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Say I am building a regression model with Gauss-Markov assumptions for predicting the quality of a movie - either a linear regression for continuous rating or logistic regression for if a customer recommends it. We are only considering movies that Al Pacino, Robert de Niro or Joe Pesci ever starred in. In the model I have the following three indicator functions:

$${X_1}=1\{Al\ Pacino\ is\ in\ it\}$$ $${X_2}=1\{De\ Niro\ is\ in\ it\}$$ $${X_3}=1\{Joe\ Pesci\ is\ in\ it\}$$

Now in addition, I am interested in the total number of actor appearances among the above three. Therefore I add a categorical variables representing the sum of them:

$$X'_4=X_1 + X_2+X_3$$

When I implement this system in R, the whole system (ignoring any constant terms) can only have four degrees of freedom. Therefore, I arbitrarily let:

$$X_4=1\{X_1 + X_2+X_3==1\}$$

I am having difficulty of understanding why exactly the resulting degree of freedom would be five instead of six since, say, $1\{X_1 + X_2+X_3==3\}$ cannot be expressed as a linear combination of $X_1, X_2, X_3$. In addition, in my formulation with $X_1, X_2, X_3, X_4$, what is the reference level?

Example:

    my_df <- tibble(
  x1 = (c(1,0,0,1,0,1,1)), 
  x2 = (c(0,1,0,1,1,0,1)), 
  x3 = (c(0,0,1,0,1,1,1)),
  z = x1 + x2 + x3,
  y = c(1,0,1,1,1,0,1)
)

summary (aov (my_df$y ~ factor(my_df$x1) + factor(my_df$x2) +factor(my_df$x3) + factor(my_df$z) ))

Edit: Thanks to Alex's comment, I realised that I had one constraint that $X_1 + X_2 + X_3 >= 1$, i.e. only considering movies that any one of the three actors starred in. The edit has been added in the original question.

I was able to figure out formally but still not intuitively I think. Formally I will show that the system described above can have only 5 degrees of freedom (df). Suppose we have 6 df instead. Without loss of generality, let: $${X_1}=\{1,0,0,1,1,0,1\}$$ $${X_2}=\{0,1,0,1,0,1,1\}$$ $${X_3}=\{0,0,1,0,1,1,1\}$$ $${Z_1}=\{1,1,1,0,0,0,0\}$$ $${Z_2}=\{0,0,0,1,1,1,0\}$$ $${C_0}=\{1,1,1,1,1,1,1\}$$ Notice $Z_1 = (X_1+X_2+X_3 == 1)$, $Z_2 = (X_1+X_2+X_3 == 2)$, and $C_0$ is a constant. Further, we can express $$Z_2 = 3C_0 - (X_1+X_2+X_3) -2Z_1$$ The system is singular. Therefore it cannot have six df.

In spite of that, I still have trouble intuitively understand what the reference level would be if, say, I am using $X_1, X_2, X_3, Z_1, C_0$.

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You don't have enough distinct levels of z in your dataframe (z only takes the values 1,2 and 3). If you add a new row with x1=x2=x3=0 (hence z=0) you get 5 df.

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