Linear regression analysis 線性迴歸分析

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1 臨床研究迴歸分析的應用温淑惠慈大公衛系 2015/8/20 1

2 迴歸分析 (regression analysis) 性別年齡教育程度生化檢驗值 Response ( 研究感興趣的變數, Y) 1. 連續型 e.g. BMI (Multiple linear regression) 2. 兩分類型 e.g. 胖 (Logistic regression) 3. 存活時間 ( 追蹤研究 ) e.g. time to death (Cox regression) QoL 2015/8/20 2

3 Multi-predictor analysis (regression model) Multiple linear regression Y=α+β 1 X 1 +β 2 X β k X k +error A continuous outcome (Y, e.g. BMD) is related to a risk factor (X 1 e.g. 停經 ) But the risk factor of interest might be related to other factors (X 2, e.g. age, BMI, smoke ) which also predict the outcome. 2015/8/20 3

4 Scatter plot ( 散佈圖 ) Which line best describes the relationship between X and Y? Birthweight Observed Linear Estriol /8/20

5 Lease squares method ( 最小平方法 ) Minimize the sum of the squared deviations of the sample points from the line Fitted model )) ( ( min ) ( x y y y e SSE n i i i n i i n i i 2015/8/20 5 ˆ ˆ, ˆ, ˆ ˆ X Y s s r X Y X Y i i

6 Å kesson, A., et al. (2006). Cadmium-Induced Effects on Bone in a Population-Based Study of Women. Environmental Health Perspectives, 114(6), doi: /ehp /8/20 6

7 以文獻 (Å kesson et al. 2006) 為例 2015/8/20 7

8 Statistical analyses In multiple linear regression models, each bone-related variable was evaluated in relation to cadmium, potential confounders. Y=α+β 1 X 1 +β 2 X β k X k +error Residual analyses indicated no deviation from a linear pattern in the regression models. All tests were two sided, and statistical evaluation was performed using SPSS (version 12.01; SPSS Inc., Chicago, IL, USA). Å kesson et al. 2006

9 以文獻 (Å kesson et al. 2006) 為例 for regression topic 探討鎘暴露對骨頭的影響骨骼相關變數 (Y): e.g. BMD 骨骼密度 (g/cm 2 ) 評估鎘的暴露量 (X) 和身體含量時, 血中的鎘含量可表示最近的暴露量, 尿中的鎘可表示身體的含量採用 multiple linear regression 一併考慮其他可能影響因素 (X2, X3,,Xk) Y=α+β 1 X 1 +β 2 X β k X k +error

10 提供散佈圖看主要感興趣變數之相關請問相關程度為何? Å kesson et al. 2006

11 變數多時相關係數可整理成矩陣 Å kesson et al. 2006

12 迴歸分析結果 Y=BMD X: U-Cd ; Other factors: weight, age,wp Fitted regression model: BMD=constant-0.011*Cd *Weight -.004*Age

13 The meaning of regression coefficient Per unit increase in covariate X, the amount of Y increases β on average. (X 每增加一單位,Y 平均值增加 β ) e.g. BMD=constant-0.011*Cd *Weight *Age The slope is , it implies that for each one unit increase in Cd, a woman s BMD decreases by on average, while controlling for other factors /8/20

14 Model assumption 迴歸係數推論結果正確性需依賴假設 (LINE) 成立 Y i =α+βx i +ε i, ε i ~N(0, σ 2 ), i=1,,n i.e. BMD=α+β*Cd+ε Linear relationship between mean of response and covariates Error term ε i ~N(0, σ 2 ) independently. That is Y i X i ~N(α+βX i, σ 2 ), conditional normality of response Equal variances of response for different value of covariates /8/20

15 Case study on FEV data: SPSS implementation for multiple linear regression 2015/8/20 15

16 Example: FEV data (from Rosner (2006) ) 一秒最大呼氣量 (FEV) FEV 與抽菸的關係? Other related factors, e.g. age, gender 2015/8/20 16

17 FEV data 2015/8/20 17

18 FEV vs. smoke 抽菸者 FEV 值大!!!...??? 可能的原因是抽菸者多為男性或者年齡較大 ( 體型較大 ) Confounder? 2015/8/20 18

19 Analysis steps Step1: Present the descriptive the clinical features for FEV and other influencing factors. Step2: Explore the correlation between FEV and X1 Xk. Step3: Build up the multiple linear regression model and check for model adequacy. Step4: Model revision. Step5: Interpretation the result (model). 2015/8/20 19

20 Step3: Build up the multiple linear regression model Now, we want to build the model as FEV=α+β 1 age+β 2 sex+β 3 Hgt+β 4 smoke 2015/8/20 20

