PROPHET StatGuide: (Simple) Linear Regression
Simple linear regression fits a straight line to X-Y data
by the method of least squares.
The fit may then be used
to test the null hypothesis that
the slope is 0.
The simple linear function
Y[i] = b0 + b1*X[i] + e[i] is the correct model,
where Y[i] is the ith observed value of Y, X[i]
is the ith observed
value of X, and e[i] is the
error term. Equivalently, the expected value
of Y for a given value of X is b0 + b1*X.
The intercept is b0, the expected value of Y when X is 0.
the slope is b1, the amount by which the expected
value of Y increases when X increases by a unit amount.
The X variable (predictor variable) values are fixed
(i.e., X is not a random variable).
The e[i] are independent,
with mean 0 and the same variance.
The Y variable (response variable) observations are
The variable Y is
with the same variance as the e[i].
For a given value of X, the variable Y has constant mean.
The normality assumption is required for
hypothesis tests, but not for estimation.
The X variable is also known as the independent variable.
The Y variable is also known as the dependent variable.
Ways to detect before performing the
linear regression whether your data violate any
Ways to examine linear regression results to detect
Possible alternatives if your data or
linear regression results indicate assumption violations.
To properly analyze and interpret the
results of simple linear regression, you should be familiar with the following terms and
Failure to understand and properly apply
simple linear regression may result in drawing erroneous conclusions from your data.
If you are not familiar with these terms and concepts, you may wish to
consult with a statistician.
You may also want to consult the following references:
- Brownlee, K. A. 1965. Statistical Theory and Methodology
in Science and Engineering. New York: John Wiley & Sons.
- Daniel, Wayne W. 1995. Biostatistics. 6th ed.
New York: John Wiley & Sons.
- Draper, N. R. and Smith, H. 1981.
Applied Regression Analysis. 2nd ed. New York: John Wiley & Sons.
- Hoaglin, D. C., Mosteller, F., and Tukey, J. W. 1985.
Exploring Data Tables, Trends, and Shapes. New York: John Wiley & Sons.
- Miller, Rupert G. Jr. 1986. Beyond ANOVA, Basics of Applied
Statistics. New York: John Wiley & Sons.
- Neter, J., Wasserman, W., and Kutner, M.H. 1990. Applied
Linear Statistical Models. 3rd ed. Homewood, IL: Irwin.
- Rosner, Bernard. 1995. Fundamentals of Biostatistics.
4th ed. Belmont, California: Duxbury Press.
- Sokal, Robert R. and Rohlf, F. James. 1995. Biometry. 3rd. ed.
New York: W. H. Freeman and Co.
- Zar, Jerrold H. 1996. Biostatistical Analysis. 3rd ed. Upper Saddle River, NJ:
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Last modified: February 20, 1997
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