• relationship = association

• contingency table = crosstabulation = cross-classification

• intersecting frequency distributions

• rows, columns
• cells
• marginals - row and column totals
• r x c designation of a table - # rows x # columns
• computing row or column percentages (conditional percentages)

• goal of bivariate analysis: measure association or relationship

1) is there a relationship?

statistical independence/dependence

• detecting relationship by comparing percentage distributions for one variable across different categories of a second

• almost always some relationship

• simple assessment: compare percentage difference across the different categories of i.v.

• cases' values on one variable go up, values on other variable go up (positive) or down (negative)

• e.g., positive = height and weight

• e.g., negative = size of vehicle and gas mileage

• can be defined for ordinal and interval variables only

• inverse = negative

2 x 2 tables
 

relative risk (RR) = % in reference category of dv for one category of iv divided by % in reference category of dv for other category of iv

odds ratio (OR) = odds of falling in reference category to not falling in reference category for one category of iv to odds for other category of iv

if row variable is i.v. and column variable is d.v., and right column is reference category of d.v. (this format is different from text), then

increased/decreased risk =
(RR - 1) x 100%
 

OR = (b/a) / (d/c)
 
 
 

Perry Preschool (Schweinhart, Barnes, Weikart)

randomized experiment: preschool vs. control (no preschool)

outcome at age 27:

Association can be interpreted as causation in experimental studies
 
 

be careful w/ percentage increases--also important to consider absolute percentage to interpret fully (issue of baseline risk)
 
 
 

Pitfalls in analyzing associations

Simpson's paradox

• direction or strength of relationship changes when aggregating across natural groups that should be treated individually

Ignoring denominators/Incomplete crosstabulations

• "More pedestrians are killed crossing with the signal than against it."

• "More people get killed crossing streets with crosswalks than those without."

"In 1993, juveniles arrested in Orange County, CA, were tested for drug use (urinalysis). Forty-three percent (43%) tested positive, and a newspaper stated that 'the results seemed to bolster beliefs that drug use and juvenile crime are related.'"

Multiple interpretations of associations in observational studies

• "spurious" relationship (confounding)- two variables related by their common association with another variable that each causes them, but the two original variables not causally linked, even indirectly

• e.g., ice cream sales and murder - positive association (weather = key)

• intervening relationship - third variable links the independent variable to the dependent variable

• e.g., race and traffic tickets

• conditional/interactive relationship - strength and/or direction of relationship between two variables differs for different values on third variable

• e.g., lightning strikes and forest fires - positive when dry, weakly positive or none when wet

• causal direction - can go either/both directions

• e.g., friends' behavior x own behavior

Elaboration/partial tables

• examine third variables that could account for association (either as spurious or intervening relationship) or explore conditional relationships

• create tables between two variables for cases that fall into particular categories of a third variable

elaboration: partial tables re seriousness of abuse (conditional relationship?)
 

less serious abuse (e.g., fondling)

	rejected	prosecuted
medical evidence	8 (100%)	0 (0%)	8
no med. evidence	20 (67%)	10 (33%)	30
	28	10	38

 
 

more serious abuse (e.g., penetration)

	rejected	prosecuted
medical evidence	11 (31%)	24 (69%)	35
no med. evidence	23 (56%)	18 (44%)	41
	34	42	76

 
 

Be careful about three criteria for causality - association, time order, and nonspuriousness (no confounding)