How the Maximal Evidence of<i>P</i>-Values Against Point Null Hypotheses Depends on Sample Size

Held, Leonhard; Ott, Manuela

doi:10.1080/00031305.2016.1209128

Cited by 71 publications

(60 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Interestingly, the fitted minBFs increase with increasing sample size for all types of non‐asymptotic p ‐values considered (see Figures and ), so we observe the same qualitative relationship between sample size and minBFs as in the linear model (Held & Ott, ). For the class of local normal priors, we compare the fitted minBFs for 2 × 2 tables to the sample‐size adjusted minBFs for the linear model with one degree of freedom ( d = 1).…”

Section: Empirical Relationship Between P‐values and Minimum Bayes Fasupporting

confidence: 65%

“…The corresponding large‐sample minBF can be written as

{minTBF}_{1} false(p false) = ({, \begin{matrix} array \sqrt{z} \exp (- z / 2) \sqrt{e} & array for z = z (p) > 1 \\ array 1 & array otherwise, \end{matrix})

where

e = \exp false(1 false)

is Euler's number. However, the case d = 2 is also of interest because it corresponds to a popular and easily applicable minBF, which directly calibrates a two‐sided p ‐value p as follows (Vovk, , section 9):

{minTBF}_{2} false(p false) = ({, \begin{matrix} array - e p \log (p) & array for p < 1 / e \\ array 1 & array otherwise. \end{matrix})

A simple derivation of can be found in Sellke et al () and is outlined in Held & Ott (, appendix B). This minBF will be termed the ‘

- e p normallog false(p false)

’ calibration in the following.…”

Section: Large‐sample Minimum Bayes Factorsmentioning

confidence: 99%

“…Under these priors, there is a closed‐form expression for the Bayes factor (Liang et al , ), which is minimised if the prior variance factor g is estimated by empirical Bayes. This yields a closed‐form expression for the minBF, which depends only on the sample size n , the dimension d of β and the coefficient of determination R 2 (Held & Ott, , equation 9). A p ‐value from the F ‐test, a one‐to‐one transformation of R 2 , can hence be mapped to this minBF, see figure 1 in Held & Ott () for illustration.…”

Section: Sample‐size Adjusted Minimum Bayes Factorsmentioning

confidence: 99%

“…This yields a closed‐form expression for the minBF, which depends only on the sample size n , the dimension d of β and the coefficient of determination R 2 (Held & Ott, , equation 9). A p ‐value from the F ‐test, a one‐to‐one transformation of R 2 , can hence be mapped to this minBF, see figure 1 in Held & Ott () for illustration. In the case d = 2, there is even a closed‐form expression for the minBF as a function of the p ‐value and the sample size (Held & Ott, , equation 12).…”

Section: Sample‐size Adjusted Minimum Bayes Factorsmentioning

confidence: 99%

“…A p ‐value from the F ‐test, a one‐to‐one transformation of R 2 , can hence be mapped to this minBF, see figure 1 in Held & Ott () for illustration. In the case d = 2, there is even a closed‐form expression for the minBF as a function of the p ‐value and the sample size (Held & Ott, , equation 12). For the other dimensions d , the conversion from the F ‐test p ‐value to R 2 has to be done numerically.…”

Section: Sample‐size Adjusted Minimum Bayes Factorsmentioning

confidence: 99%

See 4 more Smart Citations

Bayesian Calibration ofp‐Values from Fisher's Exact Test

Ott

Held

2018

Int Statistical Rev

Self Cite

View full text Add to dashboard Cite

Summary p‐Values are commonly transformed to lower bounds on Bayes factors, so‐called minimum Bayes factors. For the linear model, a sample‐size adjusted minimum Bayes factor over the class of g‐priors on the regression coefficients has recently been proposed (Held & Ott, The American Statistician 70(4), 335–341, 2016). Here, we extend this methodology to a logistic regression to obtain a sample‐size adjusted minimum Bayes factor for 2 × 2 contingency tables. We then study the relationship between this minimum Bayes factor and two‐sided p‐values from Fisher's exact test, as well as less conservative alternatives, with a novel parametric regression approach. It turns out that for all p‐values considered, the maximal evidence against the point null hypothesis is inversely related to the sample size. The same qualitative relationship is observed for minimum Bayes factors over the more general class of symmetric prior distributions. For the p‐values from Fisher's exact test, the minimum Bayes factors do on average not tend to the large‐sample bound as the sample size becomes large, but for the less conservative alternatives, the large‐sample behaviour is as expected.

