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Continuous Outcomes Making Comparisons

Continuous Outcomes Making Comparisons. Chapter 2. Outline. Describing: Numerical summaries Graphical summaries One-Sample comparisons: Historical controls Paired differences Two-Sample comparisons: Equal variances Unequal variances Multiple-Sample comparisons: ANOVA

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Continuous Outcomes Making Comparisons

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  1. Continuous Outcomes Making Comparisons Chapter 2

  2. Outline • Describing: Numerical summaries Graphical summaries • One-Sample comparisons: Historical controls Paired differences • Two-Sample comparisons: Equal variances Unequal variances • Multiple-Sample comparisons: ANOVA Multiple comparisons

  3. Summary Histograms are helpful for describing the shape of a distribution. Box-and-whisker plots can be used for comparing the distribution between groups. Means plots provide a graphical comparison of means. Numerical and graphical summaries are essential for describing continuous variables.

  4. One-Sample Comparisons

  5. Description of the Sample • A sample of pregnant women exposed to herbicides in drinking water: 270 women drank only tap water during pregnancy • Tend to be young (average age 25 years, SD = 1.9) • Report a median household income of $33K with a range of $8.7K • Have a low educational attainment (only 45% have a high school degree)

  6. Why a One-Sample Study? • Obtaining an additional group or sample for comparisons may not be practical. Comparisons involve historical control(s). • The change/difference is of interest. Variability between subjects is too large to make meaningful comparisons. Paired differences.

  7. Historical Controls • Want to compare what you found in the sample with something: Do your results differ from what has been previously published/reported? • Historical controls: Control data are not collected concurrently within the same study. • Different time period • A different region • Different population • Different kind of exposure • Seems economical—why not use historical controls all the time?

  8. Paired Differences • Subjects in the sample serve as their own control: Having an independent control group does not provide a good comparison. More variability is expected between subjects. Subjects are too different to make useful comparisons. • Examples

  9. One-Sample StudyHemoglobin Change • Historical control: Do pregnant women exposed to herbicides have the same mean hemoglobin change as women exposed to mercury ? Mean hemoglobin change in pregnant women exposed to mercury has been reported as 2.0 g/dL. • Paired difference: Do pregnant women exposed to herbicides have a mean change in hemoglobin? Mean change different from 0 g/dL would suggest a change.

  10. Inferences for the One-Sample Study • Hypothesis tests Assume the null parameter is the true parameter • Historical control study: Null parameter = Historical value • Paired difference study: Null parameter = 0 Decide whether the data support this assumption • Confidence intervals Estimate the true parameter using interval Can use the interval estimate to determine if assumptions about the parameter are reasonable

  11. Inferences for the One-Sample StudyHistorical Controls Research hypothesis: The true mean () hemoglobin change for pregnant women exposed to herbicides is not 2.0 g/dL. Null hypothesis: The true mean () hemoglobin change for pregnant women exposed to herbicides is 2.0 g/dL.

  12. Inferences for the One-Sample StudyHistorical Controls

  13. Inferences for the One-Sample StudyPaired Differences Research hypothesis: The true mean () hemoglobin change for pregnant women exposed to herbicides is not 0 g/dL. Null hypothesis: The true mean () hemoglobin change for pregnant women exposed to herbicides is 0 g/dL.

  14. Inferences for the One-Sample StudyPaired Differences

  15. Nonparametric Tests • The hypothesis tests and confidence intervals assumed that the outcome was normally distributed. What if this is not reasonable? What if the mean is not a good summary of the center? • Nonparametric tests allow for comparisons without assuming the outcome is normally distributed.

  16. Signed Rank Test Research hypothesis: The hemoglobin values for week 9 and week 36 have different distributions. Null hypothesis: The hemoglobin values for week 9 and week 36 have the same distribution. Essentially, a one-sample hypothesis test about the median instead of the mean.

  17. Signed Rank Test

  18. Two-Sample Comparisons

  19. Why a Two-Sample Study? • Provides an independent comparator group: Treatment vs control Exposed vs unexposed • Different outcomes between the groups may mean that the group is associated with the outcome.

  20. Difference in Means • Research questions typically involve the means being different. • If the means are different, then the difference between the means is 0. • Research question: Does exposure to herbicides in drinking water impact changes in hemoglobin? Is the difference in mean hemoglobin change for the two groups equal to 0?

  21. Means and Variances • We know that we cannot just look at the mean; we have to also look at the variance. • The means can easily be subtracted, but what about the variances? If the two variances are equal, then we can pool them together. If the two variances are not equal, then we cannot pool them together.

  22. Are the Variances Equal? • Consider the graphical and numerical summaries. • Hypothesis test for equal variances: Test statistic: Ratio of the two-sample variances Use F-distribution with numerator and denominator degrees of freedom. Null hypothesis: Variances are equal.

