678
L. Lyons
Fig. 15.7 Plot of p 0 against p 1 for comparing a data statistic t with two hypotheses H 0 and H 1 ,
whose expected pdf ’s for t are given by two Gaussians of peak separation μ, and of equal
width σ . For a given pair of pdf ’s for t, the allowed values of (p 0 , p 1 ) lie on a curve or straight
line (shown solid in the diagram). The expected density for the data along a curve is such that its
projection along the p 0 -axis (or p 1 -axis) is expected to be uniform for the hypothesis H 0 (or H 1
respectively). As the separation increases, the curves approach the p 0 and p 1 axes. Rejection of
H 0 is for p 0 less than, say, 3 ∗ 10 −7 ; here it is shown as 0.05 for ease of visualisation. Similarly
exclusion of H 1 is shown as p 1 < 0.1. Thus the (p 0 , p 1 ) square is divided into four regions: the
largest rectangle is when there is no decision, the long one above the p 0 -axis is for exclusion of H 1 ,
the high one beside the p 1 -axis is for rejection of H 0 , and the smallest rectangle is when the data
lie between the two pdf ’s. For = 3.33, there are no values of (p 0 , p 1 ) in the “no decision”
region. In the CL s procedure, rejection of H 1 is when the t statistic is such that (p 0 , p 1 ) lies below
the diagonal dotted straight line
15.7.4 Comparing Two Hypotheses Via χ 2
Assume that there is a histogram with 100 bins, and that a χ 2 method is being used
for fitting it with a function with one free parameter. The expected value of χ 2 is
99 ± 14. Thus if p 0 , the best value of the parameter, yields a χ 2 of 85, this would be
regarded as very satisfactory. However, a theoretical colleague has a model which
predicts that the parameter should have a different value p 1 , and wants to know what
the data have to say about that. This is tested by calculating the χ 2 for that p 1 , which
yields a value of 110. There appear to be two contradictory conclusions:
• p 1 is satisfactory: This is based on the fact that the relevant χ 2 of 110 is well
within the expected range of 99 ± 14.
L. Lyons
Fig. 15.7 Plot of p 0 against p 1 for comparing a data statistic t with two hypotheses H 0 and H 1 ,
whose expected pdf ’s for t are given by two Gaussians of peak separation μ, and of equal
width σ . For a given pair of pdf ’s for t, the allowed values of (p 0 , p 1 ) lie on a curve or straight
line (shown solid in the diagram). The expected density for the data along a curve is such that its
projection along the p 0 -axis (or p 1 -axis) is expected to be uniform for the hypothesis H 0 (or H 1
respectively). As the separation increases, the curves approach the p 0 and p 1 axes. Rejection of
H 0 is for p 0 less than, say, 3 ∗ 10 −7 ; here it is shown as 0.05 for ease of visualisation. Similarly
exclusion of H 1 is shown as p 1 < 0.1. Thus the (p 0 , p 1 ) square is divided into four regions: the
largest rectangle is when there is no decision, the long one above the p 0 -axis is for exclusion of H 1 ,
the high one beside the p 1 -axis is for rejection of H 0 , and the smallest rectangle is when the data
lie between the two pdf ’s. For = 3.33, there are no values of (p 0 , p 1 ) in the “no decision”
region. In the CL s procedure, rejection of H 1 is when the t statistic is such that (p 0 , p 1 ) lies below
the diagonal dotted straight line
15.7.4 Comparing Two Hypotheses Via χ 2
Assume that there is a histogram with 100 bins, and that a χ 2 method is being used
for fitting it with a function with one free parameter. The expected value of χ 2 is
99 ± 14. Thus if p 0 , the best value of the parameter, yields a χ 2 of 85, this would be
regarded as very satisfactory. However, a theoretical colleague has a model which
predicts that the parameter should have a different value p 1 , and wants to know what
the data have to say about that. This is tested by calculating the χ 2 for that p 1 , which
yields a value of 110. There appear to be two contradictory conclusions:
• p 1 is satisfactory: This is based on the fact that the relevant χ 2 of 110 is well
within the expected range of 99 ± 14.