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Biostatistics

Biostatistics. Unit 9 – Regression and Correlation. Regression and Correlation. Introduction Regression and correlation analysis studies the relationships between variables. This area of statistics was started in the 1860s by Francis Galton (1822-1911) who was also Darwin’s Cousin.

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Biostatistics

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  1. Biostatistics Unit 9 – Regression and Correlation

  2. Regression and Correlation Introduction Regression and correlation analysis studies the relationships between variables.

  3. This area of statistics was started in the 1860s by Francis Galton (1822-1911) who was also Darwin’s Cousin.

  4. Nature of Data The data are in the form of (x,y) pairs.

  5. Graphical Representation A scatter plot (x-y) plot is used to display regression and correlation data. The regression line has the form y = mx + b In actual practice, various forms are used such as y = ax + b and y = a + bx.

  6. General Regression Line y = a + bx + e a is the y-intercept b is the slope e is the error term

  7. Calculations For each point, the vertical distance from the point to the regression line is squared. Adding these gives the sum of squares.

  8. Regression Analysis Regression analysis allows the experimenter to predict one value based on the value of another.

  9. Data Data are in the form of (x,y) pairs.

  10. Regression Equation

  11. Using the regression equation • Interpolation is used to find values of points between the data points. • Extrapolation is used to find values of points outside the range of the data. Be careful that the results of the calculations give realistic results.

  12. Significance of regression analysis It is possible to perform the linear regression t test. In this test: b is the population regression coefficient r is the population correlation coefficient

  13. Hypotheses H0: b and r = 0 HA: b and r 0

  14. Calculations and Results Calculator setup

  15. Calculations and Results Results

  16. Correlation Correlation is used to give information about the relationship between x and y. When the regression equation is calculated, the correlation results indicate the nature and strength of the relationship.

  17. Correlation Coefficient The correlation coefficient, r, indicates the nature and strength of the relationship. Values of r range from -1 to +1. A correlation coefficient of 0 means that there is no relationship.

  18. Correlation Coefficient Perfect negative correlation, r = -1.

  19. Correlation Coefficient No correlation, r = 0.

  20. Correlation Coefficient Perfect positive correlation, r = +1.

  21. Coefficient of Determination The coefficient of determination is r2. It has values between 0 and 1. The value of r2 indicates the percentage of the relationship resulting from the factor being studied.

  22. Graphs Scatter plot

  23. Graphs Scatter plot with regression line

  24. Data for calculations

  25. Calculations Calculate the regression equation

  26. Calculations Calculate the regression equation

  27. Calculations Calculate the regression equation y = 4.53x – 1.57

  28. Calculations Calculate the correlation coefficient

  29. Coefficient of Determination The coefficient of determination is r2. It indicates the percentage of the contribution that the factor makes toward the relationship between x and y. With r = .974, the coefficient of determination r2 = .948. This means that about 95% of the relationship is due to the temperature.

  30. Residuals The distance that each point is above or below the line is called a residual. With a good relationship, the values of the residuals will be randomly scattered. If there is not a random residual plot then there is another factor or effect involved that needs attention.

  31. Calculate the residual variance

  32. Calculate the residual variance

  33. Results of linear regression t test

  34. Results of linear regression t test

  35. fin

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