A biostatisztika alapjai gyakorlati alkalmazsokkal Matematikai modellek a termszettudomnyokban Szeged, 2011. mjus 19-21.Boda Krisztina PhDSZTE OK Orvosi Informatikai IntzetTeaching Mathematics and Statistics in Sciences HU-SRB/0901/221/088

*Egy kis bevezetsEgy matematikus, egy alkalmazott matematikus s egy statisztikus mindhrman egy bizonyos munkra jelentkeztek. A felvteli beszlgetsen megkrdeztk tlk, hogy mennyi 1+1?Matematikus: Be tudom bizonytani, hogy ltezik, de azt nem, hogy csak egy megolds van.Alkalmazott matematikus: A vlasz kzelten 1.99, a becsls szrsa 0.01.Statisztikus (kilpett a szobbl, majd hirtelen visszatrt s rdekldtt): teht mit szeretntek, mennyi legyen?http://www.ilstu.edu/~gcramsey/Gallery.htmlBiostatisztikai alapismeretek Boda Krisztina

*Ajnlott irodalomReiczigel Jen, Harnos Andrea, Solymosi Norbert: Biostatisztika nem statisztikusoknak. Pars Kft. Nagykovcsi, 2007.Biostatisztikai alapismeretek Boda Krisztina

*Ajnlott irodalomDinya Elek: Biometria az orvosi gyakorlatban. Medicina KiadBiostatisztikai alapismeretek Boda Krisztina

*Ajnlott irodalomGachlyi Bla (szerk.) - Lakner Gza (szerk.): Klinikai farmakolgia a gyakorlatban - A humn klinikai gygyszerfejleszts mdszertana. Statisztikai alapelvek fejezet (Singer Jlia)SpringMed KiadBiostatisztikai alapismeretek Boda Krisztina

*Amirl sz leszHipotzisvizsglatok, kiemelve a kvetkezket:Populci, minta, ler statisztikkA vletlen ingadozs vizsglata, a statisztikai prbk elveKonfidenciaintervallum s a statisztikai szignifikanciaEgyvltozs statisztikk, mikor mit? t-prbk, 2 prbk, korrelci/regressziTbbszrs sszehasonltsok problmiVarianciaanalzisDiagnosztikus tesztekTbbvltozs mdszerek, kockzati tnyezkElemszmbecsls, statisztikai er szmtsaBiostatisztikai alapismeretek Boda Krisztina

*Populci (sokasg), mintaPopulci: azoknak az egyedeknek, objektumoknak az sszessge, amelyrl egy vizsglat sorn informcit kvnunk nyerni.Minta: a sokasg azon rszhalmaza, amelyet ppen vizsglunk A minta kivlasztsakor arra treksznk, hogy lehetleg reprezentlja az egsz populcit, vagy legalbbis kvetkeztetni lehessen a populcira. Kvetelmny a mintaelemek fggetlensge is.

Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*PldkAdathalmazokMintaGygyszerszhallgatk egy csoportja ltal kitlttt krdvek20 egszsges n vrnyomsrtkeiSokasgGygyszerszhallgatk hallgatk ltalban az egszsges nk vrnyomsaBiostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*MintaSokasgKategris vltoz lehetsges rtkeinek gyakorisgai, relatv gyakorisgai A vltoz (sokasg) eloszlsa (megkzelti)Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*MintaSokasgEgy folytonos vltozrl ksztett hisztogramA folytonos vltoz eloszlst (srsgfggvnyt) (megkzelti)Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*MintaSokasgtlag (x)Standard devici (SD)MedinSokasg-tlag (ismeretlen) A sokasg standard devicija (ismeretlen) A sokasg medinja (ismeretlen)

(megkzelti)Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Az eloszls kzepnek jellemzse

tlag:

Mdusz: a leggyakrabban elfordul rtk(ek)Medin: az a szm, amelynl az adatok fele kisebb, vagy egyenl (amely teht megfelezi az adatsort). A medin szmtsa: elszr sorba lltjuk az adatokat nagysg szerint. Pratlan elemszm esetn a medin a kzps elem, pros elemszm esetn a medin a kt kzps elem tlaga Pldaadatok: 1 2 4 1tlag=(1+2+4+1)/4=8/4=2Mdusz=1MedinElszr sorba lltjuk az adatokat nagysg szerint: 1 1 2 4Pros az elemszm, a kt kzps elem 1 s 2, tlaguk 1.5. A medin rtke 1.5Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*A szrds mrszmaiA terjedelem a maximum s a minimum kztti klnbsgKvartilisek, percentilisek: 25%-os percentilis (els kvartilis): az a szm, aminl az adatok 25%-a kisebb.A 25%-os, 50%-os s 75%-os kvartilis ngy rszre osztja az adatokat.A szrds megadsra hasznljk a 25%-os s 75%-os kvartilistA variancia

A standard devici: az adatok szrdsa az tlag krlBiostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Plda. A szrds jellemziAdatok: 1 2 4 1, rendezve: 1 1 2 4Terjedelem: max-min=4-1=3Kvartilisek: Standard devici:

Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

11-2=-1111-2=-1122-2=0044-2=24sszeg06

*Az tlag szrsa, standard error of mean, SE, SEMAzt fejezi ki, hogy az tlag, amit a mintbl szmoltunk, mennyire megbzhat. Ha tbbszr is meg tudnnk ismtelni a mrst (vgtelen sokszor), akkor mindegyik mrs-sorozat tlagnak a szrdst mutatja a populci tlag krlSzmtsa: SE=SD/nBiostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Standard devici vagy standard error??Standard devici, SD: a minta szrsa, a mintaadatok szrdsa az tlag krl. Normlis eloszls esetn az tlag 2SD-n bell van az adatok kb. 95%-aStandard error (SE=SD/n): az tlag megbzhatsga, a mintatlag szrdsa az (ismeretlen) populci tlag krl. Normlis eloszls esetn az tlag 2SE-n bell van az igazi tlag kb. 95%-os valsznsggel.

Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Normlis eloszlsok Jells: N(, )A hisztogramot kisimt grbe gyakran szimmetrikus, egycscs, harang alak grbe.Ez a grbe egyrtelmen lerhat kt paramterrel: a kzppel s a standard devicival (szrs).Az brn a kk vonalat a mintbl szmolt tlag s szrs alapjn rajzoltuk be.

Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*A 68-95-99.7 szablyEgy s paramterekkel meghatrozott normlis eloszls esetn:A megfigyelsek 68% -a esik a kzptl egyszeres tvolsgraA megfigyelsek 95% -a esik a kzptl ktszeres tvolsgraA megfigyelsek 99.7% -a esik a kzptl 3-szoros tvolsgra

Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Az eloszls elkpzelse adott tlag s szrs (SD) alapjn(normlis eloszlst felttelezve)A cikkekben a tblzatok leggyakrabban az tlagot s a szrst ismertetik. Ezek alapjn el tudjuk kpzelni, milyen lehet az eloszlsPl. letkor (v) 55.2 15.7

86.623.8Ebben az intervallumban van az adatok 95.44%-aBiostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Az SD ferde eloszlsok esetnStent length per lesion (mm): 18.8 10.5Ezekkel a paramterekkel a kvetkez eloszls kpzelhet el:A szrs a ferde eloszls miatt lett nagy. Ezrt gyakran a standard devici helyett a standard errort adjk meg a tblzatokban vagy brkon. Az valban kisebb, de mst jelent. Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*A kzp s a szrds jellemzinek prostsaBiostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

KzpSzrdsKzls cikkekbentlagStandard devici,Standard errortlag (SD)tlag SDtlag SEtlag SEMMedinMin, max5%-os, 95%-os percentilis25 % , 75% (Kvartilisek)Med (min, max)Med(25%, 75%)

*bratpusok a szmolt jellemzk alapjntlag-szrs bratlag + SDtlag + SEtlag + 95% CI

tlag SEtlag SDtlag 95% CIBiostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*bratpusok a szmolt jellemzk alapjnDoboz-bra (box diagram)Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*HipotzisvizsglatokA hipotzisvizsglat sorn a rendelkezsre ll adatok (statisztikai minta) alapjn az egsz jelensg (populci) tulajdonsgaira kvetkeztetnk. Azt vizsgljuk, hogy a tapasztalt eredmny (klnbsg) nagyobb-e, mint amit a vletlen nmagban okoz.

Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Mintavtel, szimulciLegyen a populci 120 tlag, 10 szrs normlis eloszls, ebbl vesznk 50 elem mintkatBiostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*120 tlag, 10 szrs populcibl szrmaz 50 elem mintk (ismtelt mrsek) tlagai s szrsaiBiostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

Diagram1

122.03664967569.6145338288

120.192582547210.2279396339

121.952832008111.6173741779

118.23474217929.7922597768

121.489957185211.2268953144

120.299256498710.2493901746

117.70406702749.839210258

118.39577117211.7091395022

121.779801095810.1354140359

120.04718640419.8477165177

118.03037326279.8235483775

121.49395668819.0624746357

118.752941792210.8976417915

119.626919816411.3073958094

119.02511694939.8150008366

119.48073832469.1025635038

118.219916278710.7103635863

121.15283205569.9125157043

118.18064861769.9944573325

119.79446556619.5199313699

ismtls

tlag + SD

Munka5

s1s2s3s4s5s6s7s8s9s10s11s12s13s14s15s16s17s18s19s20

118.4551095622110.7774508756123.2820139495142.6522388402108.3501686782121.1812062439132.80457127109.0859271545119.9621309144128.6245108832116.8235670167125.4550582718121.0988060239110.6158791418118.8850277077129.1386027634118.0628672347104.4956164452125.2467271416135.4461304192

128.9520426627125.8100795286141.3554812944100.9909240867121.5441173673133.8052655529123.1306171877128.3449094745104.556416166117.0927774484108.7321791903104.8231254873114.7216451801116.4169433042126.7120481617127.3928958955134.4505520439136.7654889083118.2985286806124.589639957

114.6652906146124.604953574117.8049686433114.0672296361131.1218014354131.6362571134100.8587745065135.8871444130.6110064735116.4813127917136.3935965247131.0736664283122.16996340997.7563061798119.5856455826127.6118340075117.5282548803123.9394080886106.3049652049101.3771864865

121.9347908164137.3761691258126.8932990871103.33350959107.4432887736124.9964000937119.0465312294123.8733560359128.5052761278101.6061972524113.7653001325116.0159129841122.7559281079114.9827838464114.1064061204103.9569227333119.8963403414131.2332600111108.2972360865116.1777120916

132.9500676849118.3156044436104.3385012355125.6864450926106.2775894144123.6034634832118.2076474225130.2085778053114.8766571833108.8054844632121.3138105714121.6650005819113.0707645035133.1708475237127.2957163866115.8347075335121.9184312805113.6318772598126.6689835876135.6225269166

105.2268501854106.1788410271127.0827582022115.2268353809110.1108139966127.191306394899.3519236846109.2263986112116.9696546032121.1958377399128.0283371062108.7061676418121.2266582416121.6533704184117.4216279952123.1981699067109.4291931923116.2130800668129.1212086772119.7402596819

127.0719352152108.2972360865138.0051301868129.5602217698101.155997375121.9956246433116.4691028254122.6092266125138.0322786036121.7325191948116.5917822919122.7297119232105.7168859307114.9871266836121.3732460502134.1485998029112.0851905699120.4610683391119.233887138114.5120748635

120.0960085345112.1893959254116.6039354149120.5775063983115.0961182549118.9842763171108.2393694861117.2551904648123.3491005524128.5923602455135.1190306497108.0023779042128.2456381278128.3319264377102.811141308133.3198000185134.8142589751112.806715454298.3030943945124.2204192141

103.7254563317120.3477566679130.8642552732113.9940835247111.6918795861129.0196181191121.7558249965123.7524273466121.4512806956118.7532874002112.9447540126134.0166321216131.6272531159106.5886366024119.2914126778134.9672814587112.9309979052112.2958522802109.8939222476119.143324203

