Introduction to Statistical Methods for Life and Health Sciences
Introduction to Statistical Methods for Life and Health Sciences STATS 13
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Package multcompView April 17 2009 Type Package Title Visualizations of Paired Comparisons Version 010 Date 20060806 Author Spencer Graves and HansPeter Piepho With help from Sundar DoraiRaj Maintainer Spencer Graves ltspencergravespdfcomgt Description Convert a logical vector or a vector of pvalues or a correlation difference or distance matrix into a display identifying the pairs for Which the differences were not signi cantly different Designed for use in conjunction With the output of functions like TukeyHSD diststats simint simtest csimint csimtestmu1tcomp friedmanmc kruskalmcpgirmess License GPL Depends grid Suggests multcomp pgirmess Repository CRAN Datequblication 20060819 070535 R topics documented multcompVieWpackage 2 multcompBoxplot 3 multcompLetters 5 multcomst 7 plotmu1tcomp 9 plotDisplay 12 printmu1tcompLetters l4 vec2mat 15 vec2mat2 16 Index 18 2 multcompVieWpackage multcompviewipackage Summarize multiple paired comparisons Description Convert a logical vector or a vector of pvalues or a difference or distance matrix into a display identifying the pairs for Which the differences were not signi cantly different or for Which the difference exceeded a threshold Details Pack age multcompVieW Type Package Version 010 Date 20060806 License GPL Convert a logical vector or a vector of pvalues or a difference or distance matrix into either a letterbased display using quotmultcompLettersquot or a graphic roughly like a HT using quotmultcomstquot to identify factor levels or similar groupings that are or are not signi cantly different Designed for use in conjunction With the output of functions like TukeyHSD diststats simint simtest csimint csimtestmultcomp friedmanmc kruskalmcpgirmess Auth0rs Spencer Graves and HansPeter Piepho With help from Sundar DoraiRaj Maintainer Spencer Graves ltspencergravespdfcomgt References Piepho HansPeter 2004 HAn Algorithm for a LetterB ased Representation of AllPairwise Com parisons Journal of Computational and Graphical Statistics 132456466 John R Donaghue 2004 quotImplementing Shaffer s multiple comparison procedure for a large num ber of groups pp 123 in Benj amini Bretz and Sarkar eds Recent Developments in Multiple Comparison Procedures Institute of Mathematical Statistics Lecture NotesMonograph Series vol Examples dif3 lt7 CFALSE FALSE TRUE namesdif3 lt7 Cquotaibquot quot37cquot quotbicquot multcomst dif3 multcompLetters dif3 library MASS multcompBoxplotPostthTreat dataanorexia multcompBoprot multcompBoxplot boxplot with multcomp graphics Description Create boxplots With multcomst and or multcompLetters Usage multcompBoxplotformula data horizontalTRUE compFnquotTukeyHSDquot sortFnquotmeanquot decreasingTRUE plotListlistboxplotlist figc0 075 0 1 multcomstlistfigc07 085 O 1 multcompLetterslist figc 087 097 003 098 fontsize20 fontfacequotboldquot Arguments formula a two sided formula like yz Where both y and z are columns of the dataframe Hdata y is numeric and z is a factor This Will be passed as the rst argument for both boxplot and compFn and so must work in both contexts NOT Any more complicated formula may produce errors or unan ticipated results data A dataframe for evaluating formula horiz ontal TRUE for horizontal boxplots and vertical multcomst and or multcompLet ters FALSE for the opposite compFn a function Whose output Will serve as the the only nondefault input to either multcomst or multcompLetters The default quotTukeyHSDquot actually trans lates to TukeyHSDaovformula datal Hp adj sortFn If sortFn is a function or a character string naming a function it is used to sum marize the subset of y corresponding to each level of 2 into a single number Which Will then be used to sort the levels of 2 according to the argumment de creasing This step is skipped if sortFn is NULL or NA or if it is neither a func tion nor a character string that might name a function If sortFn is a character string but a function by that name is not found in the search path multcomp Boxplot stops With Error in docallsortFn listx x could not nd function decreas ing If the levels of z are to be sorted using the output of sortFn this is uses as the decreasing in order to sort the levels of z for plotting plotList A list With names in c boxplot Hmultcomst quotmultcompLettersquot Repli cates are allowed If present they produce eg multiple HmultcomstH side by side This can be used to compare the visual effects of different arguments to multcompBoxplot quotplotmultcomst Each component of plotList is itself a list of arguments to pass to either Hboxplot quotplotmultcomst or quotplotmultcompLetters Place ment can be controlled via g arguments passed indirectly of the form cxl x2 yl y2 lfhorizontalTRUE g gives the coordinates of the gure re gion in the display region of the plot device as described on the par help page ifhorizontalFALSE gc34l2 gives cxl x2 yl y2 for placement of that portion of the plot Details For formula 2 yz if sortFn is a function or the name of a function multcompBoxplot starts by applying sortFn to the subsets of y corresponding to each level of z and then sorting those summaries in increasing or decreasing order per decreasing If sortFn is NULL or NA this sort step is skipped multcompBoxplot then creates boxplot as speci ed in plotList Next compFn is called to generate comparisons to feed to the functions mu ltcomst and or multcompLetters Whose output is then passed to plot multcomp for plotting Components of the relevant sub lists of plotList are made available to par or for plot multcompLetters to gpar Value This function invisibly retunis a list With one component for each component of plotList containing the output of the appropriate quotplotmultcomp call plus the output of quotcompFnquot Au th0r s Spencer Graves See Also boxplotnmltcomstnultcompLettersplotmultcompTukeyHSDpargpar Examples Example from helpquotTukeyHDSU multcompBoxplotbreakSNtension datawarpbreakm 39sortFn39 can be either a function or a function name default order is 39decreasingTRUE39 multcompBoxplotbreakSNtension datawarpbreakm sortFnmedian decreasingFALSE librarymultcomp datarecovery Horizontal boxplots with both multcomp Ts and Letters on the right Example from vignettequotRmcU multcompBoxplotminutesblanket recovery Vertical boxplots with both multcomp Ts and Letters on the top multcompLetters multcompBoxplotminutesNblanket recovery horizontalEALsm Horizontal boxplots with 2 different displays of the quotTsquot on the left multcompBoxplotminutesNblanket recovery plotListlist boxplotlistfigc03 l 0 l multcomstlistfigc0 015 0 l orientationquotreversequot multcomstlistfigc015 03 0 l typequotboxesquot orientationquotreversequot marc52 4 