Feb. 13

• Chapter 12, Try 1-9

• Read Ch. 15 for next Monday

• No meeting Friday

Quiz from end of last time

• 40 of 100 men have high blood pressure 50 of 200 women have high blood pressure

• For men, risk of high b.p. = 40/100 = .40

• Relative risk for men compared to women = .40 / (50/200) = .40 / .25 = 1.6

• For women odds of high b.p. = 50 to 150, which could be reduced to 1 to 3

Simpson’s Paradox

• Nature of relationship is different for whole group than it is for each subgroup

• CAUSE – Confounding effect of a third variable

Graduate Admissions Example

• In graduate academic program A:– 400 of 650 men applicants admitted (61.5%)– 50 of 75 women applicants admitted (66.7%)

• In graduate academic program B:– 50 of 350 men applicants admitted (14.3%)– 125 of 425 women applicants admitted (29.4%)

• Women had higher acceptance rate in both programs

Total of programs A and B

• Men: (400+50) / (650+350) = 450/1000 = 45% admitted

• Women: (50+125) / (75+425) = 175/500 = 35% admitted

• Overall, acceptance rate is higher for men even though women had higher acceptance in each program.

• What’s going on?

Confounding

• Program B is harder to get into

• Most women apply to program B

• Program A is easier to get into

• Most men apply to program A

Stat 200 survey question

• Have you ever driven under the influence of alcohol or drugs?

Approximate Results by Gender

Have Have Not Total

Male 208 (52%) 192 400

Female 242 (40%) 358 600

Total 450 (45%) 550 1,000

A Research Question

• Is there a “statistically significant” relationship?

• Does the relationship observed in the sample also hold in the population?

Chi-Square Procedure

• A Chi-square test is used to analyze statistical significance.

The idea of Chi-Square

• Chi-square measures the difference between the observed counts and “expected counts”

• Expected counts = the counts that would occur if there were no relationship.

Properties of Expected Counts

• Same row and column totals as observed counts

• Row percentages are the same in each row.

Expected Counts

Have Have Not Total

Male 400

Female 600

Total 450 (45%) 550 1,000

Expected Counts

Have Have Not Total

Male 180 (45%) 220 400

Female 270 (45%) 330 600

Total 450 (45%) 550 1,000

Chi-square Statistic

• Sum of (obs.-exp)2/exp where sum is over all cells.

• For our example, Chi-square=13.2

Chi-Square and Statistical Significance

• Guideline: A chi-square value is statistically significant if it is over 3.84

• Why? – Values over 3.84 will occur less than 5% of the time “just by luck”

In our example -

• 13.2 is larger than 3.84

• CONCLUDE= there is a statistically significant relationship

• So, we believe there is a relationship in the larger population

Example

• In Stat 200, students classified by handedness and gender.

• 56 of 545 (10.3%) of females are left-handed

• 43 of 355 (12.1%) of males left-handed.• Not significant, Chi-square=0.741 • So, apparently no relationship between

handedness and gender

Example

• In Stat 200, students classified by whether they smoke cigarettes and whether they’ve smoked marijuana in last 6 months.

• 217 of 735 (29.5%) of non-smokers of cigs have smoked marijuana

• 109 of 160 cig smokers (68%) have smoked marijuana.

• Significant, Chi-square=84.5 ; So, there is a relationship between cig and marijuana smoking