A good fit of the model but the estimates are very low

Question

I'm trying to model soil quality using the soil quality index (SQI) as a function of earthworm abundance and elevation. The SQI is restricted to values between 0 and 1. I used multiple soil indicators to elaborate the index (soil chemical indicators and biological indicators). More details can be found in those articles(Leul et al., 2023; Obriot et al.,2016). The model summary shows a coefficient of determination of 0.93, but the estimates are very low. here are the results of the summary:

Call:
lm(formula = SQI ~ Elevation * Earthworm, data = SQI_test)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.07640 -0.01925  0.01111  0.02691  0.04396 

Coefficients:
                         Estimate Std. Error t value Pr(>|t|)    
(Intercept)             3.584e-01  2.893e-02  12.391 8.36e-08 ***
Elevation               1.772e-04  8.242e-05   2.150  0.05460 .  
Earthworm              -2.774e-04  2.827e-04  -0.981  0.34755    
Elevation:Earthworm     1.909e-06  4.946e-07   3.860  0.00265 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.04102 on 11 degrees of freedom
Multiple R-squared:  0.939, Adjusted R-squared:  0.9223 
F-statistic: 56.41 on 3 and 11 DF,  p-value: 5.75e-07

can I still use a model like this to predict SQI values? The data here represent the averages of three replicates, since there is only one value for soil chemical properties per parameter (composite sample) the averages have been calculated to correspond to the soil chemical data

Answer 1

It seems that the low numeric values of the coefficient estimates from your model are your primary concern. That's not a problem on its own, as the numeric values of those estimates depend on the scale in which the data were coded.

Coding Elevation in meters instead of in kilometers leads to a thousand-fold difference in the numeric values of the coefficient estimates. That carries over into the interaction term involving Elevation. Predictions from models based on meters or kilometers as the scale of Elevation would be the same, however, provided that the scale of Elevation values used for predictions corresponded to that used to build the model.

A substantive potential problem, pointed out in comments on the question, is the linear model used for predicting the soil quality index (SQI) outcome variable, which is limited to values between 0 and 1. Such an outcome variable is generally inappropriate for a simple linear model such as you show.

First, with adequately extreme predictor values, a simple linear model can make predictions outside of the allowed range of [0,1]. That might not be a problem in practice, however.

Second, the distribution of error terms often doesn't meet the assumptions of a standard linear regression. That model might work adequately if observations and predictions are all near the middle of that range. You need at least to check the adequacy of the model with standard quality control measures, like the plots provided by plot.lm() in R.

There are established ways to model outcome variables that are limited to that range. If values of exactly 0 or 1 aren't possible, beta regression is one choice. It's also possible to adapt methods used to model probabilities, which are also restricted to that range. You have to be careful with that approach, however, as those generalized linear models by default assume a binomial variance that probably doesn't apply to your data. This page and this page provide details, with links to more information.

In response to data added to the question

As the SQI values are mostly near the middle of the range between 0 and 1, a simple linear regression might work well enough with SQI data in practice that you can avoid beta regression or other methods designed to work with range-limited values. Problems arise when model estimates come close to (or beyond) the theoretically possible values of an outcome variable like SQI; you certainly shouldn't try to apply such a model outside the range of observed predictor values.

Even if that's the case in general, however, your particular model is probably overfit, as @dipetkov noted in a comment on an earlier version of this question.

A rule of thumb with this type of biological data is that you should at least have about 15 independent observations per coefficient you are estimating. That some of your observations are technical replicates doesn't help with that. With 15 independent observations, you should either restrict your model to a single predictor (e.g., altitude) or use a penalized model like ridge regression.

Your model seems to be heavily influenced by the data from the Sp5 plot of land, which has the highest values of earthworms and of altitude and of SQI in your data set. Although I haven't done the calculation myself, you will probably find very high leverage for that observation when you do quality control plots of your linear model fit. High leverage for a single observation in such a small data set is a warning that something might be wrong.