What's the problem? They don't seem to be controlling for day-level fixed effects or, even better, day-city fixed effects. Suppose there's a big hockey game on Saturday night that both brings a pile of folks onto the street and increases alcohol purchases. You can get a correlation between increased alcohol sales (relative to the week prior) and assaults entirely as artefact of the underlying variable driving both assaults and alcohol sales. A big hockey game, a holiday long weekend, even a big concert in town - none of those are addressed by comparing alcohol sales with those a week prior.
Methods and FindingsWe performed a population-based case-crossover analysis of all persons aged 13 years and older hospitalized for assault in Ontario from 1 April 2002 to 1 December 2004. On the day prior to each assault case's hospitalization, the volume of alcohol sold at the store in closest proximity to the victim's home was compared to the volume of alcohol sold at the same store 7 d earlier. Conditional logistic regression analysis was used to determine the associated relative risk (RR) of assault per 1,000 l higher daily sales of alcohol. Of the 3,212 persons admitted to hospital for assault, nearly 25% were between the ages of 13 and 20 y, and 83% were male. A total of 1,150 assaults (36%) involved the use of a sharp or blunt weapon, and 1,532 (48%) arose during an unarmed brawl or fight. For every 1,000 l more of alcohol sold per store per day, the relative risk of being hospitalized for assault was 1.13 (95% confidence interval [CI] 1.02–1.26). The risk was accentuated for males (1.18, 95% CI 1.05–1.33), youth aged 13 to 20 y (1.21, 95% CI 0.99–1.46), and those in urban areas (1.19, 95% CI 1.06–1.35).
How do you fix this? Controlling for simultaneous alcohol sales in a similar part of town that's far enough away that it's unlikely to have had effects on the part of town in question would be a start, but might not catch localized effects of events that drive both alcohol sales and violence.