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I wish to explore the effects of the component percentages (independent variables) on fruit sweetness $S$ (dependent variable). Assume for simplicity that our fruit only contains 3 components, respectively taking up $A$, $B$, and $C$ of the total weight. Thus, we have $$A +B+C=1.$$

Now, I apply multiple linear regression $$S=\beta_0+\beta_1A+\beta_2B+\beta_3C$$ and am interested in if the 3 components as a whole have effects on the sweetness at all.

In this case, knowing any two gives me the third component. Should I include only two components in the regression? Any two will do?

How can I derive the overall $p$-value? MATLAB returns the $p$-values for each regressor. Can I derive the overall $p$ from them?

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You have a couple of choices:

(i) you could eliminate the constant, or

(ii)( you could eliminate one of the components (in which case the intercept would represent the estimated sweetness of 100% of whichever was the eliminated component).

Typically, the second is the better choice.

(Well, there are other choices, but these would be the most obvious ones.)

Matlab will give you the overall p-value. From here:

 [b,bint,r,rint,stats] = regress(y,X) returns a 1-by-4 vector stats that contains, 
 in order, the R2 statistic, the F statistic and its p value, and an estimate of 
 the error variance.
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  • $\begingroup$ Thanks. I notice that the overall $p$-value from regress is always (in my case) quite small. I am wondering if it is due to the fact that I always have a near-to-zero $p$-value for the constant term's coefficient? In other words, is the overall $p$-value "dragged down" by the very small $p$-value for the constant term's coefficient? $\endgroup$
    – Sibbs Gambling
    Commented Mar 11, 2015 at 5:58
  • $\begingroup$ Hi Sir, may I ask a follow-up? What can you give me some pointer on how to test if $\beta_1$, $\beta_2$, and $\beta_3$ are pairwise different? It confuses me when I wish to have 3 pairwise test, but only 2 of them is in the linear model. $\endgroup$
    – Sibbs Gambling
    Commented Mar 31, 2015 at 5:29
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    $\begingroup$ (i) overall p-value is not related to the p-value for the constant. (ii) There are a variety of ways to do it, but it's too long for a comment; you should probably ask a new question. $\endgroup$
    – Glen_b
    Commented Mar 31, 2015 at 6:22
  • $\begingroup$ Thanks for the advice! I would greatly appreciate if you could help me with that too. No pressure though. Thank you! :-) stats.stackexchange.com/questions/144177/… $\endgroup$
    – Sibbs Gambling
    Commented Mar 31, 2015 at 6:41

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