I-Bonferroni vs. Benjamini-Hochberg: Ukukhetha Ukulungiswa Kwenani Lakho le-P

kube isihloko esibucayi. Mhlawumbe kugwenywe kangcono lapho ubonana okokuqala nochwepheshe bezibalo. Isimo esibheke esihlokweni siholele esivumelwaneni esingasho lutho sokuthi α = 0.05 iyindinganiso yegolide—eqinisweni, 'ingqungquthela elula', umthetho wesithupha owabekwa uRonald Fisher ngokwakhe.

I-WHO?? Awumazi? Ungakhathazeki.

Ube ngowokuqala ukwethula I-Maximum Likelihood Estimation (MLE), i-ANOVA, kanye ne-Fisher Information (yakamuva, okungenzeka ukuthi uyiqagele). UFisher wayengaphezu komuntu ofanele emphakathini, uyise wezibalo. Wayenesithakazelo esijulile kufuzo lwe-Mendelian nesayensi yokuziphendukela kwemvelo, ayezokwenza iminikelo eminingana ebalulekile. Ngeshwa, uFisher naye wayenesikhathi esidlule esinameva. Wayezibandakanye ne-Eugenics Society kanye nenqubomgomo yayo yokuvala inzalo ngokuzithandela kulabo “abanengqondo ebuthakathaka.”

Yebo, ayikho into ebizwa nge-Statistician edumile.

Kodwa umthetho wesithupha obekwe ubaba wezibalo ngezinye izikhathi ungenziwa iphutha njengomthetho, kanti umthetho awunjalo.

Kunesibonelo esisodwa esibalulekile lapho ungaphoqelekile kuphela kodwa kufanele shintsha leli zinga le-alpha, futhi lokho kwehlela kukho ukuhlolwa kwe-hypothesis eminingi.

Ukuze wenze izivivinyo eziningi ngaphandle kokusebenzisa i- I-Bonferroni Ukulungisa noma i Benjamini-Hochberg inqubo ingaphezu kokuba yinkinga. Ngaphandle kwalokhu kulungiswa, singafakazela noma iyiphi i-hypothesis:

H₁: Ilanga liluhlaza okwesibhakabhaka

Ngokumane siqalise kabusha ukuhlola kwethu kuze kufike inhlanhla. Kodwa zisebenza kanjani lezi zilungiso? futhi iyiphi okufanele usebenzise? Azishintsheki!

P-values ​​kanye nenkinga

Ukuze siqonde ukuthi kungani, sidinga ukubheka ukuthi i-p-value yethu isitshelani. Ukuliqonda ngokujulile kunokuncane kuhle, nokukhulu kubi. Kodwa ukwenza lokhu, sizodinga ukuhlolwa, futhi akukho okujabulisa – noma okuphikiswana ngakho – njengokuthola izinto ezinzima kakhulu.

Lezi zici ngendlela emangalisayo engazinzi futhi idalwe kuma-accelerator ezinhlayiyana, i-athomu eyodwa ngesikhathi. Iphawundi ngephawundi, into ebiza kakhulu eyake yakhiqizwa. Ikhona kuphela kumicimbi ye-cosmic efana ne-supernovae, ehlala isikhathi eside kuphela izinkulungwane noma izigidi okwesibili.

Kodwa ukungazinzi kwabo kuba yinzuzo yokuthola, njengoba isici esisha esisinda kakhulu singabonisa ukubola okuhlukile komsakazo. Ukulandelana kokubola okushuthwe izinzwa ku-reactor kungasitshela ukuthi i-elementi entsha ikhona yini.

Njenge-hypothesis yethu eyize, sithi:

H₀ = Ukulandelana ukubola komsindo wangemuva. (Ayikho into entsha)

Manje sidinga ukuqoqa ubufakazi bokuthi i-H₀ akulona iqiniso uma sifuna ukufakazela ukuthi sidale into entsha. Lokhu kwenziwa ngezibalo zethu zokuhlola T(X). Ngokuvamile, lokhu kuthatha umehluko phakathi kwalokho okubonwa izinzwa nalokho okulindeleke emisebeni yangemuva. Konke izibalo zokuhlola ziyisilinganiso 'sokumangala' phakathi kwalokho esilindele ukukubona uma i-H₀ iyiqiniso nalokho okushiwo idatha yethu yesampula. Okukhudlwana T(X)ubufakazi obengeziwe esinabo bokuthi i-H₀ ingamanga.

Yilokhu kanye okushiwo yi- Izibalo zokuhlola ze-Schmidt yenza ngokulandelana kwezikhathi zokubola kwe-radioactive.