0l Example from vignettequotRmcU library MASS anorx lt7 multcompBoxplotPostthTreat dataanorexia Not run Confirm than sortEnNULL or NA leaves the order unchanged librarymultcomp datacholesteroh cholesteroltrt3 lt7 withcholesterol factor ascharactertrt levelslevelstrtc5 41 3 multcompBoxplotresponse N trt3 cholesterol sortEnNULL multcompBoxplotresponse N trt3 cholesterol sortEnNA EndNot run mu lt compLett e r s Letter summary ofsimila ties and differences Description Convert a logical vector or a vector of pvalues or a correlation or distance matrix into a character based display in Which common characters identify levels or groups that are not signi cantly dif ferent Designed for use With the output of functions like TukeyHSD diststats simint simtest csimint csimtestmultcomp friedmanmc kruskalmcpgirmess Usage multcompLettersX compare threshold005 Letterscletters LETTERS quot quot 6 multcompLetters Arguments X One of the following 1 A square symmetric matrix with row names 2 A vector with hyphenated names which identify individual items or factor levels after quotstrsplitquot 3 An object of class quotdistquot If x or xl is not already of class quotlogicalquot it is replaced with docallcompare listx threshold which by default converts numbers typically pvalues less than 005 to TRUE and everything else to FALSE If x is a matrix its diagonal must be or must convert to FALSE compare function or biliary operator not used if classx is quotlogicalquot th re sho l 01 Second reference argument to quotcomparequot Lette rs Vector of distinct characters or character strings used to connect levels that are not signi cantly different They should be recogizable when concatonated The last element of quotLettersquot is used as a pre x for a reuse of quotLettersquot if more are needed than are provided For example with the default quotLettersquot if 53 distinct connection colums are required they will be quotaquot quotzquot quot 39 39 If 54 are required the last one will be quotbquot If 105 are required the last one will be quotaquot etc If the algorithm generates that many distinct groups the display may be too busy to be useful but the algorithm shouldn t break Zquot and quotaquot Details Produces a quotLetterBased Representation of All Pairwise Comparisonsquot as described by Piepho 2004 The present algorithm does NOT perform his quotsweepingquot step Value An object of class multcompLetters which is a list with the following components Lette rs character vector with names 2 the names of the levels or groups compared and with values 2 character strings in which common values of the function argu ment quotLettersquot identify levels or groups that are not signi cantly different or more precisely for which the corresponding element of quotxquot was FALSE or was converted to FALSE by quotcomparequot monospacedLette rs Same as quotLettersquot but with spaces so the individual grouping letters will line up with a monspaced type font LetterMatriX Logical matrix with one row for each level compared and one column for each quotLetterquot in the quotletterbased representationquot The output component quotLettersquot is obtained by concatonating the column names of all columns with TRUE in that row Au th0r s Spencer Graves and HansPeter Piepho mullcomp Ts References Piepho HansPeter 2004 HAn Algorithm for a LetterB ased Representation of AllPairwise Com parisons Journal of Computational and Graphical Statistics 132456466 See Also multcompBoxplotplotmultcompLettersprintmultcompLettersmultcomst veCZmat Examples dif3 lt7 CFALSE FALSE TRUE namesdif3 lt7 CquotAiBquot quotAicquot quotBicU dif3L lt7 multcompLettersdif dif3L print dif3L printdif3L TRUE dif4 lt7 c01 02 03 l namesdif4 lt7 Cquotaebquot quot37cquot quotbedquot quotaedU diff4T lt7 multcompLettersdif4 dif4Ll lt7 multcompLettersdif4 Letterscquotquot quotquot quotLettersquot can be any character stringm but they should be recognizable when concatonated x lt7 arrayl9 dimc33 dimnameslistNULL LETTERS1 3 d3 lt7 distx multcompLettersd3 threshold mu 1t compT s T depiction of undi fentiated classes Description Convert a logical vector or a vector of pvalues or a correlation or distance matrix into a matrix With an associated visual display to show Whether the differences between items exceed a threshold Designed for use With the output of functions like TukeyHSD diststats simint simtest csimint csimtestmultcomp friedmanmc kruskalmcpgirmess Usage multcomstX comparequotltquot threshold0 05 sepu quot 8 multcomst Arguments X One of the following 1 A square symmetric matrix with row names 2 A vector with hyphenated names which identify individual items or factor levels after quotstrsplit 3 3 An object of class Hdist Ifx or xl is not already of class logical it is replaced with docallcompare listx threshold which by default converts numbers typically pvalues less than 005 to TRUE and everything else to FALSE If x is a matrix its diagonal must be or must convert to FALSE compare function or biliary operator not used if classx is quotlogicalquot th re sho l 01 Second reference argument to Hcompare sep Concatonation character for names of objects with identical similarity dissim ilarity patterns The output of multcomst is matrix for which the number of rows number of columns number of uses of the quotsepquot character Details Produces a matrix of class quotmultcomstquot describing the quotundifferentiated classes that identify the other factor levels or items that are not distinct or not signi cantly different from the base of the T if two or more levels have the same pattern of signi cant differences the two are combined into one T with two Hbases The resulting T s are similar to the quotundifferentiated classes discussed by Donaghue 2004 Value An object of class quotmultcomst which is a matrix of values 1 0 l with one row for each level compared and one column for each T read as follows 1 2 base of the T represented by that column 0 levels not signi cantly different and 1 levess signi cantly different If two or more levels have the same pattern of signi cant and insigni cant differences they are combined into a single column that can be represented by a T with multiple bases The column name will be a character string concatonating all row names with H1 in that column separated by the quotsepquot character Thus the matrix should have as many 1 s as it has rows Also the lower triangular portion should have as many Hl s as there are TRUE eg signi cant differences among the comparisons Au th0r s Spencer Graves and HansPeter Piepho References John R Donaghue 2004 quotImplementing Shaffer s multiple comparison procedure for a large num ber of groups pp 123 in Benj amini Bretz and Sarkar eds Recent Developments in Multiple Comparison Procedures Institute of Mathematical Statistics Lecture NotesMonograph Series vol See Also