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, and identically normally distributed with mean 0 and the same variance.The Y variable (

**response variable**) observations are independent.The variable Y is normally distributed 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 assumptions.**Ways to examine**linear regression results to detect assumption violations.**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
concepts:

- simple linear regression
- correlation
- linear functions
- method of least squares
- residuals
- Gaussian (normal) distribution assumption
- equality of variance (homoscedasticity) assumption
- violation of assumptions
- transformations

- 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: Prentice-Hall.

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