[
sigma_{obs} = sqrt{frac{1}{n-1} sum_{i=1}^{n} (ln t_i – overline{ln t})^2}
]

Izibalo zokuhlola ze-Schmidt zisetshenziswe ekutholweni kwe: I-Hassium (108), I-Meitnerium (109) phakathi 1984 I-Darmstadtium (110), Ingxenye 111 I-Roentgenium (111), I-Copernicium (112) kusukela 1994 kuya ku-1996 I-Moscow (115), I-Tennessee (117). kusuka 2003 kuya ku-2016

Kubalulekile ukucacisa ukusatshalaliswa kwe-H₀ ukuze sikwazi ukubala amathuba okuthi izibalo zokuhlola zidlulele njengezibalo zokuhlola zedatha ebhekiwe.

Sicabanga ukuthi ukubola komsindo kulandela i ukusatshalaliswa kwe-exponential. Kunezizathu eziyisigidi zokuthi kungani lokhu kuwukucabanga okuhle, kodwa masingagxili lapha. Uma singenakho ukusatshalaliswa kwe-H₀, ukwenza ikhompuyutha inani lethu lamathuba ngeke kwenzeke!

[
H_0^{(Schmidt)}​:t_1​,…,t_n​ i.i.d. ∼ Exp(λ)
]

Inani lika-p bese liyithuba elingaphansi kwemodeli eyinull yokuthola izibalo zokuhlola okungenani ngokwedlulele njengaleso esibalwe kudatha yesampula. Uma mancane amathuba okuba izibalo zethu zokuhlola, maningi amathuba okuthi i-H₀ ingamanga.

[
p ;=; Pr_{H_0}!big( T(X) ge T(x_{mathrm{obs}}) big).
]

Yebo, lokhu kuveza udaba oluthakazelisayo. Kuthiwani uma sibona izinga lokubola elingavamile, izinga lokubola elimane lifane nelezinhlayiyana ezibolayo ezingatholakali? Kuthiwani uma izinzwa zethu zithola ukulandelana kokubola okungalindelekile, noma kungenzeka, okuletha izibalo zokuhlola ezinkulu? Ngaso sonke isikhathi uma siqhuba isivivinyo mancane amathuba okuthi uthole i-outlier ngenhlanhla nje. Lo mthengisi uzonikeza izibalo ezinkulu zokuhlola njengoba zizohluka kakhulu kunalokho esilindele ukukubona lapho i-H₀ iyiqiniso. I-T(x) enkulu izoba semisileni yokusabalalisa kwethu okulindelekile kwe-H₀ futhi izokhiqiza inani elincane lika-p. Amathuba amancane okubona noma yini eyeqisayo kunalokhu okungaphandle. Kodwa ayikho into entsha ekhona! sisanda kuthola ama-31 abomvu ngokudlala i-roulette izikhathi eziyisigidi.

Kubonakala kungenakwenzeka, kodwa uma ukhumbula ukuthi ama-protons akhanyisa izinhlayiya eziqondiwe izinyanga ngesikhathikungenzeka kumile. Ngakho silandisa kanjani ngakho?

Kunezindlela ezimbili: indlela elandelanayo kanye nencane kakhulu. Okukhethayo kuncike ekuhlolweni. Singasebenzisa:

• Isilinganiso Sephutha Sokuhlakanipha Komndeni (FWER) nokulungiswa kwe-Bonferroni
• Isilinganiso Sokutholwa Kwamanga (FDR) kanye nenqubo ye-Benjamin-Hochberg

Lezi azishintsheki! Udinga ukucubungula ngokucophelela isifundo sakho bese ukhetha okulungile.

Uma unentshisekelo ku-physics yayo:

Izakhi ezintsha zidalwa ngokusheshisa ama-ion alula ngo-10% ngesivinini sokukhanya. Le misebe ye-ion ibhomba ama-athomu aqondiswe esindayo. Isivinini esimangalisayo namandla e-kinetic ayadingeka ukuze kunqotshwe umgoqo we-coulomb (amandla amakhulu anyanyekayo phakathi kwezinhlayiya ezimbili ezishajwe kahle.

Isici Esisha	I-Beam (Protons)	Ithagethi (Amaphrothoni)
I-Nihonium (113)	I-Zinc-70 (30)	I-Bismuth-209 (83)
I-Moscovium (115)	I-calcium-48 (20)	I-Americium-243 (95)
I-Tennessee (117)	I-calcium-48 (20)	I-Berkelium-249 (97)
U-Oganesson (118)	I-calcium-48 (20)	I-Californium-249 (98)

Isilinganiso Sephutha Sokuhlakanipha Komndeni (Bonferroni)

Lena indlela yethu yokulandela, futhi yini okufanele isetshenziswe uma singakwazi ukuvuma noma yimaphi amanga. Le ndlela igcina amathuba okuvuma okungenani eyodwa Thayipha iphutha ngaphansi kwezinga lethu le-alpha.

