Comparison of Simultaneous Confidence Bands for Quadratic Regression Over a Finite Interval

Spurrier, John D.

doi:10.1080/00401706.1993.10485327

Cited by 5 publications

(6 citation statements)

References 8 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%

“…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. 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 %

).…”

Section: Numerical Examples In Dose Response Studiesmentioning

confidence: 99%

“…On the other hand, for the cornmeal example given in Spurrier (, pp. 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.…”

Section: Numerical Examples In Dose Response Studiesmentioning

confidence: 99%

“…For the special case of

k = 2

and

(a, b) = (- \infty, \infty)

, Wynn & Bloomfield () utilised geometric intuition to express the simultaneous confidence level as an integral over a three‐dimensional region of a trivariate

t

density following Dunnett & Sobel (), which simplifies to a two‐dimensional integral after a change of variables. Spurrier (, ) expressed the simultaneous confidence level as a three‐dimensional integral of a trivariate

t

density when

k = 2

and

(a, b)

and

X

satisfy certain conditions (cf. condition (17) of Spurrier ).…”

Section: Introductionmentioning

confidence: 99%

“…Spurrier (, ) expressed the simultaneous confidence level as a three‐dimensional integral of a trivariate

t

density when

k = 2

and

(a, b)

and

X

satisfy certain conditions (cf. condition (17) of Spurrier ). Another approach hinges on the results for the volume of a tube about a one‐dimensional manifold in the unit sphere in

{double-struckR}^{k + 1}

, which have been studied by Naiman (), Knowles & Siegmund (), Johnstone & Siegmund (), Johansen & Johnstone (), and Sun & Loader (), among others.…”

Section: Introductionmentioning

confidence: 99%

See 4 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.

show abstract

“…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%

“…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. 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 %

).…”

Section: Numerical Examples In Dose Response Studiesmentioning

confidence: 99%

“…On the other hand, for the cornmeal example given in Spurrier (, pp. 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.…”

Section: Numerical Examples In Dose Response Studiesmentioning

confidence: 99%

“…For the special case of

k = 2

and

(a, b) = (- \infty, \infty)

, Wynn & Bloomfield () utilised geometric intuition to express the simultaneous confidence level as an integral over a three‐dimensional region of a trivariate

t

t

density when

k = 2

and

(a, b)

and

X

satisfy certain conditions (cf. condition (17) of Spurrier ).…”

Section: Introductionmentioning

confidence: 99%

“…Spurrier (, ) expressed the simultaneous confidence level as a three‐dimensional integral of a trivariate

t

density when

k = 2

and

(a, b)

and

X

satisfy certain conditions (cf. condition (17) of Spurrier ). Another approach hinges on the results for the volume of a tube about a one‐dimensional manifold in the unit sphere in

{double-struckR}^{k + 1}

, which have been studied by Naiman (), Knowles & Siegmund (), Johnstone & Siegmund (), Johansen & Johnstone (), and Sun & Loader (), among others.…”

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

View full text Add to dashboard Cite

show abstract

Simultaneous confidence bands for multiple comparisons of several percentile lines

Zhou,

Zhang

2024

Comput Stat

View full text Add to dashboard Cite

One-sided hyperbolic simultaneous confidence bands for multiple and polynomial regression models

Zhou

2016

Metrika

View full text Add to dashboard Cite

Comparison of Simultaneous Confidence Bands for Quadratic Regression Over a Finite Interval

Cited by 5 publications

References 8 publications

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

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

Simultaneous confidence bands for multiple comparisons of several percentile lines

One-sided hyperbolic simultaneous confidence bands for multiple and polynomial regression models

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