118.4512442097121.5263140085115.8046487336114.3925104162138.1066752702126.5509539127114.5138597468121.0265011952139.7221197595102.6099156053119.5802909325119.9758983904109.7384441283117.6267190505121.3894577933112.2772985883130.8243057184123.170805484896.0188075015108.9164428168

120.326326699130.1674459074125.9976628108106.7688302402121.1604015526114.250492818587.9333996028133.7479673867133.5212303576116.429176008112.3225914245136.9685336004119.8535031409106.5187419346131.2750512926126.0582124206116.3229811328136.2544438354145.7336068898108.455078892

112.4259100226104.8448624107128.6001136879113.082428773107.7842810747137.6165485755121.3354110706116.972860572119.7869394974123.9642259253116.2524952934110.974629327134.9931111082117.3345438775107.2283524383120.9227278497122.1535242922129.1246874945108.8668298809110.9998905423

118.4520172802128.3384065874128.8090700956123.1482841223126.9516318035127.2121338235124.0554141378115.1219933791122.2381527742128.5934743765112.2246388451125.5743953453127.3227056419114.2946715237114.7585365589111.2684870651114.8626964397114.7374590192126.4602090788116.1291223372

121.7651473172126.245863915116.7228291098126.7484961619111.6518390819111.476033756115.3746282699123.3749984141107.2645730647123.3070591599115.5552311803122.7471969588119.0380842974130.6594825411111.6247134024132.3343397718112.1081780485131.2418774734107.4416062084105.7105194678

128.7989292297108.0305949773128.742722457197.337938617126.0904085331122.0721699912121.7402953745127.9525307228123.5357857087136.358535504127.9189931057116.7647681842122.5649228519122.4906739782107.5933099247136.0403033078112.7690168988126.9273028203108.5675606492117.8879600349

112.6972805042121.2536474969115.5670432428112.0548134469109.8286887401113.8163227853118.7494447851129.7319798442132.288637663122.3087977752132.7268322103111.8773369245125.343531484106.4366600377117.3329522618117.2257910486113.5727373668126.678533282119.8527300704123.4439153751

121.3323301573130.1060777524109.028856362116.3344407661108.3184500504132.1819084598132.198006516102.2222889634119.4562131178140.8723577089101.6722265678122.0971810955109.5514520176115.395046426109.923207977110.9100552031112.3246377875110.4167907516117.3931494424111.0969563643

129.7307520264100.207849148495.3741576569103.9486008568119.4845552465115.7092313707117.0863882481131.2692760013121.1858219295119.4270979187112.8906845525119.879514689499.073254508115.433290678299.9003489898130.8794438347125.1914980759109.6852159509115.4409304337118.7556066117

133.4170022648111.2606426733115.9495653457119.551187102103.367160894122.4402083909108.064973877130.4076889329132.6907934828107.3160504649124.1878365664115.7628460834108.7925923758110.3645857557124.6602963266108.0696577748131.8466005006131.4929434858139.0044374904108.6320210862

128.6256250142116.726057816113.1609295345114.3027660265120.8789470485108.9938862907118.8573108567107.4213699513134.7936589201129.9442331702114.8653112369127.2012198871102.4230144441107.5850335229109.0022763793108.2271595197110.2421905063134.4810655911128.1556891018125.704441719

136.509056877198.9835783304123.7795189201115.5349494485132.5013684738134.680381355118.6514694683103.8732492208111.0548012849109.0148728809108.4714497966119.9521833161114.2703425404135.5120233103113.104756868105.6013801036109.0092794885118.2690269462112.4584926703137.2766704054

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Munka4

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-2.62325556830.23997017711.51271706275.612173481536.7852180716-0.54783413365.638207767319.81929926843.037803253416.13548192844.9325080909-12.65257196792.3145275918-27.614987629919.2508878172-4.52563426735.541676273413.176338598-16.3441200129-6.9453335527

1.31162778426.67968151910.46025772679.7237034424-19.313779375811.9362084661-3.9464566726-10.103485692513.7147139867-18.357945919-19.1062099475-0.3971422302-5.9507442529-3.610443855-3.3750779949-8.067286216819.038066057-0.2491333362-5.2487166613-14.7664650285

4.77932644574.079765858514.2752242027-10.234475666931.1082203552-14.229431144510.2461626739-17.7558604264-10.44580812960.3009745342-13.35511115033.859065600419.06164470717.87135832077.2025386544-25.7934971162.01630427918.715142030310.346741419214.4241312228

4.964772415418.2825033335-9.553195923218.17479642342.1970663511-11.641895980612.1547827803-22.235201413414.2640601553-15.4711187861-14.511238077825.916392587510.2744252217-6.0329512053-0.6642494554-18.80362106029.404016054819.12096649919.671165318910.0848751572

13.166595636-3.609079612912.675786820223.4347226069-16.27498704691.009539119-13.65244770570.394993549-9.84985035767.04737885831.592627541-1.3035332813-11.15913619293.235913936810.72345412477.9975052358-10.663995908530.3499291476-5.74718797-5.9931380747

-8.2580754679-4.786215868127.68429112626.057791779325.54825186966.474692781924.3666818278-17.430056686915.869704839112.2710389405-11.41524990095.56867689742.6321799851-13.082308214536.2315859093-5.1453071137-15.4461304192-9.86210579873.54664280179.8416649053

8.323013389610.3633396975-6.0751403907-17.3151875060.3971308615-38.0530764232-5.468325525724.8358674071-25.2146264756-12.3453901324-13.446845059725.5904524238-2.0936681722-11.1541567094-28.170211408-7.39798906584.374919626626.80312718442.2434051061-7.4700437835

-4.2145302359-9.3101789389-12.8005353872-2.755416517220.0791987481-1.9230583348-5.598508323610.203257261314.75991666672.02928731612.4166717796-15.155865185119.56989308384.2652118282-32.3771018884-1.2655505088-12.3488462123-10.9675738713-5.59853106115.6116698236

