# Analysis of varying scenarios in least-squares regression?

First question: Is there a name for the type of analysis that is described below? (the second question is italicized in sentence below).

I have a dependent variable $$y$$ that is related to the independent variable $$x$$ according to $$\tag{1} y=kx$$ where $$k$$ is a proportionality coefficient linking $$x$$ to $$y$$. Variables $$x$$ and $$y$$ are measured variables. Due to measurement uncertainty, it is best practice to measure multiple values of $$y$$ at varying values of $$x$$ and then apply a least-squares regression to find the best fit line, and hence the best estimate of $$k$$. Suppose we make measurements at 10 different values of $$x$$, and thus have 10 $$(x,y)$$ pairs.

Then one might ask, how might the estimate of $$k$$ vary if all the different possible regression scenarios of these 10 data pairs were evaluated, and what would (or could) the variation in $$k$$ tell us about the data?

E.g., in one scenario (say) the pairs $$(x_i, y_i), i=1,2$$ are used to evaluate $$k$$. In another scenario the pairs $$(x_1, y_i), i=3,4,8,10$$ are used to evaluate $$k$$, and so on until all the different possible scenarios of the 10 data pairs are realized.

• Presumably "measurement uncertainty" applies both to $y$ and $x,$ in which case this approach will not obtain the "best estimate of $k.$" Regardless, what exactly is a "regression scenario"? For instance, would regression using pairs $(x_i,y_i)$ with $i=7,1,9,2,4,2,2,5,8,8$ constitute a "regression scenario"? (This is an example of a bootstrap iteration; it is extraordinarily useful for addressing questions like yours.)
– whuber
Commented Jan 6, 2020 at 21:55
• @whuber yes, uncertainty applies to both $y$ and $x$ since they are measured. The answer to your question is "yes", however I was not considering the use of a data pair in a particular scenario more than once. That is, in your example you used the pairs $i=2, 8$ more than once, and in for my inquiry this would not be done. Commented Jan 6, 2020 at 22:32
• That's an important issue, because limiting the resampling to subsets of the original data produces more variation in the estimates than is inherent in the original procedure (of using all the data). When subsets are used, some adjustment needs to be made to the estimates. One example of this is the Jackknife in which all the subsets are as large as possible (without equaling the entire dataset).
– whuber
Commented Jan 6, 2020 at 23:45