C. Statistics. X = n geometric the n th root of the product of numerical data ln X GM = or ln GM = X 2. X n X 1

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1 C. Statstcs a. Descrbe the stages the desg of a clcal tral, takg to accout the: research questos ad hypothess, lterature revew, statstcal advce, choce of study protocol, ethcal ssues, data collecto ad processg. b. Expla cocepts statstcs such as: dstrbuto of data ad frequecy dstrbutos, measures of cetral tedecy ad dsperso of data ad the approprate selecto ad applcato of o-parametrc ad parametrc tests statstcal ferece. data types omal a lst of possble results e.g. death/dscharge/trasfer to aother sttuto ordal a ordered groupg of results o a scale wth dscrete pots e.g. ASA status, Duke s stagg umercal terval equal tervals betwee values but o absolute zero e.g. temperature C rato a lear scale from a absolute zero e.g. mea arteral pressure parametrc data whch are dstrbuted ormally varable a measuremet of a sample parameter a measuremet of the populato measuremet of cetral tedecy mea arthmetc X the average of umercal data: X = geometrc the th root of the product of umercal data l X GM = X 2 X or l GM = ot applcable to omal or ordal data affected by outlers meda the mddle result rak order mode the most commo result data o a dscrete scale dstrbutos may have more tha oe mode measuremet of varablty rage the dfferece betwee largest ad smallest values terquartle (25%-75%) or 5%-95% rages are sometmes quoted sutable for ordal or terval data varace (X X ) 2 var = Statstcs 2.C.1 James Mtchell (December 24, 2003)

2 sutable for terval data oly stadard devato (X X ) 2 s = 1 s used for descrptos of a sample ad -1 for descrptos of a populato derved from a sample both varace ad stadard devato are a fucto of the populato ad do ot chage wth sample sze comparg varaces of samples s a test to see f they are from the same populato: F test coeffcet of varato the stadard devato as a percetage of the mea allows comparso of the degree of varace betwee measuremets of dfferet quattes depedet of sample sze ad mea value CV = s X F-test varaces of data sets ca also be compared usg the F-test larger varace s dvded by smaller varace tables provde a cofdece lmt for the rato obtaed that the data sets are from the same populato stadard error of the mea descrbes the relatoshp betwee the sample mea ad the populato mea SEM = s for a ormally dstrbuted data set, the populato mea has a probablty of 96% of fallg wth 2xSEM of the sample mea ths s used to provde cofdece tervals from sample data cofdece tervals are ow preferred to p values SEM falls wth creasg sample sze hypothess testg ull hypothess (H 0 ) there s o dfferece betwee groups studed alterate hypothess (H 1 ) there s a dfferece betwee groups studed type I error false cocluso that H 0 s false based o sample (spurous sgfcat result) probablty of a type I error a gve study s called α probablty that a type I error has occurred s called p type II error false cocluso that H 0 s true based o sample (mssg a real dfferece) probablty of a type II error a gve study s called ß ß vares versely wth α, depedg o study desg sgfcace a arbtrary decso as to the maxmum p value acceptable as evdece that H 0 s false typcally 0.05 for bologcal studes power probablty of fdg H 0 s false gve that t really s false (fdg a real dfferece) power = 1 - ß creases wth sample sze, α, parametrc (>o-parametrc) aalyss dstrbutos bomal Statstcs 2.C.2 James Mtchell (December 24, 2003)

3 descrbes the probablty dstrbuto for a fxed umber of depedet evets wth two possble outcomes of costat probablty probablty of x succeses each of probablty π from trals s P(x;, π) = C x π x (1 - π) -x for large values of, ths approaches ormal dstrbuto posso descrbes dstrbuto of results where a evet occurs wth a kow frequecy (λ) at radom tervals ormal a symmetrcal dstrbuto represetg the lmt of the bomal dstrbuto as approaches parametrc tests requre ormal dstrbuto smlar varace betwee data sets (F-test) depedet data sets Studet s t-test compares two groups from the same populato to detect a dfferece meas at a specfed level of sgfcace t = X 2 SE 2 + SE X 2 2 or t = X 2 var p ( ) where var p = var 1 + var 2 ( 1 1) +( 2 1) prcple, t s a mea dvded by a SEM t s a expresso of the spread of the SEM dstrbuto for a gve sample sze (degrees of freedom). There s a 95% probablty that the populato mea les wth t x SEM of the sample mea. t approaches 1.96 as sample sze approaches, ad s approxmated to 1.96 for >30. the result t s compared wth tables gvg mmum values of t for specfed sgfcace ad degrees of freedom ca be performed o pared (depedet) or upared data, oe- or two-taled depedg o whether the alteratve hypothess postulates a drecto for chage of the mea ANOVA (aalyss of varace) compares the varace of multple groups agast the depedet varable (oe-way ANOVA) or agast each other (Multple ANOVA) by calculatg a F-rato = betwee-groups varace wth-groups varace ths detfes the presece of a dfferece but ot whch groups cause t multple comparso tests used to detfy dfferet groups from ANOVA based o t-test but modfed to dmsh the rsk of type I error may epoymous varetes Boferro, Newma-Keuls, Duca, Duett, Du, Tukey, Scheffe, Least sgfcat dfferece etc. o-parametrc tests sutable for ordal data ad data whch s ot ormally dstrbuted less power tha parametrc tests Wlcoxo sged raks test pared t-test Ma-Whtey u test 2-taled t-test two data sets to compare all data are raked 1 to rak sums R 1 ad R 2 Statstcs 2.C.3 James Mtchell (December 24, 2003)

