> RSS = 3:10 #Right shoe size
> LSS = rnorm(RSS, RSS, 0.1) #Left shoe size - similar to RSS
> cor(LSS, RSS) #correlation ~ 0.99
[1] 0.9994836
> 
> weights = 120 + rnorm(RSS, 10*RSS, 10)
> 
> ##Fit a joint model
> m = lm(weights ~ LSS + RSS)
> 
> ##F-value is very small, but neither LSS or RSS are significant
> summary(m)

Call:
lm(formula = weights ~ LSS + RSS)

Residuals:
      1       2       3       4       5       6       7       8 
 4.8544  4.5254 -3.6333 -7.6402 -0.2467 -3.1997 -5.2665 10.6066 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  104.842      8.169  12.834 5.11e-05 ***
LSS          -14.162     35.447  -0.400    0.706    
RSS           26.305     35.034   0.751    0.487    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 7.296 on 5 degrees of freedom
Multiple R-squared:  0.9599,	Adjusted R-squared:  0.9439 
F-statistic: 59.92 on 2 and 5 DF,  p-value: 0.000321

> 
> ##Fitting RSS or LSS separately gives a significant result. 
> summary(lm(weights ~ LSS))

Call:
lm(formula = weights ~ LSS)

Residuals:
   Min     1Q Median     3Q    Max 
-6.055 -4.930 -2.925  4.886 11.854 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  103.099      7.543   13.67 9.53e-06 ***
LSS           12.440      1.097   11.34 2.81e-05 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 7.026 on 6 degrees of freedom
Multiple R-squared:  0.9554,	Adjusted R-squared:  0.948 
F-statistic: 128.6 on 1 and 6 DF,  p-value: 2.814e-05

>