multcompBoxplot multcompLetters plot multcomst veCZmat plotmultcomp 9 Examples dif3 lt7 CEALSE FALSE TRUE namesdif3 lt7 Cquotaebquot quot37cquot quotbecquot multcomst dif3 dif4 lt7 c01 02 03 l namesdif4 lt7 Cquotaebquot quot37cquot quotbedquot quot3701quot diff4T lt7 multcomstdif4 plotdiff4T plot multcomp plot multcomp graphics Description Plot graphics for multcomst or multcompLetters objects Usage S3 method for class multcomst plotX horizontalFALSE coll6 typecquotTsquot quotboxesquot orientationcquotstandardquot quotreversequot addFALSE at width figc0 l O l lwd3 labellevelsifaddNA else 005 labelgroupsNA Tbase04 H S3 method for class multcompLetters plotX horizontalFALSE 30116 typecquotLettersquot quotboxesquot addFALSE at width figc0 l O l labellevelsifaddNA else 005 labelgroupsNA H Arguments X an object of class multcomst or multcompLetters horiz ontal A logical scalar indicating Whether the list of items compared reads left to right horizontal TRUE or top to bottom horizontal FALSE If this multcomp graphic accompanies boxplots for different levels or groups compared the box plot argument horizontal is the negation of the multcomp plot horizontal argument C01 The color for each group of items or factor levels The colors Will cross the different items or factor levels and Will therefore have the orientation speci ed via horizontal If the number of columns exceeds lengthcol col is recycled For alternative choices for col see HColor Speci cation in the par help page type orientation width lwd label levels label groups Tbase Details plotmultcomp An alternative display for either multcomst or multcompLetters is boxes or rectangles If typequotboxesquot with quotmultcomstquot the quotbasesquot of each quotTquot will be indicated by a triangle The quotstandardquot orientation has the multcomst pointing towards the names of the items or factor levels with the quotreversequot orientation the bases of the quotTsquot point away By default the names are on the left or below unless the mean of the relevant g range is less than 05 TRUE to add to an existing plot FALSE to start a new plot The names of the factor levels or items compared will be plotted only if addFALSE A numeric vector or matrix or a list with components quotatquot and quotwidthquot If a list both components must be either a numeric vector or matrix The numeric vector quotatquot whether the function argument or quotatquot component of the quotatquot list must be either a numeric vector or matrix giving the locations where the quotTsquot or quotLettersquot graphics should be drawn lengthat is 1 2 or 3 times the number of the number of factor levels or items compared If lengthat is twice the number of levels or items compared it gives the range of the display for that level the base of a quotTquot will be at the midpoint If lengthat is three times the number of items compared the intermediate number will be the center of the base of the quotTquot A numeric vector or matrix with as many rows as quotTsquot or quotgroupsquot and with up to three columns With one column it will be the quotcenterquot of the plot range for that group With two columns they will delimit the range With three they will provide quotbottomquot quotcenterquot and quottopquot of the range for that set of grouping indicators If quotatquot is a list the argument quotwidthquot is ignored and is taken from the list quotatquot A numerical vector of the form cx1 x2 yl y2 giving the coordinates of the gure region in the display region of the plot device as described on the par help page width of line to connect elements of quotTquot graphics that might not otherwise be connecte NA for no labels or distance away from the plot for the labels as a proportion of the plot range NA for no labels or distance away from the plot for the labels as a proportion of the plot range A numeric scalar giving the proportion of the available space devoted to the base of the Ts used only when typequotTsquot graphical parameters can be given as described on the plot help page or for plotmultcompLetters as describe on the gpar help page The requested graphic is either plotted by itself or added to an existing plot as speci ed by the arguments The placement can be controlled by g and at The fontsize and fontface of a plot of a multcompLetters object with type 2 quotLettersquot can be adjusted as describe on the gpar help page plotmultcomp Value A list With two components at A matrix With three columns giving the middle and extremes of the display for each of the factor levels or items compared width A matrix With as many rows as Ts or comparitor levels and With two columns giving the plot range for that comparitor level Auth0rs Spencer Graves See Also multcomst multcompLetters multcompBoxplot gpar Examples plotmultcomst dif4 lt7 cl 02 03 l namesdif4 lt7 CquotAiBquot quotAicquot quotBicquot quotAiDU ch4 lt7 multcomstdif4 Standard plot base of quotTsquot point left ch41 lt7 plotch4 labelgroups00 Redo using quotatquot plotch4 labelgroups005 atch4l Same plot with group labels plotch4 labelgroups002 closer to the figure Base of quotTsquot point right plotch4 labelgroupsTRUE Base of quotTsquot point down plotch4 horizontalTRUE Base of quotTsquot point up plotch4 horizontalTRUE orientationquotrU orientationquotrU labelgroups00 labelgroups00amp Same 4 plots but with boxes amp triangles plotch4 labelgroups005 typequotbU plotch4 labelgroups005 orientationquotrM not Ts plotch4 horizontalTRUE labelgroups00amp plotch4 horizontalTRUE labelgroups00amp orientationquotrM typeubu plotmultcompLetters 12 plotDisplay using dif4 from above mcL4 lt7 multcompLettersdif4 LettersLETTERS Standard plot Not run Requires grid mcL4l lt7 plotmcL4 labelgroups00 Redo using quotatquot list plotmcL4 labelgroups005 atmcL4l With bold face and larger font plotmcL4 labelgroups00amp fontsize28 fontfacequotboldU Horizontal rather than Vertical plotmcL4 horizontalTRUE labelgroups00 EndNot run Same as boxes rather than letters plotmcL4 labelgroups005 typequotbU plotmcL4 horizontalTRUE labelgroups00amp typequotbquot plotDisplay plot multcomp displays Description Helper functions for plotmultcomst and plotmultcompLetters These not intended to be called directly and are hidden in a namespace You can use getAnyWhere to see them Usage plothobj at width horizontal col add lwd label levels label groups Tbase orientation plotLettersobj at horizontalcoladd label levels font family quotmonoquot figpar quotfigquot marparquotmarquot plotBoxesobj at width horizontalcoladd label levels label groups orientation Arguments obj a matrix describing Which levels rows Will be plotted With Which groups columns For ploth and plotBoxes obj is a matrix of numbers from 1 O 1 For plot et ters obj is a logical matrix TRUE if that quotletterquot group or column of obj is to be plotted With that level row of obj pIOLDisplay