Even Statisticians Love Geometry

Charles Burd, April 16, 2014Advisor: Dr. Chauhan

Objective• There may be multiple ways of estimating an unknown

value.

• The results obtained from multiple methods may not be the same.

• In such situation, is there a way to determine which method may be more appropriate, and under what conditions?

Background Estimating proportion

𝒑 �̂�

Population proportion :unknown Random sample proportion - known

95% Confidence Interval (CI) of :

margin of error �̂� ±1.96 √ �̂� (1−�̂� )𝑛

ComparingProportions of two populations

Unknown

Objective:

𝑝1−𝑝2Estimate

Estimation of Two approaches: Which is better?

Overlap:

Compute CI for each sample

Decision rule: If the two intervals overlap, population proportions may be the same.

𝑝1𝑝 2

Standard:

Compute one CI for difference ():

Decision rule: If the interval contains zero, the proportions may be the same.

𝑝1−𝑝2 0

ExampleOverlap Approach

Population 1 Population 2

margin of error

95% CI for 95% CI for

0.3 0.4 0.5 0.6 0.7

𝑝1𝑝 2

Overlap Approach: The intervals overlap, so the proportions may be the same.

Example Continues Standard Approach

Population 1 – Population 2

margin of error

95% CI for

0.1 0 0.1 0.2 0.3

Standard Approach: Interval does not contain zero, so the proportions are not the same.

𝑝1−𝑝2

Result: The overlap method concludes the population proportions not different while the standard method finds a difference.

A Closer Look

Equal population proportions by overlap method implies equal by standard method, but not vice-versa (ratio greater than 1).

• Overlap method is more conservative and less powerful.• If two populations differ, standard method will detect it.

Individual intervals of overlap iff: () contains zero

Only difference between the two methods then is the width of the intervals. narrower width less chance zero included proporotions different

𝐸1+𝐸2

√𝐸12+𝐸22√𝐸12+𝐸22𝐸1

𝐸2

Standard approach interval:

A Closer Look

0.04960.0351

0.0351

• Max of when • Min as one adj. side

So, overlap approach is expected to be more deficient when and nearly equal.

What does this geometric relationship tell us about overlap method’s deficiencies?

Simulation

Do simulation results confirm analytical expectations?

Percentage of time finding a difference between populations

Overlap Standard

, sample size , 4000 draws

Conclusion

• We can always get better results with the standard method.

• Overlap method is at its worst when the two margin of errors are close.

• Overlap is simple, convenient to use, but for formal testing, use standard method.

√𝐸12+𝐸22𝐸1

𝐸2

Reference

Nathaniel Schenker and Jane F. Gentleman : On judging the significance

of differences by examining the overlap method between confidence

intervals, The American Statistician 55 (Aug., 2001) 182-186.

Charles Burd, April 16, 2014Advisor: Dr. Chauhan