I'm reading a journal article which applies Poisson distribution in determining how many factors can be regarded as beyond the poverty threshold.
My question is: why applying Poisson distribution and comparing the model with actual distribution can do the job? And, any test of significance can be applied to confirm the deviation exists?
Let me choose one part of the article to explain my question.
The author explains that:
... the Poisson framework can demonstrate the extent to which the empirical distribution of deprivation can be modelled against the expected frequency of distribution given the population average number of deprived items. It follows that any positive deviation from the Poisson distribution would indicate an excess incidence of deprivation, capturing the size of the population that are systemically experiencing undue concentrations of deprivation (Babones et al., 2016).'
Also,
Where empirical concentrations of unmet need can be explained by pure random chance, there is no need to problematize the situation by identifying a poverty threshold. It is only where empirical concentrations of unmet need are systematically greater than chance that needs should be identified as "poverty".
My understanding is that Poisson distribution can be applied to answer the probability of the times of occurrence of an event given the lambda value (mean). But I haven't heard that one can determine such frequency is due to chance or not.
Here is an example from the article: The first application concerns identifying the threshold for non-possession. In Fig. 1, a Poisson curve is plotted using the population average number of missing items, 1.16, as the value of lambda. Against this has been plotted the empirical frequency of individuals missing 1, 2, 3, …, 8 items. While the two curves demonstrate a broadly similar trend, a heavy right tail can be observed in the empirical frequency with larger numbers of missing items. This implies that there exists an excessive number of individuals that do not possess multiple sets of essential items than would normally arise from the Poisson expectation.
Table 2 presents the data plotted in Fig. 1 to estimate the cumulative incidence of excess non-possession relative to the baseline counterpart. According to the Poisson distribution, it is expected that around 2.39% of the population (584,275) will fail to possess 4 items with the value of lambda at λ = 1.1647. Yet, the actual data suggests that around 4.22% of the population (1,031,853) have reported missing 4 essential items. Importantly, such a discrepancy arises at k = 4 where the empirical distribution of non-possession surpasses the baseline incidence onwards. It follows that the k ≥ 4 threshold results in a nominal deprivation rate of 5.6 per cent, which translates into a total excessive number of 1,365,830 individuals.
In short, "the thresholds allow one to identify the level at which the empirical concentrations of unmet need are greater than the statistical expectancy".
I'm very interested in knowing extra function of Poisson distribution.