at width horizontal add lwd label levels label groups Tbase orientation font family fig mar Details 13 an array With one row for each level and 3 columns giving low middle and high levels for the display for that level an array With one row for each group of levels in the display and 3 columns giving low middle and high levels for the display for that group A logical scalar indicating Whether the list of items compared reads left to right horizontal TRUE or top to bottom horizontal FALSE If this multcomp graphic accompanies boxplots for different levels or groups compared the box plot argument horizontal is the negation of the multcomp plot horizontal argument The color for each group of items or factor levels The colors Will cross the different items or factor levels and Will therefore have the orientation speci ed via horizontal If the number of columns exceeds lengthcol col is recycled For alternative choices for col see HColor Speci cation in the par help page TRUE to add to an existing plot FALSE to start a new plot The names of the factor levels or items compared Will be plotted only if addFALSE line Width for the display outline Distance from the plot region to print the names of the levels as a proportion of the plot range NA for no level labels Distance from the plot region to print the names of the groups as a proportion of the plot range NA for no level labels A numeric scalar giving the proportion of the available space devoted to the base of If reversed the bases of each T or traingle indicating the master levels of that quotundifferentiated class Will point right or up depending on horizontal rather than down or left character string naming the font family used by quotplotLettersquot This function plots the different quotLettersquot in different colors by plotting one color at a time It s currently not smart enough to align the letters properly except by assuming a monospaced font gure region 2 x0 x1 y0 yl as a proportion of the device region margin 2 lower left upper right in lines graphical parameters can be given as described on the plot help page or the gpar help page The requested graphic is either plotted by itself or added to an existing plot as speci ed by the arguments quotplothquot and quotplotBoxesquot use traditional R graphics and Will not be discussed further here quotplotLettersquot uses grid graphics because it seems to provide more support for controlling the side byside placement of Letters of possibly different colors and Widths The Letters display Will be positioned in the quotplot region de ned by g and mar assuming the entire device region is 37 lines both Wide and tall Thus the plot region is diff gl237 lines Wide and diff gl237 lines high If for example g 2 c09 l O 1 this makes the plot region 37 lines Wide With the default 14 plintmultcompLetters marc5 4 4 2Ol lines the quotWid quot of the plot region is therefore 37 412l 25 lines quotplotLettersquot initially ignores this contradictory negative Width and centers the plot at the midpoint of h0 g1mar237 h1 g2 mar437 v0 g3mar137 and v1 g4mar337 The quotLettersquot for the different levels compared are rescaled from at quotcenterquot to t inside Atrng ifhorizontal ch0 hl else cv0 v1 With quot11quot levels compared and atn1g rangeat quotcenterquot at quotcenterquot is expanded to atn1g O5 and rescaled to match Atn1g ifdiffAtrngltO an error message is issued Meanwhile the quotLettersquot are centered at the midpoint of Wn1g ifhorizontal cv0 v1 else vh0 hi the opposite of Atn1g the argument quotWidthquot used by ploth and plotBoxes is not used and not even accepted by plotLetters If labellevels these are positioned in the midpoint of the right margin in the quotWquot direction Value uDoneu Au th0r s Spencer Graves See Also plotmultcomstplotmultcompLettersgpar Examples Designed to be called from plotmultcomst or plotmultcompLetters NOT directly by users printmultcompLetters print a multcompLetters object Description print method for an object of class multcompLetters Usage S3 method for class multcompLetters printX allFALSE Arguments X an object of class multcompLetters al 1 FALSE to print only the character vector representations of the multcompLet ters comparison summary TRUE to print also the matrix representation Other optional print parameters as described on the p ri nt help page vec2mat 15 Details Prints only the Letters component of the multcompLetters list unless allTRUE Value xLetters the named character vector representation of the multcompLetters evaluation of the distance relationships Au th0r 5 Spencer Graves See Also multcompLetters Examples dif3 lt7 CEALSE FALSE TRUE namesdif3 lt7 CquotAiBquot quotAicquot quotBCquot dif3L lt7 multcompLettersdif3 dif3L printdif3L printdif3L TRUE vechat Convert a vector with hyphenated names into a matrix Description Convert a vector With hypehnated names into a symmetric matrix With names obtained from vect2mat2namesx Usage veCZmat X sepquot7quot Arguments X Either 1 a vector With hyphenated names indicating pairs of factor levels groups or items that are and are not signi cantly different or 2 a matrix in dicating same If x is already a matrix it is checked for symmetry NAs are not allowed sep quotstrsplitquot character to apply to namesx Details x must have names each of Which contains exactly one hyphen if not vec2mat issues an error message If the same comparison is present multiple times the last value is used no check is made for duplicates 16 vec2mat2 Value A symmetrix matrix of the same class as the input With names obtained from uniquestrsplitnamesx All nonspeci ed elements Will be 1 if classx is numeric FALSE if logical and if character Used by the functions multcompLetters and multcomst Au th0r s Spencer Graves See Also multcompLettersnmltcomst Examples dif3 lt7 CFALSE FALSE TRUE namesdif3 lt7 Cquotaibquot quot37cquot quotbicquot Vechat dif3 dif3 lt7 13 namesdif3 lt7 Cquotaibquot quot37cquot quotbicquot Vec2m3tdif3 difch lt7 cquotthisquot39is3939it39 namesdifch lt7 Cquotaibquot quot37cquot quotbicquot Vechat difch Vechat array 1 dimc 2 2 vechatZ Convert a vector of hyphenated names into a character matrix Description Convert a vector of hyphenated names into a character matrix With 2 columns containing the names split in each row Usage veCZmatZ X sepquot4v Arguments X Vector of hyphenated names sep quotstrsplitquot character to apply to namesx vec2maIZ 17 Details If each element of x does not contain exactly 1 quotsepquot character an error is issued Value A character matrix With rownames x and With the character string preceeding the quotsepquot character in the rst column and the character string following the quotsepquot character in the second column Auth0rs Spencer Graves See Also veCZmat multcompLetters Examples VechatZ c quotaib quot37cquot quotbicquot VechatZ c quot343quot vvbau