9. Hypothesis Test

9. : Two Sample Test- paired t-test- t-test- modified t-test

1Two Sample Test 0 .. 1 2 . : Two Sample Test . 1 2 (unknown) .

2 , .1 and 2 came from two dependent data (= a paired data)1 and 2 came from two independent data

3Paired data . (paired or dependent data)4An example of pairing 1: (self-paring) . 10 , . . 1 , ( , ) . 5Sleep duration under pill ASleep duration under PlaceboA sleep duration under A and the sleep duration under placebo are from the same participant.Participant 1Participant 1Participant 2Participant 2Participant 3Participant 3Participant 4Participant 4Participant 10Participant 106An example of pairing 2: matching , .We want to know if BP is higher in renal syndrome patients than in healthy persons.Blood pressure was measured from patients with renal syndrome and healthy controls. We want to compare the mean BP of patients with renal syndrome and that of healthy controls.By design, we match (or pair) a patient with a control person based on his/her sex, age and race. 7Patients with renal syndromeHealthy control personsMale, age=29, WhiteMale, age=25, WhiteFemale, age=44, AsianFemale, age=49, AsianMale, age=31, BlackMale, age=31, WhiteMale, age=35, BlackMale, age=27, WhiteMatching Variable:-Sex-Age(5)-Race8Why do we pair? . Ex1) We want to control the variations in BP due to other factors such as age, sex, race, obesity, genetic composition etc. By paring BPs from the same patient, we are able to control the variation for everything other than the treatment A.Ex2) By paring BPs from a patient and a control with the same sex, age, and race, we are able to control the variation for sex, age, and race other than the disease status.9Paring ?Example 1 , 2 ., . Treatment A placebo .

10Hypothesis test for paired data11Hypothesis for paired dataIf =0, the pill has no effect.H0 : = 0 = 0H1 : 0 .i , .

12 . , 0 t- .The test method becomes the same as the one-sample t-test.

13 t Paired t-test Sample mean of the is,

And the standard deviation is,

where n is number of pairs.Test statistics is , d.f.=n-1

If or then we reject H0.

14Example 10 , . =0.56 . t= 9 t .

with d.f.=9

15T-distribution table 9 t (two-sided) 5% 2.26, 2.26 t 5% . 2% 2.82, 1% 3.25. t 3.18 2.82 3.25 , 2% 1% .

16What is the degrees of freedom?

What is the p-value?

t-distribution table

t-distribution with 10 d.f.17 2% (significant at the 2% level) . 2% . P P0 .5810. Analysis of Variance(ANOVA)59 3 ?Two-sample t test 3 . (one-way analysis of variance) .

60Example . , , .1 .1, 122, 223, 3261 i (i =1, 2, 3) ni , , si . ? ?

62 1, 2, 3 . ,H0 : 1 = 2 = 3 ,H1 : i j63 t-test ? two-sample t test ? . 3 . .

64 1 = 2 = 3 , , 0.05 ,P(3 H0 ) = (1-0.05)3= 0.857 P( 1 H0 ) = 1-0.857 = 0.143= P(type I error)

65 0.05 . ( 0.05) .

66 k .H0 : 1 = 2 = . =k k . (1=2=3==) .672 2 . (1), (2) 2 . (1) (2) , .

68 .: (within group variance) (between group variance) ?

69 i ni .i j xij .ixij1x11 , x12 , x13 , x14 ,.., x1n1 2x21 , x22 , x23 , x24 ,.., x2n2 3x31 , x32 , x33 , x34 ,.., x3n3

70i . (overall mean). (grand mean) .

71 .

i i (within group variation) . i (between group variation) .

72 , ,

,SST = SSW + SSB

Total Sum of Squares(SST)Within group Sum of Squares(SSW)Between group Sum of Squares(SSB)73 , . . SSW

n=n1 + n2 + n3 ,

(within groups mean square) .

74Within groups mean square MSW two-sample t test (pooled estimate of the common variance) . MSW pooled estimate of the variance for one-way ANOVA . ANOVA . ANOVA . 2 ANOVA .

75 . SSB ,

(between groups mean square) .

76F-test in ANOVA . F .

k .

Fk-1,n-k distributionp-valueF77H0 F k-1 n-k F .F 1 .F H0 . F > Fk-1, n-k, 1- .ANOVA , .P F Fk-1, n-k .

78 . 1 : 2 : 3 : 1 .

k=3, n=42+47+42=131 .nisi142-7.23.7247-4.03.93420.63.7

79,

80 ,

H0: 1 = 2 = 3 0.05 F .

81 3-1=2, 131-3=128 F 45.4 0.001 . 9 2, 120 F1-0.001 = F0.999 = 7.32 . 45.4 7.32 p