2 Datapoints and a (1-Parameter) Model:
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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? |
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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'
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