show abstract

Section: Empirical Relationship Between P‐values and Minimum Bayes Fasupporting

confidence: 65%

“…The corresponding large‐sample minBF can be written as

{minTBF}_{1} false(p false) = ({, \begin{matrix} array \sqrt{z} \exp (- z / 2) \sqrt{e} & array for z = z (p) > 1 \\ array 1 & array otherwise, \end{matrix})

where

e = \exp false(1 false)

{minTBF}_{2} false(p false) = ({, \begin{matrix} array - e p \log (p) & array for p < 1 / e \\ array 1 & array otherwise. \end{matrix})

A simple derivation of can be found in Sellke et al () and is outlined in Held & Ott (, appendix B). This minBF will be termed the ‘

- e p normallog false(p false)

’ calibration in the following.…”

Section: Large‐sample Minimum Bayes Factorsmentioning

confidence: 99%

Section: Sample‐size Adjusted Minimum Bayes Factorsmentioning

confidence: 99%

Section: Sample‐size Adjusted Minimum Bayes Factorsmentioning

confidence: 99%

Section: Sample‐size Adjusted Minimum Bayes Factorsmentioning

confidence: 99%

See 3 more Smart Citations

Bayesian Calibration ofp‐Values from Fisher's Exact Test

Ott

Held

2018

Int Statistical Rev

Self Cite

View full text Add to dashboard Cite

show abstract

An objective Bayes perspective on p‐values

Held

2017

Biometrical J

Self Cite

View full text Add to dashboard Cite

Microwave‐induced thermoacoustic microscopy based on short‐pulse microwave and high‐frequency point‐focused ultrasonic transducer

et al. 2023

View full text Add to dashboard Cite

BackgroundAs an emerging hybrid imaging modality, microwave‐induced thermoacoustic imaging (MITAI) provides high contrast and deep tissue penetration, and has been extensively applied in cancer diagnosis, arthritis detection, and brain research. However, the previous studies had a limited spatial resolution of about 0.45–1.5 mm.PurposeHere, we describe a microwave‐induced thermoacoustic microscopy (MITAM) system to help overcome the resolution limitation of current MITAI to image more subtle tissue features. On this basis, this paper applies MITAM to the thin skin and to demonstrate the potential of MITAM in detecting scleroderma.MethodsTo achieve high resolution, short pulse width microwave (pulse width: 70 ns) and high‐frequency ultrasonic point‐focused transducer (center frequency: 25 MHz) were used to build the MITAM system. Two parallel copper wires with a diameter of 90 μm in the X/Y plane and Y/Z plane were imaged to estimate X/Y/Z resolution. Nine Balb/c mice were randomly divided into three groups and injected with different concentrations of bleomycin to induce scleroderma models. Their ex vivo skins were then imaged by our MITAM system. Visual observations were performed on the 3‐dimensional skins MITAM images. And the mean value, Standard deviation, quartile distance, and signal‐to‐noise ratio were calculated to verify the results of the qualitative observations. Hematoxylin‐Eosin (HE) and Masson staining were used to validate the findings of the MITAM.ResultsThe thickness of each imaged skin was measured to be about 450 μm. As an organ composed of multiple layers of tissues, the skin needs to be imaged at high resolution for the detection of related diseases. The results obtained showed that the improved resolution (68 μm in the Z‐axis and 135 μm in the X‐axis/Y‐axis) of MITAM over conventional MITAI allowed us to differentiate scleroderma skins from normal skins and to identify the severity of scleroderma skins, consistent with the pathological findings of these skins.ConclusionsThe preliminary results obtained indicate that the MITAM can relieve the resolution limitation of traditional MITAI and has the potential to detection scleroderma. However, the transmission‐type MITAM mentioned in this paper is difficult to image in vivo due to the narrow area between the antenna and the transducer. In the future, a reflective scanning MITAM will be constructed to detect scleroderma in vivo.

show abstract

How the Maximal Evidence ofP-Values Against Point Null Hypotheses Depends on Sample Size

Cited by 71 publications

References 33 publications

Bayesian Calibration ofp‐Values from Fisher's Exact Test

Bayesian Calibration ofp‐Values from Fisher's Exact Test

An objective Bayes perspective on p‐values

Microwave‐induced thermoacoustic microscopy based on short‐pulse microwave and high‐frequency point‐focused ultrasonic transducer

Contact Info

Product

Resources

About