Statistics 541: Two-Sample

Admistrivia

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Two sample t-test

• Scientific contex
  • Suppose we compare two groups
  • one typically called X other typically called Y.
  • Or scientifically: control and treatment group
  • EG: Do people with ACL replacements do "better" than those without ACL replacement?
  • interested in deciding which group is better/larger/higher mean.

• How can we compare two group?
  • Traditional answer: two sample t-test (keyword = pooled)
  • Also called simple regression
  • Let indicator represent each group
  • Run regression of outcome on indicator variable

• What if the two groups have different variances?
  • Traditional answer: Welchs approximation (also large sample)
  • Also called simple regression using the sandwich estimator
  • Done automatically by all statistical packages

• What if we take two measurement on each person?
  • Traditional answer: paired t-test
  • Also called multiple regression
  • First indicator called treatment (1=treatment, 0=control)
  • Add one indicator for each person (called "row effect")
  • Don't look at row effects, only at column effect
  • Easy to represent in a table

ANOVA

• Scientific context
  • Suppose we compare more than two groups
  • EG: Which surgery is best, ACL reconstruction, cadavorious replacement, none, Teflon replacment
  • interested in deciding which group is best.

• How can we compare many groups?
  • Traditional answer: one-way ANOVA
  • Also called multiple regression
  • have one indicator for each group (no constant)
  • Multiple comparisons come up

• What if the two groups have different variances?
  • Traditional answer: Oops, don't like doing that
  • Easy with sandwich estimator

• What if we take several measurements on each person?
  • Traditional answer: two way ANOVA (keyword randomized block design)
  • Also done as multiple regression

The interesting new issue is multiple comparisons

• Compare to grand mean (Bonferroni = sqrt(2 log p))
• Compare all pairs (balanced design Bonferroni = 2 sqrt(2 log p), work out both ways of thinking about this)
• Improvement: Tukey-Cramer, uses dependencies in situation and hence get a slight improvement over Bonferroni
• Easier problem: HSU, uses the fact that we don't care which is best, only if each category COULD be best or not.