Biostatistics

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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
  Darwins 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 a bx e
• a is the y-intercept
• b is the slope
• e 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.

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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

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Calculations and Results

• Calculator setup

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Calculations and Results

• Results
• Conclusion p lt .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