Multiple linear regression fits
a response variable as a linear combination of multiple X variables
by the method of least squares.
Assumptions:
The linear functionYi = b0 + b1*X1i + b2*X2i + ... + bk*Xki + ei
is the correct model,
where Yi is the ith observed value of Y, Xji
is the ith observed
value of the jthX variable, and ei is the
error term. Equivalently, the expected value
of Y for a given value of X is
Y = b0 + b1*X1 + b2*X2 + ... + bk*Xk.
The intercept is b0, the expected value of Y when
the value for each X variable is 0.
The Xj variable (predictor variable) values are fixed
(i.e., none of the Xj is a random variable).
The Y variable (response variable) observations are
independent.
The variable Y is
normally distributed
with the same variance as the ei.
For a given set of X variable values, the variable Y has constant mean.
The normality assumption is required for
hypothesis tests, but not for estimation.
The X variables are also known as the independent variables.
The Y variable is also known as the dependent variable.
The coefficients are bj, the amount by which the expected
value of Y increases when Xj increases by a unit amount,
when all the other X variables are held constant.
This interpretation of the coefficients does not hold
if some of the X variables are functions of the others,
such as an interaction term Xj*Xk.
Note that it is not assumed that the X variables are
independent of each other.
Guidance:
Ways to detect before performing the
multiple linear regression whether your data violate any
assumptions.
Ways to examine multiple linear regression results to detect
assumption violations.
Possible alternatives if your data or
multiple linear regression results indicate assumption violations.
To properly analyze and interpret the
results of multiple linear regression, you should be familiar with the following terms and
concepts:
If you are not familiar with these terms and concepts, you are advised to
consult with a statistician. Failure to understand and properly apply
multiple linear regression may result in drawing erroneous conclusions from your data.
Additionally, you may want to consult the following references:
Belsley, David A., Kuh, Edwin, and Welsch, Roy E.
1980. Regression Diagnostics. New York: John Wiley & Sons.
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.
Neter, J., Kutner, M.H., Nachtsheim, C.J., and Wasserman, W. 1996.
Applied Linear Regression Models. 3rd ed. Chicago: Irwin.
Neter, J., Wasserman, W., and Kutner, M.H. 1990. Applied
Linear Statistical Models. 3rd ed. Homewood, IL: Irwin.
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