  23. Inferences for the Two-Sample Study • Hypothesis tests Assume the null parameter is the true parameter • The groups are the same. • The true mean difference is 0. Decide whether the data support this assumption • Confidence intervals Estimate the true parameter using interval Can use the interval estimate to determine if assumptions about the parameter are reasonable

  24. Inferences for the Two-Sample Study • The main issue is whether or not the means are equal. Formulas for the test and confidence interval are different depending on equal or unequal variances. Hypotheses involving the mean are the same. Conclusions still focus on the true mean difference (not the variance). • Hypotheses: Null: The true mean difference is 0. Research: The true mean difference is not 0.

  25. Inferences for the Two-Sample StudyEqual Variances Research hypothesis: The average difference in the hemoglobin changes in pregnant women who are exposed to herbicides (via tap water) vsthose who are not (bottled water) is not 0 g/dL, tap – bottle ≠ 0 g/dL. Null hypothesis: The average difference in the hemoglobin changes in pregnant women who are exposed to herbicides (via tap water) vsthose who are not (bottled water) is 0 g/dL, tap – bottle = 0 g/dL.

  26. Inferences for the Two-Sample StudyEqual Variances

  27. What if variances were unequal?

  28. Inferences for the Two-Sample StudyUnequal Variances

  29. Nonparametric Tests • The hypothesis tests and confidence intervals assumed that the outcome was normally distributed. What if this is not reasonable? What if the mean is not a good summary of the center? • Nonparametric tests allow for comparisons without assuming the outcome is normally distributed.

  30. Wilcoxon Test AKA Mann-Whitney test Research hypothesis: The hemoglobin change values for pregnant women exposed to herbicides (tap-only) have a different distribution than the hemoglobin change values for pregnant women who are not exposed (bottle-only). Null hypothesis: The hemoglobin change values for pregnant women exposed to herbicides (tap-only) have the same distribution as the hemoglobin change values for pregnant women who are not exposed (bottle-only).

  31. Multiple-Sample Comparisons

  32. Description of the Sample • Tap-only drinkers (n = 270), bottle-only drinkers (n = 315), and tap-and-bottle drinkers (n = 394) Tap-only and tap-and-bottle drinkers have similar ages and tend to be younger than bottle-only drinkers. Tap-and-bottle drinkers and bottle-only drinkers are similar when it comes to income, having median income $20K higher than tap-only drinkers. Bottle-water drinkers only have the highest high school graduation rates (85%), followed by tap-and-bottle drinkers (64%) and by tap-only drinkers (45%). Bottle-only drinkers have the lowest pre-pregnancy smoking rates (17%), followed by tap-and-bottle drinkers (25%) and tap-only drinkers (32%). Most (84%) of the bottle-only women have had adequate prenatal care, unlike tap-and-bottle drinkers (58%) and tap-only drinkers (35%), who have lower percentages with adequate prenatal care.

  33. Description of the SampleHemoglobin

  34. Why a Multiple-Sample Study? • Provides for multiple comparisons: Placebo vs active placebo vs treatment Unexposed vs marginally exposed vs exposed • Different outcomes between the groups may mean that group is associated with the outcome.

  35. Difference in Means • Research question: Does exposure to herbicides in drinking water impact changes in hemoglobin? • Now there are multiple means to compare. Are any of the means different? Which means are different?

  36. Means and Variances We know that we cannot just look at the mean; we have to also look at the variance. We assume that the variances between the groups are equal.

  37. Analysis of Variance • The ANOVA is used when the research question involves the comparisons of means from more than two independent groups. It is assumed that The groups are independent The variance for each of the groups is the same The outcome comes from a normal distribution • When there are only two groups, a two-sample t-test (with pooled variance) and an ANOVA are equivalent.

  38. Analysis of Variance? • Are not we interested in differences in means? • Why talk about variance? Variance in the outcome can be divided into variance between the groups and variance within the groups. ANOVA partitions the variability and determines whether enough variability comes from between the groups.

  39. ANOVA Table

  40. Overall Test of Means Research hypothesis: The true mean hemoglobin changes are different for at least two of the groups: tap ≠ bottle or tap ≠ both or both ≠ bottle. Null hypothesis: The true mean hemoglobin changes are the same for all the groups: tap = bottle = both.

  41. Overall Test of Means

  42. Comparisons Between Groups

  43. Nonparametric Tests • The hypothesis tests and confidence intervals assumed that the outcome was normally distributed. What if this is not reasonable? What if the mean is not a good summary of the center? • Nonparametric tests allow for comparisons without assuming the outcome is normally distributed.

  44. Kruskal-Wallis Test Research hypothesis: At least one of the three groups comes from a different population. Null hypothesis: All of the three groups come from the same population.

  45. Kruskal-Wallis Test

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