17.45731879052.922922703927.7768776868-5.5898453866-17.68430593076.1895093495-0.96663370640.207364792-0.1396074367-10.1648197415-2.065462467726.5862354354-4.505091055822.2569724428-10.8366180029-17.9800622391-7.7553295341-0.4976755008-9.5003770184-3.9825522435

-24.089058570110.50591436077.2683860708-18.0000370165-12.539612726-14.684326288315.8726606969-12.422015061-9.7823544827.6344235822-3.067304987820.779202714-5.349306775325.926283342412.24668721997.928588274826.0794195128-6.055552148612.2521669255-29.0545813186

6.6064330895-9.369773579217.2001364263-1.557464202131.32722667947.045571237514.8002300193-18.92940417759.605787454311.998110949216.2162450579-0.026045654517.954994291437.2416252503-1.04291757455.0158519116-31.2477595799-12.960344975-33.7225628755-3.872560228

23.79817942714.137780251816.6083282238-15.9607452588-3.3529204302-12.93108198311.4430577014-9.5390987553-5.467472874410.3730144474-2.651609065717.79260401249.503969522522.953167899716.407500425-19.694118691425.8619138549-18.9047455024-14.53514641982.2604695005

0.61944138-2.13589260061.5544335383-1.19437140710.5410743142-0.57815123-0.810128995-1.89118350140.71992121781.8516623186-1.62547530642.7982234769-0.8195106602-1.4678330444-0.8602878552-1.5084215192-2.56124121733.0468831937-3.4316660731-1.3695444068-0.4561034189

12.593854298614.247027306615.017038910414.434200585717.366292384914.024637103415.273651902814.720203811816.403048304612.585145380816.0174192613.780380948214.485330040415.754726209615.076336357615.618381282314.767007762814.241733260616.013873868213.214411645514.7817350312

Munka2

Munka3

*Mekkora lehet a vletlen ingadozs?A mintk tlagai 120 krl ingadoznak, ha nem trtnik semmi, csak sima ismtlsKt mrs klnbsgnek tlaga a 0 krl ingadozikMekkora az a klnbsg, amit mr nem a vletlen okoz?Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Hipotzisek

Nullhipotzis: vletlen ingadozst mrtem, semmi nem trtnt.A klnbsg 0 krl ingadozikAlternatv hipotzis: a vletlen ingadozsnl nagyobbat mrtem, valami trtntA klnbsg 0-tl eltr szm krl ingadozik???

0Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*A nullhipotzis fennllsa esetn ismerjk a klnbsg-tlag eloszlstHa igaz a nullhipotzis (igazbl nincs klnbsg a populci tlagok kztt), a t-rtk nagy valsznsggel (95%) ide esik.A kimarad valsznsg 5% ()0Pontosabban: az abbl szmolt t-statisztika eloszlst ismerjkt=klnbsg tlag/klnbsg SEBiostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Statisztikai prba: kiszmtjuk adatainkbl a t=tlag/SE prbastatisztikt s megnzzk, hova esikDnts: az eltrs nem szignifikns|t|ttblaBiostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Dntsi szably p-rtk alapjnp-rtk: a mi ltalunk szmtott t-rtk ltal az eloszlsbl levgott terlet nagysgaAnnak valsznsge, hogy ha igaz a nullhipotzis (=nincs hats), a tapasztalt eltrst vagy annl mg nagyobb eltrst kapjunkAnnak valsznsge, hogy a puszta vletlen legalbb ekkora klnbsget okoz

p, a klnbsg nem szignifikns adott szintenBiostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Mirt ppen 5%?Oka*:Fisher a nullhipotzis ellen szl bizonytk erssgt mr indexnek tekintette a p-rtket A p

*A konfidenciaintervallum fogalma(bizonyossgi intervallum)Olyan, a mintaelemekbl szmolt intervallum, amely nagy valsznsggel tartalmazza a populci-paramter valdi (ismeretlen) rtktPl. 95%-os bizonyossgi intervallum az tlagra: olyan, a mintaelemekbl szmolt intervallum, ami 95% valsznsggel tartalmazza a populci valdi tlagt.A megbzhatsg mrtkt jelz valsznsg (megbzhatsgi szint) tlnk fgg. Szoksos rtkei: 0.90, 0.95, 0.99 )A becsls hibja (-val jelljk) a megbzhatsgi szint fggvnyben 1-0.90=0.1, 1-0.95=0.05, 1-0.99=0.01Leggyakrabban hasznlt megbzhatsgi szint 95% (0.95), teht -ra leggyakrabban =0.05 rtket alkalmazzk.

Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Az tlagra vonatkoz 95%-os bizonyossgi intervallum szemlltetse az adott ksrlet kpzeletbeli ismtlseivelHa a ksrletet kpzeletben 100-szor megismtelnnk, a 100 kapott 95%-os konfidencia intervallum kzl vrhatan 95 fogja tartalmazni a populci tlagt, s 5 nem.

http://www.kuleuven.ac.be/ucs/java/index.htm

Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Szignifikancia s a konfidenciaintervallumA mintt az adott (tlag) populcibl vettk. A populcitlag =0. Ez lehet pl. annak a nullhipotzisnek megfelel eloszls, hogy a kezels eltti s a kezels utni populcitlag ugyanaz. A konfidenciaintervallum tartalmazza az adott tlagot jelen esetben a nullt. Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Szignifikancia vizsglatok s a konfidenciaintervallum kapcsolata (H0: 1= 2, azaz 1- 2 =0, Ha: 1 2)

p-rtkszignifikancia 95% CI (p1 p2)-rep 0.05nem szign. 5%-os sz.pl. (-1.72, 5.81)0 benne van a konf. intervallumban

Megjegyzs. Ha relatv kockzatot vagy eslyhnyados vizsglunk, akkor a konfidenciaintervallumban az 1-et keressk0Nem szignifikns, p>0.05Szignifikns, p

*Ha van kt adatsorom, mikor kell t-prbt (s akkor melyiket), khi-ngyzet prbt, vagy korrelcit stb szmtani???Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*A prba megvlasztsa fggAz analzis cljtl (sszehasonlts vagy kapcsolat)Az adatok tpustlA ksrleti elrendezstl.Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*t-prbatlagokat hasonlt. Akkor alkalmazzuk, ha kt csoportban folytonos vltozink vannak, amelyekbl van rtelme tlagot szmolni (pl. kor, vrnyoms, stb.)Felttele: Normalits

