How to find the most sensitive parameter to an output?

Question

I have 4 different parameters that can affect my output. I am running a linear regression model to find out which of these parameters have the largest effect on the output. I have the following sample data:

> print(S.E.S)
      Output      Deck      Dia   Girder  Bearing
1  0.7130376 1.1700306 4693.794 6756.606 3405.554
2  0.7882629 1.0427384 3879.201 7084.479 3420.701
3  0.7922394 0.8064505 4012.528 7662.097 3397.028
4  0.7755186 0.9507799 4198.280 7965.731 3481.417
5  0.7788124 0.9664673 4446.782 7383.764 3472.230
6  0.7847128 0.6803113 4341.931 6950.527 3432.176
7  0.7785227 0.6720781 4082.003 6599.963 3454.938
8  0.7850641 1.1011206 3663.842 7731.346 3484.941
9  0.7704714 1.5822466 3779.936 7241.310 3382.051
10 0.7855922 1.0402248 3303.753 7429.381 3394.371

I use the following code:

Model_S.E.S <- lm(Output~Bearing+Dia+Girder+Deck, S.E.S)
summary(Model_S.E.S)

> summary(Model_S.E.S)

Call:
lm(formula = Output ~ Bearing + Dia + Girder + Deck, data = S.E.S)

Residuals:
        1         2         3         4         5         6         7         8         9        10 
-0.023443  0.012088  0.003735 -0.006959  0.013808  0.010280 -0.001921 -0.006984  0.014205 -0.014808 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)  
(Intercept)  7.427e-01  5.920e-01   1.254   0.2651  
Bearing      3.731e-05  1.863e-04   0.200   0.8492  
Dia         -3.587e-05  1.650e-05  -2.174   0.0817 .
Girder       1.292e-05  1.601e-05   0.807   0.4565  
Deck        -4.458e-02  2.486e-02  -1.794   0.1328  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.01748 on 5 degrees of freedom
Multiple R-squared:  0.6726,    Adjusted R-squared:  0.4107 
F-statistic: 2.568 on 4 and 5 DF,  p-value: 0.1644

This summary tells me that Deck is the most sensitive parameter as it has the lowest coefficient. But if I just look at lm of 1 parameter at the time I get:

summary(lm(formula = Output~Deck, data = S.E.S))
Call:
lm(formula = Output ~ Deck, data = S.E.S)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.057137 -0.000939  0.006869  0.012354  0.014281 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.80517    0.02940   27.39  3.4e-09 ***
Deck        -0.02991    0.02848   -1.05    0.324    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.02264 on 8 degrees of freedom
Multiple R-squared:  0.1212,    Adjusted R-squared:  0.01137 
F-statistic: 1.104 on 1 and 8 DF,  p-value: 0.3242

If I repeat this for the remaining 3 variables I see that the Deck coefficient is still the largest, but the R-squared of the Dia is the largest. I am not sure what this means.

Answer 1

So if i understand you question correctly. You wonder why the R-squared is the highest for the model with all four parameter? While the "Deck" has the highest beta-value when its the single predictor?

First of all. 10 observations are way to small to get any reliable estimations. Get more data.

Second The R-squared doesn't get penalized for adding predictors, since every new predictor removes unexplained variance. Adjusted R-squared takes this into account. So for comparing models use the latter.

Thirdly, when you add a predictor to a model you can only compare that predictors estimates in consideration to the model. and since in your example the p-value is low non of them matters. but to illustrate a point lets say that "Deck" had a significant influence on y by it self, but looses that in a multiple model. that means that "Deck" is not significant when other predictors are in the model. It can still have a influence by it self, but not when controlled for other predictors. That's why it's estimates can differ between models.

Edit: If you add a confidence interval to your beta-values you will see, since they have high p-values, that they range from between +/-. This since you cant say of they have a positive beta-value or negative.

So in conclusion, get more data. Don't make any decisions on a sample of 10.