Lecture 4 Last time We hmshed up our discussion or some srmoie graohroai toois we rrrst considered extensions or box piots then the soncaHed modrrred box piot We therr considered the registrar s data there are some moe geherai brmcroies atWork here hatwe reyrewed rt aiso set you up foriab as you are gomg to get more questions today about these But rst Today i had a coupie comments to make from iast time SpecificaHy about the registrar S data recaii that we had to reshape ahd reprocess the data to get it We WiH begin with a discussion of rarrdomrzed cirmcai trraisy we WiH reyrew he mto forms that are more meamrrgtui to us dark ages rrr cirmcai trrais arrd therr introduce Austm Bradford HiH of anra h ai rnan wasiedto an L UH doyetaried moeiy wrth srmriar rdeasrrom R A Fisher Aiso remember that toward the end or ciass someone asked What happened to it i and the response was that that student oniy had UNSCHED ciasses We hen soeii out a frameWork ror statrstroai rnrerenoe in ma context expioring and they were droooe rrom the anaiysrs iook outror things We that when how the design or the experiment how the data were ooiiected proyrdes 39 39 a 39 39 39 yairdrty to our statrs rcai tests that you might be interested in We dose with an examine roh or a more modern cirmcai trrai F H h d y a data rt s hnm seeing yourself and your actions re ected in data 5 o U U U bbbbbwwwNNNNNNNHr A dept number bldg room start MGMT 0466B 000000 MGMT 474 000000 STATS 0013 HUMANTS A00065 110000 SOCIOL 0101 DODD 00 1 090000 SOCIOL 0101 PUB AFF 01337 14390000 SOCIOL 0020 BROAD 0216013 14 0000 SOCIOL 0020 BUNCHE 03143 13 0000 STATS 0013 BOELTER 09413 12 0000 STATS 0013 128 12 0000 HIST 0107A DD 0 078 11 0000 HIST 0097C ROLFE 03120 13 0000 ARMENIA 0106A PUB AFF 02214 12 0000 PLC 0 19C DODD 00175 14 0000 PUB PLC 0219C PUB AFF 06362 15 4500 PUB PLC 90 UB AFF 02343 18 0000 PUB PLC 0 09 PUB AFF 02343 10 0000 POL SCI 0222 UN HE 0 13 0000 ENGR 0183 WGYOUNG CS00076 08 00 00 183 BOELTER 05419 14 Types of studies 0000 days UNSCHED UNSCHED MW In the heal h and life sciences we are faced with two kinds of studies hat diifer 39 d in terms the condi Ions under which data are collecte In an experimental study we impose some change or treatment and se measure the result or respon changes by our presence The kinds of inference you can make will depend on the type of studyyou conduct as well as its overall design technical questions you should ask of a data set or program for how data are to be vil equot Risk Surveillance System and a list of courses from the registrar what kinds of r studies do these represen cccmmmmmccccccccccmmw 1d 1eV numiclasses m1nistart 2 U 3 9 3 U 3 11 4 G 4 10 5 U 2 8 6 U 3 10 7 U 3 8 8 U 3 10 9 G 5 9 10 G 1 13 11 G 4 8 12 U 4 8 13 U 4 8 14 U 3 9 15 U 3 11 16 G 3 10 17 U 4 8 18 U 4 9 19 U 2 9 20 G 8 8 21 U 3 10 22 U 3 11 23 U 3 8 24 U 3 8 25 U 3 10 26 G 4 9 Clinical trials t1meispent daysionicampus 730 w o o wwwNU bu bU U bU wU U U r bu wwbbbb quota 39 39 39 in all areas of biology have been grea ly informed by procedures used in clinical trials The rliniralirial hnwe er I heugold standar quoto r 39 39 who studied lung cancer noted that before 1946 1 1 4 r 1 yr m my in than50 MM n r r I u consultant or by the same investigator previously Under these conditions 39 39 39 I rhznrn 39 important differences in the reguits obtained consequently there were many competing new treatments Clinical trials in an attempt to improve the evaluation or dlfferenttreatrnentsi Austin Bradford Hill began advocating a more systematic ap roach o designing clinical trials like Doll he was trustrated with he quality or research at he time going so tar as to dues ion the ethics or the existing system Hill was the son or a dis inguished physiologist his hope or a medical career was hwarted by the onset or tuberculosis in t9t7i and instead while an invalid he completed a degree in economics by correspondence lrl1927 Hlll moved to the London School of Hyglerle and Troplcal Medlclrle and Clinical Trials 39 39 39 39 appyig statistics in a clinical setting ratherthan theoretical minutiae it seems hat his advice while orten sta istically sound was motivated by practical concerns in terms or clinical trials Hill argued rorweiispecined study aims or comes and he consistent use or controls 77 Patients were to be divided into two groups the treatment group would receive a new rug or procedure while the control group would be prescribed the standard therapy Upon completion orthetrial researchers would examinetne dirrerences treatment is superiorto the existing herapy duringthe tgsos he researched mainly in occupational epidemiology is with renown in medical statis ics started in l937 with the publication or nlstextbooki how pai n divided into he treatment and control groups iert soieiyto Principles otllledical statistics based on a series or ar icles in the Lancet physicians he reitthere could be a problem What Was he Worrled about Clinical trials in 1948 Hill published a groundbreaking study on Clinical Trials the errectiveness or streptomycin an antibiotic in treating pulmonary tuberculosis here is how he assigned patients to the treatrnent and control groups rn method that ii as patients appear at a clinic or study center researchers alternately assign hem to treatment or control til i lllllldl her ini ials or even heir binndate taking Hill s very practical siancei do hese memde comple e V remove 90 9quot 5 quot557 be an s are case and this information was then given to the rnedtcal oniceror the centre special study they were intact treated as they would have been in the past the sole maintained Medicaiastmrcsurci Stitvhmvclntieatmemvivulmmrawmbeiculosls aw 1943 2 reeves Clinical Trials An aside Some histor y The tuberculosis study was the rst time randomization of treatments was a clinical trial after its publication Hill wrote a series ofarticles describing its use m m a 5 Following the immense success of penicillin here was a great deal of research activity around detecting other potential an ibiotics In these articles I had set out the need for controlled experiments in clinical medicine with groups chosen at random At the outset I think I pleaded that Also tuberculosis was the most important cause of deathquot ofyoung adults trials should be made using alternate cases suspect Wand its a WW 39quot Europe and North Amenca at the 39me large IF ifthat in fact were done strictly they would be random I dellberatelyleft out the w rd quotrand ization39 and quotrando sampli g n er quot tt ttrme caus s g to persuade the doctors to CONSIderable laboratory W 39k arid some early e fPerlmems 0 Pa ents come into controlled trials in the verysimplest form and I might have suggestedthat Streptomycm WOUId be 3 S elee treatment for PUImOnaW scared them off I think the concepts of quotrandomizationquot and