Using statistics to determine a trigger/threshold value between continuous variables

Question

I'm working with a few continuous variables, as below:

# A tibble: 6 x 3
     pH `Log Chl-a` `Log Toxin`
  <dbl>       <dbl>       <dbl>
1  8.1        1.30         3.15
2  8.36       0.968        3.30
3  8.03       1.21         3.90
4  7.86       1.10         2.79
5  7.81       1.19         3.30
6  8.04       1.46         3.41

I know from literature there is relationship between pH and chl-a, and both would be related to toxic algae levels in a water body.

I want to establish a trigger using these variables. So, if chl-a and/or pH goes above a certain level, I can be sure that toxin levels in the water body are elevated.

Unfortunately, after looking at regression between pH vs toxin and chl-a vs toxin, the relationship looks relatively poor (R=0.26 and 0.29 respectively).

My question is, how could I set up a threshold/trigger of either chl-a, pH, or the combination of the two, to predict high toxic algae? Will regression help yield this trigger value?

Thanks!

Answer 1

One method is to learn the "threshold-response" function $$ v \mapsto E_WE[Y|A \geq v, W]$$ where $Y$ is your outcome, $A$ is your continuous variable of interest, and $W$ are variables to adjust for. You could plot this as a function of the threshold $v$ and see if there is any natural threshold or choose a threshold that leads to a sufficiently high average outcome.

An easy way to implement this is to estimate $E[Y|A \geq v, W]$ for each $v$ separately using for instance linear regression or generalized additive models. Then average the predictions across all observations to get your estimate. Specifically, perform the regression of $Y$ on $W$ using only observations with $A\geq v$ for each threshold $v$. If inference is wanted, there are ways to get this as well.

If you have multiple thresholds for different continuous variables, you can also estimate $$ (v_1,v_2) \mapsto E_WE[Y|A_1 \geq v_1, A_2 \geq v_2, W]$$ for a grid of thresholds $v_1,v_2$ and then visualize the estimates with a 2d plot.

It may also be worth looking at logic regression or decision trees which allow for adaptive learning of thresholds.