Significance test for ordinal data (Kendall Tau)?

I have a litte problem I'm trying to figure out. I have a set of 4 variables, with an output value, and I want to see if these variables have any affect on the output as I we add them. I'll try to explain more in depth:

We'll set the output as Y, which we call a quality measure. The higher the better.

Variable one, x1, is a kind of "layer". Based on the info known about layer 1, we can expect different qualities value. As we add more layers (2,3,4), I'm interested in seing if this additional information provide higher quality measures.

We'll start out with a simple model, where the x is only binary and we have 2 layers.

If we only know he info in the first layer, we have quality 50 for both 0/1. (no matter if layer one is 1 or 0 we expect quality to be 50). As we add another layer, i'm interested to see if these values from layer 2 gives a different expected value. Let's say we get the following results:

y x1 x2

75 1 1

25 1 0

25 0 1

75 0 1

It's obvious that if layer 2 (x2) equals layer 1 (x1), our quality value is higher. This I can prove with a Chi-square test.

Now, my problem is that this is an extreme simplification of my actual problem. I have 4 layers, where each layer can be -3 to 3 (7 integer steps). This gives me 7^4 different combinations, hence I need to find another solution... after browsing around online it seems like I can use Kendall Tau, but haven't been able to make much sense out of it.

Anyone who can point me in the right direction? It's not necessary to evaluate each different combination of all layers, merely if the quality value gets higher as the layer values higher - and if so, which layer that is the most important that provide the higher quality value.

Appreciate all possible help!