quotrandom t er 5395 sampling numbersquot are slightly odd to the layman or for that matter to the lay doctor w 39 statistics I thought it would be better to get run hen It omes to m m WW Minimum m is my a W m M cimiimiimi cm doctors to walk first before I tried to get them to new mmmemcgaxepmugmusu Memoiies oflhe Biilisri stieplomyciii liial iii luoeiculosis The riist iaiidomized clinicalliial SirAustin Bradford Hill Clinical Trials Through randomiz ion and the blinding of the physicians Hill achieved Souncpopulazton his goal of reducing bias by allocating the patien s 0 treatment and l a control groups in such a way that the two groups are initially equivalent in all respects relevant to the inquiryquot he wntes Randomized controlled trials It ensures that neither our personal idiosyncrasies our likes or dislikes consciously or unwittingly applied nor our lack of balanced judgement has A5 an experiment he39ll the deSigquot is entemd t u u m straigh orward par lclpan s are assigned randomly to either receive a treatment under study or a control perhaps a placebo or a standard herapy allocation has been outside our control and the groups are therefore unbiased prostlm Lomoonron immi39mmn torinol it removes the danger inherent in an allocation based on personal judgement that believing we may be biased in our judgements we endeavour to allow for that bias to exclude it and that in doing so we may 3323103225 adr gnag tfnoggmz overcompensate and by thus leaning over backward introduce a lack of cases scientists ape eva39l atih 9 W em er 0mm balance mm the other direct39onquot a drug elps with a particular condition 3 say and having used a random allocation the sternest critic is unable to say when we eventually dash into print that quite probably the groups were differentially biased through our predilections or through our stupidity Exampizs m cnntmiizd tIIais at snciai vrunvzmmes carrtzu out In the United States uvinn 195591 I Vim tm mm m autumn mssst n iamw ms msts s HIMWm and r mm cnrrtmtrm nXanditti Emuas m 5371 InsttnitIrms39nrtIrntttnrzmmr Ramnmaunra nn ztsinwtmms Patctamnt Wm en ni lilezflmer rrtrrsrttstmmm t rm mm It mm m m I t and trim mm nan Rumrimmwavimiemxtxceii39nl39il moz inuvi hn mm mm Stmtttmmjo39imiinmtinn EBB wt rm st tstrrmrrsrttmrcr tramsaw D iN39aizmszi39IMHmzm trmmmmttrstnsmmmrmaI cnniummmnxmnmtmc cnntIni a39nuvs Ismttrs mt rsmts at mm was mm and Mw umrw smut mannaIna ems rtmtsutrrm tsrts tumstrstssrtmrrw mmmmrmrmrtrts pmtttrmsssrrms rst rtrsrrm I mm rm rm m me my raw mt rm rams csrrmtm mm tam m mm t m mm tmts m mm m WW mt WW t mm mm m Ittmsrtsmt t m stmrm Th f d 39 f L39 39 euseo ran omIza Ion urmssm mmmrm Iain i rtsrmtsrmstratum smtm mtmtthm pt tormsstrrms Wm Wm mm H tssm m mm mm rth t strmsm WWW mam msstsrmttr Lemlinursntm llmiWainIn miis wha cimicnm nhnnzd mmrmtm acmmu39 toInIsn smsmtmmtor shvw iivls mus ldi quotInnizmn nnm muIIsziim M Hmi U JSuhuim m39IIm nun This kind oftrial is not unique to the medical sciences in fact a number of linuxinunilnmmls intervention studies took place in the 1970s and BUS all based on this idea of jnxmmgvnj m WW 39 Wi mquotquotquot quotquot EggsVzcamfggnmmv randomiza ion II mm nmrwrts In nsntw Wm rt 39Inumnqa m st mrtrrstrtmvs rttrsmtm he next slide we Present a few exam Plest and thequot a Piece of a large list of 237223 m sliti f rrsnitiimiri sr ts itmilifilil lff iirrsttil quottt i tquotirs such studies 245 in all published in the late 80s EEMWM WM husIakuMustrnvanqmm isen tsus Randnm ai arm at 22st mm in Eam39Iqs trsmtmm il mm mm tmttmms m Iarmum unamvimm nenz tt mm mm in 1Imemmtnn m i stmm avmzls an t sttr and t minimumquot and i ttmmt mum at 2 Sim a rim umimmtinn m rst mm m Iwg aIchmItx Issr mt st rtI39t tm In39ense I39Iinmvnn 1 m mm H t Imuvlil v iimnrivcaw Jinn m7qu at Dinerwrit am mm cunt at an mm mu r t trststmmtmrrsrstmmmtttnvr m5 Ynsndk t lmsnnvzaimxzn stmttssrsmmsttmrtmmstz tmrrmmttnmsnrtm rsstrtrrstttstt tsttmtmtmrnmrturt Inardartsnnurstnlirkmntinn nn 1gtnceNnicIlztinn mt39 Imsutms r s mnunsvmnwntmttnnzotattieum izhnurtumhuaithmdiivv g varnsmtrmss HMSB mgmm as m menus WINDS and m ttmsrt amines It s L sou wruucs tsswonx lmL slV INMiIANCL AND mun DR quotNIPYTION AND MPOSITIOI V svrPoRl PROGRAMR I THE BlaiJDhRAI HY ImtItrtg J H Anrxpmmtnlul rmdt to measure the c r n t39 cm n Etrtcrtmcrtts lmcd ilrrcznl dn tdcdutlu ll Ill jorriilcguncswnch mm mm Cnlnmbm 0m Mumquot Comm warm uivrmplmd N the mm at plugum undrrgnlim tests TtIc lnclml Dcpanmcm Iw ltlcnl39cr M IuItrt 1 E St39rving rhtvagr39mt Anuprnmml in A Wmct tIr t39rtrrttmIttttttL tI Itttttss xm39ml le and pillhlir hrallflt numnf New York Commttntty I Menlni IItttItIt V Y a Tln39 mullrplohlrm dIrmmu Metuchett VJ m rrstmn Irm mutantquot 39 rtt I quotquot quot 39Rquotquot quot39 Emu E A mm m hrrrrrrrtttrrrrthtp Rt39purl 1m LIIIDl Sun 5 M39 quotquotquot quot Franmcu CnlIf San Fianciscn Dnclopmtlu thd Irtc Dtrcmhu 7 Cummcuc Iltdlml and l uhitu Ultituu I m tttt smut erlIm gt m with IHs nAml Mnltml Inuulism Gmman L L Elsa B Muck lGerpm U a funk H Earl ttm Isrrtttt tttmrm xuIpml u llm I luv Meluchcn M hcarucmw i9 Hcdnck oms c a Schmullc a 0m nfxchnnl ymnh Employ mcm dcmunitraunn progut 19777it Akron Ohio Akron Mcdim nrnr Fmplnymcm n5 Unsurl ttm Im Avaitahlc from the t ample CrtmtmrI tttttI CIvIl Iusttuc Irr substation Ittr ctrrr authors Kent Slale L39nutrslly Kern Ohio trcmrrtcm Jll 39ctrI pro unrest d llalnl safely H r39 Hm I a at Vulcy I L Kansas Blue CrassBin Shtcld Oulyatitm benci39m experiments Mullml 39ult39r N70 24 l43 mm P II Tim Furrrrlrmrm I IIUIL39I andm39linn mumh mt mrlully ripmud familm Copcnhngcn Denmurk n Dnnlsh Nnnun l lnslimtc or strum Rexarch Im Marin R c 4 mmprrherulw magi1m for mullipmhlrm amt117 one returns Is suppttsd or rt gutm rcsearch mch 39l Itut rcrtrcrtr ma Ittr cxanlplc rcrsr m uni In esl m a Iongsrrtct uric IJUI39IE by tit am whtclt wt bclxcvu hut uhaructcntcs quot1c Isls the calf ortct we use are mIslczIdIn v lo the cxtcni lhtu toms ex crl R Wquot quot quotWWW quot quot quotquotquotquot39quot39quot Km P39Cd s39lquot5quot F 5 U P Rico umvemtt at Puma le Irtstttutc at Cartbtmrt Smdlk s quotm 39 quot r quotquotquot McCa RSeIItgmartA Pyrke M Betkowin I Kngnnl mass to n I i m dunno iquot Nth we we 6mm It a Pettttord P The pursm39rapmmuc A deyaflhemmllerlualb39 ex crtmcnt In accord wtth witttt w r r s lhc ntatrt putpos or tupznar ht11m u J ol iallV deprived mu cw York Cantuturtin munch so I test or 5mm Strch lhcn ttrc listud tn Sccllon 3 on Same Sucxct at Nun York Tratrtt g ttt attention l or hrcviiy39x stth alnne we nltiihcr crtrse Mudlcussysh lu Curpuraliun Imm39mrednddAyrornvaw Final rcfctcncc cmncs nor phlce any entries I lD man than on Calcgur Revert Chxcazn 111 Mcdlt 39r I978 Fisher and randomization It would be incorrect to suggest hat the idea of randomization is due to Hill Hill was working