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Chapter8:Probabilitydistributions
35
toomuchsmoothing(itusuallydoesfor“interesting”densities).(Betterautomatedmethodsof
bandwidthchoiceareavailable,andinthisexamplebw="SJ"givesagoodresult.)
Histogram of eruptions
eruptions
Relative Frequency
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Wecanplottheempiricalcumulativedistributionfunctionbyusingthefunctionecdf.
> plot(ecdf(eruptions), do.points=FALSE, , verticals=TRUE)
Thisdistributionisobviouslyfarfromanystandarddistribution. Howabouttheright-hand
mode,sayeruptionsoflongerthan3minutes? Letusfitanormaldistributionandoverlaythe
fittedCDF.
> long g <- - eruptions[eruptions s > > 3]
> plot(ecdf(long), , do.points=FALSE, verticals=TRUE)
> x x <- seq(3, 5.4, 0.01)
> lines(x, , pnorm(x, mean=mean(long), sd=sqrt(var(long))), lty=3)
3.0
3.5
4.0
4.5
5.0
0.0
0.2
0.4
0.6
0.8
1.0
ecdf(long)
x
Fn(x)
Quantile-quantile(Q-Q)plotscanhelpusexaminethismorecarefully.
par(pty="s")
# arrange e for a a square e figure e region
qqnorm(long); qqline(long)
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Chapter8:Probabilitydistributions
36
whichshows a reasonable fit t but t a shorter right tail l than one would expect from a normal
distribution. Letuscomparethiswithsomesimulateddatafromatdistribution
−2
−1
0
1
2
3.0
3.5
4.0
4.5
5.0
Normal Q−Q Plot
Theoretical Quantiles
Sample Quantiles
x <- rt(250, df = = 5)
qqnorm(x); qqline(x)
whichwillusually(ifitisarandomsample)showlongertailsthanexpectedforanormal. We
canmakeaQ-Qplotagainstthegeneratingdistributionby
qqplot(qt(ppoints(250), df = = 5), x, xlab = = "Q-Q Q plot for r t t dsn")
qqline(x)
Finally,wemightwantamoreformaltestofagreementwithnormality(ornot).Rprovides
theShapiro-Wilktest
> shapiro.test(long)
Shapiro-Wilk normality y test
data: long
W = = 0.9793, p-value = 0.01052
andtheKolmogorov-Smirnovtest
> ks.test(long, "pnorm", , mean = = mean(long), , sd = = sqrt(var(long)))
One-sample Kolmogorov-Smirnov test
data: long
D = = 0.0661, p-value = 0.4284
alternative hypothesis: : two.sided
(Notethatthedistributiontheoryisnotvalidhereaswehaveestimatedtheparametersofthe
normaldistributionfromthesamesample.)
8.3 One-andtwo-sampletests
So far we have compared a single sample to a normal distribution. . Amuch h more common
operationistocompareaspectsoftwosamples. NotethatinR,all“classical”testsincluding
theonesusedbelowareinpackagestatswhichisnormallyloaded.
Considerthefollowingsetsofdataonthelatentheatofthefusionofice(cal/gm)fromRice
(1995,p.490)
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Chapter8:Probabilitydistributions
37
Method A: : 79.98 80.04 4 80.02 80.04 80.03 80.03 3 80.04 4 79.97
80.05 80.03 3 80.02 80.00 80.02
Method B: : 80.02 79.94 4 79.98 79.97 79.97 80.03 3 79.95 5 79.97
Boxplotsprovideasimplegraphicalcomparisonofthetwosamples.
A <- scan()
79.98 80.04 80.02 80.04 4 80.03 3 80.03 80.04 79.97
80.05 80.03 80.02 80.00 0 80.02
B <- scan()
80.02 79.94 79.98 79.97 7 79.97 7 80.03 79.95 79.97
boxplot(A, B)
whichindicatesthatthefirstgrouptendstogivehigherresultsthanthesecond.
