1. Section 9.5
Paired Comparisons

2. ?

3. Key Question ?

4. Do we really have two independent samples
Key Question
Do we really have two independent samples or

5. Do we really have two independent samples
Key Question
Do we really have two independent samples or
do we have only one sample of paired data?

6. Percentage of Women in the Labor Force
City
New York, NY
Los Angeles, CA
Chicago, IL
Philadelphia, PA
Detroit, MI
San Francisco, CA
Boston, MA
Pittsburgh, PA
St. Louis, MO
Hartford, CT
Washington, DC
Cincinnati, OH
Baltimore, MD
Newark, NJ
Minneapolis-St. Paul, MN
Buffalo, NY
Houston, TX
Paterson, NJ
Dallas, TX

7. Percentage of Women in the Labor Force not independent samples
City
New York, NY
Los Angeles, CA
Chicago, IL
Philadelphia, PA
Detroit, MI
San Francisco, CA
Boston, MA
Pittsburgh, PA
St. Louis, MO
Hartford, CT
Washington, DC
Cincinnati, OH
Baltimore, MD
Newark, NJ
Minneapolis-St. Paul, MN
Buffalo, NY
Houston, TX
Paterson, NJ
Dallas, TX
not independent samples

8. Paired Comparisons
Can greatly reduce variation over independent samples

9. Paired Comparisons
Can greatly reduce variation over independent samples
Can produce much more powerful test and more precise confidence interval estimate of true mean difference

10. Three Different Designs for Comparisons

11. Three Different Designs for Comparisons
1) Completely randomized (two independent samples)

12. Three Different Designs for Comparisons
1) Completely randomized (two independent samples)
2) Matched pairs (paired observation)

13. Three Different Designs for Comparisons
1) Completely randomized (two independent samples)
2) Matched pairs (paired observation)
3) Repeated measures (paired observation)

14. Three Different Designs for Comparisons
1) Completely randomized (two independent samples)
2) Matched pairs (paired observation)
3) Repeated measures (paired observation)

15. Completely Randomized Design
Arbitrary pairing of values from two independent samples

16. Completely Randomized Design
Arbitrary pairing of values from two independent samples
Example: Compare sitting and standing pulse rates by randomly assigning half of class to sit and half to stand

17. Correlation
What is correlation?

18. Correlation
Correlation measures strength and direction of a linear relationship between two variables

19. Completely Randomized (p. 642)
Correlation is near 0, as it should be, because these are independent measurements taken in arbitrary order.

20. Completely Randomized Design
To analyze data from completely randomized design
of two independent samples,
use 2-SampTInt and
2-SampTTest techniques

21. Three Different Designs for Comparisons
1) Completely randomized
2) Matched pairs (paired observation)
3) Repeated measures (paired observation)

22. Matched Pairs Design Example:
Match pairs of students on preliminary measure of pulse rate.
Then randomly assign sitting to one student in each pair and standing to the other.

23. Matched Pairs (p. 642)
Hint of linear trend ( r = 0.48) because these are dependent measurements based on pairing people with similar pulse rates.

24. Matched Pairs Design
Matched pairs design has dependent observations within a pair, so the two-sample t-procedures are not a viable option.

25. Matched Pairs Design
Look at the difference between each pair and estimate the mean difference with a one-sample t-procedure.

26. Three Different Designs for Comparisons
1) Completely randomized
2) Matched pairs (paired observation)
3) Repeated measures (paired observation)

27. Repeated Measures Design
Example:
Have each student sit and stand with the order randomly assigned
Each student gets both treatments.

28. Repeated Measures Design (p. 642)
Strong linear trend (r = 0.93) as
these are paired measurements
from same person.

29. Repeated Measures Design
Dependent measurements occur here also, so the two-sample t-procedures are not a viable option.
Look at the difference between each pair and estimate the mean difference with a one-sample t-procedure.

