Analysis of Covariance (ANCOVA) Comparing simple linear regression lines
Analysis of covariance (ANCOVA) combines features of simple linear regression with one-way analysis of variance Both a quantitative variable X and an ANOVA grouping variable are used to describe the measurement (Y) variable.
Simple linear regression fits a straight line to X-Y data.
One-way analysis of variance fits a mean to each group.
One-way analysis of covariance fits a straight line to each group of X-Y data, such that the slopes of the lines are all equal.
This fitted model may then be used to test the null hypotheses:
- 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.
To properly analyze and interpret the results of analysis of covariance (ANCOVA), you should be familiar with the following terms and concepts:
Failure to understand and properly apply analysis of covariance (ANCOVA) 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.
- 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.