Regression and Correlation Biostatistics Unit 9 Regression and Correlation

Regression and Correlation 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.

Data for Regression and Correlation Data are in the form of (x,y) pairs. 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, two forms are used which are y = ax + b and y = a + bx.

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

Calculations For each (x,y) point, the vertical distance from the point to the regression line is squared. Adding these gives the sum of squares. Regression analysis allows the experimenter to predict one value based on the value of another. A similar procedure is used in biochemistry with standard curves.

Data Data are in the form of (x,y) pairs. List L1 contains the x values and List L2 contains the y values.

Calculation of regression equation using TI-83 The Linear Regression test is used. Conclusion: The equation of the regression line is y = 4.54x – 1.57

Using the regression equation Interpolation is used to find values of points between the data points. This is a relatively safe and accurate process. Extrapolation is used to find values of points outside the range of the data. This process is more risky especially as you get further and further from the ends of the line. Be careful to make sure that the calculations give realistic results.

Significance of regression analysis It is possible to perform the linear regression t test to give a probability. In this test: b is the population regression coefficient r is the population correlation coefficient The hypotheses are: H0: b and r = 0 HA: b and r 0

Calculations and Results Calculator setup

Calculations and Results Conclusion: p < .001 (.000206)

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.

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.

Correlation Coefficient Perfect negative correlation, r = -1.

Correlation Coefficient No correlation, r = 0.

Correlation Coefficient Perfect positive correlation, r = +1.

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.

Graphs Scatter plot

Graphs Scatter plot with regression line

Data for calculations

Calculations Calculate the regression equation

Calculations Calculate the regression equation Result: The regression equation is y = 4.54x – 1.57

Calculations Calculate the correlation coefficient

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.

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.

Calculate the residual variance

Calculate the residual variance Result: The residual variance is 56.1366. Residual SD is 7.4924 which TI-83 gives.

Results of linear regression t test

Results of linear regression t test

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