In chapter 3.3, we have modelled a scenario where a political party commits the electoral fraud in all polling divisions of the selected electoral districts. 8% of the influenced observations were located in 10% of the electoral districts. Thinking about a proper modelling scenario to check the outcomes Chapter 4, we have to reject the described scenario because if we perform some manipulation on all observations in the electoral district, the modelled data will inherit the geographical pattern of the original data. Instead, we have to introduce the manipulated data sporadically. For example, we could randomly select from polling divisions which have Conservative party share less than 33% in 2008. This is 28951 out of 63416 polling divisions. We could get 6300 polling divisions from that list to get the right sample size.
Again, distribution of voter turnout and quantity of empty ballots is normal, with turnout mostly between 40 and 60% and the empty ballot count mainly around 200. This data enables modelling of ballot stuffing. So, we extracted a data sample of size 6300 and performed modelling with the same conditions like the first scenario (50-75% from the empty ballot count in 2008 is used in ballot stuffing in 2011), but this time for Conservative party. Density scatterplots confirm the interference, as for the previous scenario:
Conservative Liberal New Democratic
Figure 37. Density scatterplots for voter turnout and party shares (modelled data).
This confirms the conclusions already made in Chapter 3.3, and now the main question is how did the simulation affect the results of the spatial analysis. We can
see that Global Moran’s Indices changed dramatically for Conservative party share and voter turnout. The structure of the graph matrix is as before:
a) histogram of Moran’s index values for territory units;
b) scatterplot of Moran’s index values against the number of observations;
c) scatterplot of Moran’s index values against the interquartile ranges of variable.
We can see how different are the values of Local Moran’s I, and how the simulation also affected the relationship between the index and the interquartile ranges:
Real data
Modelled data
a) b) c)
Figure 38. Global Moran’s Index for Conservative party share (Canada, 2011).
Thus, Moran’s index looks like a sensitive measure of the interference. Less than 10% of the observations were affected, but most of the indices shifted into a range between 0 and 0.1 which means spatial randomness. Local Moran’s test shows the same differences for voter turnout and Conservative party share, for which the percentage of statistically significant clusters decreased. This can be seen below:
Real data
Modelled data
a) b) c)
Figure 38. Percentage of significant results of Local Moran statistics for voter turnout (Canada, 2011).
Looking at the exploratory plots, we can see a significantly shifted group of observations for voter turnout (12 in total, marked by red circle):
Real data Modelled data
Figure 39. Exploratory plots of Local Moran statistics for voter turnout (Canada, 2011).
They are likely the observations where we have introduced ballot stuffing. We can see that they have higher own values on x axis, while neighborhood means on y axis
are the same as for most of the observations. In fact, there are 14 polling divisions where the data were influenced, so the graph can not detect all of them. As we see from the graph, significant Local Moran’s result are occasional and they do not reflect that outstanding group. Also, there are no statistically significant HL and LH associations indicated, as before. So, we can say than even if Local Moran statistics indicate the interference in general, it can not detect specific examples. There can be three possible explanations of this:
there are no outliers indeed (which we would like to reject because we can see points in LH and HL zones of the exploratory plot that have quite big difference between their values and neighborhood means);
algorithms were used for calculations (probably, poor methodology for calculating spatial weights matrix based only on common border lengths);
methodological approach (Local Moran’s I is not a suitable method for detection of such data).
To check the second hypothesis, we have run Local Moran’s index tool in ArcGIS.
We mapped voter turnout values for real and modelled data using the same color scheme. Then, we ran Anselin Local Moran’s I tool and overlaid identifiers of local spatial association types upon this map to see the changes:
Real data Modelled data
Figure 40. Local Moran statistics for voter turnout (electoral districts #53022, Canada, 2011).
First, we can clearly recognize only 14 out of 19 simulated changes by dark orange and red color. Some of the observations that were classified as LL and were not changed during the modelling process remained being LL, while some were changed and transformed from LL to HL and insignificant. One observation in the center, which was a LH-outlier, was transformed into HH after modelled ballot stuffing. This happened because the real value was low and the neighboring values were high, while after modelling the value increased (see the map on Figure 40). At the same time, some observations changed their class when neither their values nor the neighbors values had changed. But the most surprising thing was that some of the changed observations did not appear among significant results neither in real data nor in modelled data, though it is clear that their values are very different from neighboring values. Looks like this happens because both the observation and the neighbor need to have values different from population mean. In our case, only the value has is different, while the neighborhood is close to an average. Thus, even if Local Moran’s statistics help to detect the interference, like shown above, it does only the general estimate if the data was changed. To enable detection of specific observations that were manipulated, we need to look for another approach.
Coming back to Figure 39, we can expect that comparing values and neighborhood means without comparing them with population mean, like Local Moran’s I does, could give us better results. The workflow is the next:
get values and neighborhood means for selected electoral district from the database;
get id list of polling divisions affected by modelling for the same district, this will be a model list;
calculate ratio between value and neighborhood mean for each observation;
calculate distances between these ratios;
get the list of observations which make up an upper decile (10-quantile) with largest distances, this will be a method list;
match the model list with the method list.
For voter turnout in the electoral district we were looking at above, the number of matches was 12 – the same as the number of outstanding points on the plot. For 14 modelled observations, this is a good number, but at the same time method list contained 19 observations because it was made up from 10-quantile. So, the true effect of the procedure is about 63% for that polling division. If we do the same for remaining polling divisions, we get results which detect the affected observations much better than Local Moran’s I did:
Figure 41. Percentage of detected observations for the electoral fraud modelling scenario in the electoral districts (Canada, 2011).
Thus, we can confirm that spatial analysis of geographic distribution of the electoral data can give valuable information regarding the electoral fraud detection. After modelling a real-life scenario we could reveal the places of interference by using statistical methods.
CONCLUSION AND FUTURE WORK
During the analysis, we have applied a set of statistical techniques to answer the question about the presence of spatial patterns of the electoral data. The analysis provided strong evidences of large-scale regional patterns in voting behavior, and gave an insight about the small-scale local patterns of neighborhood similarity. Simulation studies confirmed significance of the obtained results.
The offered methods and techniques have a lot of space for improvement.
First, the given approach should be tested on other countries and territories.
Second, computational methods might be improved, especially for time-series data analysis. Third, large cities can be analyzed as sets of neighborhoods, which gives new aggregation levels for the study. And finally, the next elections in Canada are coming in 2015, and for us it means new data which might be used as a control for the current results.
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