推論統計

1. 推論統計 台北醫學大學公共衛生學科 葉錦瑩

2. §統計方法 描述統計（Descriptive Statistics） 推論統計（Inferential Statistics） • 參數估計（Estimation） • 假說檢定(Hypothesis Testing)

3. §統計推論 參數估計（Estimation） 點估計 區間估計 假說檢定(Hypothesis Testing) 有母數分析（利用常態分布的特性） 無母數分析

4. §假說檢定的源起及概念 • 由描述統計資料觀察到樣本間的不同，而欲驗證其差異的可能性。 • 藉由否證‘無差異’以間接驗證‘有差異’。 • 結果判斷無論是‘有差異’或‘無差異’，均須承擔判斷錯誤的機率，即-error或-error。 • 希望結果判斷是‘有差異’的，而且是正確的機率越大越好，即檢(定)力(power of testing)。

5. Hypothesis Testing • Scientific Hypothesis（科學假說） • Statistical Hypothesis（統計假說） Null Hypothesis（虛無假說；無效假說； H0 ；Hn）：對立假說的餘假說。 Alternative Hypothesis（對立假說；H1 ；H a）：量化以後的科學假說。 • 例： H0 : μ1＝μ2 H0 : μ1μ2 H0 : μ1μ2 H1 : μ1≠μ2 H1 : μ1 μ2 H1 : μ1 μ2 H0 : ρ＝0 H0 : β＝0 H1 : ρ≠0 H1 : β≠ 0

6. 單尾檢定(one-tailed test)與雙尾檢定(two-tailed test) H0: μ1≦μ2 H0: μ1≧μ2 H0: μ1＝μ2 H1: μ1＞μ2 H1: μ1＜μ2 H1: μ1≠μ2

7. 第一類型誤差TypeⅠerror (α error)與p-value • α：significant level; rejection level • α area：critical region; region of rejection • p value ＝Observed Significant Level; O.S.L. ＝Pr { the fact occurred│H0} ＝Pr { the observed and more extreme value obtained│H0} ＝Pr { the error made if rejected H0} if p ≧α then ：do not reject H0 if p ＜α then ：reject H0

8. 第二類型誤差TypeⅡerror (β error )與檢力(power of testing) • β error＝Pr {don't rejected H0│H1 is true} ＝Pr { the error made if don't rejected H0} • 1－β：power of testing（檢定力；檢力） ＝Pr { correctly rejected H0│H1 is true}

9. 第一類型誤差TypeⅠerror (α error)第二類型誤差TypeⅡerror (β error )

10. The steps of statistical inference • nature of data • assumption • scientific hypothesis • statistical hypothesis • test statistics（統計數） • significant level（顯著水準） • rejection region（拒絕區） • test statistics calculation under null hypothesis • decision • 當P≧α，則不能拒絕需無假說，故二者不呈統計上有意義之差異。 • 當P＜α，則拒絕需無假說，故二者呈統計上有意義之差異。 • conclusion

11. §假說檢定的步驟(一)：假說之建立 • 描述統計的觀察 • 以文字寫出新的想法，即科學假說 • 量化成為統計假說中之對立假說 (Alternative Hypothesis) • H1: μ1>μ2 【單尾檢定】 • 對立假說的餘假說即為虛無假說 (Null Hypothesis) • H0: μ1≦μ2 • 若能拒絕H0，則H1即可接受，故科學假說即可成立

12. §假說檢定的步驟(二)：p值與值 • 研究者通常希望藉由否證（拒絕）H0，而使H1成立時，不要造成太大的錯誤，其機率一般訂為5%，即稱為顯著水準（ error；第一類型誤差） • 根據資料顯示之觀察值，選擇適當的統計數，瞭解其抽樣分布，以選擇適當的統計方法 • 在H0:μ1=μ2之下，計算統計數，以求得觀察值及其更極端值出現的機率，即為p值，亦即拒絕H0實際可能錯誤的機率

13. §假說檢定的步驟(三)：結果判斷 • 當p>時，則表示若H0為真，而拒絕H0的機率（p值）大於我們所訂的標準（值），故不可拒絕H0，即H1不被接受，結果判斷為：兩者呈現統計上無意義之差異或不呈統計上有意義之差異。 • 當p<時，則表示拒絕H0時所可能造成錯誤的機率比所訂標準小，故可以拒絕H0，而H1可以接受，結果判斷為：兩者呈現統計上有意義之差異。

14. 假說檢定-----統計數與母群參數的差異 • 連續性機率分布的檢定 • 二項分布的檢定（大樣本）

15. 假說檢定-----統計數與母群參數的差異 • 卜瓦松分布X的檢定（大樣本） • 連續性機率分布S2的檢定

16. 母群變異數（2 ）已知

17. Example 7.2.1 • Researchers are interested in the mean age of a certain population. Let us say that they are asking the following question: Can we conclude that the mean age of this population is different from 30 years?

19. 母群變異數（2 ）未知

20. Example 7.2.3 • Castillo and Lillioja (A-1) describe a technique they developed for peripheral lymphatic cannulation in humans. The authors claim that their technique simplifies the procedure and enables the collection of adequate volumes of lymph for kinetic and metabolic studies. The investigators’ subjects were 14 healthy adult males representing a wide range of body weight. One of the variables on which measurements were taken was body mass index (BMI)＝weight (kg)/height2 (m2). The results are shown in Table 7.2.1. We wish to know if we can conclude that the mean BMI of the population from which the sample was drawn is not 35.

