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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.

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## Biostatistics

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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 = a + bx + e a is the y-intercept b is the slope e 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.**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: b is the population regression coefficient r is the population correlation coefficient**Hypotheses**H0: b and r = 0 HA: b and r 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 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**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 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.

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