Testing Distance-Decay with Spearman's Rank: What r_s = 0.52 Really Shows
Two students test whether the number of cafés and restaurants in 14 Maltese localities is related to distance from Valletta. Distances in kilometres were measured as straight lines from each locality centre to Valletta on a 1:50,000 map, and catering establishments were counted from a business directory published in 2019. Spearman's rank correlation gives r_s = 0.52. For n = 14 the critical value at the 0.05 level (two-tailed) is 0.456. Two places stand out: a south-eastern fishing and waterfront village 9 km from Valletta has 31 establishments, and a dormitory suburb 6 km out has only 4. The students conclude that 'distance-decay explains the pattern of catering in Malta' and recommend their result as a model for the island.
1. Interpret the value r_s = 0.52 against the critical value, and explain what it does and does not allow the students to claim.
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Statistical significance: 0.52 exceeds the critical value of 0.456 at the 0.05 level for n = 14, so the null hypothesis of no relationship can be rejected. The result is unlikely to arise by chance in more than 1 in 20 samples. That is all significance means.
Strength: 0.52 is a moderate relationship at best. Squaring it gives r² ≈ 0.27, so distance from Valletta accounts for only about 27% of the variation in establishment numbers. Roughly three-quarters of the variation is explained by something else - tourist footfall, seafront location, local population, planning permissions, a marina or a square.
Direction: the students must state it. If the ranks run so that nearer localities have more establishments, the relationship is negative (distance-decay). The sign matters and is easy to lose when ranks are reversed.
What it does not allow: 'distance-decay explains the pattern' is too strong. A moderate correlation over 14 cases does not constitute a model for the island, and correlation is not causation - an unmeasured third factor such as tourism or seafront access could drive both distance and establishment numbers. The two named outliers are the clue: their ranking is set by function, not by distance, and a single outlier can shift r_s substantially with n = 14.
2. Identify the weaknesses in the data and method, explaining how each would bias the result.
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1. A 2019 directory used today. Establishments open and close constantly; a five-year-old list undercounts recent growth and counts closed businesses. The error is not random - it will be greatest in rapidly changing suburbs, biasing the ranks.
2. Raw counts, not rates. A locality of 20,000 people will have more cafés than one of 2,000 whatever its distance from Valletta. Counting establishments rather than establishments per 1,000 residents or per 1,000 visitors builds population size into the variable and will manufacture a spurious pattern.
3. Straight-line distance from a 1:50,000 map. This ignores the road network and travel time. Two places may be 6 km apart in a straight line but 20 minutes apart by road, and the harbour separates them. Perceived distance is what matters to a customer.
4. Ties in the ranks. Establishment counts contain many repeated values, and Spearman's rank handles ties crudely; ties reduce the coefficient and distort significance.
5. Small n with outliers. With 14 cases, removing the waterfront village could change r_s markedly. No outlier analysis was done.
6. Definition drift. Does a hotel restaurant count? A pastizzeria? An inconsistent definition adds error.
Each of these weakens the conclusion independently; together they make 'distance-decay explains catering in Malta' unsupported.
3. Redesign the investigation to test distance-decay more convincingly.
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Hypothesis: the density of catering establishments per 1,000 residents decreases with road travel time from Valletta.
Variables: dependent = establishments per 1,000 residents (or per 1,000 visitor-nights in tourist areas); independent = road distance and, better, peak travel time in minutes, measured from a routing service and spot-checked by map measurement on a 1:25,000 extract.
Sampling frame: all localities above a stated population threshold, drawn from the most recent official list, and a current establishment list with a written inclusion rule (licensed to serve food or drink to the public, open at the time of survey). A field check of at least 20% of counts by visiting the locality would validate the directory.
Sample size: aim for 25–30 localities rather than 14, so a single outlier cannot dominate; stratify by region (harbour, north, south, Gozo) to avoid clustering.
Analysis: Spearman's rank is acceptable but the many ties argue for Pearson on the log of counts, or a scatter graph with a line of best fit and residuals examined. Report r_s with and without outliers and state the effect.
Control: check whether the pattern survives when tourism is controlled for, by separating localities with seafronts or marinas from inland ones.
Reporting: give the sign, the value, r², the critical value and the conclusion - 'moderate, statistically significant, but distance explains only about a quarter of the variation'.