30. Page 642
For each of these designs, find the 95% confidence interval for the difference between mean pulse rates for standing and sitting (standing – sitting).
CRD: n = 14, MPD: n= 14, RMD: n = 28

31. Completely Randomized Design
To analyze data from completely randomized design of two independent samples, use 2-SampTInt and
2-SampTTest techniques
n = 14

32. Completely Randomized Design
2-SampTInt
Inpt: Data Stats
x1: (stand) C-Level: .95
sx1: Pooled: No Yes
n1: 14 Calculate
x2: (sit)
sx2: 13
n2: 14

33. Completely Randomized Design
2-SampTInt
Inpt: Data Stats
x1: (stand) C-Level: .95
sx1: Pooled: No Yes
n1: 14 Calculate
x2: (sit)
sx2: 13 ( , )
n2: Stand - Sit

34. Matched Pairs Design
n = 14
mean SD

35. Matched Pairs Design
Matched pairs design has dependent observations within a pair, so the two-sample t-procedures are not a viable option.
Look at the difference between each pair and estimate the mean difference with a one-sample t-procedure.

36. Matched Pairs Design TInterval Inpt: Data Stats x: 3.71 sx: 12.38
C-Level: .95
Calculate
(-3.438, )

37. Use Data for Input

38. Use Data for Input L1 L2

39. Use Data for Input L1 L2
L3 = L2 – L1

40. Repeated Measures Design
mean SD

41. Repeated Measures Design
Dependent measurements occur here also, so the two-sample t-procedures are not a viable option.
Look at the difference between each pair and estimate the mean difference with a one-sample t-procedure.

42. Repeated Measures Design
TInterval
Inpt: Data Stats
x: 8.36
sx: 5.28
n: 28
C-Level: .95
Calculate
(6.3126, )

43. Paired Observations 1) Completely randomized (- 8.9645, 14.664)
2) Matched pairs (paired observation)
(-3.438, )
3) Repeated measures (paired observation)
(6.3126, )

44. Paired Observations 1) Completely randomized
( , ) possibly no difference
2) Matched pairs (paired observation)
(-3.438, ) possibly no difference
3) Repeated measures (paired observation)
(6.3126, ) conclude:difference

45. Paired Observations 1) Completely randomized (-14.66, 8.9645)
2) Matched pairs (paired observation)
(-3.438, )
3) Repeated measures (paired observation)
(6.3126, )
Can produce much more powerful test and more precise confidence interval estimate of true mean difference between standing and sitting

46. Paired Observations
What must we do to construct a confidence interval for the mean difference from paired observations?

47. Paired Observations
What must we do to construct a confidence interval for the mean difference from paired observations?
1) Check conditions

48. Paired Observations
What must we do to construct a confidence interval for the mean difference from paired observations?
1) Check conditions
2) Do computations

49. Paired Observations
What must we do to construct a confidence interval for the mean difference from paired observations?
1) Check conditions
2) Do computations
3) Give interpretation in context

50. Check Conditions

51. Check Conditions
1) Randomness: For survey, must have a random sample from one population where a “unit” may consist of a pair of twins or same person’s two feet for example.

52. Check Conditions
1) Randomness: For survey, must have a random sample from one population where a “unit” may consist of a pair of twins or same person’s two feet for example.
For experiment, treatments randomly assigned within each pair. If same subject assigned both treatments, treatments must be assigned in random order.

53. Check Conditions
2) Normality: The differences, not the two original samples, must look like it’s reasonable to assume they came from normally distributed population or sample size must be large enough that sampling distribution of sample mean difference is approximately normal.
15/40 guideline can be applied to the differences.

54. Check Conditions
3) Sample size: For survey, size of the population of differences should be at least ten times as large as the sample size.
Condition does not apply to experiment.

55. Do Computations
Confidence interval for mean difference, , is given by:
Where n is sample size, d is mean of differences between pairs of measurements, sd is sample standard deviation, and t* depends on confidence interval desired and degrees of freedom (df = n-1)

56. Give Interpretation in Context
Interpretation is of form, “I am 95% confident that the mean of the population of differences, , is in this confidence interval.
In context includes describing the population you are talking about.

57. Questions?