I would like to formalise some operations I am doing, however it is unclear how I should deal with categorical variables.
Imagine a dataset with 15 distinct couples (ID). Each couple was observed 3 times (time).
Each partner has responded to two questions: p and act. p is a dummy vector {0, 1} while act is a categorical variable with 4 levels {a,b,c,d}. _m refers to partner male and _w to partner female.
ID time p_m p_w act_m act_w
1 A 1 1 1 c b
2 A 2 1 1 b c
3 A 3 1 1 c d
4 B 1 1 1 b b
5 B 2 0 1 a a
6 B 3 1 1 b b
7 C 1 1 1 b b
8 C 2 1 1 c c
9 C 3 1 1 c b
10 D 1 1 1 c b
11 D 2 1 0 b a
12 D 3 1 1 c b
13 E 1 1 1 d d
14 E 2 1 1 b c
15 E 3 1 1 c c
First, I am interested in formalising the matches on p. Because p is a dummy variable, it seems that I can simply write:
$joint_{jt} = (p_{jt}^{m} \times p_{jt}^{w})$
where $t$ denotes time, $p_{jt}^{m}$ denotes partner $m$ response at time $t$ belonging to couple number $j$ (and vice versa for partner $w$).
ID time p_m p_w act_m act_w joint_j
1 A 1 1 1 c b 1
2 A 2 1 1 b c 1
3 A 3 1 1 c d 1
4 B 1 1 1 b b 1
5 B 2 0 1 a a 0
6 B 3 1 1 b b 1
7 C 1 1 1 b b 1
8 C 2 1 1 c c 1
9 C 3 1 1 c b 1
10 D 1 1 1 c b 1
11 D 2 1 0 b a 0
12 D 3 1 1 c b 1
13 E 1 1 1 d d 1
14 E 2 1 1 b c 1
15 E 3 1 1 c c 1
However, it is unclear to me if I can use the $\times$ operator for categorical variables. Basically, what I am interested in is when $joint_{jt} = 1$ and $p_m == p_w$, then 1.
My question is, how do you formalise $p_m == p_w$, when $p$ is categorical?
I have been told that I could not use logical operators in my papers (econ, sociology field), but that I should use the arithmetic operators. So, how do you express TRUE/FALSE mathematically?
Now I have
$jointact_{jt} = (joint_{jt} \times act_{jt}^{m} \times act_{jt}^{w})$
But this seems wrong to me.
Could I for instance do, first define a vector $act$ if $act_{jt}^{m} = act_{jt}^{j}$ is true
$$ act_{jt} = \left\{\begin{array}{ll} 1 & act_{jt}^{m} = act_{jt}^{w}\\ 0 & otherwise \end{array}\right. $$
and then combine $act_{jt}$, with $join_{jt}$, like
$jointact_{jt} = (joint_{jt} \times act_{jt})$
ID time p_m p_w act_m act_w join_j joinact_j
1 A 1 1 1 c b 1 0
2 A 2 1 1 b c 1 0
3 A 3 1 1 c d 1 0
4 B 1 1 1 b b 1 1
5 B 2 0 1 a a 0 0
6 B 3 1 1 b b 1 1
7 C 1 1 1 b b 1 1
8 C 2 1 1 c c 1 1
9 C 3 1 1 c b 1 0
10 D 1 1 1 c b 1 0
11 D 2 1 0 b a 0 0
12 D 3 1 1 c b 1 0
13 E 1 1 1 d d 1 1
14 E 2 1 1 b c 1 0
15 E 3 1 1 c c 1 1
Not being able to use logical operators does not make sense. They are hard to avoid (your suggestion for $act_{jt}$ uses them to define cases) and their use makes the paper easier to read.
Fortunately, there is a notationally convenient alternative using orthogonality. Let $\langle \cdot, \cdot \rangle$ be an inner product and let $\{a,b,c,d\}$ be an orthonormal basis. Then, by definition, $\langle a, a \rangle = \langle b, b \rangle = \langle c, c \rangle = \langle d,d \rangle = 1$, while other inner products such as $\langle a,b \rangle$ are $0$. Using $joint_j = p_j^m \cdot p_j^w$, you can write: $$jointact_j = joint_j \langle act_j^m, act_j^w \rangle$$