[
Pr(text{at least one Type I error in the family}) leq alpha
]

Lokhu futhi ukulungisa okulula. Vele uhlukanise ileveli ye-alpha ngenombolo yezikhathi ukuhlolwa okuqaliswe ngazo. Ngakho-ke kukho konke ukuhlolwa uyenqaba i-null hypothesis uma futhi kuphela uma:

[
p_i leq frac{alpha}{m}
]

Ngokulinganayo, ungalungisa amanani akho we-p. Uma ugijima m ukuhlolwa, thatha:

[
p_i^{text{adj}} = min(1, m p_i)
]

Futhi wenqabe i-null hypothesis uma:

[
p_i^{(text{Bonf})} le alpha
]

Konke esikwenzile lapha kwaba ukuphindaphinda izinhlangothi zombili zokungalingani ngakho m.

Ubufakazi balokhu futhi umugqa owodwa omncane. Uma sivumela u-Aᵢ kube isehlakalo sokuthi kukhona okungelona iqiniso ekuhlolweni i. Khona-ke ithuba lokuba okungenani libe nephozithivu elilodwa elingelona iqiniso liyoba ithuba lokuhlangana kwazo zonke lezi zenzakalo.

[
text{Pr}(text{at least one false positive}) = text{Pr}left(bigcup_{i=1}^{m} A_iright) le sum_{i=1}^{m} text{Pr}(A_i) le m cdot frac{alpha}{m} = alpha
]

Lapha sisebenzisa isibopho senyunyana. umqondo oyisisekelo emathubeni othi amathuba okuthi A₁, noma A₂, noma Aₖ enzeka kufanele abe ngaphansi noma alingane nesamba samathuba omcimbi ngamunye owenzekayo.

[
text{Pr}(A_1 cup A_2 cup cdots cup A_k) le sum_{i=1}^{k} text{Pr}(A_i)
]

Isilinganiso Sokutholwa Kwamanga (Benjamini-Hochberg)

Inqubo ye-Benjamin-Hochberg nayo ayinzima kakhulu. Kalula nje:

• Hlunga amanani akho we-p: p₁ ≤ … ≤ pₘ.
• Yamukela u-k wokuqala lapho pₖ > α/(m−k+1)

Ngale ndlela, umgomo uwukulawula izinga lokutholwa okungamanga (FDR).

[
text{FDR} = Eleft[ frac{V}{max(R, 1)} right]
]

Kuphi R inombolo yezikhathi lapho senqaba khona i-null hypothesis, futhi V inombolo yokwenqatshwa (ngeshwa) okuyizinto ezingamanga (amaphutha ohlobo I). Umgomo uwukugcina le metric ingaphansi komkhawulo othize q = 0.05.

Izimpawu ze-BH ziyi:

[
frac{1}{m}q, frac{2}{m}q, dots, frac{m}{m}q = q
]

Futhi silahla amanani okuqala amancane kakhulu we-p lapho:

[
P_{(k)} leq frac{k}{m}q
]

Sebenzisa lokhu uma ulungile ngemibono engamanga. Lapho okukukhathazayo okuyinhloko kuwukunciphisa uhlobo lwe-II isilinganiso sephuthaokusho ukuthi, ufuna ukwenza isiqiniseko sokuthi kumbalwa okungalungile okungamanga, azikho izimo lapho samukela i-H₀ lapho i-H₀ empeleni ingamanga.

Cabanga ngalokhu njengocwaningo lwe-genomics lapho uhlose ukuhlonza wonke umuntu onofuzo oluthile olumenza abe sengozini yokuhlaselwa umdlavuza othile. Kungaba yingozi kangako uma siphatha abantu abangenalo isakhi sofuzo kunokuba sibe sengcupheni yokuvumela othile onalo ahambe engaselashwa.

Ngokushesha side-by-side

I-Bonferroni:

• Ilawula izinga lephutha elihlakaniphile lomndeni (FWER).
• Iqinisekisa amathuba esilinganiso esisodwa sokutholwa okungamanga ≤ α
• Izinga eliphezulu lokuphikisa okungamanga ⇒ Amandla ezibalo aphansi
• Ukungabekezelelani engozini

Benjamini-Hochberg

• Ilawula izinga lokutholwa okungamanga (FDR)
• iqinisekisa ukuthi phakathi kwakho konke okutholiwe, imibono engamanga ithi ≤ q
• Ambalwa amanga amanga ⇒ Amandla ezibalo aphezulu
• Ukubekezelela ubungozi okuthile

I-p encane kakhulu ye-athomu esinda kakhulu

Asikwazi ukuba nama-elementi angekho kuthebula le-periodic, ngakho-ke uma kuziwa ekutholeni i-elementi entsha, ukulungiswa kwe-Bonferroni kuyindlela efanele. Kepha uma kukhulunywa ngedatha ye-decay chain eqoqwe ama-silicon ezwela isikhundla, ukukhetha m akulula kangako.

