STA 312 Fall 2010

Categorical Data Analysis

(Discrete random variables)

Vitamin C and Colds

Cold No Cold

Placebo 31 109

Vitamin C 17 122

Race of Prisoner and Victim

White Victim Black Victim

White Prisoner 151 9

Black Prisoner 63 103

Graduate School Admissions

Not Admitted Admitted

Dept. A 332 601

Dept. B 215 370

Dept. C 596 322

Dept. D 523 269

Dept. E 437 147

Dept. F 668 46

Issues to consider

• Sometimes there is a clear independent variable and dependent variable, sometimes not -- different statistical models?

• Of course some variables have more than 2 categories.

• Beyond 2-dimensions (Prisoner race by Victim race by Death penalty, Admission by Sex by Department)

• Structural zeros - Sex by Cause of death, but one cause is childbirth.

Two treatments for Kidney Stones

Treatment A Treatment B

Effective 273 (78%) 289 (83%)

Ineffective 77 61

Total 350 350

Simpson’s Paradox

Treatment A Treatment B

Small Stones 93% (81/87) 87% (234/270)

Large Stones 73% (192/263) 69% (55/80)

Both 78% (273/350) 83% (289/350)

Distributions

• Bernoulli

• Binomial

• Multinomial

• Poisson (process)

Poisson Process• Events happening randomly in space or time• Independent increments• For a small region or interval,

– Chance of 2 or more events is negligible– Chance of an event roughly proportional to the

size of the region or interval

• Then (solve a system of differential equations), the probability of observing x events in a region of size t is