22. (df=13)

23. 一個母群成功率的檢定 • (7.5.1)

24. Example 7.5.1 • In a survey of injection drug users in a large city, Coates et al. (A-17) found that 18 out of 423 were HIV positive. We wish to know if we can conclude that fewer than 5 percent of the injection drug users in the sampled population are HIV positive.

25. 變異數的檢定 • (7.7.1)

26. Example 7.7.1 • The purpose of a study by Gundel et al. (A-25) was to examine the release of preformed and newly generated mediators in the immediate response to allergen inhalation in allergic primates. Subjects were 12 wild-caught, adult male cynomolgus monkeys meeting certain criteria of the study. Among the data reported by the investigators was a standard error of the sample mean of .4 for one of the mediators recovered from the subjects by bronchoalveolar lavage (BAL). We wish to know if we may conclude from these data that the population variance is not 4.

28. One-Sided Test

29. 假說檢定-----兩個母群參數的差異 • 連續性機率分布(X1－X2)的檢定 • 若σ未知、為大樣本，則： • 若σ未知、為小樣本，且σ12＝σ22則：

30. 假說檢定-----兩個母群參數的差異 • 若σ未知、為小樣本，且σ12≠σ22則： • 若(X1, X2)為配對樣本，則：

31. 假說檢定-----兩個母群參數的差異 • 二項分布(π1－π2)的檢定（大樣本） • 若π1、π2未知，則： • 卜瓦松分佈(μ1－μ2)的檢定（大樣本） • 若μ1、μ2未知，則：

32. 假說檢定-----兩個母群參數的差異

33. 未知 (X1-X2)-(μ1-μ2) (X1-X2)-(μ1-μ2) 12 22 + n1 n2 S12 S22 + n1 n2 §假說檢定-----兩個母群參數的差異 • 假設兩個樣本是來自同一母群體的獨立樣本，即μ1=μ2，亦即μ1-μ2 =0 • 若(X1-X2)的分布為常態，則可經由標準化Z值求p值，即～N{(μ1-μ2),(12 /n1 + 22 /n2 )} • Z= t(dfc)=

34. (7.3.1)

35. Example 7.3.1 • Researchers wish to know if the data they have collected provide sufficient evidence to indicate a difference in mean serum uric acid levels between normal individuals and individuals with Down’s syndrome. The data consist of serum uric acid readings on 12 individuals with Down’s syndrome and 15 normal individuals. The means are x1＝4.5 mg/100 ml and x2＝3.4 mg/100 ml.

37. (7.3.2)

38. § t 檢定（範例） • 男性31人膽汁過飽和度(%)平均值及標準差分別為84.5及24.0；女性29人之平均值及標準差分別為88.5及27.6；是否男女有別？ • Ｈ0：μ1=μ2 Ｈ1：μ1μ2 【雙尾檢定】 • 自由度﹦31+29-2﹦58 • 共同母群變異數估計值=兩樣本變異數估計值的加權平均﹦[(n1-1)*S12+(n2-1)*S22]/(n1+n2-2) =(30*24.0*24.0+28*27.6*27.6)/58=644.2 • t=(84.5-88.5)/ 644.2/(1/31+1/29) = -0.61 • ∣t值∣< ｔ.975（58） =2.0017，p>，不能拒絕Ｈ0 • 男女的膽汁過飽和度(%)不呈統計上有意義之差異

39. Example 7.3.2 • The purpose of a study by Eidelman et al. (A-6) was to investigate the nature of lung destruction in cigarette smokers before the developments of marked emphysema. Three lung destructive index measurements were made on the lungs of lifelong nonsmokers and smokers who died suddenly outside the hospital of nonrespiratory causes. A larger score indicates greater lung damage. For one of the indexes the scores yielded by the lungs of a sample of none nonsmokers and a sample of 16 smokers are shown in Table 7.3.1. We wish to know if we may conclude, on the basis of these data, that smokers, in general, have greater lung damage as measured by this destructive index than do nonsmokers.

41. (7.3.5)

43. (7.3.3) • (7.3.4)

44. Example 7.3.3 • Researchers wish to know if two populations differ with respect to the mean value of total serum complement activity (CH50). The data consist of CH50 determinations on n2＝20 apparently normal subjects and n1＝10 subjects with disease. The sample means and standard deviations are • x1＝62.6, s1＝33.8 x2＝47.2, s2＝10.1 Example 7.3.3

45. Example 7.3.3

46. Example 7.3.3 • Researchers wish to know if two populations differ with respect to the mean value of total serum complement activity (CH50). The data consist of CH50 determinations on n2＝20 apparently normal subjects and n1＝10 subjects with disease. The sample means and standard deviations are x1＝62.6, s1＝33.8 x2＝47.2, s2＝10.1

47. 配對 t 檢定 • (7.4.1)

48. Example 7.4.1 • Nancy Stearns Burgess (A-11) conducted a study to determine weight loss, body composition, body fat distribution, and resting metabolic rate in obese subjects before and after 12 weeks of treatment with a very-low-calorie diet (VLCD) and to compare hydrodensitometry with bioelectrical impedance analysis. The 17 subjects (9 women and 8 men) participating in the study were from an outpatient, hospital-based treatment program for obesity. The women’s weights before and after the 12-week VLCD treatment are shown in Table 7.4.1. We wish to know if these data provide sufficient evidence to allow us to conclude that the treatment is effective in causing weight reduction in obese women.

50. (7.6.1) • (7.6.2)