Biostatistics

Biostatistics

Unit 9

Regression and Correlation

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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.

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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.

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General Regression Line

y = + x + is the y-intercept

is the slope

is the error term

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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.

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Data

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

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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

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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.8

Significance of regression analysis

It is possible to perform the linear regression t test to give a probability. In this test: is the population regression coefficientis the population correlation coefficient

The hypotheses are:

H0: and = 0

HA: and 0

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Calculations and Results

Calculator setup

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Calculations and Results

Results

Conclusion: p < .001 (.000206)

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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.

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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.

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Correlation Coefficient

Perfect negative correlation, r = -1.

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Correlation Coefficient

No correlation, r = 0.

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Correlation Coefficient

Perfect positive correlation, r = +1.

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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.

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Graphs

Scatter plot

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Graphs

Scatter plot with regression line

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Data for calculations

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Calculations

Calculate the regression equation

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Calculations

Calculate the regression equation

Result: The regression equation is

y = 4.54x – 1.57

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Calculations

Calculate the correlation coefficient

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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.

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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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Calculate the residual variance

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Calculate the residual variance

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

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Results of linear regression t test

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Results of linear regression t test

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fin

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