Regression , Types of Regression, Application of Regression, methods

Biopharmaceutics and research methodology. B pharm 8th semester

1. HI TECH COLLEGE OF PHARMACY CHANDRAPUR NAME – MUKESH V KAPSE CLASS – B PHARM 4 th YEAR 8 th SEM ROLL NO – 32 SUBJECT – Biostatistic and Research methodology TOPIC –REGRESSION

2. contents Regression Types of Regression Application of Regression Regression Line Regression Equation Methods of determine regression Line Standard Errors of Regression References

3. Regression It is a statistical tools used to calculate a continuous dependent variable from various independent variable and is commonly used for prediction and forecasting . Calculate unknown value from known values. According to Blair, regression analysis is define as the measure of average relationship between two or more variable in term of the original units of the data. This technique is used for modeling and analysis of mathematical data. Regression analysis makes an equation to describe the statistical relationship between one or more predictor variable and response variable.

4. Types of regression Simple Multiple Linear N on Liner

5. Application of regresion It is use to know the relationship between different variables. To find out the value of dependent variable from value of independent variables. To find out coefficient of correlation , coefficient of determination. r = ± √ bzy x byz b=regression coefficient r= correlation coefficient In corporate sector it is useful to check quality. To determine the statistical curve.

6. Regression line The line of the best fit between two variable and stating the common average relationship are known as regression lines. E.g.Variables X,Y Regression lines X on Y Y on X Consider X dependent variable Y-independent variable R line is called as X on Y {used to estimate value of X that are Unknown from known value of Y.} Consider Y dependent variable X-incident variable R line is called as Y on X {used to estimate value of Y that are Unknown from known value of X.}

7. Regression equation These equations are use to show regression lines Regression equations of X on Y = x= a+by , ∑x = Na + b∑y Regression equation of Y on X = y= a+bx (b- slop, a- constant) b xy = r σ x / σ y for X and Y regression coefficient ( bxy ) σ – standard deviation r – correlation of coefficient b yx = r σ y / σ x

8. The regression line of Y on X gives the finest possible values of Y for the given values of X . So this line is the sum of the square of deviations of the calculated value of Y and observed value of Y is minimum. ∑ ( Yc - Yo )² = minimum For X and Y ∑ ( Xc - Xo)²=minimum Where , c= calculated o=observed If only one regression line exist between two variables then r= ±1 (coefficient of correlation) If r=0 then both variables are independent variables i.e. the line are perpendicular and values become average.

9. Properties of regression coefficient Coefficient of correlation is calculated by determining the geometric mean of two regression coefficient . r =√ bxy-byx r= +1 to –1 Regression coefficient is independent from origin but not from scale. If one regression coefficient is greter than 1 then remaining must be smaller than 1. Both regression coefficient have sign (+ ve /- ve ) The value of coefficient of correlation is less than the mean value of regression coefficient.

10. Methods to determine regression lines Scattered diagram Least square method (curve fitting)

11. Curve fitting method Process of constructing a curve or mathematical function that has best fit to a series of data point possibly subjective to constants. Curve fitting can involve either interpolation when fit to the end data is required to smoothing in which smooth function is constructed that approximately fits the data. Fitted curve can be used as an aid for visualization to interfere values of a function where no data is available. To summarize the relationship among the two or more variable.

12. Method of least squares Uses to predict the behavior of dependent variables Provides the overall rationale for the placement of the line squares of deviation ∑( Yc-Yo )²=minimum We minimize the sum of square of errors Application of this methods is to create a stright line that minimise the sums of squares of errors that are generated by the results of the associated equation. a)Linear/Ordinary least square problem b) Non-linear least square problem. Regression line Y on X is (y-yˉ )= byx (x-xˉ) byx = r( σ y)/( σ x)

13. Multiple regression One dependent variable (unknown) Two or more independent vaeiable (known) Eg : No of tablets /hr - dependent(unknown) speed of machine no of punches - independent(known) Y on X1 and X2 Y=a+b1x1 + b2x2 for N no of variables Y=a+b1x1 + b2x2…….. bnxn Where, b1 and b2- regression coefficient a=constant

14. applications Use to handle multiple point at a point To determine a mathematical relationship between random values (variables) To study relationship of multiple independent variable. Use in biological, social , pharmaceutical science to study possible relationship between variables

15. Standard errors of regression It is the measure of variation of an observation made around the computed regression line. It is represents the average distance (difference)that the observed values fall from the regression line. It should be smaller. If the value are smaller the observations are very close to regression line. value =0 indicates, variation corresponding to the regression line , the correlation is perfect. S yx = √∑(y-ye)²/N Where, Syx = Standard error of estimation of Y on X, estimated value = Ye , N=No of Observations

16. applications For calculation of confidence intervals and prediction intervals. The larger the value of Syx or Sxy then greter the scatter on the line of regression. In this case the degress og correlation is very poor. In term of standard deviation , Syx = σ y ² (1-r ² ) Sxy = σ x ² (1-r ² ) In term of variance Syx /(1-r ² ) = σ y ² = variance

17. Reference “ Biostatistic and R esearch methodology” by Dr Vinod Kumar, Dr Sanjay Sharma, Deepak Kumar, Pee Vee Publication. “ Biostatistic and Research methodology” by Prof. Chandrakant Kokare , Nirali Prakashan .

18. Thank You