biostatistics unit 9 regression and correlation 1
Post on 19-Dec-2015
219 views
TRANSCRIPT
![Page 1: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/1.jpg)
Biostatistics
Unit 9
Regression and Correlation
1
![Page 2: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/2.jpg)
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.
2
![Page 3: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/3.jpg)
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.
3
![Page 4: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/4.jpg)
General Regression Line
y = + x + is the y-intercept
is the slope
is the error term
4
![Page 5: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/5.jpg)
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.
5
![Page 6: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/6.jpg)
Data
Data are in the form of (x,y) pairs. List L1 contains the x values and List L2 contains the y values.
6
![Page 7: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/7.jpg)
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
7
![Page 8: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/8.jpg)
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
![Page 9: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/9.jpg)
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
9
![Page 10: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/10.jpg)
Calculations and Results
Calculator setup
10
![Page 11: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/11.jpg)
Calculations and Results
Results
Conclusion: p < .001 (.000206)
11
![Page 12: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/12.jpg)
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.
12
![Page 13: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/13.jpg)
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.
13
![Page 14: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/14.jpg)
Correlation Coefficient
Perfect negative correlation, r = -1.
14
![Page 15: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/15.jpg)
Correlation Coefficient
No correlation, r = 0.
15
![Page 16: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/16.jpg)
Correlation Coefficient
Perfect positive correlation, r = +1.
16
![Page 17: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/17.jpg)
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.
17
![Page 18: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/18.jpg)
Graphs
Scatter plot
18
![Page 19: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/19.jpg)
Graphs
Scatter plot with regression line
19
![Page 20: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/20.jpg)
Data for calculations
20
![Page 21: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/21.jpg)
Calculations
Calculate the regression equation
21
![Page 22: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/22.jpg)
Calculations
Calculate the regression equation
Result: The regression equation is
y = 4.54x – 1.57
22
![Page 23: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/23.jpg)
Calculations
Calculate the correlation coefficient
23
![Page 24: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/24.jpg)
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.
24
![Page 25: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/25.jpg)
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.
25
![Page 26: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/26.jpg)
Calculate the residual variance
26
![Page 27: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/27.jpg)
Calculate the residual variance
Result: The residual variance is 56.1366. Residual SD is 7.4924 which TI-83 gives.
27
![Page 28: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/28.jpg)
Results of linear regression t test
28
![Page 29: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/29.jpg)
Results of linear regression t test
29
![Page 30: Biostatistics Unit 9 Regression and Correlation 1](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d3e5503460f94a177a6/html5/thumbnails/30.jpg)
fin
30