Simple Linear Regression

Transcription

1 Simple Linear Regression Assoc. Prof Dr Sarimah Abdullah Unit of Biostatistics & Research Methodology School of Medical Sciences, Health Campus Universiti Sains Malaysia

2 Regression Regression analysis is a statistical tool that utilizes the relationship between variables so that one variable can be predicted from the other, or others Uses a variable (x) to predict some outcome variable (y) values in y change as a function of changes in values of x

3 Simple Linear Regression SIMPLE (SLR) Only ONE independent variable vs ONE dependent variable Age Cholesterol Duration of Jogging BMI Amount salt intake Blood pressure Mothers weight Birthweight of babies

4 Simple Linear Regression To explore the nature of the relationship between two continuous variable To investigate the change in response How much the value Y (dependent variable) varies with one unit of change in value X (independent variable) Y = a+bx Y = ß 0 +ß 1 x 1 a: an intercept of the regression line b: a slope of the line - amount of change in Y for a unit change in X B (slope)

5 Regression coefficient (r) is the constant (a) that represents the rate of change of one variable (y) as a function of changes in the other (x) the slope of the regression line

6 Regression coefficient (r) r>0 r<0 r=0

7 Coefficient of determination (r 2 ) It provides a measure of how well future outcomes are likely to be predicted by the model how much the independent x is explained by dependent y = Correlation coefficient in SLR ranges from 0 to 1

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9 Steps in analysis Simple Linear Regression 1. Data exploration - descriptive statistics 2. Fit least square line 3. Check fitness of the regression model (r 2 ) 4. slope of the regression line (b or slope) 5. Check residual diagnostic (LINE assumption) 6. Interpretation and Presentation 7. Prediction

10 Simple linear Regression Hands on Please open Cholesterol_SLR.sav RQ: What is the relationship between age and cholesterol level among patients with familial hyperlipidemia?

11 Steps in analysis : Step 1. Data exploration (scatter plot) Check for: - Minimum - Maximum - Distribution - Relationship - outliers

12 Step 2. Fit least square line Double clicks cholesterol in mmol/l age in year

13 Step 3. Check fitness of the regression model (r 2 )

14 Step 3. Check fitness of the regression model (r 2 ) 10.9% of variability of cholesterol is explained by age

15 Step 4. slope of the regression line (b or slope)

16 Step 4. slope of the regression line (b or slope) Model 1 Model 1 Regression Residual Total ANOVA b Sum of Squares df Mean Square F Sig a a. Predictors: (Constant), age in year b. Dependent Variable: cholesterol in mmol/l (Constant) age in year Unstandardized Coefficients a. Dependent Variable: cholesterol in mmol/l Coefficients a Standardized Coefficients 95% Confidence Interval for B B Std. Error Beta t Sig. Lower Bound Upper Bound Regression equation: Y = ß 0 +ß 1 x 1 a: 5.90 b: 0.06 Cholesterol = (0.06*age)

17 Step 5: Check residual diagnostic (LINE assumption) ASSUMPTIONS CHECKING i. Linearity - overall fitness Scatter plot ii. Equal variance Residual Vs predicted iii. Normality of residuals Histogram iv. Linearity of each independent variable. Scatter plot Residual Vs each independent variable v Independent observation Design Other assumptions: *Random sample *Independent observation

18 i. Overall linearity and Equal variance Before checking overall linearity Need to create the predicted values and residuals from dataset 18

19 i. Overall linearity and Equal variance 19

20 ii. Normality or residuals 20

21 i. Linearity and Equal variance 21

22 Simple linear Regression - Interpretation and Presentation Independent variable SLR a b* (95% CI) P value Age (years) 0.06 (0.02, 0.09) r 2 =10.9 % a simple linear regression ( dependent variable: cholesterol mmol/l) *b = crude regression coefficient Increasing of one year of age will increase 0.06 mmol/l of serum cholesterol (0.02, 0.09, p=0.002)

23 Prediction Y = a+bx Y = ß 0 +ß 1 x 1 a: 5.90 b: 0.06 Cholesterol = (0.06*age) If age 60, the predicted cholesterol = 5.9+(0.06*30) = 7.7

24 Simple linear Regression PRACTICE Please open SLR2.sav

25 Simple Linear Regression Duration of exercise Regression equation: Y = ß 0 +ß 1 x 1 a: 10.8 b: Cholesterol = (-0.62*exercise) Increasing of one unit of exercise will reduce 0.62 mmol/l of serum cholesterol (-0.77, -0.46, p<0.001)

26 Simple Linear Regression Diet inventory score Regression equation: Y = ß 0 +ß 1 x 1 a: 6.0 b: 0.45 Cholesterol = (0.45*diet) Increasing of one unit of diet will increase 0.45 mmol/l of serum cholesterol (0.30, 0.60, p<0.001)

27 Simple Linear Regression Socio-economic index Regression equation: Y = ß 0 +ß 1 x 1 a: 5.8 b: 0.21 Cholesterol = (0.21*diet) Increasing of one unit of SE index will increase 0.21 mmol/l of serum cholesterol (0.17, 0.25, p<0.001)

28 Simple Linear Regression SIMPLE (SLR) Only ONE independent variable age Cholesterol diet Cholesterol Exercise Cholesterol Socio-economic Cholesterol

29 Simple linear Regression - Interpretation and Presentation Independent variable SLR a b* (95% CI) P value Age (years) 0.06 (0.02, 0.09) Duration of exercise (-0.79, -0.46) <0.001 (hrs/wk) Diet inventory score 0.45 (0.30, 0.61) <0.001 Socio-economic index 0.21 (0.17, 0.25) <0.001 a simple linear regression ( dependent variable: cholesterol mmol/l) *b = crude regression coefficient Y = ß 0 +ß 1 X 1

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