int e 194 s an 0 s and became andomiza ion on fairly practical grounds reducing bias In the 1920 s and 1930 s R A Fisher who we met in the rst lecture leaning houghtfully over his calculator was promoting the idea of randomization 39om a technical perspective to is er randomization gave rise to valid statistical procedures Another aside Fisher and Hill There is in fact an interesting story that connects hese two researchers both were active in roughly the same time period and they were certainly aware of ach o her s work They exchanged correspondence starting in 1929 Dear Sirquot and then in 1931 Dear Fisherquot and Dear Bradford Hillquot and then in 1 0 My ear Fisherquot and My dear Bradford Hillquot and then by 1952 My dear Ronquot and My dear Tonyquot Hill went by Tony another long story 0 But by 1958 they were back to Dear Fisherquot and Dear Bradford Hillquot as the two Doll a signi cant coinvestigator wi h Hill were on opposites sides in a dispute as to whether or not smoking caused lung cancer From the point ofview of our discussion one of Fishers main criticisms ofthe studies sugges ing that smoking caused lung cancer was the fact that they were entirely observational he wanted a properly randomize experiment which of course would be dif cult as you can t force people to start 0 ing We will speak more about causation and what you can conclude from different types of studies over he next couple of lectures Fisher and randomization The theory of estimation presupposes a process of random sampling All our conclusions within that theory rest on this basis without it our tests of significance ould b worthes In controlled experimentation it has been found not difficult to introduce explicit and objective randomisation in such a way that the tests of s39 nifica c nstrably correct In other cases we must still act in faith that Nature has done e randomisation for us We now recognise randomisation as a postulate necessary to the vali ity of our conclusions and the modern experimenteris careful to make sure that this postulate is justified Fisher velopmenl ofthe theory ofexuerimenlal design Proceedings of the international Statistical Conferences 1 9473 434733 Hill s tuberculosis trial But back to the task at hand here is a simple summary ofthe patients enrolled in Hill s tuberculosis trial as Hill hoped the groups seem relatively well balanced in terms oftheir measured condition Tm l 39muliliuu uir Admission tuning 3 l Sudimcma 395 2 Tcmn quot E u 3 lion In E U E in First Week i O o l 390 0 Good n 1433 v F u 4 o m n l a r 39 l FJII I7 20 99 999 F l 12 11720 J 2 571 37 7S C pm My 14 mo IUD9 17 15 I7 21su I6 20 mama25w IquotF33 3 C 24 l 5 36 1 Tatar 55 52 ll Total WF Sit Medical Research Council SlVEpluMcintreatmentufpuimunarytubercuiusis aw 19482 7597782 Hill39s tuberculOSiS trial Some analysis with Hill s data And here are Hills original resu39ts from his 1948 paper What do We see Here we create a 2x2 table for Hill s data we will focus on whether or not pa ients survived to the end of the trial TABLE lL Axsenmenr 0 Radiological Appearancz at Six ManI15 as Compared with Appearance on Adminian Treatment Radiological Assessmem Streptomycin Group Control Group C S Considerable improvement 28 51 4 5quot5 Moderate or slight improvemenr m 18 13 25quot 8 No material change 2 4 3 8 2 Moderate or slight deterioration s 9 12 23 E 38 51 89 Considerable dereriorarion 6 11 6 11 quot 0 3 r 4 7 14 27 g 0 Total 55 100 52 100 a c 9 14 4 18 D p BMJ 1948 Z 7BBV7EZ 52 55 1 07 Some analysis with Hill s data Another view Hills tuberculosis study To work this out we see that 1452 or 27 of the pa ients receiving the control died 00quot whereas 455 73 of those receiving Streptomycin died Here is a mosaic plot of Hill s tuberculosis study its wor h taking a second look at the computations hat go into his plot What do we think Treatment Ehrvivsd Recall that the columns are sized according to he proportion of people receiving Streptomycin and he Control slightly more received the treatment 38 51 89 Survived Status Then within each column the proportion g of participants hat survived is shaded yellow we will call his the conditional proportion who survived given that a participant received ei her Streptomycin or the control 52 55 107 Died A A A 4 oo Some analysis with Hill s data Statistical anal sis When you read about these kinds oftrials in he medical literature it is not y 39 39 39 quot meliilamel condmonal pmbabm les39 quot 395 cusmmary m IOOK 3 mequot fracmquot On the face of it things look promising for Streptomycin relative to the standard herapy bed rest but is that where our analysis stops In this case he ratio ofthe proportion of pa ients that died in the Streptomycin group 73 to those hat died in the Control group 27 is 0 27 Streptomycin reduced the rate of mortality by nearly a quarter Efx gdm fgfree y a jzgf 3quot e ecn In par ICUIar39 comd hese resuns have epidemiological studies where treatmentquot is really exposure to some toxic And Wha is me model for Chance here i u y u a u I horrible happens to you Randomized controlled trials Let39s go back to the cartoon ofa statis ical inference problem that we started with in the rst lecture Hill s tuberculosis study 1 quot g anlau ihle m in 39 r 39 39 uammauelul the purposes of argument a good null hypo hesis is a statement that would t 39 r 39 he context on 39 39 be imeresuhg m rejec trial whehiieisglng the ef cacy ofa new medical procedure the natural null nypume 2 We hen de ne a test statistic some quantity calculated from our data hat Undermis m delwe assume hm me We hammer are me same so a is used to evaluate how com atible the results are with hose expected 39 under he nu hypmhesis if he hypmhesized S a emem or model or patients would have had he same chance ofsurvval under eI her put another scenario was We way their ou come wh er hey lived or dIed wou d have been the sam regardless of which group they were placed 39 E 3 We hen simulate values ofthe test statistic using the null hypothesis today Under this hypothesis the table we see is merely the result of random vvllm gmup we YW MS39S 395 quotquote39 3quot f Ch 39 P quotquot9 quot 9 les les 39 assigned them to and the fact that we saw4 in the Streptomycin grou abom his more formally when we review probabimy and 14 in the control group was purely the result of chance 4 And nally we compare the value ofthe test statis ic calculated for our data and r 39 39 39 39 39 mey 39 1 Simulating random assignments In this simulated table we have 1152 or 21 chance of dying under the control and a 75 or A chance under Streptomycin the treatment reduced the mortality rate Hill s tuberculosis study among the participants by nearly 60 Therefore under the null hypothesis ifwe had chosen a different random assignment 0 patients wewoul s i ave 18 eople who died and 