Uniform Confidence Bands for a Quadratic Model

Trout, J. Richard; Chow, Bryant

doi:10.1080/00401706.1973.10489088

Cited by 6 publications

(4 citation statements)

References 7 publications

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“…For this example involving quadratic regression,

c_{h}

cannot be computed by using the method of Spurrier () and

c_{c}

cannot be computed by using the method of Trout & Chow () since the required constraint on

X

is not satisfied. On the other hand, for the cornmeal example given in Spurrier (, pp.…”

Section: Numerical Examples In Dose Response Studiesmentioning

confidence: 99%

“…318–319) with two replications at each of the five percentages of sucross and

α = 0.05

, our program gives

c_{h} = 3.3038

(taking less than 2 seconds) which agrees with the entry 3.304 of Spurrier , Table , with

m = 4

k = 2

and

1 - α = 95 %

). For the numerical example given in Trout & Chow (, p. 622) with

(a, b) = (8, 16)

and

α = 0.05

, our Matlab program gives

c_{c} = 1.0923

(taking less than 3 seconds) while Trout and Chow gave

c_{c} = 1.08

using their integration method and

c_{c} = 1.10

from their interpolation. As pointed out in the Introduction, the methods of Spurrier () and Trout & Chow () involve a three‐dimensional integral and so are slower computationally than our method which involves a two‐dimensional integral.…”

Section: Numerical Examples In Dose Response Studiesmentioning

confidence: 99%

“…For the numerical example given in Trout & Chow (, p. 622) with

(a, b) = (8, 16)

and

α = 0.05

, our Matlab program gives

c_{c} = 1.0923

(taking less than 3 seconds) while Trout and Chow gave

c_{c} = 1.08

using their integration method and

c_{c} = 1.10

Section: Numerical Examples In Dose Response Studiesmentioning

confidence: 99%

“…Another frequently quoted simultaneous confidence band is of constant width over

x \in (a, b)

and given by

x^{⊤} b \in x^{⊤} \overset{false^}{bold-italicb} \pm c_{c} \overset{false^}{italicσ} \forall x \in false(a, b false),

where

c_{c}

is a critical constant suitably chosen so that the simultaneous confidence level of the band is equal to

1 - α

. For

k = 2

, Trout & Chow (, ) expressed the simultaneous confidence level of this band as a three‐dimensional integral of a trivariate

t

density and computed the critical constant

c_{c}

only under the restrictive assumption that the sum of the cubed deviations of the observed

x

‐values about its mean is equal to zero.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Exact Simultaneous Confidence Bands for Quadratic and Cubic Polynomial Regression with Applications in Dose Response Study

Liu

Zhou

Bretz

2013

Aus NZ J of Statistics

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Exact simultaneous confidence bands (SCBs) for a polynomial regression model are available only in some special situations. In this paper, simultaneous confidence levels for both hyperbolic and constant width bands for a polynomial function over a given interval are expressed as multidimensional integrals. The dimension of these integrals is equal to the degree of the polynomial. Hence the values can be calculated quickly and accurately via numerical quadrature provided that the degree of the polynomial is small (e.g. 2 or 3). This allows the construction of exact SCBs for quadratic and cubic regression functions over any given interval and for any given design matrix. Quadratic and cubic regressions are frequently used to characterise dose response relationships in addition to many other applications. Comparison between the hyperbolic and constant width bands under both the average width and minimum volume confidence set criteria shows that the constant width band can be much less efficient than the hyperbolic band. For hyperbolic bands, comparison between the exact critical constant and conservative or approximate critical constants indicates that the exact critical constant can be substantially smaller than the conservative or approximate critical constants. Numerical examples from a dose response study are used to illustrate the methods.

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“…For this example involving quadratic regression,