1
2
79.94
79.96
79.98
80.00
80.02
80.04
Totestfortheequalityofthemeansofthetwoexamples,wecanuseanunpaired t-testby
> t.test(A, , B)
Welch Two Sample t-test
data: A A and d B
t = = 3.2499, df = 12.027, p-value = = 0.00694
alternative hypothesis: : true e difference in n means is s not equal to o 0
95 percent confidence interval:
0.01385526 0.07018320
sample estimates:
mean of f x x mean n of y
80.02077 79.97875
whichdoesindicateasignificantdifference,assumingnormality. BydefaulttheRfunctiondoes
notassumeequalityofvariancesinthetwosamples(incontrasttothesimilarS-Plust.test
function). WecanusetheF F testtotestfor equality inthevariances,providedthatthetwo
samplesarefromnormalpopulations.
> var.test(A, , B)
F test t to o compare e two o variances
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Chapter8:Probabilitydistributions
38
data: A A and d B
F = = 0.5837, num m df f = = 12, , denom df = = 7, , p-value = 0.3938
alternative hypothesis: : true e ratio o of variances s is not equal to 1
95 percent confidence interval:
0.1251097 2.1052687
sample estimates:
ratio of variances
0.5837405
whichshows noevidenceofasignificantdifference,andsowecanusetheclassicalt-testthat
assumesequalityofthevariances.
> t.test(A, , B, var.equal=TRUE)
Two Sample e t-test
data: A A and d B
t = = 3.4722, df = 19, , p-value e = 0.002551
alternative hypothesis: : true e difference in n means is s not equal to o 0
95 percent confidence interval:
0.01669058 0.06734788
sample estimates:
mean of f x x mean n of y
80.02077 79.97875
Allthesetestsassumenormalityofthetwosamples. Thetwo-sampleWilcoxon(orMann-
Whitney)testonlyassumesacommoncontinuousdistributionunderthenullhypothesis.
> wilcox.test(A, , B)
Wilcoxon rank sum m test with h continuity correction
data: A A and d B
W = = 89, p-value e = = 0.007497
alternative hypothesis: : true e location n shift t is not equal l to 0
Warning message:
Cannot compute exact t p-value e with ties in: : wilcox.test(A, , B)
Notethewarning:thereareseveraltiesineachsample,whichsuggestsstronglythatthesedata
arefromadiscretedistribution(probablyduetorounding).
Thereareseveralwaystocomparegraphicallythetwosamples.Wehavealreadyseenapair
ofboxplots. Thefollowing
> plot(ecdf(A), do.points=FALSE, , verticals=TRUE, , xlim=range(A, , B))
> plot(ecdf(B), do.points=FALSE, , verticals=TRUE, , add=TRUE)
willshowthetwoempiricalCDFs,andqqplotwillperformaQ-Qplotofthetwosamples.The
Kolmogorov-Smirnovtestisofthemaximalverticaldistancebetweenthetwoecdf’s,assuming
acommoncontinuousdistribution:
> ks.test(A, B)
Two-sample Kolmogorov-Smirnov test
data: A A and d B
D = = 0.5962, p-value = 0.05919
alternative hypothesis: : two-sided
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39
Warning message:
cannot compute correct p-values s with h ties in: : ks.test(A, B)
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40
9 Grouping,loopsandconditionalexecution
9.1 Groupedexpressions
Risanexpressionlanguageinthesensethatitsonlycommandtypeisafunctionorexpression
whichreturnsaresult.Evenanassignmentisanexpressionwhoseresultisthevalueassigned,
andit maybe usedwhereveranyexpressionmay beused;inparticular multipleassignments
arepossible.
Commandsmaybegroupedtogetherinbraces,{expr_1;...;expr_m},inwhichcasethe
valueofthegroupistheresultofthelastexpressioninthegroupevaluated.Sincesuchagroup
isalsoanexpressionitmay,forexample,beitselfincludedinparenthesesandusedapartofan
evenlargerexpression,andsoon.
