Do parking lots with gate attendants, for example, have fewer thefts from vehicles than parking lots without gate attendants? Are apartment buildings with on-site managers at less risk of having drug dealing than ones without such managers? To answer questions like these you will need to determine if there is a statistical association between some characteristic of persons, places or events and some measure of the problem (crime, injury, etc.).
There are many ways to calculate association. Often a correlation coefficient is used. Correlation coefficients range from -1 to 1. A negative correlation means an increase in one characteristic is associated with a decline in the other (and a decline is one associated with an increase in the other). A positive correlation means that an increase in one characteristic is associated with an increase in the other (and a decline in one is associated with a decline in the other). Big coefficients mean strong associations (positive or negative). If a correlation coefficient is near zero, there is an absence of association - a change in one characteristic is unrelated to a change in the other. Any spreadsheet or statistical analysis program can perform the calculations.
The Crime Analysis Unit of the Jacksonville-Duval County (FL) Sheriff's Department examined apartment complexes over 50 units. They found that for the 269 apartment complexes, the correlation between number of units and number of crimes is about .57: a modest positive correlation. There is a very high positive correlation (.91) between the number of property crimes and the number of violent crimes in these apartment complexes.
You cannot use a correlation coefficient to measure association in a case-control study (Step 32). Instead, you should use an odds ratio.
Odds ratios can be any number greater than zero. When an odds ratio is equal to one, there is no association between the characteristic and the outcome. That is, the risk of the outcome is the same whether or not the characteristic is present. If the odds ratio is between 0 and 1, risk is higher when the characteristic is absent than when it is present (a negative association). An odds ratio of .1 indicates the risk of the outcome when the characteristic is present is a tenth of that when the characteristic is absent. If an odds ratio is greater than 1, the risk is higher when the characteristic is present than when it is absent (a positive association). An odds ratio of 3 means that the risk of the outcome is three times as large when the characteristic is present than when it is absent.
To use an odds ratio both the outcome and the characteristic must have only two values. For example, for the outcome, 1 means that a bar is a high crime bar and 0 means that it is a low crime bar. For the characteristic, 1 means that the staff has been trained how to prevent assaults, and 0 means that the staff has not been so trained. The odds ratio would tell you whether there is an association between bars that have staff trained to prevent assaults and a bar having a great deal of crime. Here we would expect a negative association, so the odds ratio would have to be less than one to meet our expectations.
Table 1 shows how to calculate an odds ratio. The outcome is in the row and the characteristic is in the column. The number of cases with the appropriate value for both outcome and characteristic is in each cell. Cell A contains the number of cases that have the characteristic in question. Cell C contains the number of cases without the characteristic. Cell B contains the number of controls that have the characteristic. Cell D contains the number of controls without the characteristic. The odds ratio can be computed with a hand calculator using the formula at the bottom of the table, but many statistical software packages will also calculate it.
Table 2 illustrates the application of odds ratios in a case control study of drug dealing places in San Diego (see Step 32). The outcome is persistent cocaine or heroin dealing. There were 58 apartment buildings in the study that had indicators of persistent dealing (cases). There were also 47 apartments in the study that showed no indication of any drug dealing (controls). Does the presence of locked or unlocked gates or on-site managers influence whether a drug dealer will select the location? When apartments with unlocked gates were compared to those locked or with no gates the odds ratio was greater than one, but this was not significantly different from one (see Step 53), so we cannot rule out the possibility that there is no relationship between unlocked gates and drug dealing (recall, when an odds ratio is one, there is no association). The association between locked gates and drug dealing is significantly positive. Buildings with locked gates were almost three and a half times more likely to have cocaine or heroin dealing than other apartment buildings. Apartment buildings with on-site managers had about three-tenths the chance of becoming dealing sites as the apartments without on-site managers.
This is a statistically significant negative association. But remember, correlation is not the same as causation. A correlation suggests that the characteristic in question might be a contributing cause, but by itself a correlation is insufficient to demonstrate causation.
Table 1: Calculating Odds Ratios
|Yes(1) - Cases||A||C||A+C|
|No(0) - Controls||B||D||B+D|
|Odds Ratio = (A/B)/(C/D) = (A*B)/(C*B)|
Table 2: Apartment Building Security and Drug Dealing
|Unlocked Gates||Locked Gates or No Gates||Odds Ratio|
|Locked Gates||Unlocked or No Gates||Odds Ratio|
|On-site Manager||No On-site Manager||Odds Ratio|