こんにちは由美子です

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つづるそめや
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3 . sum Variable Obs Mean Std. Dev. Min Max var

4 x 3 C 3 = x 0.29 x 3 C 2 = x x 3 C 1 = x 3 C 0 = Bernoulli P(X=x) = ncx p x (1-p) n-x Mean = np = 0.29 x 10 = 2.9 SD = np(1-p) = = 1.4 P 0.5 SD 0 1 SD P = 0.5 p = 0.2 p =

5 10 5 STATA tablesq B B(10,0.29) = 0 Pr(k == 0) = Pr(k >= 0) = Pr(k <= 0) = tablesq B B(10,0.29) = 1 Pr(k == 1) = Pr(k >= 1) = Pr(k <= 1) = tablesq B B(10,0.29) = 2 Pr(k == 2) = Pr(k >= 2) = Pr(k <= 2) = tablesq B B(10,0.29) = 3 Pr(k == 3) = Pr(k >= 3) = Pr(k <= 3) = tablesq B B(10,0.29) = 4 5

6 Pr(k == 4) = Pr(k >= 4) = Pr(k <= 4) = tablesq B B(10,0.29) = 5 Pr(k == 5) = Pr(k >= 5) = Pr(k <= 5) = tablesq B B(10,0.29) = 6 Pr(k == 6) = Pr(k >= 6) = Pr(k <= 6) = tablesq B B(10,0.29) = 7 Pr(k == 7) = Pr(k >= 7) = Pr(k <= 7) = tablesq B B(10,0.29) = 8 Pr(k == 8) = Pr(k >= 8) = Pr(k <= 8) = tablesq B B(10,0.29) = 9 Pr(k == 9) =

7 Pr(k >= 9) = Pr(k <= 9) = tablesq B B(10,0.29) = 10 Pr(k == 10) = Pr(k >= 10) = Pr(k <= 10) = (skewed) C K (0.05) k (0.95) 20-k, K = 0, 1, 2, , 1, 2, 20C 0 (0.05) 0 (0.95) 20 = C 1 (0.05) 1 (0.95) 19 = C 2 (0.05) 2 (0.95) 18 =

8 1 ( ) = cut off X ,500, X STATA. bitesti N Observed k Expected k Assumed p Observed p Pr(k >= 36) = (one-sided test) Pr(k <= 36) = (one-sided test) Pr(k <= 14 or k >= 36) = (two-sided test) 0.05 Pr(k >= 36) Person-years 28,010 19,017 47,027 28,010/47,027 p = 28,010/47,027. bitesti /47027 N Observed k Expected k Assumed p Observed p 8

9 Pr(k >= 41) = (one-sided test) Pr(k <= 41) = (one-sided test) Pr(k <= 25 or k >= 41) = (two-sided test) Two-sided test pmaximal likelihood estimate A/Nvariance A(N A)/N 3 Ncohort size) (A Doll&Hill British physicians study independence and homogeneity assumption binomial distribution cohort (case)variability pmaximal likelihood estimate A/N=1582/28698 = variance A(N A)/N 3 = 1582 x 27116/(28698) 3 95% CI = p ± 1.96var = (0.0243, ) pmaximal likelihood estimate A/N=166/5796 = variance A(N A)/N 3 = 166 x 5630/(5796) 3 95% CI = p ± 1.96var = (0.0243, ) 9

10 Binomial distribution ()

11 8 4 active, 5 inactive 0.2 active 0.0, 0.1, 0.2, , 1.0 accept N=8, p = 0.2 Pregnant probability of accept cumulative P0.2 active (0.99) P=0 P=0.1 P=0.2 P=0.3 P=0.4 P=0.5 P=0.6 P=0.7 P=0.8 P=0.9 P=

12 1 accept true 8 4 accept accept OK accept accept accept accept Operating Characteristic Curve (OC) OC (two stage screening) 12

13 Poisson Distribution binomial situation binomial distribution 0 1-p 1 Poisson person-time Poisson distribution Poisson distributioin 2 independence assumption B A Poisson Stationary assumption Poisson 1 1 Poisson Hazard model Poisson PXx e - λ λ x /x! 0 λe= p 1 variance np λ = np = 10,000 x = 2.4 P(X=4) = e -2.4 (2.4) 4 / 4! = λ = np = 3 P(X=x) = (x 3) / 3 > (p=0.05) X =