Szoksos brkBiostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*2 (khi-ngyzet) prbaGyakorisgokat (%-okat),(gyakorisgi eloszlsokat hasonlt)

Szoksos brkBiostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

Eljrs

Sikeres

Sikertelen

Sikeres %

Hagyomnyos

10

20

33.33 %

j eljrs

19

11

63.30 %

Diagram4

0.333

0.633

Diagram2

49

58

55

48

50

54

49

60

58

48

50

54

50

53

52

55

45

56

55

55

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65

53

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60

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56

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kvnatos sly

Diagram3

52

58

53

52

57

56

47

65

60

48

53

54

50

56

60

57

40

60

60

54

44

50

68

51

60

64

55

62

66

50

52

59

sly

testly

testmagassg

Munka1

nemmagassgsly

lny165.0052.00hagyomnyos, n=3033.30%

lny171.0058.00j, n=3063.30%

lny171.0053.00

lny160.0052.00

lny160.0057.00

lny175.0056.00

lny160.0047.00

lny173.0065.00

lny174.0060.00

lny162.0048.00

lny165.0053.00

lny171.0054.00

lny165.0050.00

lny173.0056.00

lny160.0060.00

lny170.0057.00

lny150.0040.00

lny170.0060.00

lny162.0060.00

lny167.0054.00

lny156.0044.00

lny164.0050.00

lny169.0068.00

lny173.0051.00

lny170.0060.00

lny169.0064.00

lny168.0055.00

lny169.0062.00

lny165.0066.00

lny172.0050.00

lny170.0052.00

lny172.0059.00

fi183.0072.00

fi170.0073.00

fi182.0075.00

fi179.0080.00

fi174.0070.00

fi184.0075.00

fi172.0065.00

fi181.0079.00

fi180.0089.00

fi184.0085.00

Munka1

Munka2

Munka3

Diagram6

0.333

0.633

Diagram2

49

58

55

48

50

54

49

60

58

48

50

54

50

53

52

55

45

56

55

55

45

50

65

53

58

60

56

58

56

55

52

58

kvnatos sly

Diagram3

52

58

53

52

57

56

47

65

60

48

53

54

50

56

60

57

40

60

60

54

44

50

68

51

60

64

55

62

66

50

52

59

sly

testly

testmagassg

Munka1

nemmagassgsly

lny165.0052.00hagyomnyos, n=3033.30%

lny171.0058.00j, n=3063.30%

lny171.0053.00

lny160.0052.00

lny160.0057.00

lny175.0056.00

lny160.0047.00

lny173.0065.00

lny174.0060.00

lny162.0048.00

lny165.0053.00

lny171.0054.00

lny165.0050.00

lny173.0056.00

lny160.0060.00

lny170.0057.00

lny150.0040.00

lny170.0060.00

lny162.0060.00

lny167.0054.00

lny156.0044.00

lny164.0050.00

lny169.0068.00

lny173.0051.00

lny170.0060.00

lny169.0064.00

lny168.0055.00

lny169.0062.00

lny165.0066.00

lny172.0050.00

lny170.0052.00

lny172.0059.00

fi183.0072.00

fi170.0073.00

fi182.0075.00

fi179.0080.00

fi174.0070.00

fi184.0075.00

fi172.0065.00

fi181.0079.00

fi180.0089.00

fi184.0085.00

Munka1

Munka2

Munka3

*Korrelci-regresszi Kt folytonos vltoz lineris kapcsolataBiostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

Diagram2

49

58

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54

49

60

58

48

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54

50

53

52

55

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56

55

55

45

50

65

53

58

60

56

58

56

55

52

58

kvnatos sly

jelnelegi sly

kvnatos sly

y = 0.6144x + 19.706R2 = 0.7374

Munka1

nemslykvnatos sly

lny52.0049.00

lny58.0058.00

lny53.0055.00

lny52.0048.00

lny57.0050.00

lny56.0054.00

lny47.0049.00

lny65.0060.00

lny60.0058.00

lny48.0048.00

lny53.0050.00

lny54.0054.00

lny50.0050.00

lny56.0053.00

lny60.0052.00

lny57.0055.00

lny40.0045.00

lny60.0056.00

lny60.0055.00

lny54.0055.00

lny44.0045.00

lny50.0050.00

lny68.0065.00

lny51.0053.00

lny60.0058.00

lny64.0060.00

lny55.0056.00

lny62.0058.00

lny66.0056.00

lny50.0055.00

lny52.0052.00

lny59.0058.00

fi72.0084.00

fi73.0072.00

fi75.0075.00

fi80.0078.00

fi70.0070.00

fi75.0080.00

fi65.0066.00

fi79.0071.00

fi89.0090.00

fi85.0084.00

Munka2

Munka3

*A hipotzisvizsglat meneteHipotzisek fellltsaNullhipotzis: semmi nem trtntAlternatv hipotzis: valami vltozs vanA dnts megbzhatsga (vagy a hiba) rgztse: =0.05Dntsi szably fellltsa (fgg: a ksrleti elrendezstl, -tl, az elemszmtl)DntsA nullhipotzist elfogadjuk (nincs szignifikns klnbsg szinten, nincs elegend informci a klnbsg (hats) kimutatsra)A nullhipotzist elvetjk, a klnbsg szignifikns %-os szinten. A tapasztalt klnbsg nem csupn a vletlen mve, valami ms hats (kezels??) is kzbejtszott.

Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Normlis eloszlst felttelezve,az tlagok sszehasonltsra hasznlhat prbkEgy minta esete: egymints t-prbaKt minta esete:sszetartoz mintk: (eltt-utn, baloldal-jobboldal): pros t-prba= egymints t-prba a klnbsgekreFggetlen mintk (placebo-kezels, frfi-n, beteg-egszsges): ktmints t-prbaAzonos szrsok esetn klasszikusKlnbz szrsok esetn mdostott (Welch, D)Szrsok egyezsnek tesztelse: F-prba, Levene-prbaTbb (>2) minta esete: varianciaanalzisBiostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Egymints t-prbaEgy kezels sorn szksgess vlt annak ellenrzse, hogy az milyen hatssal van a vrnyomsra. A vizsglt paciensek korcsoportjban a systols vrnyoms normlrtke 120. n=9 szemlyt megmrve a kvetkez rtkeket kaptk: 182.00 152.00 178.00 157.00 194.00 163.00 144.00 114.00 174.00 (tlag=162, SD=23.92 .Mondhatjuk-e a mintaadatok alapjn, hogy az adott korcsoport populcijban az tlagos vrnyoms 120 Hgmm? Nullhipotzis (HO): A populci tlag 120, =120Alternatv hipotzis (Ha): A populci tlag nem 120 , 120 (ktoldalas)Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Dntsi szably a konfidencia intervallum alapjnKonfidencia intervallum alapjn: adjuk meg a populci-tlagra vonatkoz 95%-os konfidencia intervallumot! A konfidencia intervallum: (tlag - t*SE, tlag + t * SE )= (162-2.306*23.92/9, 62+2.306*7.97)=(143.61,180.386)Dntsi szably a konfidencia intervallum alapjn: benne van-e az adott szm a konfidencia-intervallumban? Ha igen: a klnbsg nem szignifikns adott szintenHa nem: a klnbsg szignifikns adott szinten

Esetnkben 120 nincs benne a konfidencia intervallumban, teht a klnbsg szignifikns 5%-os szinten Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Dntsi szably a t-rtk alapjnSzmtsuk ki a t-rtket: t= (tlag - c)/SE=(162-120)/7.97=5.26. szabadsgfok: n-1=9-1=7s hasonltsuk az abszolt rtkt a t8,0.05=2.306 tblzatbeli rtkhez :5.26>2.306Dntsi szably: ha |t|>ttbla, a klnbsg szignifikns adott szintenAz elfogadsi tartomny a vltoz azon rtkeinek halmaza, amelyekre elfogadjuk a nullhipotzist (- ttbla ,ttbla)A kritikus tartomny ennek ellentettje. A kritikus tartomny rtkeire a nullhipotzist nem fogadjuk el.Esetnkben a klnbsg szignifikns 5%-os szinten

t=5.26Elfogadsi tartomnyBiostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Dntsi szably a p-rtk alapjnp-rtk: a mi ltalunk szmtott t-rtk ltal az eloszlsbl levgott terlet nagysgaAnnak valsznsge, hogy ha igaz a nullhipotzis (=nincs hats), a tapasztalt eltrst vagy annl mg nagyobb eltrst kapjunkHa a p

*Pros t-prbankontrollos ksrlet, vagyMs mdon sszetartoz adatok:Jobb oldal-bal oldalIllesztett prok- matched pairs (klnbz szemlyek, de a ksrlet szempontjbl prba llthatk)Nullhipotzis: a kt minta-tlag ugyanannak a populci-tlagnak a kzeltse, (nincs kezels-hats, a tapasztalt klnbsg vletlen)Alternatv hipotzis: a kt minta-tlag kt klnbz populci-tlagnak a kzeltse (van hats)Dntsi szably: Konfidenica intervallum a klnbsgret-rtk szmts s sszehasonts a tblzattalp-rtk (szoftver)Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Pros t-prba, plda (folyt).Gondolatmenet: ha a kezels nem hatsos, az tlagos klnbsg kicsi (kzel 0). Ha a dita hatsos, az tlagos klnbsg nagy. A populcira nzve ez a kvetkez hipotziseket jelenti:HO: eltt= utn or klnbsg= 0 (c=0)!!HA: eltt utn or klnbsg 0Legyen =0.05. A szabadsgfok=10-1=9, ttblzat=t0.05,9=2.262tlag=4, SD=3.333SE=3.333/10=1.054Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls Dnts a konfidenciaintervallum alapjn: 95%CI: (4-2.262*1.054, 4+2.262*1.054)=(1.615, 6.384)Ha H0 igaz, akkor a 0 benne van a konfidencia-intervallumbanMost0 nincs benne a 95%-os konfidencia-intervalluman, ezrt dntsnk az, hogy a klnbsg szignifikns 5%-os szinten, a kezels hatsos voltAz tlagos slyvesztesg 4 kg, ami akr 6.36 is lehetne, de minimum 1.615, 95% valsznsggel.

*Pros t-prba, plda (folytats)Dnts a prbastatisztika alapjn (t-rtk:

Azt hasonltjuk a tblabeli kritikus rtkhez.|t|=3.795>2.262(=t0.05,9), a klnbsg szignifikns 5%-os szinten Dnts p-rtk alapjn: p=0.004, p

*Plda az orvosi irodalomblV. Lindn: Vitamin D and Myocardial Infarction. BMJ 1974,3,647-650Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Ksrleti elrendezsekBiostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls Pros t-prbaMinden egyedet ktszer vizsglunk1.2.x1y1x2 y2 xn yn

Ktmints t-prbaMinden egyedet csak egyszer vizsglunk, s mindegyik egy s csak egy csoportban lehet.CsoportMrsek1x11x21xn2y12y22ymAz elemszm nem felttlenl azonos a kt mintban

*Ktmints t-prbaKt fggetlen minta sszehasonltsaFelttelek:A mintk fggetlenek: normlis eloszls populcibl szrmaznak:az xi-k N(1,) s az yi--k N 2, ) eloszls populciblH0: 1=2, Ha: 12 PrbastatisztikaKlnbz variancik esetn:

Dnts:Ha |t|>t,szab.fok, a klnbsg szignifikns szinten, H0-t elvetjk

.Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Ktmints t-prbaKt fggetlen minta sszehasonltsaFelttelek:A mintk fggetlenek: normlis eloszls populcibl szrmaznak:az xi-k N(1,) s az yi--k N 2, ) eloszls populciblH0: 1=2, Ha: 12 PrbastatisztikaAzonos variancik esetn:

