2 Datapoints and a (1-Parameter) Model:


Context
y x
- -
2 1
8 2
One has 2 indep'nt observations from the 'no-intercept' model E[y|x] = B x.
 
The y's might represent the total numbers of typographical errors on x randomly sample pages of a large document, and the data might be y = 2 errors in total in a sample of x = 1 page, and y = 8 errors in total in a separate sample of x = 2 pages. The B in the model represents the mean number of errors per page of the document.

Or, the y's might represent the total weight of x randomly sample pages of a document, and the data might be y = 2 units of weight in total for a sample of x = 1 page, and y = 8 units for a separate sample of x = 2 pages. The B in the model represents the mean weight per page of the document.

We gave this 'estimation of B' problem to several statisticians and epidemiologists, and to several grade 6 students, and they gave us a variety of estimates, such as B-hat = 3.6/page, 3.33/page, and 3.45!

HOW CAN THIS BE?
applet **IF** Flash applet still works: Click on different (y,x) locations to obtain different fitted values given by different B's (different y/x ratios), and keep track of the various 'fit' criteria.
Left: Vertical lines (red): the residuals measured on the y-scale; Blue lines: the Poisson probabilities of obtaining these 2 y values.
Right: red=trace of sum of squares blue=trace of logL

Shiny App R command to run Shiny App that allows user to specify trial parameter values and monitor various fit criteria:

shiny::runUrl('http://www.biostat.mcgill.ca/hanley/2DatapointsAndaModel/App.R.zip')

For the 1 parameter (beta-only) models in the first 2 radio buttons, Click on different locations within the beige rectangle to specify trial beta (slope) values and the corresponding fitted values.

For the 2 parameter (beta-sigma) models in the last 2 radio buttons, click on different locations within the beige rectangle specify beta (slope) values and within the L[beta,sigma] rectangle to specify sigma values. The L[beta,sigma] contours and the profileL[beta] functions will be displayed only after you have tried 15 parameter pairs.



Where do you stand? Notions of the statistical 'centre'


Context
You are delivering a heavy article to a high-rise building which is served by 3 elevators. None of them has an indicator light to show its current location, and there is but one button for all three. Once you have pushed the button, any one of the 3 elevators may respond - with equal probability. You must take whichever one arrives next. These elevators are unequally spaced along one wall. You will need to carry the heavy article to the elevator from wherever along the wall you stand and wait. You can take as long as is required to reach the elevator. Thus the distance to the elevator is your only concern.

USE THIS APP TO HELP YOU OPTIMIZE R command to run Shiny App that allows user to try various locations and monitor various fit criteria:

shiny::runUrl('http://www.biostat.mcgill.ca/hanley/2DatapointsAndaModel/ElevatorApp.R.zip')


Updated: Jan 22, 2020