# How do you determine the effect of a simple predictor variable after a PLS analysis?

So, I am running PLS on a genetic dataset with phenotypic and genotypic information. I have about 1000 binary predictors (X), representing molecular markers, for each individual. My indicator variables (Y) are yield in pounds per acre for each individual.

I am predicting yield (Y) using molecular markers (X) with about 3 latent variables. I have made the predictions and am satisfied with the model's ability to predict based on genotype.

What I want to know is: How do I determine the effect that each marker is having on the Y prediction within each latent variable? Preferably in units of Y (lbs per acre).

I should add that I know that there are more appropriate models to model the effects of individual markers, but I am primarily interested in comparing this model with other models, as well as knowing a few predicted marker effects for explanatory purposes.

## 1 Answer

You can look at the PLS' coefficients (the β of $\hat Y = β (X - \bar X)$, not the loadings), but take into account the meaning of the center $\bar X$ of the model: the PLS models influence of deviations from this center.

If you took the column averages of your data matrix: does this $\bar X$ have any meaning? If not, you may want to build another PLS model where $\bar X$ has an interpretible meaning

• e.g. the X average pattern for wild type
• all-zero (i.e. PLS wihtout centering X)

You can then check whether this gain in interpretability costs too much performance.

An alternative is probing the markers singly. First, predict the outcome for all-zero input. Then have in turn each marker being the only one set 1 and look how much the predicted yield changes.

Side note: if your input were continuous, you'd also need to take into account the spread of inputs for each column.

• Thanks for your help. I have done the all-zero option (not centering X) because I have the predictors as a 0/1 presence/absence matrix. If I have done this, then my $\Beta$ should directly correspond to my predicted values, correct? – Nick Adams Nov 19 '12 at 16:18
• @NickAdams: yes. But I'm a little bit a suspicious person in such questions, so I'd actually have a quick look whether they are the same... – cbeleites unhappy with SX Nov 19 '12 at 17:19