Szabadsgfok: n+m-2Dnts:Ha |t|>t,szab.fok, a klnbsg szignifikns szinten, H0-t elvetjk

.Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*A variancik sszehasonltsaH0: 21=22Ha:21 > 22 vagy 21 > 22 (egyoldal prba)A prbastatisztika (F): a nagyobbik standard devici ngyzett osztjuk a kisebbel:

Szabadsgfokok: nagyobb SD-hez tartoz minta elemszma-1Kisebb SD-hez tartoz minta elemszma-1Dnts: F tblzat alapjnHa F>F,tblzat, a kt variancia szignifiknsan klnbzik szintenBiostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Az F-eloszls tblzata (rszlet)=0.05Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*PldaBiostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

Kontroll csoport

Kezelt csoport

170

120

160

130

150

120

150

130

180

110

170

130

160

140

160

150

130

120

n=8

n=10

=162.5

=128

sx=10.351

sy=11.35

sx2=107.14

sy2=128.88

A szmtott t rtk = 6.6569 ,a kritikus t rtk, t0.025,16=2.12. Mivel 6.6569>2.12, elvetjk a nullhipotzist s azt lltjuk, hogy a kt tlag kztti klnbsg szignifikns 5% -os szinten.

,

A szmll szabadsgfoka 10-1=9, a nevez 8-1=7, a 9,7 szabadsgfokokhoz tartoz kritikus F rtk F

*Eredmny SPSS-selBiostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls A variancik sszehasonltsa.p=0.93>0.05, nem szignifikns.Elfogadjuk a variancik azonossgtAz tlagok sszehasonltsa (t-prba).Els sor: egyenl variancik esete. t=6.657, df=16, p

*Kategrikus vltozk elemzse2 prbk (khi-ngyzet)Kt kategris vltoz kapcsolata.Pl. 2 kzti ellenrzs sorn az ittasok arnya 1.5% ill. 0.5 %Pl. a sikeres mttek gyakorisga 2 fle eljrs esetn 33.3% s 66.6%*.

Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

Eljrs

Sikeres

Sikertelen

Sikeres %

Hagyomnyos

10

20

33.33 %

j eljrs

19

11

63.30 %

sszes vizsglt

Ittas

Nem ittas

Ittas%

I.

600

9

1.5%

II.

400

2

0.5%

*2x2-es tblzatokA gyakorisgok kontingencia tblzatba rendezhetk.2 formula knnyen szmthatYates korrekci: pontosabb kzeltsHa a vrt gyakorisgokra vonatkoz felttel nem teljesl:Fisher fle egzakt prbaBiostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

Vizsglt tnyez

1. csoport

2.csoport

sszes

Van +

a

b

a+b

Nincs -

c

d

c+d

sszesen

a+c

b+d

N

_1176816025.unknown

_1176816134.unknown

*2 prba az 1*. plda adatairaNullhipotzis: Az ittassg arnya azonos a kt vizsglatban, az ittassg fggetlen attl, hogy mikor nztk.Gyakorisgi tblzat (amibl szmolunk)

A fggetlensg esetn vrhat gyakorisgok

2=2.205, p=0.137 , nem szignifikns 5%-os szinten. Felttel: (vrt gyakorisg

*2 prba a 2. plda adatairaNullhipotzis: A siker valsznsge azonos a kt eljrs esetn.Az eljrs sikeressge fggetlen attl, hogy j vagy hagyomnyos eljrsrl van-e sz.Gyakorisgi tblzat (amibl szmolunk)

2=5.406, p=0.02 , szignifikns 5%-os szinten. Felttel: (vrt gyakorisg

*2x3-as kontingencia tblzatKt kategris vltoz kapcsolata.Pl. diabetes gyakorisga a 3 csoportban csoportban: 31%, 27% s 25%*.Krds: klnbzik-e a diabetes gyakorisga az egyes csoportokban?

Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

DIABCS1Cs2Cs3Totalyes31272583no697375217Total100100100300

*A 2 prba vgrehajtsa, feltteleiH0 teljeslse esetn a vrt gyakorisgok szmtsa (Ei=sorsszeg*oszlopsszeg/total)2 statisztika szmtsa: 2 =(Oi-Ei)2/EiSzabadsgfok: sorok szma-1)*(oszlopok szma-1)Dnts tblzat alapjn: 2 > 2 tblzat, , dfFelttel: 5-nl kisebb vrt gyakorisgot tartalmaz cellk szma max. 20%Egzakt tesztek (Fisher): nincs felttele, a pontos p-rtket adjk

2 =0.933Df=(3-1)*(2-1)=2

0.933

*Korrelci-regresszi: kt folytonos vltoz kztti (lineris) kapcsolat vizsglata. A kapcsolat vizsglathoz elszr ksztsnk brt (pont bra vagy szrds-bra).Egy ilyen brn a kapcsolat irnyt s szorossgt vizsgljuk, valamint az ltalnos alakzatot.A kt vltoz mrt rtkeivel mint koordintkkal berajzoljuk a megfelel pontokat.

Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

*Lehetsges kapcsolatokPozitv korrelciNegatv korrelciNincs korrelciBiostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

