Dear members of the R-SIG-ME list,


I am new to linear models and struggling with interpreting some of the R output but hope to get some advice from here.

I created the following dummy data set:

scores <- c(2,6,10,12,14,20)

weight <- c(60,70,80,75,80,85)

height <- c(180,180,190,180,180,180)

The scores of a game/match should be dependent on the weight of the player but not on the height. 

For me the output of the following two linear models make sense:

> (lm1 <- summary(lm(scores ~ weight)))

Call:
lm(formula = scores ~ weight)

Residuals:
       1        2        3        4        5        6 
 1.08333 -1.41667 -3.91667  1.33333  0.08333  2.83333 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept) -38.0833    10.0394  -3.793  0.01921 * 
weight        0.6500     0.1331   4.885  0.00813 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

Residual standard error: 2.661 on 4 degrees of freedom
Multiple R-squared: 0.8564,	Adjusted R-squared: 0.8205 
F-statistic: 23.86 on 1 and 4 DF,  p-value: 0.008134 

> 
> (lm2 <- summary(lm(scores ~ height)))

Call:
lm(formula = scores ~ height)

Residuals:
         1          2          3          4          5          6 
-8.800e+00 -4.800e+00  1.377e-14  1.200e+00  3.200e+00  9.200e+00 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept)  25.2000   139.6175   0.180    0.866
height       -0.0800     0.7684  -0.104    0.922

Residual standard error: 7.014 on 4 degrees of freedom
Multiple R-squared: 0.002703,	Adjusted R-squared: -0.2466 
F-statistic: 0.01084 on 1 and 4 DF,  p-value: 0.9221 

The p-value of the first output is 0.008134 which makes sense as scores and weight have a high correlation

and therefore, the scores "can be explained" by the explanatory variable/factor weight very well. Hence, the R-squared

value is close to 1. For the second example it also makes sense that the p-value is almost 1 (p=0.9221) as there is

hardly any correlation between scores and height.

What is not clear to me is shown in my 3rd linear model which includes both weight and height.

> (lm3 <- summary(lm(scores ~ weight + height)))

Call:
lm(formula = scores ~ weight + height)

Residuals:
         1          2          3          4          5          6 
 1.189e+00 -1.946e+00 -2.165e-15  4.865e-01 -1.081e+00  1.351e+00 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept) 49.45946   33.50261   1.476  0.23635   
weight       0.71351    0.08716   8.186  0.00381 **
height      -0.50811    0.19096  -2.661  0.07628 . 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

Residual standard error: 1.677 on 3 degrees of freedom
Multiple R-squared: 0.9573,	Adjusted R-squared: 0.9288 
F-statistic:  33.6 on 2 and 3 DF,  p-value: 0.008833 

It makes sense that the R-squared value is higher when one adds both explanatory variables/factors to the linear model as 

the more variables are added the more variance is explained and therefore the fit of the model will be better. However, I do NOT

understand why the p-value of height (Pr(> | t |)  = 0.07628) is now almost significant? And also, I do NOT understand why the overall

p-value of 0.008833 is less significant as compared to the one from model lm1 which was p-value: 0.008134.

The p-value of weight being low (p=0.00381) makes sense as this factor "explains" the scores very well.



After fitting the 3 models (lm1, lm2 and lm3) I wanted to compare model lm1 with lm3 using the anova function to check whether the factor height

significantly improves the model. In other words I wanted to check if adding height to the model helps explaining the scores of the players.

The output of the anova looks as follows:

> lm1 <- lm(scores ~ weight)
> 
> lm2 <- lm(scores ~ weight + height)
> 
> anova(lm1,lm2)
Analysis of Variance Table

Model 1: scores ~ weight
Model 2: scores ~ weight + height
  Res.Df     RSS Df Sum of Sq      F  Pr(>F)  
1      4 28.3333                              
2      3  8.4324  1    19.901 7.0801 0.07628 .
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

In my opinion the p-value should be almost 1 and not close to significance (0.07) as we have seen from model lm2

height does not at all "explain" the scores. Here, I thought that a significant p-value means that the factor height adds

significant value to the model.


I would be very grateful if anyone could help me in interpreting the R output.

Best regards

 







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