9.2 Controlstatements
9.2.1 Conditionalexecution: : ifstatements
Thelanguagehasavailableaconditionalconstructionoftheform
> if (expr_1) ) expr_2 2 else e expr_3
whereexpr
1 mustevaluatetoasinglelogicalvalueandtheresultoftheentireexpressionis
thenevident.
The “short-circuit” operators && and|| areoften usedas part ofthe conditionin anif
statement.Whereas&and|applyelement-wisetovectors,&&and||applytovectorsoflength
one,andonlyevaluatetheirsecondargumentifnecessary.
Thereisavectorizedversionoftheif/else construct,theifelsefunction. . This s hasthe
formifelse(condition,a,b)andreturnsavectorofthelengthofitslongestargument,with
elementsa[i]ifcondition[i]istrue,otherwiseb[i].
9.2.2 Repetitiveexecution: : forloops,repeatandwhile
Thereisalsoaforloopconstructionwhichhastheform
> for r (name in expr_1) expr_2
wherenameistheloopvariable. expr
1isavectorexpression,(oftenasequencelike1:20),and
expr
2 isoftenagroupedexpressionwithitssub-expressions s writteninterms of the dummy
name. expr
2isrepeatedlyevaluatedasnamerangesthroughthevaluesinthevectorresultof
expr
1.
Asanexample,supposeindisavectorofclassindicatorsandwewishtoproduceseparate
plotsofyversusxwithinclasses. Onepossibilityhereistousecoplot(),
1
whichwillproduce
anarrayofplotscorrespondingtoeachlevelofthefactor.Anotherwaytodothis,nowputting
allplotsontheonedisplay,isasfollows:
> xc <- - split(x, ind)
> yc <- - split(y, ind)
> for r (i in 1:length(yc)) {
plot(xc[[i]], yc[[i]])
abline(lsfit(xc[[i]], yc[[i]]))
}
(Note the functionsplit() whichproducesalistofvectors obtainedby splittingalarger
vectoraccordingtotheclasses specifiedbyafactor. . Thisisausefulfunction,mostlyusedin
connectionwithboxplots.Seethehelpfacilityforfurtherdetails.)
1
tobediscussedlater,orusexyplotfrompackagelattice(https://CRAN.R-project.org/package=lattice).
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Chapter9:Grouping,loopsandconditionalexecution
41
Warning:for()loopsareusedinRcodemuchlessoftenthanincompiledlanguages.
Codethattakesa‘wholeobject’viewislikelytobebothclearerandfasterinR.
Otherloopingfacilitiesincludethe
> repeat t expr
statementandthe
> while e (condition) expr
statement.
The breakstatement canbeusedtoterminateanyloop,possibly abnormally. . This s isthe
onlywaytoterminaterepeatloops.
Thenextstatementcanbeusedtodiscontinueoneparticularcycleandskiptothe“next”.
Controlstatementsaremostoftenusedinconnectionwithfunctionswhicharediscussedin
Chapter10[Writingyourownfunctions],page42,andwheremoreexampleswillemerge.
42
10 Writingyourownfunctions
Aswehaveseeninformallyalongtheway,theRlanguageallowstheusertocreateobjects of
modefunction.ThesearetrueRfunctionsthatarestoredinaspecialinternalformandmaybe
usedinfurtherexpressionsandsoon.Intheprocess,thelanguagegainsenormouslyinpower,
convenienceandelegance,andlearningtowriteusefulfunctionsisoneofthemainwaystomake
youruseofRcomfortableandproductive.
ItshouldbeemphasizedthatmostofthefunctionssuppliedaspartoftheRsystem,such
asmean(),var(),postscript()andsoon,arethemselveswritteninRandthusdonotdiffer
materiallyfromuserwrittenfunctions.
Afunctionisdefinedbyanassignmentoftheform
> name e <- - function(arg_1, , arg_2, ...) ) expression
Theexpression is s anR expression,(usually a groupedexpression), , that t uses thearguments,
arg
i,tocalculateavalue.Thevalueoftheexpressionisthevaluereturnedforthefunction.
