 Biostatistics Unit 9 – Regression and Correlation.

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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.

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

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.

General Regression Line y =  +  x +   is the y-intercept  is the slope  is the error term

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

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

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

Regression Equation

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.

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

Hypotheses H 0 :  and  = 0 H A :  and  0

Calculations and Results Calculator setup

Calculations and Results Results

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 r 2. It has values between 0 and 1. The value of r 2 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

Calculations Calculate the regression equation y = 4.53x – 1.57

Calculations Calculate the correlation coefficient

Coefficient of Determination The coefficient of determination is r 2. 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 r 2 =.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

Results of linear regression t test

fin