# Interaction in ANCOVA

I'm doing a study with frogs in which I see that there are differences in jumping ability between juveniles and adults, I get that adults jump more than juveniles.

I want to check if these differences are really due to the fact that there is an effect of the age class or if they are due to the effect of the size of the individuals (SVL), since the smaller individuals jump less than the adults.

or this I have tried an ANCOVA in which I have --> jump ~ Age class * SVL

The interaction--> Age class: SVL is significant.

My question is this:

Does this indicate that the differences in jumping performance between juveniles and adults are due to the effect of size and not really for the effect of being juvenile or adult? or this indicates that SVL has not effect on jumping performance and this differences are due to a real effect of the age class?

Thanks a lot :D

• how to measure jumping ability, the height of the jump or frequency of jumps in given time period (1 min)? – user158565 Oct 31 '18 at 20:31
• Measuring The distante that the frog archived marking the initial position of the frog before jump and the final position after land. – Mr frog Oct 31 '18 at 23:12
• Please register &/or merge your accounts (you can find information on how to do this in the My Account section of our help center), then you will be able to edit & comment on your own question. – gung - Reinstate Monica Nov 1 '18 at 0:35
• Sorry for the inconvenient, I maded this question without create an account because I did no have one , then I recieved a mail which suggested me to verify my account, so I created one. Sorry again for the inconvenient – Mr.Frog Nov 2 '18 at 15:54

Let $$Y$$ be distance of frog jump, $$X_1$$ be SVL, and $$X_2$$ be the age group with 1 meaning juvenile, and 0 for adult.

The complicated-est model is: $$Y=\beta_0 +\beta_1 X_1 +\beta_2 X_2 + \beta_3X_1X_2+\epsilon$$
If $$\beta_3$$ is not zero, i.e., interaction between SVL and age group exists, we fit two straight lines for two age groups in fact. For adults, the model is
$$Y=\beta_0 +\beta_1 X_1 +\epsilon$$ For juvenile, the simple linear regression for juvenile is $$Y=(\beta_0 +\beta_2) +(\beta_1+\beta_3)X_1+\epsilon$$
Under this situation, these is no common slope and no common intercept, so it is meaningless to talk about common slope and common intercept which were called main effects. We need to check the regression lines separatedly.

If interaction between SVL and age group does no exist, the model becomes $$Y=\beta_0 +\beta_1 X_1 +\beta_2 X_2+\epsilon$$
It means the effect of age does not depend on SVL. Regardless what SVL is, the effect of age group is $$\beta_2$$. Similarly, the effect of SVL is the same among two age groups, i.e., SVL increase 1 unit, the jump distance will increase by $$\beta_1$$ unit.

It is easy to explain other situations, such as $$\beta_2=0$$, $$\beta_1=0$$,....

Let discuss the situation that interaction between age and SVL exists specifically.

At first, we need to say that both age and SVL have effect on jump distance. Secondly, the effect of SVL is not the same between age groups, and the difference of jump distance between age groups depends on the SVL level.

Let $$\beta_0 =1, \beta_1 =2, \beta_2 =3, \beta_3 =4$$. Then we have two lines describe as $$Y= 1 + 2X_1+\epsilon \text { for adults}$$
$$Y= 4 + 6X_1+\epsilon \text { for juvenile }$$

where $$X_1$$ is SVL. It means when SVL increase by 1 unit, adult will jump 2 unit far and juvenile will just 6 unit far. When SVL = 1, adult jumps 3 units and juvenile jumps 10 units, and the difference between age groups is 10 - 3 = 7 units. If SVL =2, adult jumps 5 units and juvenile jumps 16 units, and the difference between age groups is 16 - 5 = 11 units. So the effect of SVL depends age group, and effect of age group depends on SVL values.

• Thank you very much for your answer. Then, if I understood correctly the effect that Age has on Jumping is due to the influence of SVL due to their interaction is significant, not? – Mr.Frog Nov 2 '18 at 15:58
• Maybe we should talk about one model at first. Which one you want to talk, the model with interaction? – user158565 Nov 2 '18 at 18:03
• Yes, I'm talking about the model with interaction – Mr.Frog Nov 2 '18 at 18:48
• See last section in the Answer i just added for explanation of interaction. Is it clear enough? – user158565 Nov 2 '18 at 19:23