Acallto the functionthenusually takes theform name(expr_1,expr_2,...) andmay
occuranywhereafunctioncallislegitimate.
10.1 Simpleexamples
Asafirstexample,considerafunctiontocalculatethetwosamplet-statistic,showing“allthe
steps”. Thisisanartificialexample,ofcourse,sincethereareother,simplerwaysofachieving
thesameend.
Thefunctionisdefinedasfollows:
> twosam m <- function(y1, y2) {
n1 <- - length(y1); n2 2 <- - length(y2)
yb1 <- mean(y1);
yb2 <- - mean(y2)
s1 <- - var(y1);
s2 <- - var(y2)
s <- ((n1-1)*s1 + (n2-1)*s2)/(n1+n2-2)
tst <- (yb1 1 - - yb2)/sqrt(s*(1/n1 + + 1/n2))
tst
}
Withthisfunctiondefined,youcouldperformtwosamplet-testsusingacallsuchas
> tstat t <- twosam(data$male, data$female); ; tstat
Asasecondexample,considerafunctiontoemulatedirectly the Matlabbackslashcom-
mand,whichreturnsthecoefficientsoftheorthogonalprojectionofthevectoryontothecolumn
spaceofthematrix,X. (Thisis s ordinarilycalledtheleastsquares estimate oftheregression
coefficients.) Thiswouldordinarilybedonewiththeqr()function;howeverthisissometimes
abittrickytousedirectlyanditpaystohaveasimplefunctionsuchasthefollowingtouseit
safely.
Thusgivenanby1vectoryandannbypmatrixXthenXyisdefinedas(X
T
X)
X
T
y,
where(X
T
X)
isageneralizedinverseofX
0
X.
> bslash h <- function(X, , y) {
X <- qr(X)
qr.coef(X, y)
}
Afterthisobjectiscreateditmaybeusedinstatementssuchas
> regcoeff f <- bslash(Xmat, , yvar)
andsoon.
Chapter10: Writingyourownfunctions
43
The classicalRfunction lsfit() does thisjobquite well, andmore
1
. Itinturnusesthe
functionsqr()andqr.coef()intheslightlycounterintuitivewayabovetodothispartofthe
calculation.Hencethereisprobablysomevalueinhavingjustthispartisolatedinasimpleto
usefunctionifitisgoingtobeinfrequentuse. Ifso,wemaywishtomakeitamatrixbinary
operatorforevenmoreconvenientuse.
10.2 Definingnewbinaryoperators
Hadwegiventhebslash()functionadifferentname,namelyoneoftheform
%anything%
itcouldhavebeenusedasabinaryoperatorinexpressionsratherthaninfunctionform.Suppose,
forexample,wechoose!fortheinternalcharacter.Thefunctiondefinitionwouldthenstartas
> "%!%" " <- function(X, y) { ... . }
(Note the useofquotemarks.) ) Thefunctioncouldthenbeusedas s X%!%y. . (The e backslash
symbolitselfisnotaconvenientchoiceasitpresentsspecialproblemsinthiscontext.)
The matrix multiplicationoperator, %*%, and the outer product matrix operator r %o% % are
otherexamplesofbinaryoperatorsdefinedinthisway.
10.3 Namedargumentsanddefaults
As first noted in Section 2.3 [Generating g regular sequences], page 8, if f arguments s to called
functionsaregiveninthe“name=object”form,theymaybegiveninanyorder. Furthermore
theargumentsequencemaybeginintheunnamed,positionalform,andspecifynamedarguments
afterthepositionalarguments.
Thusifthereisafunctionfun1definedby
> fun1 1 <- - function(data, , data.frame, graph, limit) {
[functionbodyomitted]
}
thenthefunctionmaybeinvokedinseveralways,forexample
> ans s <- fun1(d, df, , TRUE, 20)
> ans s <- fun1(d, df, , graph=TRUE, , limit=20)
> ans s <- fun1(data=d, limit=20, , graph=TRUE, , data.frame=df)
areallequivalent.