Izazi zefiziksi zivame ukusebenzisa inombolo elindelekile yamaketango angahleliwe akhiqizwa yilo lonke ukusesha kudathasethi yonke:

[
Pr(ge 1 text{ random chain}) approx 1 – e^{-n_b}
]

[
1 – e^{-n_b} leq alpha_{text{family}} Rightarrow n_b approx alpha_{text{family}} quad (text{approximately, for rare events})
]

Inani lamaketango angahleliwe livela ekubhekeni idatha yasemuva lapho kungekho kuhlolwa okwenzekayo. kusuka kule datha singakha ukusabalalisa okuyize H₀ ngokusebenzisa ukulingisa kwe-monte carlo

Silinganisela inani lamaketango angahleliwe ngokwenza imodeli yezilinganiso zomcimbi wangemuva futhi sihlole kabusha imicimbi yangemuva ebhekiwe. Ngaphansi kwe-H₀ (alukho uchungechunge lokubola kwento esindayo), sisebenzisa i-Monte Carlo ukuze silingise ukugcwaliseka okuyize okuningi futhi sibale ukuthi i-algorithm yosesho ikhiqiza kangaki iketango elidlulele kakhulu njengeketango eliphawuliwe.

Ngokunembe kakhulu:

H₀​: izehlakalo zangemuva zifika njengenqubo ye-Poisson enenani elithi λ ⇒ izikhathi zokufika phakathi nokufika ziExponential.

Bese kuthi uchungechunge lwengozi lube ngu-k ukushaya okulandelanayo esikhathini esingu-τ. Siskena idatha sisebenzisa izibalo zethu zokuhlola ukuze sinqume ukuthi ingabe iqoqo elidlulele likhona yini.

lambda_rate = 0.2   # events per second
T_total = 2_000.0   # seconds of data-taking (mean events ~ 400)
k = 4               # chain length
tau_obs = 0.20      # "observed extreme": 4 events within 0.10 sec

Nmc = 20_000
rng = np.random.default_rng(0)

def dmin_and_count(times, k, tau):
    if times.size < k:
        return np.inf, 0
    spans = times[k-1:] - times[:-(k-1)]
    return float(np.min(spans)), int(np.sum(spans <= tau))

...

I-Monte-Carlo Simulation ku-GitHub

Uma unentshisekelo ezinombolweni, ekutholweni kwesici 117 I-Tennessee (Ts), inani lika-p 5×10−16 yasetshenziswa. Ngicabanga ukuthi uma kungekho ukulungisa okuke kwasetshenziswa, ithebula lethu le-periodic, ngeshwa, lingeke libe usayizi wephosta, futhi ikhemistry izoba sengozini.

Isiphetho

Lo mqondo wonke wokucinga okuthile ezindaweni eziningi, bese uphatha i-blip ebaluleke ngokukhethekile njengokungathi ivela kokuphawuliwe okukodwa, uvame ukubizwa ngokuthi Bheka-kwenye indawo Umphumela. futhi kunezindlela ezimbili eziyinhloko esingalungisa ngazo lokhu:

• I-Bonferroni Ukulungisa
• Benjamini-Hochberg Inqubo

Ukukhetha kwethu kuncike ngokuphelele ekutheni sifuna ukonga kangakanani.

Kodwa noma nge-p-value ye 5×10−16ungase uzibuze ukuthi kunini lapho inani lika-p lika-10^-99 lisafanele lilahlwe nini. Futhi konke lokho kwehlela kuVictor Ninov, isazi sefiziksi eLawrence Berkeley National Laboratory. Kwakungubani – okwesikhashana – indoda eyathola isici 118.

Nokho, uphenyo lwangaphakathi lwathola ukuthi wakha iketango le-alpha-decay. Kulokhu, mayelana nokungaziphathi kahle kocwaningo kanye nedatha engamanga, ngisho nenani lika-p 10^-99 ayikuthetheleli ukwenqaba i-null hypothesis.

Izikhombo

Bodmer, W., Bailey, RA, Charlesworth, B., Eyre-Walker, A., Farewell, V., Mead, A., & Senn, S. (2021). Usosayensi ovelele, u-RA Fisher: imibono yakhe nge-eugenics nohlanga. Ufuzo, 126(4), 565-576.

Khuyagbaatar, J., Yakushev, A., Düllmann, CE, Ackermann, D., Andersson, LL, Asai, M., … & Yakusheva, V. (2014). I-Ca 48+ Bk 249 fusion reaction eholela kusici Z= 117: I-α-decaying Db 270 ephile isikhathi eside kanye nokutholakala kwe-Lr 266. Izincwadi zokubukeza ngokomzimba, 112(17), 172501.

Okuhle, Ukuqhathanisa Okuningi kwe-HMF: Ukulungiswa Kwe-Bonferroni kanye Namazinga Okuthola Amanga.