89 who Treatment survived but they would appear in different cells of he table C S 5 We can simulate under this odelquot pretty easily hat is we take the 18 people 2 who died an e w o survive an we rerandomize assigning 52 of a 41 48 89 them to the control group and 55 to the treatment group 3 a m 7 5 Let39s see what that produces g 11 7 18 52 55 107 Simulating random assignments In this simulated table we have the opposite wi h 652 or 12 chance of dying un er the control and a 1255 or 22 chance under Streptomycin the treatment almost doubled the mortality rate among the par icipants Treatment Simulated table Simulated table 0 s Emmi svewmwn Danni 5119an U 3 E 46 43 89 m t t to E E 0 39c E n i 5 6 12 18 52 55 107 Proportion oi s39mulated tables with n deaths under Streptomycin Simulating random assignments 85 1 em uiai me nu en in ueauis unuei 39 other entries in the table Using he language of hypothesis testing we will take the number of patients in g the Streptomycin group that died as our test statistic 0 Therefore the question becomes under the random assignment patients to 8 treatments how common is it for us to see 4 or fewer deaths 39 Streptomycin group H 8 mm m o Howwould we gure this out o 1 2 3 A 5 e 7 e 9 1o 11 12 13 14 15 161718 Hypothesis testing Simulatin random assi nments g g The value 0 006 the proportion of random tables wi h 4 or fewer deaths in the m r a more In mi pm we see mm a value as small or a 7199mmamnmomewmndamaum mmm 39 smaller than four is fairly rare to be precise g enreme than the one you computed for your data r fewer deaths in the Streptomycin group 5 1 e 39 r In wei u u 39 39 null hypothesis the smaller the value the stronger the evidence it was 5 b 39 39 source i m i a This then provides us WI h evidence hat there is something more at work here than he phenomenon you re smdymg random assignment 5 Keep in mind however that rare things do happen but only rarely it is lfwe believed the null hypo hesisthatthere a 1 11 1 1a 1 1 1i 1 13 r mor was no difference between treptomycin and bed rest the results Hlll observe wou ve been extremely rare coming up a very small fraction of he time group This is the nature of statistical reasoning and this is why Fisher advocated performing many experiments as you study a phenomenon Hill s tuberculosis trial To sum up 1 The nuii hypothests ror Hiii s experiment was that Streptomycin andthe standard therapy bed rest wouid perrorrn the sarne when treattng puirnonary tuoercuiosts 2 We took as our test Sta istic the number of pa tents that died in he Streptomycin group 3 Under he nuii hypo hests or no difference we repeated Hiii s randorntzatton a iarge nurnoerorttrnes wtth each one we recorded the nurnoeror dea hs assigned to the Streptomycin group We hen iooked at therractton or randorn asstgnrnents that gaye us 4 o Vioxx Randomized Controiied mais are Common in medicai research iet s have a iook at a more recent case Vi0gtltgtlty an antttnriarnatory agent was tntroduced to the market tn the iate 19905 and was prescnoed tor the treatrnent or arthntts and acute pain in 2000 the New Engiand uournai or Medtctne controiied tnai destgned to examine whether pattents recetytng rorecowo ytoxx wouid haye rewer upper gastrotntes that events perrorattons uicers bieedirig han arthritis were randomized into two treatment rou S 50 rng or rorecowo whtte he other recetyed 500 rng or naproxeri e we oo ts as eytdence that the nuii hypo hests sornethtng o her han strnpie chance asstgnrnent couid coiiected wrong that ts r expiairi the data he The Nut Engtttttt coMmmsoN 0F UPPER GASTKOIN 39AI ROXEN IN PATIENTS t CLAiRE EDMEAaDiER M on Laugh ane MDrr RUBEN Eussusrvmms NLD may Dayts M D PM D CNN nzn u Hawtzv M DMaacc awn reams J sanrtnzzn M Dr AasrRA r L Build Each year ehrrreat uppergastrmmehr ttnat events occur tn 2 to 4 Derrem er nettents who setectrye inhibitorclcyulaoxygeliaaer2wuuid be 3y secreted mm a tower tneruence orehntcetty importh NSAID napruxen amung petterrts wrth vheuinnmld arthntts rt111d We rendentty assrgned 3075 pettent who were at iedsl 50 years at age tor at least 4n veers or age and reeeryrng iungriemi giticocurii tirerepyt ntt who had rheurnetmd arthntts ta reaetye erther uerty The primary end poim was Loniimied ciintcai upper gestretntestrrret events gdsiruuuudeiidl gene r hon ordustruetran uppergasiruiniestmai hteeutng Ihmllx memo and Miamer had simiiar glitter n t r re uwruu 1 so ntentns 2 eenttrnteu gastroinleahnni events uer ton patienhEars neeurred with rereeaxrtt as compared wmr 45 per ttm paltenlryears wrttr nae proxew reiative new 95 95 percent can ueme Intern yet pate no Report The resneetrye rate uiuumr Generalities 39 39 anew i validity he way we coiiect data dtctates the kinds or tnrerences we are aiiowed to make Howeyer we have not said anything about what the effect of streptomycin a r assurnpttons about how peopie were recrutted tnto the tnai more on thts next trne rhts rneansthattn addttton to air he dues tons i had you ask about where data corne from you can now add a rew techntcai ones we are going to start paying attention to the role that randomness plays Ftnaiiy he hypothests testingframeworkis orten rererred to as significance testing tn hat we are atternpttng to estaoitsh the stgntrtcance or sorne errect he nuii hypothests typtcaiiy oetng that here ts no errect Vi oxx in addttton to or prooierns the researchers constdered a yanety or posstoie Eider errects rrorn tawng rorecowo R or naproxeri wt here we present a Wobyetwo tape or pattents who experienced cardtoyascuiar adyerse eyents CE Treatment Lu 3 4010 4002 8012 g Tu a 8 19 45 64 4029 4047 oxx Vioxx L L quot 1940290rquot 39 39 d39 d t h39l d f 39b 454047 11 f t39 t h d carutovas ar a verse even S w le uquot er m sea r 0 Pa len 5 a As With HIll39s data thIs number seems convIncIng and yet we should ask r r r under tWIce as hIgh whether or not these results could be produced by pure chance Treatment 139 39 N R r quot m 3 39A I Eunder 39 r I we can resul ing tables u g 4010 4002 8012 m E r mm u uu hat are as 3 strong or stronger than what we observed in this case under different g randomizations ofthe patients how often do we see tables with 45 or more deaths under Vioxx 3 19 45 64 Well we can simulate 4029 4047 oxx nthis case quot r quot 39 k quot Vioxx pr p m quot 1 39ab39es The evidence here is seems very strong Merck the manufacturer of Vioxx however 39 39 39 As the research community debated he meaning ofthis particular trial several ongoing trials 39 39 39 39 wmem 39 39 39 n market c 39ncreased risk of cardiovascular adverse events 139 wnemel viu wa scientists and senior management knew ofthese hazards and when they knew it next ime we will use this example to study how signi cance tests are applied in practice and some ofthe common hazards nr HHl lm 24 32 40 48 56 84 count 0t CE and Ft