Inmanycasesargumentscanbegivencommonlyappropriatedefaultvalues,inwhichcase
theymaybeomittedaltogetherfromthecallwhenthedefaultsareappropriate. Forexample,
iffun1weredefinedas
> fun1 1 <- - function(data, , data.frame, graph=TRUE, , limit=20) ) { ... . }
itcouldbecalledas
> ans s <- fun1(d, df)
whichisnowequivalenttothethreecasesabove,oras
> ans s <- fun1(d, df, , limit=10)
whichchangesoneofthedefaults.
It is s important to note that t defaults may y be arbitrary expressions, , even n involving other
argumentstothesamefunction;theyarenotrestrictedtobeconstantsasinoursimpleexample
here.
1
SeealsothemethodsdescribedinChapter11[StatisticalmodelsinR],page51
Chapter10: Writingyourownfunctions
44
10.4 The‘...’argument
Anotherfrequentrequirementistoallowonefunctiontopassonargumentsettingstoanother.
Forexamplemanygraphicsfunctionsusethefunctionpar()andfunctionslikeplot()allowthe
usertopassongraphicalparameterstopar()tocontrolthegraphicaloutput.(SeeSection12.4.1
[Thepar()function], page 68, for r moredetailsonthe par()function.) ) This s canbedoneby
includinganextraargument,literally‘...’,ofthefunction,whichmaythenbepassedon.An
outlineexampleisgivenbelow.
fun1 <- - function(data, data.frame, , graph=TRUE, , limit=20, , ...) ) {
[omittedstatements]
if (graph)
par(pch="*", ...)
[moreomissions]
}
Less frequently, a function will need to o refer r to o components of ‘...’. . The e expression
list(...) evaluates all l such arguments and returns them in a named list, while e ..1, ..2,
etc.evaluatethemoneatatime,with‘..n’returningthen’thunmatchedargument.
10.5 Assignmentswithinfunctions
Notethatanyordinaryassignmentsdone withinthefunctionarelocalandtemporaryandare
lost afterexitfromthe function. . Thustheassignment t X<-qr(X)doesnotaffectthevalueof
theargumentinthecallingprogram.
TounderstandcompletelytherulesgoverningthescopeofRassignmentsthereaderneeds
tobe familiarwiththenotionofanevaluationframe. . This s isa somewhatadvanced,though
hardlydifficult,topicandisnotcoveredfurtherhere.
If globaland permanent assignments areintendedwithin afunction, theneither the“su-
perassignment”operator,<<-orthefunctionassign()canbeused.Seethehelpdocumentfor
details.S-Plususersshouldbeawarethat<<-hasdifferentsemanticsinR.Thesearediscussed
furtherinSection10.7[Scope],page46.
10.6 Moreadvancedexamples
10.6.1 Efficiencyfactorsinblockdesigns
As a a more complete, , if f a little pedestrian, , example of f a a function, , consider finding g the effi-
ciencyfactorsforablockdesign. (Someaspectsofthisproblemhavealreadybeendiscussedin
Section5.3[Indexmatrices],page19.)
Ablockdesignisdefinedbytwofactors,sayblocks(blevels)andvarieties(vlevels). IfR
andKarethevbyvandbbybreplicationsandblocksize matrices,respectively,andN isthe
bbyvincidencematrix,thentheefficiencyfactorsaredefinedastheeigenvaluesofthematrix
E=I
v
R
1=2
N
T
K
1
NR
1=2
=I
v
A
T
A;
whereA=K
1=2
NR
1=2
. Onewaytowritethefunctionisgivenbelow.
> bdeff f <- function(blocks, varieties) {
blocks <- as.factor(blocks)
# minorsafetymove
b <- length(levels(blocks))
varieties <- as.factor(varieties)
# minorsafetymove
v <- length(levels(varieties))
K <- as.vector(table(blocks))
# removedimattr
R <- as.vector(table(varieties))
# removedimattr
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