Part 15: Hypothesis Tests15-1/18

Statistics and Data Analysis

Professor William Greene

Stern School of Business

IOMS Department

Department of Economics

Part 15: Hypothesis Tests15-2/18

Statistics and Data Analysis

Part 15 – Hypothesis Tests: Part 3

Part 15: Hypothesis Tests15-3/18

A Test of Independence

In the credit card example, are Own/Rent and Accept/Reject independent?

Hypothesis: Prob(Ownership) and Prob(Acceptance) are independent

Formal hypothesis, based only on the laws of probability: Prob(Own,Accept) = Prob(Own)Prob(Accept) (and likewise for the other three possibilities.

Rejection region: Joint frequencies that do not look like the products of the marginal frequencies.

Part 15: Hypothesis Tests15-4/18

A Contingency Table Analysis

Part 15: Hypothesis Tests15-5/18

Independence Test

Step 2: Expected proportions assuming independence: If the factors are independent, then the joint proportions should equal the product of the marginal proportions.

[Rent,Reject] 0.54404 x 0.21906 = 0.11918 (.13724) [Rent,Accept] 0.54404 x 0.78094 = 0.42486 (.40680) [Own,Reject] 0.45596 x 0.21906 = 0.09988 (.08182) [Own,Accept] 0.45596 x 0.78094 = 0.35606 (.37414)

Part 15: Hypothesis Tests15-6/18

Comparing Actual to Expected

22

Rows Columns

The statistic is N times the sum over the four cells

(Observed-Expected) = N ×

Expected

If this is large (because the observed proportions don't

look like the expected ones) then rej

2 2

2

2 2

ect the hypothesis.

(This is a "chi squared statistic.")

(0.13724 0.11918) (0.40680 0.42486)

0.11918 0.4248613,444(0.08182 0.09988) (0.37414 0.35608)

0.09988 0.35608 = 103.33013

Part 15: Hypothesis Tests15-7/18

When is Chi Squared Large?

For a 2x2 table, the critical chi squared value for α = 0.05 is 3.84.

(Not a coincidence, 3.84 = 1.962) Our 103.33 is large, so the hypothesis of

independence between the acceptance decision and the own/rent status is rejected.

Part 15: Hypothesis Tests15-8/18

Computing the Critical Value

CalcProbability Distributions Chi-square

The value reported is 3.84146.

For an R by C Table, D.F. = (R-1)(C-1)

Part 15: Hypothesis Tests15-9/18

Analyzing Default

Do renters default more often (at a different rate) than owners?

To investigate, we study the cardholders (only)

We have the raw observations in the data set.

DEFAULTOWNRENT 0 1 All 0 4854 615 5469 46.23 5.86 52.09

1 4649 381 5030 44.28 3.63 47.91

All 9503 996 10499 90.51 9.49 100.00

Part 15: Hypothesis Tests15-10/18

Part 15: Hypothesis Tests15-11/18

Part 15: Hypothesis Tests15-12/18

2 2

2

2 2

[.4623 (.9051 .5209)] [.0586 (.0949 .5209)]

(.9051 .5209) (.0949 .5209)10499

[.4428 (.9051 .4791)] [.0363 (.0949 .4791)]

(.9051 .4791) (.0949 .4791)

Part 15: Hypothesis Tests15-13/18

Hypothesis Test

Part 15: Hypothesis Tests15-14/18

In my sample of 210 travelers between Sydney and Melbourne, it appears that there is a relationship between income and the decision whether to fly or not. Do the data suggest that the mode choice and income are independent?

Part 15: Hypothesis Tests15-15/18

Treatment Effects in Clinical Trials

Does Phenogyrabluthefentanoel (Zorgrab) work?

Investigate: Carry out a clinical trial. N+0 = “The placebo effect” N+T – N+0 = “The treatment effect” Is N+T > N+0 (significantly)?

Placebo Drug Treatment

No Effect N00 N0T

Positive Effect N+0 N+T

Part 15: Hypothesis Tests15-16/18

Part 15: Hypothesis Tests15-17/18

Confounding Effects

Part 15: Hypothesis Tests15-18/18

What About Confounding Effects? Normal Weight Obese

Nonsmoker

Smoker

Age and Sex are usually relevant as well. How can all these factors be accounted for at the same time?