# Effect of covariates that have been removed from a model

I am fitting a GAM to data of capture success in trapping sites using the mgcv package in R. The independent variables are proportions of land cover types. Here is the output of my model:

Family: Gamma

Formula:
capt_succ ~ s(p_urban, k = -1, bs = "cs") +
s(p_crop, k = -1, bs = "cs") + s(p_forest, k = -1,
bs = "cs") + s(p_wetland, k = -1, bs = "cs") + s(p_grassland,
k = -1, bs = "cs")

Parametric coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  2.55181    0.01436   177.7   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Approximate significance of smooth terms:
edf        Ref.df    F        p-value
s(p_urban)        8.256      9         21.508   6.70e-16 ***
s(p_crop)         7.077      9         8.855    3.22e-07 ***
s(p_forest)       8.523      9         20.536   3.18e-15 ***
s(p_wetland)      8.068      9         5.782    0.000124 ***
s(p_grassland)    8.195      9         8.585    1.12e-06 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

R-sq.(adj) =  0.907   Deviance explained = 97.1%
GCV = 0.040875  Scale est. = 0.011957  n = 58


Multicollinearity was low in this model as the variance inflation factor of each covariate was < 5.

Given that the proportion of wetlands induced autocorrelated residuals, I removed this covariate from the model. However, when I used the model without the proportion of wetlands to do predictions of capture success over large areas, I noted that high values of capture success were present in areas with many wetlands. Is it possible that the model kept the effect of wetlands while this covariate were removed from the model ?

• Possibly, yes; Consider the proportions of sand, silt, clay in a sample, with the sum of these proportions equal to 1. If I remove clay I can still work out it's contribution because that is 1 - (sand + silt). By definition, sites with no wetlands will have proportionally higher amounts of one or more of the other land-use types because the land use must be something other than wetland. The opposite is true too: high wetland % -> low everything else. I presume these proportions don't add to one here otherwise I'm not sure how the model even fitted? – Gavin Simpson Jun 3 '16 at 17:24
• A VIF is not very useful if you think covariates have smooth effects. VIF measures colinearity. Concurvity is the corresponding issue for models with smooth terms. See ?concurvity. I would fit the model with method = "REML" as the concurvity can exacerbate under smoothing with GCV. – Gavin Simpson Jun 3 '16 at 17:26
• Thank you very much Gavin for your advice. The sum of proportions is not equal to 1 in the model. For example, I don't include the proportion of open areas, water bodies, some types of crops, and other types of land cover types (named "Other"). All these land cover types (except "Others") cover a larger proportion of the study area than wetlands. So, I find that it is difficult to work out the contribution of wetlands as easily as clay (i.e., 1-(sand + slit)). – Nell Jun 3 '16 at 20:20
• I have verified the concurvity between smooths concurvity(mod, full=FALSE)\$estimate and all values were < 0.5. – Nell Jun 3 '16 at 20:32