Diagram1

550

535

535

520

455

420

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math

math score

language

adatok

Nameretailtheatermathlanguage

Pat5130525550

Sue5560515535

Inez5890510535

Amie6350495520-0.2157385322

Gene85304304550.7413007121

Bob9590400420

Joe1608004100.9989149375

-0.258397254

adatok

math

math score

theater

Munka2

math

math score

theater

Munka3

math

math score

language

math

math score

language

math

math score

retailing

xy

-3-0.1411200081

-2.8-0.3349881502

-2.6-0.5155013718

-2.4-0.6754631806

-2.2-0.8084964038

-2-0.9092974268

-1.8-0.9738476309

-1.6-0.999573603

-1.4-0.98544973

-1.2-0.932039086

-1-0.8414709848

-0.8-0.7173560909

-0.6-0.5646424734

-0.4-0.3894183423

-0.2-0.1986693308

00

0.20.1986693308

0.40.3894183423

0.60.5646424734

0.80.7173560909

10.8414709848

1.20.932039086

1.40.98544973

1.60.999573603

1.80.97384763090.7964355021

20.9092974268

2.20.80849640380.1574204489

2.40.6754631806

2.60.5155013718

2.80.3349881502

30.1411200081

y

Diagram2

51

55

58

63

85

95

math

math score

retailing

adatok

Nameretailtheatermathlanguage

Pat5130525550

Sue5560515535

Inez5890510535

Amie6350495520-0.2157385322

Gene85304304550.7413007121

Bob9590400420

Joe1608004100.9989149375

-0.258397254

adatok

math

math score

theater

Munka2

math

math score

theater

Munka3

math

math score

language

math

math score

language

math

math score

retailing

xy

-3-0.1411200081

-2.8-0.3349881502

-2.6-0.5155013718

-2.4-0.6754631806

-2.2-0.8084964038

-2-0.9092974268

-1.8-0.9738476309

-1.6-0.999573603

-1.4-0.98544973

-1.2-0.932039086

-1-0.8414709848

-0.8-0.7173560909

-0.6-0.5646424734

-0.4-0.3894183423

-0.2-0.1986693308

00

0.20.1986693308

0.40.3894183423

0.60.5646424734

0.80.7173560909

10.8414709848

1.20.932039086

1.40.98544973

1.60.999573603

1.80.97384763090.7964355021

20.9092974268

2.20.80849640380.1574204489

2.40.6754631806

2.60.5155013718

2.80.3349881502

30.1411200081

y

Diagram3

30

60

90

50

30

90

math

math score

theater

adatok

Nameretailtheatermathlanguage

Pat5130525550

Sue5560515535

Inez5890510535

Amie6350495520-0.2157385322

Gene85304304550.7413007121

Bob9590400420

Joe1608004100.9989149375

-0.258397254

adatok

math

math score

theater

Munka2

math

math score

theater

Munka3

math

math score

language

math

math score

language

math

math score

retailing

xy

-3-0.1411200081

-2.8-0.3349881502

-2.6-0.5155013718

-2.4-0.6754631806

-2.2-0.8084964038

-2-0.9092974268

-1.8-0.9738476309

-1.6-0.999573603

-1.4-0.98544973

-1.2-0.932039086

-1-0.8414709848

-0.8-0.7173560909

-0.6-0.5646424734

-0.4-0.3894183423

-0.2-0.1986693308

00

0.20.1986693308

0.40.3894183423

0.60.5646424734

0.80.7173560909

10.8414709848

1.20.932039086

1.40.98544973

1.60.999573603

1.80.97384763090.7964355021

20.9092974268

2.20.80849640380.1574204489

2.40.6754631806

2.60.5155013718

2.80.3349881502

30.1411200081

y

*Mrszm a lineris kapcsolat szorossgnak mrsre: a korrelcis egytthat (r ) s tulajdonsgaiA korrelcis egytthat rtke mindig -1 s +1 kztt van; -1 s 1 jelzi a tkletes lineris kapcsolatot. -1r 1.a) Ha r kzel van +1-hez vagy -1-hez, azt mondjuk, hogy szoros (magas) korrelci van a kt vltoz kztt.

b) Ha r=1, tkletes pozitv korrelci Ha r= -1, tkletes negatv korrelci.

c) Ha r=0, nincs korrelci, vagyis nincs lineris kapcsolat. Ha r kzel van 0-hoz, akkor alacsony korrelcirl beszlnk.

Biostatisztikai alapismeretek Boda Krisztina Ler statisztika A vletlen ingadozs Konfidenciaintervallum Egyvltozs mdszerek Tbbszrs sszehasonltsok Varianciaanalzis Diagnosztikus tesztek Tbbvltozs mdszerek Elemszmbecsls

Diagram4

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theater

adatok

Nameretailtheatermathlanguage

Pat5130525550

Sue5560515535

Inez5890510535

Amie6350495520-0.2157385322

Gene85304304550.7413007121

Bob9590400420

Joe1608004100.9989149375

-0.258397254

adatok

math

math score

theater

Munka2

math

math score

theater

Munka3

math

math score

language

math

math score

language

math

math score

retailing

xy

-3-0.1411200081

-2.8-0.3349881502

-2.6-0.5155013718

-2.4-0.6754631806

-2.2-0.8084964038

-2-0.9092974268

-1.8-0.9738476309

-1.6-0.999573603

-1.4-0.98544973

-1.2-0.932039086

-1-0.8414709848

-0.8-0.7173560909

-0.6-0.5646424734

-0.4-0.3894183423

-0.2-0.1986693308

00

0.20.1986693308

0.40.3894183423

0.60.5646424734

0.80.7173560909

10.8414709848

1.20.932039086

1.40.98544973

1.60.999573603

1.80.97384763090.7964355021

20.9092974268

2.20.80849640380.1574204489

2.40.6754631806

2.60.5155013718

2.80.3349881502

30.1411200081

y

Diagram5

550

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math

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language

adatok

Nameretailtheatermathlanguage

Pat5130525550

Sue5560515535

Inez5890510535

Amie6350495520-0.2157385322

Gene85304304550.7413007121

Bob9590400420

Joe1608004100.9989149375

-0.258397254

adatok

math

math score

theater

Munka2

math

math score

theater

Munka3

math

math score

language

math

math score

language

math

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retailing

xy

-3-0.1411200081

-2.8-0.3349881502

-2.6-0.5155013718

-2.4-0.6754631806

-2.2-0.8084964038

-2-0.9092974268

-1.8-0.9738476309

-1.6-0.999573603

-1.4-0.98544973

-1.2-0.932039086

-1-0.8414709848

-0.8-0.7173560909

-0.6-0.5646424734

-0.4-0.3894183423

-0.2-0.1986693308

00

0.20.1986693308

0.40.3894183423

0.60.5646424734

0.80.7173560909

10.8414709848

1.20.932039086

1.40.98544973

1.60.999573603

1.80.97384763090.7964355021

20.9092974268

2.20.80849640380