4 U = / 2 ( 1 ( 1 + 1)) - R 1 U s compared wth values for a specfed α ad degrees of freedom Kruskall Walls test ANOVA Fredma s test Multple ANOVA Spearma rak order r χ 2 test used for omal data compares rates of depedet evets for sgfcat dfferece requres expected rates of more tha 5 evets 80% of cells (otherwse use Fsher s Exact Test) χ 2 (O E) 2 = where E s the expected umber ad O the observed umber of E evets tables of χ 2 values for levels of sgfcace ad degrees of freedom degrees of freedom = (tervetos - 1) x (outcomes - 1) Vomtg No Vomtg Total Atemetc Placebo Total E vomtg = 20 each group, E o vomtg = 80 each group χ 2 = = 12.5, df = 1 Yates correcto for 2x2 matrces wth fewer tha 40 trals reduces the magtude of (O - E) by 0.5 for each term χ 2 Fsher s exact test for small sample szes 2x2 matrces calculates p regardless of df usg the bomal dstrbuto p = R 1!R 2!C 1!C 2!! 11! 12! 21! 22! for the example p= oerous to calculate for large McNemar s test s used for matched data. odds rato the rato of the cdece of a outcome a exposed group versus a cotrol group a case-cotrol study called a rsk rato a prospectve cohort study a cofdece terval assocated wth a odds rato does ot spa 1 f the correlato s sgfcat odds rato descrbes the stregth of assocato as well as ts presece regresso ad correlato regresso s a mathematcal descrpto of the assocato betwee two varables the depedet varable s plotted o the Y axs lear regresso produces a relatoshp y=ax+b such that (Y - ax +b) 2 for the set of data pots s mmzed correlato s a descrpto of the tghtess wth whch a data set matches a regresso (X a = X )(Y Y ) (X X ) 2 b = Y ax the coeffcet of determato, r 2 expresses stregth of correlato (from 0 to 1) (ax r 2 = + b Y ) 2 (Y Y ) 2 the correlato coeffcet s r (varyg from -1 to +1) r 2 expresses the proporto of the value of the depedet varable attrbutable to the depedet varable Regresso assumes a costat varace over the rage of data. If varace s ot Statstcs 2.C.4 James Mtchell (December 24, 2003)

5 costat, a correcto factor s added to the least squares calculato. Where the ature of the relatoshp betwee depedet ad depedet varables s kow to be of a o-lear ature (e.g. dose-respose), a trasform ca be appled to the data to reder the relatoshp lear (e.g. Hll plot or Leweaver-Burke plot). sequetal aalyss sutable for pared subjects gvg omal results e.g. drug vs placebo: vomtg or o vomtg a curve ca be draw o a graph of umber of uted pars versus excess outcomes, the boudares of whch mark statstcal sgfcace at a specfed p value ad H 0 demostrated: H 1 demostrated excess results umatched pars H 0 demostrated Rato Rate allows termato of a tral as soo as sgfcace s acheved largely obsolete techque, replaced by term aalyss (wth a low p) the umber of oe of two possble results dvded by that of the other used for comparso betwee data sets the umber of oe result dvded by the total umber of data pots applcable to ordal ad omatve data c. Expla the prcples of errors of statstcal ferece ad descrbe techques to mmze such errors through good study desg. d. Descrbe the features of a dagostc test, cludg the cocepts: sestvty, specfcty, postve ad egatve predctve value ad how these are flueces by the prevalece of the dsease questo. sestvty a property of a test, regardless of the populato tested the proporto of subjects havg the attrbute beg tested for who show a postve result for the test e.g. proporto of hepatts B fected patets testg postve for hepatts B surface atge (HBsAg) = true postves (true postves + false egatves) specfcty the proporto of subjects ot havg the attrbute beg tested for who show a egatve result for the test e.g. proporto of patets wthout hepatts B testg egatve for HBsAg = true egatves (true egatves + false postves) Statstcs 2.C.5 James Mtchell (December 24, 2003)

Lecture Notes Types of economic variables

Lecture Notes Types of economic variables Lecture Notes 3 1. Types of ecoomc varables () Cotuous varable takes o a cotuum the sample space, such as all pots o a le or all real umbers Example: GDP, Polluto cocetrato, etc. () Dscrete varables fte