## Chi square: Goodness of fit and test of independence

Ok, so Chi square GOF tests a variable for its fit to a theoretical proportion. Yet, a Chi test of independence does this, for two variables, with the proportions coming from the set of data.

For example, take the following matrix observation and expected

x y z total
a 10 5 20 35
b 20 25 30 75
c 30 15 20 60
expected
12 9 14
25.71428571 19.28571429 30
22.28571429 16.71428571 26

this would result in 4 dof, Chi = 11.48 overall therefore significant at < .05

But, why can one not simply analyse the columns separately, given the total column as the theoretical proportion?

The estimated values come out exactly the same for each value, which makes it much easier to pin down which thing is the cause of significance.

In the above example, a chi square of independence is significant, with the largest individual chi values (c,x) and (a,z). If analysing each column separately, with the proportions given as suggested and 2 dof, then no column comes out as significant.

I understand that these have different usages in practice, but I am trying to work out whether doing this is valid or not.