios

Structural change assessment
Stephen Catterall
8th December 2009
Methods
The purpose of this section is to compare the TV distribution obtained in the survey with
thirteen reference TV distributions representing various hypothetical scenarios. A suitable
means of comparison of TV distributions is to plot quantile functions in each case. These
plots capture all relevant information in the distributions whilst avoiding the calculation
of summary statistics such as the mean, which may not be appropriate in this context. In
order to account for sampling variability, bootstrapping is used, which is the only
possible method given the available data. Nonparametric bootstrapping is used for the
survey scenario, taking 1000 samples of size 133 (with replacement) from the observed
set of TV values. Parametric bootstrapping is used for the hypothetical scenarios, taking
1000 samples of size 133 from a multinomial distribution with probabilities as specified
in Table 12. The results of the bootstrapping for a scenario comprise 1000 quantile
functions. These may be summarised by envelope curves at the 2.5, 50 and 97.5
percentiles so as to make interpretation easier.
1.0
0.5
0.0
TV median
1.5
2.0
Results
Figure 2 below shows envelope curves for the survey scenario and the thirteen
hypothetical scenarios. The light coloured lines indicate the 2.5 percentile and 97.5
percentile, while the darkly coloured line indicates the median (50 precentile). A
summary of the medians for each scenario is provided in Figure 1.
1
2
3
4
5
6
7
8
scenario
9
10
11
12
13 survey
Figure 1. TV median values for various scenarios. The circle marks the median of the medians of
the 1000 bootstrapped samples for each scenario. The vertical bar indicates the corresponding 2.5
and 97.5 percentiles (so giving a measure of the associated variability).
0.8
1.0
0.8
TV
0.0
0.2
0.4
0.6
0.8
0.0 0.5 1.0 1.5 2.0
0.0 0.5 1.0 1.5 2.0
1.0
1.0
0.0
0.2
0.4
0.6
scenario 8
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
TV
TV
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
probability
probability
scenario 9
scenario 10
scenario 11
scenario 12
0.6
0.8
1.0
0.2
0.4
0.6
0.8
probability
scenario 13
survey scenario
0.4
0.6
probability
0.8
1.0
1.0
TV
TV
0.0
probability
0.0
0.2
0.4
0.6
probability
0.8
1.0
1.0
0.8
1.0
0.8
1.0
0.0 0.5 1.0 1.5 2.0
probability
0.0 0.5 1.0 1.5 2.0
probability
0.4
0.8
0.0 0.5 1.0 1.5 2.0
scenario 7
0.0 0.5 1.0 1.5 2.0
scenario 6
0.0 0.5 1.0 1.5 2.0
scenario 5
TV
0.2
0.6
probability
TV
0.0
0.4
probability
TV
0.2
0.2
probability
0.0 0.5 1.0 1.5 2.0
0.0
TV
0.0
scenario 4
probability
0.0 0.5 1.0 1.5 2.0
0.2
0.0 0.5 1.0 1.5 2.0
TV
0.6
scenario 3
0.0
0.2
0.4
0.6
probability
0.0 0.5 1.0 1.5 2.0
TV
0.4
0.0 0.5 1.0 1.5 2.0
0.0
TV
0.2
0.0 0.5 1.0 1.5 2.0
0.0
TV
scenario 2
0.0 0.5 1.0 1.5 2.0
TV
scenario 1
0.0
0.2
0.4
0.6
0.8
1.0
probability
Figure 2. Comparison of the survey scenario with the 13 hypothetical scenarios. The hypothetical
scenarios are ordered as in Table 12.
Discussion
The survey scenario has a relatively large amount of sampling variability, in comparison
to the hypothetical scenarios. This is not really surprising as most of the hypothetical
scenarios are artificially constrained so as to represent some kind of extreme. Still,
scenarios 1, 5, 6, 9, 10, 11 and 13 have very low variability and could not be described as
realistic, though they are useful reference points. Of the remaining scenarios, scenario 3
seems to be the most similar to the survey scenario. However, none of the hypothetical
scenarios have the large proportions of TV=0 and TV=2 present in the survey scenario.