1. …. a linear regression coefficient indicates the impact of each independent variable on the outcome in the context of (or “adjusting for”) all other variables. - J.Concato, A.R. Feinstein, T.R. Holford

2. Regression analysis Regression analysis is an area of statistics that attempts to predict or estimate the value of the response variable from known values of one or more independent variables. Analysis using a single independent variable is called Simple(linear) regression. While combination of independent variables are used it is called multiple (linear) regression. Simple linear regression is used to assess the relationship between a single continuous response variable (or) the tendency of one variable to change with other.

3. Requirements for simple linear regression: Two continuous variables one-independent and one independent variables such that the distribution of their data can be summarized with mean and standard deviation.

4. Linear Regression Equation Y = a + bX Suppose we want to test whether there is any relation between birth weight (BW) of baby and Blood Pressure (BP) Dependent variable is BP and independent variable is BW So the equation will be BP = a + b (BW) i.e. Given a value of birth weight (BW) corresponding Blood Pressure (BP) can be predicted. In mathematics Y is called a function of X but in statistic the term regression is used to describe the relationship.

5. Example Growth curve of height may be considered as the regression of height on age In toxicology the lethal effects of drug are described by the regression of percent kill on the amount of drug

6. Assumptions Relation between x and y is linear over range of values Values of the independent variable x are said to be fixed The variable x is measured without error. For each value of x there is a sub-population of Y values The variances of sub populations of Y are all equal. The means of sub-population of Y all lie on the same straight line.

7. Regression Coefficients Y = a + bX b is called the slope of the equation a is called the Y intercept for which X = 0 b = Cov (X,Y)/Var (X) = 1/n∑(x-x)(y-y)/1/n ∑(x-x) 2 a = y – b * x

8. Calculation of Correlation Coefficient Sr. No 7 (1) BW8 (2) BP (3) X-mean ((1)-2.69)) (4) Y-Mean ((2)) – 76.9) (3)* (4) X-Mean * (Y-Mean) (3) 2 (X – Mean) 2 11.547-1.19-29.9035.581.42 21.950-.79-26.921.25.62 32.271-.49-5.92.89.24 42.576-.19-.9.17.04 52.776.01-.9-.01.00 62.881.114.1.45.01 73.086.319.12.82.10 83.285.518.14.13.26 93.491.7114.110.01.50 103.71061.0129.129.391.02 Tot Mean Var ∑x= 269 2.69 ∑y= 769 76.9 106.69 10.6694.21 SD.684

9. b = Cov (X,Y)/Var (X) = 10.669/.421 = 25.34

10. a = y – b* x = 76.9 – 25.34 * (2.69) = 8.74 So the regression equation will be Y = 8.74 + 25.34 * X What does these coefficients tells us? The slope b means that for each unit change in X (i.e Birth weight), Y (Blood pressure) increases by 25.34 units.

11. BPBP BW 0 10 20 30 40 50 60 1.01.52.02.53.03.5 4.0 0 o o o o o o o o 70 80 90 100 110 b a constant y = 8.74 + 25.34 x

12. Uses Measure of linear association Interpolation Prediction To identify which combination of variables best predicts response variables or outcome. Misuses Extrapolation without assurance that the trend remains same. Using the regression relationship whose slope has been shown to be not significantly different from zero Concluding that cause and effect relationship exists, while the relationship may just be statistical Applying the relationship established in one group of subject to another group without the assurance that is applicable to all groups.