The slopes are in fact equal.

Given that the slopes are equal, the intercepts for each line are also equal. This is the test for whether the group (treatment) means are equal, after making the adjustment for X.

Given that the slopes are equal, the slopes are all equal to 0.

The simple linear function

**Y[ij] = M + m[i] + b*(X[ij]-Xbar) + e[i]**

is the correct model, where**Y[ij]**is the*jth*observed value of Y in group*i*,**X[ij]**is the*jth*observed value of X in group*i*,**M**is an overall mean for all the Y values,**m[i]**is treatment effect for group*i*,**Xbar**is the overall average of the X values, and**e[i]**is the error term. Equivalently, the expected value of Y for a given value of X is

**M + m[i] + b*(X[ij]-Xbar)**.

The**slope**is**b**, 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 treatment effects

**m[i]**values are fixed.The

**e[i]**are independent, and identically normally distributed with mean 0 and the same variance.The Y variable (

**response variable**) observations are independent.**Y[ij]**is normally distributed with the same variance as the**e[i]**. For a given value of**X[ij]**in group*i*, the variable Y has constant mean.

The X variable is also known as a **covariate**, or
as a **concomitant variable**.

Analysis of covariance controls for X to make a more precise test of whether the treatment (group) means (intercepts) are equal. It can also be used to study the linear relationship between X and Y for each group.

**Ways to detect**before performing the analysis of covariance whether your data violate any assumptions.**Ways to examine**analysis of covariance results to detect assumption violations.**Possible alternatives**if your data or analysis of covariance results indicate assumption violations.

To properly analyze and interpret the
results of *analysis of covariance (ANCOVA)*, you should be familiar with the following terms and
concepts:

- Brownlee, K. A. 1965.
*Statistical Theory and Methodology in Science and Engineering.*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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