# Use of three way Anova or Ancova in R?

I doing a three way Anova (or Ancova - I'm not sure, hence my question) in R. I am testing how temperature, the development stage and the size of a carcass affect the development rate of maggots.

My response variable is Duration (a measurement of hours) and my factors are Size (2 levels = small and large), and Stage (7 levels = eggs, 1st instar, 2nd instar, 3rd instar, postfeed, pupa and total) and Temperature (4 levels = 15, 20, 25, 30).

I intend to examine how Duration varies with Temperature and Size and Stage.

I imported my data set (AnovaTWD). One function I have tried is:

model1 <-(aov(Duration~Stage*Temperature*Size, AnovaTWD))
summary(model1)


This gave me:

                        Df  Sum Sq Mean Sq F value   Pr(>F)
Stage                    6 7206782 1201130 149.059  < 2e-16 ***
Temperature              1 1924926 1924926 238.881  < 2e-16 ***
Size                     1   78491   78491   9.741  0.00205 **
Stage:Temperature        6 2500293  416716  51.714  < 2e-16 ***
Stage:Size               6  120539   20090   2.493  0.02367 *
Temperature:Size         1  140090  140090  17.385 4.43e-05 ***
Stage:Temperature:Size   6  184679   30780   3.820  0.00122 **
Residuals              214 1724431    8058
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


Am I right in thinking that R is recognising Temperature as a continuous variable here?

Another function I have tried is:

AnovaTWD$Temperature <- factor(AnovaTWD$Temperature)
model2 <-(aov(Duration~Stage*Temperature*Size, AnovaTWD))
summary(model2)


This gave me:

                        Df  Sum Sq Mean Sq F value   Pr(>F)
Stage                    6 7206782 1201130 527.084  < 2e-16 ***
Temperature              3 2399926  799975 351.048  < 2e-16 ***
Size                     1  116577  116577  51.157 1.90e-11 ***
Stage:Temperature       18 3068185  170455  74.800  < 2e-16 ***
Stage:Size               6  157139   26190  11.493 6.34e-11 ***
Temperature:Size         3  214981   71660  31.446  < 2e-16 ***
Stage:Temperature:Size  18  292783   16266   7.138 1.10e-13 ***
Residuals              186  423861    2279
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


I know that R is treating Temperature as a fixed factor now. However, I have been advised to treat Temperature as a covariate. I read "ANCOVA is easily reached using the aov() function using the syntax + variable name to indicate that the predictor variable is a covariate."

So I tried:

 summary(aov(Duration~Size*Stage+Temperature, AnovaTWD))


This gave me:

Df  Sum Sq Mean Sq F value Pr(>F)
Size          1   87904   87904   4.928 0.0274 *
Stage         6 7223925 1203988  67.497 <2e-16 ***
Temperature   3 2411455  803818  45.063 <2e-16 ***
Size:Stage    6  143505   23917   1.341 0.2400
Residuals   225 4013443   17838
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


Is this now ANCOVA and treating Temperature as a covariate? Or is it not testing an interaction with the other factors?

I could simply treat Temperature as a fixed variable as my experiment design used the 4 constant temperatures (15, 20, 25, 30). However, I know this approach is not giving all the information possible, hence I am looking into ANCOVA as it is usually used when there are both continuous and categorical predictors.

The advice you were given - use '+ variable name to indicate that the predictor variable is a covariate - is not correct. + means that you are including the variable as an additive effect. In your examples, it means you are saying you do not want to interact Temperature with the other variables. The change between continous versus categorical treatment of the variable is determined by whether it is a factor or not.

In your first model Temperature is treating temperature as a continuous variable and specified interactions between all the predictors.

In your second model Temperature is a factor, and you have specified interactions between all the predictors.

In the third model, Temperature is still a factor, but you have specified no interactions between Temperature and the other variables.

ANCOVA just means that one of the variables is not a factor.

• Thank you for your answer, it was very clear. Just to add to this, does this mean that Temperature is a 'covariate' in the first model as it's being treated as continuous? Thank you Feb 16 '17 at 10:50
• Yes, that's right.
– Tim
Feb 16 '17 at 22:21