Section 9.5 Paired Comparisons

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Key Question ?

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

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?

Percentage of Women in the Labor Force City 1972 1968 New York, NY 45 42 Los Angeles, CA 50 50 Chicago, IL 52 52 Philadelphia, PA 45 45 Detroit, MI 46 43 San Francisco, CA 55 55 Boston, MA 60 45 Pittsburgh, PA 49 34 St. Louis, MO 35 45 Hartford, CT 55 54 Washington, DC 52 42 Cincinnati, OH 53 51 Baltimore, MD 57 49 Newark, NJ 53 54 Minneapolis-St. Paul, MN 59 50 Buffalo, NY 64 58 Houston, TX 50 49 Paterson, NJ 57 56 Dallas, TX 64 63

Percentage of Women in the Labor Force not independent samples City 1972 1968 New York, NY 45 42 Los Angeles, CA 50 50 Chicago, IL 52 52 Philadelphia, PA 45 45 Detroit, MI 46 43 San Francisco, CA 55 55 Boston, MA 60 45 Pittsburgh, PA 49 34 St. Louis, MO 35 45 Hartford, CT 55 54 Washington, DC 52 42 Cincinnati, OH 53 51 Baltimore, MD 57 49 Newark, NJ 53 54 Minneapolis-St. Paul, MN 59 50 Buffalo, NY 64 58 Houston, TX 50 49 Paterson, NJ 57 56 Dallas, TX 64 63 not independent samples

Paired Comparisons Can greatly reduce variation over independent samples

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

Three Different Designs for Comparisons

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

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

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

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

Completely Randomized Design Arbitrary pairing of values from two independent samples

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

Correlation What is correlation?

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

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

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

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

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.

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

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

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

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

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

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

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.

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

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

Completely Randomized Design 2-SampTInt Inpt: Data Stats x1: 77.71 (stand) C-Level: .95 sx1: 17.04 Pooled: No Yes n1: 14 Calculate x2: 74.86 (sit) sx2: 13 n2: 14

Completely Randomized Design 2-SampTInt Inpt: Data Stats x1: 77.71 (stand) C-Level: .95 sx1: 17.04 Pooled: No Yes n1: 14 Calculate x2: 74.86 (sit) sx2: 13 (- 8.964, 14.664) n2: 14 Stand - Sit

Matched Pairs Design n = 14 mean SD

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.

Matched Pairs Design TInterval Inpt: Data Stats x: 3.71 sx: 12.38 C-Level: .95 Calculate (-3.438, 10.858)

Use Data for Input

Use Data for Input L1 L2

Use Data for Input L1 L2 L3 = L2 – L1

Repeated Measures Design mean SD

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.

Repeated Measures Design TInterval Inpt: Data Stats x: 8.36 sx: 5.28 n: 28 C-Level: .95 Calculate (6.3126, 10.407)

Paired Observations 1) Completely randomized (- 8.9645, 14.664) 2) Matched pairs (paired observation) (-3.438, 10.858) 3) Repeated measures (paired observation) (6.3126, 10.407)

Paired Observations 1) Completely randomized (- 8.9645, 14.664) possibly no difference 2) Matched pairs (paired observation) (-3.438, 10.858) possibly no difference 3) Repeated measures (paired observation) (6.3126, 10.407) conclude:difference

Paired Observations 1) Completely randomized (-14.66, 8.9645) 2) Matched pairs (paired observation) (-3.438, 10.858) 3) Repeated measures (paired observation) (6.3126, 10.407) Can produce much more powerful test and more precise confidence interval estimate of true mean difference between standing and sitting

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

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

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

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

Check Conditions

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.

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.

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.

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.

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)

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.

Questions?