is that now I get a different set of coefficients, one for each spray level.

So, it includes A, B, C, D, E and F.

It hasn't dropped any levels.

And it can do that now because it has six parameters, and six means to work with.

And these are exactly equal to the means for each spray in the data.

So, if I were to just go ahead and calculate the means for each spray, right?

It works out to be the same numbers.

14.5, 15.3, 2.08 in both, and so on.

Now, I want to emphasize this model is

no different than my model that included an intercept.

Why don't I go back to my model with my intercept just to illustrate this.

So, now it's just that the coefficients have a different interpretation.

Now, the intercept from the model, when I fit count as spray but

included the intercept, my intercept now is interpreted, 14.5,

as the mean for spray A.

And you can see that it's exactly the empirical mean for

spray A when I calculate the mean.

It works out that way.

And then, spray B, we talked about earlier,

was the comparison between the reference level spray A and spray B.

Okay? So, if I add these together,

14.5 and 0.833, I should get the mean for spray B.

Okay, and that's what you see, 14.5 plus 0.833, that gets me 15.33, and so on.

So, If I add 14.5 and minus 12, I'm gonna get 2.08.

If I add 14.5 and negative 9, I get 4.9, and so on.

So, this model, where I've included an intercept,

has all the same information as the model where I omitted the intercept,

the only difference is how the coefficients are interpreted.

In the model with the intercept now the intercept is interpreted as

the sprayA mean and all the coefficients are interpreted

as relative to sprayA differences from sprayA.

And then if I would have fit it without the intercept then I get the mean for

each spray.

And if I want differences then I have to subtract the coefficients.