14 Poisson distribution person-time µ = (person time) x (incidence rate) PXx e - µ µ x /x! λ: expected number of events per unit time µ: expected number of events over the time period t µ = λt µ maximal likelihood of estimate(mle) A incidence rate (IR) MLE A/person-time (PT) binomial distribution Doll & Hill Person-years (PY) MLE=1582/ = % CI = 1582 ± = (1504, 1660), incidence rate 95% CI = (1504/123436PY, 1660/123436PY) = (0.0122/PY, /PY) 95% CI = 166 ± = (141, 191) incidence rate 95% CI = (141/25250PY, 191/25250PY) = (0.0558/PY, P/Y) death variance 51.5 Outbreak gap variance Poisson distribution mean variance µ outbreak Poisson 14

15 list (injury)n Poisson distribution XYZ(1). list airline injuries n XYZowned poisson injuries XYZowned, exposure(n) irr Iteration 0: log likelihood = Iteration 1: log likelihood = Iteration 2: log likelihood = Poisson regression Number of obs = 9 LR chi2(1) = 1.77 Prob > chi2 = Log likelihood = Pseudo R2 = injuries IRR Std. Err. z P> z [95% Conf. Interval] XYZowned n (exposure)

16 . gen lnn=ln(n) XYZ 1.46 P = CI XYZ incidence rate ratio rate = e βo + β1xyzowned count = n e βo + β1xyzowned = e ln(n) + βo + β1xyzowned. poisson injuries XYZowned lnn Iteration 0: log likelihood = Iteration 1: log likelihood = Iteration 2: log likelihood = Poisson regression Number of obs = 9 LR chi2(2) = Prob > chi2 = Log likelihood = Pseudo R2 = injuries Coef. Std. Err. z P> z [95% Conf. Interval] XYZowned lnn _cons e = 1.98 point estimate

17 (normal distribution) 8 8 probability distribution (normal distribution/gaussian distribution/bell-shaped distribution) µ standard deviation (σ) 17

18 µ 0 standard deviation (SD) σ 1 standard normal distribution SD 68.2% 15.9% 15.9% -1SD +1SD µ 1SD

19 95.4% 2.3% 2.3% -2SD +2SD 2SD 2 5 Z

20 Z

21 Standard normal distribution curve 2.0SD 0.5 Distribution SD standard X = 3.0 standard normal distribution (Z) standard normal distribution X=3.0 Z SD Z = (X 2)/0.5 Z = (3 2)/0.5 = 2 21

22 4 100cm 10cm 80cm SD 2SD mmHg SD 2.5 z= (X129)/19.8 X=167.8 mmhg mmHg =(X + 129)/19.8 X = 90.2 mmhg 1.96SD % 2.5% Z -1.96SD 1.96SD 150mmHg Z = / 19.8 z=1.06, 14.5% mmHg 22

23 µ 1 =80.7, σ 1 =9.2, µ 2 =94.9, σ 2 = z = x 94.9 / 11.5, x = mmhg, z = / 9.2 = 0.06, mmHg

24 µ 24

25 20 25

26 µ / σ µ / σ µµ µ / σ µ/ σ/ σ σ / / 26

27 (confidence interval) 27

28 µσ µσ µσ σ µ σ σ µ σ µσ σ µ µ µ 28

29 µ µσ µ µ µ µσ σ µ σ 29

30 σ µ 30

31 211 mg/dl mg/dl 25 µ 0 25 µ Null hypothesis H 0 : µ 0 = µ SD 25 Ho 2 H 0 accept 25 p < 0.05 sample psample Type I error, type II error, power, sample size 31

32 25 Alternative hypothesis H A : µ 0 µ 1 µ µ µ µ σ 2 32

33 2 µ µµ µµ σ µ µ ασ µ σ 1 µ 33

34 34

35 . list BS ttest BS=100 One-sample t test

36 Variable Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] BS Degrees of freedom: 17 Ho: mean(bs) = 100 Ha: mean < 100 Ha: mean ~= 100 Ha: mean > 100 t = t = t = P < t = P > t = P > t = µ 1 = 100 mg/dl H mg/dlp= two sided t-test row data SD. ttesti One-sample t test Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] x Degrees of freedom: 17 36