21 Multiple linear regression 2015/8/20 21

22 Check for model adequacy. 點進圖形後選擇常態機率圖 ( 為檢驗資料是否符合常態性假設 ) 畫殘差圖 (Y axis: 殘差值, X axis:fev 值 ) 判斷同質性假設若此兩假設不符則後續檢定迴歸係數之結果可能不可信 (not valid) 2015/8/20 22

23 Results-1:Pearson Correlation matrix 除了看出 FEV 與因子 (Xs) 間相關以外,Xs 彼此也有些達統計相關 e.g. age vs. Hgt 2015/8/20 23

24 Summary of correlation analysis FEV 與抽菸的關係? Others related factors, e.g. age, gender FEV 與身高年齡都呈正相關, 且有統計上顯著相關 (p<0.05) FEV 與性別有關 (p<0.05), 男性其 FEV 值越大 FEV 與抽菸有關 (p<0.05), 抽菸者其 FEV 值越大, 但此現象可能是有 confounder 造成, 例如性別年齡身高尚未考慮 2015/8/20 24

25 Multiple linear regression Results-1: R-square FEV 的變異可被模式中所有因子共同解釋的變異比例為換句話說, 還有 22.6% 為誤差, 可能還有其他影響 FEV 因素未被考慮 2015/8/20 25

26 Residual analysis Result: Normality 圖中直線若接近 45 度直線則表示常態性假設成立通常 sample size 若夠大可不用太擔心常態性不成立如果常態性不成立, 一般會將 Y 轉換成 log(y) 重新做 regression 2015/8/20 26

27 Residual analysis Result: Equal variance (Homogeneous) 正常圖形應該看來是雜亂無 pattern 右圖看來有點扇形 (Fan shape) 可能是違反同質性另外 Y-axis 標準化殘差值落在 (-3,3) 之外的就是異常值 2015/8/20 27

28 Outliers 異常值下表即為 outliers. 一般可拿掉後重做 regression. 2015/8/20 28

29 Step4: Model revision 根據初步分析結果 FEV 可被 age, gender, smoke. Height 解釋變異之比例達 77.4% 常態性符合, 同質性雖不甚符合, 但 n 夠大有 5 個異常值 Model revision 試著將 outliers 去掉後再做一次 2015/8/20 29

30 Model revision 2015/8/20 30

31 先儲存標準化殘差, 再利用 selection 功能將 outlier 去掉執行完 regression 後請到資料選擇觀察值 2015/8/20 31

32 Delete outliers and do regression again 條件為 abs(zre_1) <=3 2015/8/20 32

33 Interpretation of regression analysis 重新做 regression 後的結果即可仿照 page 步驟檢視統計結果 N=649 ( 原本有 654 筆 ) 2015/8/20 33

34 Model assumptions hold Linearity Scatter plot Independence 獨立個體的 FEV 值測量為互不相關 Normality 符合常態機率圖接近 45 度直線 Equal Variance 殘差圖與之前一樣 Outliers 雖有 ( 很接近 3) 但很輕微故不再排除 2015/8/20 34

35 Adjusted R-square (new) R-square is 78.7%. A little larger than previous one. 2015/8/20 35

36 Interpretation of regression analysis 拿掉 outlier 後 regression model 影響不大 FEV= Age+0.131Sex-0.067Smoke Hgt 與 FEV 顯著相關變項 : Age, Sex, Height 2015/8/20 36

37 整理成 paper 之表格 ( 供參考 ) Table: Multiple linear regression analysis between FEV and factors. Beta 95% CI Factors Coefficient lower bound upper bound p-value Age(yr) <0.001* Sex <0.001* Smoke Height(cm) <0.001* Sex:0=female, 1=male. Smoke: 0=no, 1=yes. *: statistical significance 2015/8/20 37

38 References 1. M. Pagano, K. Gauvereau. Principles of Biostatistics(2nd Ed). Australia ; Pacific Grove, CA : Duxbury, ( 歐亞書局代理 ) 2. Rosner B. (2006) Fundamentals of Biostatistics (6th ed). Belmont, CA : Thomson-Brooks/Cole ( 歐亞代理 ) 3. Vittinghoff E., Glidden D.V., Shiboski S.C., McCulloch C.E. Regression Methods in Biostatistics. Spreinger Å kesson, A., et al. (2006). Cadmium-Induced Effects on Bone in a Population-Based Study of Women. Environmental Health Perspectives, 114(6), doi: /ehp /8/20 38

39 Thanks for listening 2015/8/20 39

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