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Paired samples

Paired samples . Unit 7. Types of two-sample problems. Paired samples Each point in one sample is matched to a unique point in the second sample Also called “matched-pair” and “dependent” samples Independent samples Data points samples are unrelated Two unrelated (independent) groups.

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Paired samples

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  1. Paired samples Unit 7 Paired Samples

  2. Types of two-sample problems • Paired samples • Each point in one sample is matched to a unique point in the second sample • Also called “matched-pair” and “dependent” samples • Independent samples • Data points samples are unrelated • Two unrelated (independent) groups Paired Samples

  3. Examples of paired samples • Sequential samples • Pre-test/post-test • Cross-over trials • Matched-pairs Covered this unit. Paired Samples

  4. Independent samples – separate groups Covered next unit. Paired Samples

  5. Illustrative example: oatbran.sav Person CORNFLK OATBRAN ---- ------- ------- 1 4.61 3.84 2 6.42 5.57 3 5.40 5.85 4 4.54 4.80 5 3.98 3.68 6 3.82 2.96 7 5.01 4.41 8 4.34 3.72 9 3.80 3.49 10 4.56 3.84 11 5.35 5.26 12 3.89 3.73 13 2.25 1.84 14 4.24 4.14 • Data: low density lipoprotein (“bad cholesterol”) mg/dl • Two weeks on cornflake diet  LDL cholesterol • Washout period • Two weeks on oatbran diet  LDL cholesterol • Randomize, so half start on CORNFLK, half on OATBRAN (cross-over trial) • Notice unique pairing Paired Samples

  6. Means of individual samples • Calculate means in each samples (by hand, with TI-30XIIS, with SPSS) • Mean LDL, CORNFLK (xbar1) = 4.444 • Mean LDL, OATBRAN (xbar2) = 4.081 vs. Paired Samples

  7. Calculate difference variable “DELTA” • Let DELTA = CORNFLK - OATBRAN • Order of subtraction does not materially effect results (but but does change sign) ID CORNFLK OATBRAN DELTA ---- ------- ------- ----- 1 4.61 3.84 0.77 2 6.42 5.57 0.85 3 5.40 5.85 -0.45 ↓↓↓↓ 14 4.24 4.14 0.10 Positive values in this illustration represent declines on oatbran Paired Samples

  8. Explore DELTA Here are the 14 paired differences (DELTAs): 0.77, 0.85, -0.45, -0.26, 0.30, 0.86, 0.60, 0.62, 0.31, 0.72, 0.09, 0.16, 0.41, 0.10 • Interpret: • Shape: No major departures from Normality (difficult b/c n is small) • Central location: median about 0.35 • Spread: from about -0.4 to 0.8 • 12 of 14 showed a decline Stemplot |-0|42|+0|011334|+0|667788×1 Paired Samples

  9. Calculate the mean and SD of DELTA n = 14 xbard = 0.3629 sd = 0.4060 Use plain language to describe results: “The mean decline in LDL of the 14 subjects while on the oat bran diet was 0.363 mg/dl with a standard deviation of 0.406 mg/dl.” Paired Samples

  10. 95% confidence interval for µd Same as one-sample t procedure except inferences are directed toward DELTA. Illustrative example Formula We are 95% confident μd is in this interval Paired Samples

  11. Significance test (oatbran.sav) Same as one-sample t procedure except inferences are directed toward DELTA. Claim: “no difference” • H0: µd = 0 vs. H1: µd 0 • tstat = 3.34 with 13 df • P value between .01 and .002 • Evidence against H0 is highly significant Formula Illustrative data Paired Samples

  12. SPSS output (oatbran.sav) One-sample t procedures directed at DELTA! Paired Samples

  13. Beware! • Confidence intervals and P values address random error only • Other types of practical problems (e.g., biased samples, information bias, confounding) not taken into account Statistics in practice involves much more than recipes. Paired Samples

  14. Power • If we retain H0 is false, we commit a type II error • Pr(type II error) b • Power  1−b = Pr(avoid type II error) Conditional probability notation: Pr(A|B)  probability of A given B (retain H0 | H0 false)  type II error Pr(retain H0 | H0 false)  b Pr(reject H0 | H0 false)  1 – b  power Paired Samples

  15. Power of the t test depends on: • “Difference worth detecting” () • this is the difference you are looking for • Standard deviation of data (s) • If s unavailable, use s from pilot study • Acceptable type I error rate (a) • We consider only a = .05 two-sided • Sample size (n) Paired Samples

  16. Basis of calculation (optional) Paired Samples

  17. Power (formula & example) Illustrative example: We want to determine the power of a test in which a mean difference of 2 is worth detecting. We study 30 observations, and prior studies suggests the variable being studied has a standard deviation of 6. What is the power of the study at α = .05 two-sided? Power is the probability of avoiding a type II error. A power of 44.83% is considered low; power should exceed 80%. Paired Samples

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