One-Sample Tests: Sign Test and Wilcoxon Signed Rank Test - Prof. Michae Hudgens, Study notes of Data Analysis & Statistical Methods

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One Sample Tests Bios 662

Michael G. Hudgens, Ph.D. [email protected]

http://www.bios.unc.edu/∼mhudgens

2006-09-20 16:

Outline

• Small sample, Normal
• Large sample
• Nonparametric
  • Sign test
  • Wilcoxon signed rank test

One Sample Test: Small Sample, Normal

• For small sample and Y ∼ N (μ, σ^2 ),
• Form previous lecture, test statistic

t = Y¯ − μ s/

n

∼ tn− 1

with critical region Cα = {t : |t| > tn− 1 , 1 −α/ 2 }

One Sample Test: Small Sample, Normal

• P-value: probability of obtaining test statistic as or more extreme than observed from the sample p = Pr[T ≤ −|t|] + Pr[T ≥ |t|] where T ∼ tn− 1
• Equivalently

p = 2 Pr[T ≤ −|t|] = 2 Pr[T ≥ |t|]

One Sample Test: SIDS Example

• Example: text page 281; problem 8.
• Let our random variable Y equal the weight of SIDS baby (twin) minus weight of non-SIDS baby (twin)

H 0 : μdif f = 0

HA : μdif f 6 = 0

One Sample Test: SIDS Example

• y¯ = 0.1818, s = 369.57, n = 22

t = y¯ s/

n

• Critical region at α = 0. 05

C. 05 = {t : |t| > t 21 ,. 975 = 2. 08 }

• P-value

p = 2 ∗ Pr[T ≤ − 0 .0023] = 0. 9982

One Sample Test: Small Sample

• t test assumptions:
  • Observations are independent
  • Sample is from (the same?) normal distribution

One Sample Test: Large Sample

• For large sample, use normal approximation (CLT):

Y¯ ∼ N (μ, σ

2 n

Z =

Y¯ − μ s/

n approx N (0, 1)

• Approximation improves as n → ∞
• Note: Y ’s do not need to be normally distributed

One Sample Test: Large Sample

• Example: Iron deficiency (continued)

H 0 : μ = 14.45; HA : μ 6 = 14. 45

C. 05 = {z : |z| > 1. 96 }

Y^ ¯ = 12.5; s^2 = 22.5625; n = 63

z =

√^12.^5 −^14.^45

• Reject H 0

One Sample Testing/Estimation: Large Sample

• Testing H 0 : μ = 0

Cα = {z : |z| > z 1 −α/ 2 } where z = Y¯ − μ 0 s/

n

• Estimation: confidence interval for μ

Y^ ¯ ± z 1 −α/ 2 √^ s n

• Can show: Reject H 0 iff CI excludes μ 0

One Sample Testing/Estimation: Large Sample

• Theorem: Reject H 0 iff CI excludes μ 0
• Sketch of proof: Suppose CI excludes μ 0 ie

μ 0 ∈/ [ Y¯ − z 1 −α/ 2 √^ s n

, Y¯ + z 1 −α/ 2 √^ s n

]

Without loss of generality, assume μ 0 < Y¯ − z 1 −α/ 2 √^ s n

This implies

z 1 −α/ 2 < Y¯ − μ 0 s/

n ≡ z

implying z ∈ Cα 

One Sample Test: Small Sample, Non-normal

• Suppose small sample, non-normally distributed data
• Nonparametric tests:
  • Sign test
  • Wilcoxon signed rank test
• Read van Belle et al 8.1-8.

Sign Test

• Delete obs=ζ. 5 , 0 and reduce n by 1 for each
• Let r be the number of obs > ζ. 5 , 0
• Under H 0

r ∼ Bin(n, .5)

Pr[r ≤ r′] =

∑^ r′

i=

n i

)i ( 1 −

)n−i

2 n

∑r′

i=

n i

Sign Test

• Note: H 0 : ζ. 5 = ζ. 5 , 0 ⇐⇒ H 0 : π =. 5 where π is the parameter of a binomial RV
• In order to test H 0 : π = .5 vs HA : π 6 = .5, we must find values rα/ 2 and r 1 −α/ 2 such that Pr[r ≤ rα/ 2 ] + Pr[r ≥ r 1 −α/ 2 ] ≤ α