Precision, with Imprecise Words

This morning, after my course on extreme values, some students did show me a question they got from practicals they were suppose to work on, with undergraduate students :

To be more specific, they wanted some feedback about point B. Now, let’s make it clear : I have no idea what “precision” and “variation” could mean… But let’s try and see if we can get something usefull, that might help to understand the question. In order to illustrate, consider the following regression model,

> plot(cars,pch=19,col="black",cex=.8)
> abline(lm(dist~speed,data=cars),lty=2)

Here is the summary table of the linear regression model

> summary(lm(dist~speed,data=cars))

            Estimate Std. Error t value Pr(>|t|)    
(Intercept) -17.5791     6.7584  -2.601   0.0123 *  
speed         3.9324     0.4155   9.464 1.49e-12 ***

My first idea was that “variation of the X’s” should be related to the “variance” of the explanatory variable. But it is stupid. For instance, If we transform the explanatory variable, say with a multiplicative factor of 100, then the variance of X will be 10,000 times larger. And the regression will be the same

> cars100=cars
> cars100$speed=100*cars$speed
> plot(cars100,pch=19,col="black",cex=.8)
> abline(lm(dist~speed,data=cars100),lty=2)

in the sense that

> summary(lm(dist~speed,data=cars10))

             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -17.57909    6.75844  -2.601   0.0123 *  
speed         0.39324    0.04155   9.464 1.49e-12 ***

And similarly, divide by 100. So, I guess using some affine transformation of the explanatory variable is clearly not the way we should get a variable with more “variability”. Let us try something else. And keep in mind the following quantities,

> var(cars$speed)
[1] 27.95918
> sd(cars$speed)/mean(cars$speed)
[1] 0.3433535

with the variance, and the coefficient of variation. Consider the following modified dataset,

> carsg=cars
> carsg$speed[12]=8
> carsg$speed[23]=25
> carsg$speed[34]=24
> carsg$speed[39]=12

Four values were changed, here. Observe that, somehow, there is more variability

> var(carsg$speed)
[1] 31.84694
> sd(carsg$speed)/mean(carsg$speed)
[1] 0.3640845

But if we consider the output of the regression model, we get

> summary(lm(dist~speed,data=carsg))

            Estimate Std. Error t value Pr(>|t|)    
(Intercept) -18.5681     5.3621  -3.463  0.00113 ** 
speed         3.9708     0.3254  12.201 2.55e-16 ***

It look like we got here a more precision on the slope, with a smaller variance, and a larger Student-t-value. But what if we consider the following transformation,

> carsg=cars
> carsg$speed[11]=5
> carsg$speed[21]=25
> carsg$speed[31]=25
> carsg$speed[50]=7

Again, we have more variability here, on the explanatory variable,

> var(carsg$speed)
[1] 32.9898
> sd(carsg$speed)/mean(carsg$speed)
[1] 0.3754036

But this time,

> summary(lm(dist~speed,data=carsg))

            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -1.5078     8.0498  -0.187    0.852    
speed         2.9077     0.4932   5.896 3.61e-07 ***

the estimator of the slope has more variance, and we have a smaller Student-t-value. So here, if we increase the “variability” of X, we get get… almost anything. The intuition about those two transformations is relatively simple. In the first case, I have moved observations that were far away from the regression line – but in the center of the distribution, and I put them closer to the regression line, but more on the border of the sample (to increase the variance)

(I would not call them outliers since outliers are defined as observations far away from the model, but on Y, not on X). In the second case, I did exactly the opposite.

I am not sure if I understood correctly this sentence. But it looks like it is incorrect. Since there is only one false statement here, I will go for this one. What do you think?