[R] Difference between two-way ANOVA and (two-way) ANCOVA

peter dalgaard pdalgd at gmail.com
Wed Jul 4 15:53:21 CEST 2012

On Jul 4, 2012, at 15:20 , syrvn wrote:

> Hi!
> as my subject says I am struggling with the different of a two-way ANOVA and
> a (two-way) ANCOVA.
> I found the following examples from this webpage:
> http://www.statmethods.net/stats/anova.html
> # One Way Anova (Completely Randomized Design)
> fit <- aov(y ~ A, data=mydataframe)
> # Randomized Block Design (B is the blocking factor) 
> fit <- aov(y ~ A + B, data=mydataframe)
> # Two Way Factorial Design 
> fit <- aov(y ~ A + B + A:B, data=mydataframe)
> fit <- aov(y ~ A*B, data=mydataframe) # same thing
> # Analysis of Covariance 
> fit <- aov(y ~ A + x, data=mydataframe)
> I) The 1. example is pretty clear. A simple on way ANOVA. 
> II) Is it correct to say that example 2. (which is called a Randomized Block
> Design) is a two way ANOVA? 
> III) Example 3 is like example 2. (in case I was right in II) )  a two way
> ANOVA but including an interaction term. That's why
> they call it here a Factorial Design.
> So far so good.
> IV) For me, the ANCOVA (last example) looks like a two-way ANOVA. So in what
> way is the variable "x" different to variable "B" so that it is called an
> ANCOVA and not an ANOVA??? I presume that from the type of data R knows
> whether to perform an ANCOVA or an ANOVA.
> V) Is it right to say that the ANCOVA example is a two-way ANCOVA? Or can a
> one-way ANCOVA actually exists?
> You see I am a bit confused especially how R distinguishes between the
> ANCOVA and the two-way ANOVA?
> I hope to find some useful answers here.

Well, it's not really about R, is it?

Anyways, I'd call  y~A+x a ONE-way ANCOVA, because it deals with the covariation of two variables (y and x) in a one-way layout. In the traditional applications, x is often independent of A (pre-randomization measurement like soil quality, etc.) so that the group means of y can be estimated as the value of the regression at the grand mean of x ("adjusted means"), and the mean difference between two groups is the vertical difference between the parallel regression lines.

> Cheers!
> --
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Peter Dalgaard, Professor
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com

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