ANCOVA Regression Algorithm

An ANCOVA Regression Algorithm is an GLM regression algorithm (with a mixture of continuous random variables and qualitative variables) that fits an ANCOVA Model.

• AKA: Analysis of Covariance, ANCOVA.
• Context:
  • It is an extension of the ANOVA Algorithm
  • It can be used to solve an ANCOVA Statistical Hypothesis Test.
  • It can range from being a One-Way ANCOVA to being a Two-Way Factorial ANCOVA.
  • It can be implemented into an ANCOVA Software System, such as vassarstats.net/vsancova.html
• Example(s):
  • STATA Support ANCOVA examples: http://campusguides.lib.utah.edu/c.php?g=160853&p=1054196
• Counter-Example(s):
  • an ANOVA Algorithm.
  • a Multivariate Linear Regression Algorithm.
  • a Mann-Whitney Test.
• See: Statistical Analysis Algorithm, Experiment Analysis Algorithm.

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References

2019

• (Horn, 2019) ⇒ Robert A. Horn (2019). "Umderstanding Aanalysis of Covariance". "Educational Psychology 625: Intermediate Statistics" Course Handout. Copyright: 2008 Northern Arizona University. Published Online: 2008. Retrieved: 2019-04-12.
  • QUOTE: Analysis of Covariance (ANCOVA) – an extension of ANOVA that provides a way of statistically controlling the (linear) effect of variables one does not want to examine in a study. These extraneous variables are called covariates, or control variables. (Covariates should be measured on an interval or ratio scale.) ANCOVA allows you to remove covariates from the list of possible explanations of variance in the dependent variable. ANCOVA does this by using statistical techniques (such as regression to partial out the effects of covariates) rather than direct experimental methods to control extraneous variables. ANCOVA is used in experimental studies when researchers want to remove the effects of some antecedent variable. For example, pretest scores are used as covariates in pretest-posttest experimental designs. ANCOVA is also used in non-experimental research, such as surveys or non-random samples, or in quasi-experiments when subjects cannot be assigned randomly to control and experimental groups. Although fairly common, the use of ANCOVA for non-experimental research is controversial (Vogt, 1999).

2013a

• (Wikipedia, 2013)http://en.wikipedia.org/wiki/Analysis_of_covariance
  • Covariance is a measure of how much two variables change together and how strong the relationship is between them. Analysis of covariance (ANCOVA) is a general linear model which blends ANOVA and regression. ANCOVA evaluates whether population means of a dependent variable (DV) are equal across levels of a categorical independent variable (IV), while statistically controlling for the effects of other continuous variables that are not of primary interest, known as covariates (CV). Therefore, when performing ANCOVA, we are adjusting the DV means to what they would be if all groups were equal on the CV.

2013b

• (Wikipedia, 2013) ⇒ http://en.wikipedia.org/wiki/Analysis_of_covariance#Assumptions_of_ANCOVA
  • There are five assumptions that underlie the use of ANCOVA and affect interpretation of the results:
    1. The residuals (error terms) should be normally distributed.
       
    2. The error variances should be equal for different treatment classes.
       
    3. The slopes of the different regression lines should be equal.
       
    4. The regression relationship between the dependent variable and concomitant variables must be linear.
    5. The error terms should be uncorrelated.
       
  • The third issue, concerning the homogeneity of different treatment regression slopes is particularly important in evaluating the appropriateness of ANCOVA model. Also note that we only need the error terms to be normally distributed. In fact both the independent variable and the concomitant variables will not be normally distributed in most cases.

2010

• (Seltman, 2010) ⇒ Howard J Seltman. (2010). “Experimental Design and Analysis.” Carnegie Mellon University.
  • QUOTE: An analysis procedure for looking at group effects on a continuous outcome when some other continuous explanatory variable also has an effect on the outcome. …

    The term ANCOVA, analysis of covariance, is commonly used in this setting, although there is some variation in how the term is used. In some sense ANCOVA is a blending of ANOVA and regression.

    The term ANCOVA (analysis of covariance) is used somewhat differently by different analysts and computer programs, but the most common meaning, and the one we will use here, is for a multiple regression analysis in which there is at least one quantitative and one categorical explanatory variable. Usually the categorical variable is a treatment of primary interest, and the quantitative variable is a "control variable" of secondary interest, which is included to improve power (without sacrificing generalizability).

    Consider a particular quantitative outcome and two or more treatments that we are comparing for their effects on the outcome. If we know one or more explanatory variables are suspected to both affect the outcome and to define groups of subjects that are more homogeneous in terms of their outcomes for any treatment, then we know that we can use the blocking principle to increase power. Ignoring the other explanatory variables and performing a simple ANOVA increases [math]\displaystyle{ \sigma^2 }[/math] and makes it harder to detect any real differences in treatment effects.

     ANCOVA extends the idea of blocking to continuous explanatory variables, as long as a simple mathematical relationship (usually linear) holds between the control variable and the outcome.

2008

• (Upton & Cook, 2008) ⇒ Graham Upton, and Ian Cook. (2008). “A Dictionary of Statistics, 2nd edition revised.” Oxford University Press. ISBN:0199541450
  • QUOTE: ANOCOVA (ANCOVA; analysis of covariance): ANOVA with a mixture of continuous random variables and qualitative variables. ANOCOVA models can also be thought of as multiple regression with some dummy variables.

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