37 Ho: mean(x) = 100 Ha: mean < 100 Ha: mean ~= 100 Ha: mean > 100 t = t = t = P < t = P > t = P > t =

38 µ µ µ µ µ µ µ µ µ δµ µ δ δ) δ 38

39 β. list pre post

40 ttest pre=post Paired t test Variable Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] pre post diff Ho: mean(pre - post) = mean(diff) = 0 Ha: mean(diff) < 0 Ha: mean(diff) ~= 0 Ha: mean(diff) > 0 t = t = t = P < t = P > t = P > t =

41 . ttest pre=post, unpaired Two-sample t test with equal variances Variable Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] pre post combined diff Degrees of freedom: 38 Ho: mean(pre) - mean(post) = diff = 0 Ha: diff < 0 Ha: diff ~= 0 Ha: diff > 0 t = t = t = P < t = P > t = P > t = paired t test powerful 41

42 µ µ µ µ µ µ µ µ µ µ σ σ 42

43 . list AMI Chole

44 ttest Chole, by(ami) Two-sample t test with equal variances Group Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] combined diff Degrees of freedom: 38 Ho: mean(0) - mean(1) = diff = 0 Ha: diff < 0 Ha: diff ~= 0 Ha: diff > 0 t = t = t = P < t = P > t = P > t =

46 . ttest Chole, by(ami) unequal welch Two-sample t test with unequal variances Group Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] combined diff Welch's degrees of freedom: Ho: mean(0) - mean(1) = diff = 0 Ha: diff < 0 Ha: diff ~= 0 Ha: diff > 0 t = t = t = P < t = P > t = P > t = equal variance Nonparametric Methods 46

47 . gen d=post - pre. list pre post d Null hypothesis 1/2 binomial distribution np = n/2, variance np(1-p) = n/4 n/2 Z = [+ (n/2)]/(n/4)

48 Z = [9 5]/(10/4) = 2.53 Z 1.96 null hypothesis STATA. signtest pre=post Sign test sign observed expected positive 1 5 negative 9 5 zero all One-sided tests: Ho: median of pre - post = 0 vs. Ha: median of pre - post > 0 Pr(#positive >= 1) = Binomial(n = 10, x >= 1, p = 0.5) = Ho: median of pre - post = 0 vs. Ha: median of pre - post < 0 Pr(#negative >= 9) = Binomial(n = 10, x >= 9, p = 0.5) = Two-sided test: Ho: median of pre - post = 0 vs. Ha: median of pre - post ~= 0 Pr(#positive >= 9 or #negative >= 9) = min(1, 2*Binomial(n = 10, x >= 9, p = 0.5)) = two sided test. ttest pre=post Paired t test 48

49 Variable Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] pre post diff Ho: mean(pre - post) = mean(diff) = 0 Ha: mean(diff) < 0 Ha: mean(diff) ~= 0 Ha: mean(diff) > 0 t = t = t = P < t = P > t = P > t = µ σ µ 49

50 σ β β. signrank pre=post Wilcoxon signed-rank test sign obs sum ranks expected positive negative zero all unadjusted variance adjustment for ties adjustment for zeros adjusted variance Ho: pre = post z = Prob > z = (Mann-Whitney test) 50

51 . ranksum EFV, by(drug) Two-sample Wilcoxon rank-sum (Mann-Whitney) test drug obs rank sum expected combined unadjusted variance adjustment for ties adjusted variance Ho: EFV(drug==0) = EFV(drug==1) z = Prob > z = Wilcoxon signed-rank test paired test unpaired test power 51

52 52

分布

(normal distribution) 30 2 Skewed graph 1 2 (variance) s 2 = 1/(n-1) (xi x) 2 x = mean, s = variance (variance) (standard deviation) SD = SQR (var) or 8 8 0.3 0.2 0.1 0.0 0 1 2 3 4 5 6 7 8 8 0 1 8 (probability