Testing a Fertiliser Subsidy Recommendation Built on a Choropleth and a Correlation
Three students compare rice yields across three townships to advise the regional government where to target a fertiliser subsidy. They shade each township on a choropleth by average yield per square kilometre of township area, obtaining A 3.1 t/ha, B 4.6 t/ha and C 4.2 t/ha, and conclude that A is the priority. Township A covers 1,240 km², of which 62% is reserved forest and only 8% is harvested for paddy; B covers 240 km² with 88% under paddy; C covers 610 km² with 44% under paddy. The students then plot 60 village records of fertiliser use against yield and obtain r = 0.78, and state that subsidising fertiliser would raise yields by about 60%. A separate dataset shows growing-season rainfall varying from 1,050 mm to 1,900 mm across the same villages.
1. Identify the error in the choropleth and state how the data should be presented instead.
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The choropleth measures yield against total township area, but the figure quoted is a yield per hectare of paddy. Township A is 1,240 km², of which only 8%, about 99 km², grows paddy; 62% is reserved forest. Dividing production by the whole township area therefore produces an artificially low value that reflects land use, not farming performance. This is the unit-area problem: administrative areas are unequal containers and their boundaries have nothing to do with the phenomenon being mapped.
The result is a shading pattern that looks like a statement about farmers in A when it is really a statement about how much forest A contains.
Better presentations:
- Calculate yield per hectare of harvested paddy, and map that as a choropleth only if the units are comparable; otherwise use proportional circles located at the paddy areas.
- Use a dot map of paddy area with a second symbol for production, which shows where the cropping actually is.
- Use a dasymetric or land-use base rather than administrative boundaries.
If the policy question is where to target a subsidy, the denominator should be paddy-growing households or paddy hectares, not square kilometres of township.
2. Interpret r = 0.78 correctly, including the coefficient of determination, and evaluate the claim that subsidies would raise yields by about 60%.
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r = 0.78 is a correlation coefficient across 60 village records. Squaring it gives r² = 0.61, meaning that about 61% of the variation in yield between villages is associated with variation in fertiliser use, and about 39% is associated with other factors or with chance in this sample.
The claim of a 60% yield increase is not supported, and its origin is a confusion: 0.61, a proportion of variance, has been read as a percentage increase in yield. Variance explained is not an effect size. To say how many tonnes per hectare an extra kilogram of nutrient would add, the students need the slope of the regression line, with an uncertainty range, not r or r². A strong correlation is also consistent with the reverse direction - farmers who already get high yields can afford more fertiliser - and with both being driven by a third factor.
Finally the unit of analysis matters: the correlation is between villages, not between plots, so it cannot license a claim about what would happen if an individual farmer applied more. The honest statement is that fertiliser use is strongly associated with yield and worth investigating, not that a subsidy will deliver 60%.
3. Name one confounder in this dataset and describe how the students could test whether fertiliser is really causing higher yields.
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Rainfall is the obvious confounder. It ranges from 1,050 mm to 1,900 mm across the same villages, and villages with reliable water can both afford and profit from fertiliser while also yielding more regardless of it. Soil type, rice variety, irrigation and labour availability are similar candidates.
Ways to test the causal claim:
- Control for it statistically: fit a multiple regression of yield on fertiliser use, rainfall, soil type and variety. If the fertiliser coefficient stays positive and precise after rainfall enters the model, the association is more credible; if it collapses, the original correlation was largely rainfall working through fertiliser use.
- Stratify: group villages into rainfall bands (for example 1,050–1,300 mm, 1,300–1,600 mm, above 1,600 mm) and check whether the fertiliser–yield relationship holds within each band.
- Compare like with like: matched pairs of villages with similar rainfall, soil and variety but different fertiliser use.
- If resources allow, a field trial or a before-and-after comparison in a subset of villages where the subsidy is piloted, which is the only design that directly answers the policy question.
The students should also map residuals, since a confounder that varies geographically will show up as spatial clustering of over- and under-prediction.