R Installation

1. R Installation R is an open sourcesoftwarepackageforstatisticaldataanalysis

2. R Installation • R download: http://www.r-project.org/ • Germany, Stefan Drees Bonn: http://cran.r-mirror.de/ • This isthemainprogram! • RStudio (Auxiliaryprogramforediting R files): http://www.rstudio.com/ide/download/ • Installtheprograms (R shouldbetherealready).

3. Working withthe R command prompt (withoutRStudio) • Practicalsession

4. Working withRStudio • Start RStudio • „File -> New -> R Script“ Letsyouedit a R commandor R script (= smallprogramme = severalconsecutivecommands)

5. Working withRStudio • Select • Files • Plots • Packages (foradvancedanalyses) • Help New R files The command prompt

6. Working withRStudio • Commandsandprogramscanbestored in R files • Execute onecommandline: • Ctrl+Enteror • Button „Run“ • Execute severallines: • Mark linesanduse „Ctrl+Enter“ or „Run“ button

7. WhatisStatistics?

8. Whatisstatistics? • Statisticsis a meanstoconnectempiricalknowledgeandtheoryandisconstitutedasfollows: • Data representation (Empirics) • Methodsfordescription, analysis, andinterpretationofdata, in ordertoallowpredictions, conclusionsanddecisions(Statistical Theory)

9. Whatisstatistics? • DescriptiveStatistics • Probabilitytheory • Test theory

10. DescriptiveStatistics

11. DescriptiveStatistics • Basic concepts: • Population: Collectionofobjectsforwhich a conclusionshallbemade (canbe human beingsbut also a collectionofatomswhenapplied in physics) • Sample: a representativepart/sub-set ofthepopulation • Random sample: elementsofthepopulationdrawnrandomlyandindependentlyofeachother • Example: „Mietspiegel“ (= statisticsofrents) forthecityof Bonn • Population: all rooms, flats etc. forrent in Bonn (toomanytoinvestigate all) • Sample: selectedpart; all flatsfromPoppelsdorf • Random sample: Investigation of n = 100, 200,… randomobjectsfrom Bonn

12. Attributes / traits Values qualitative quantitative Descriptivestatistics patients, bloodsamples, DNA samples, houses, atoms Observationalobjects bloodpressure, weight, age, bloodgroup, numberofsiblings, maritalstatus, rent Blood group, maritalstatus discrete Numberofsiblings continuous bloodpressure, weight, age, rent

13. Descriptivestatistics • Scaling: • Nominal scale: attributevaluesthatare not directlycomparable (sex, subjectofstudies, countryoforigin) (qualitative) • Ordinalscale: attributevaluesthathave a „natural“ order (grades, fontsizes: tiny-small-medium-large-huge) • Intervalscale: differencebetweenattributevaluesisinterpretable (temperature in °C) (quantitative) • Tobedistinguished: • Discreteattributes: Attribute valuescanbecounted • Continuousattributes: All real numbers, orat least all numbersfrom an interval, arepossible

14. Descriptivestatistics • Frequencies: • Absolute frequencyni: • Numberofobersvationswithattributevaluei (counts) • Relative frequency hi: • Portion ofelementswithattributevaluei • Tobecomputedas absolute frequencydevidedby total numberofobjectsN: ni/ N • Relative frequenciesliebetween 0 and 1 • Relative frequencieshavetoaddupto 1 (<- canbeusedto check computation)

15. tally sheet value absolute frequencyni relative frequency hi Bonn Köln Bonn Köln 0.34 0.39 1 0 17 78 0.38 0.38 2 A1 19 76 0.12 0.10 3 A2 6 20 0.10 0.09 4 B 5 18 0.04 0.03 5 A1B 2 6 0.02 0.01 6 A2B 1 2 0.00 0.00 7 other 0 0 1.00 200 Descriptivestatistics AB0 bloodgroup N = 50 1.00

16. Descriptivestatistics • Frequencies: • Cumulativefrequency: • Sumof all frequenciesupto a givenvaluei. • Denotedasfor absolute frequenciesanddenotedasifor relative frequencies • Oftenusedwhenvaluesaresubdividedintoclasses • Classification: • Arrangement ofattributevaluesintodisjointgroups, so called „classes“ • Classesaredisjoint, i.e. non-overlapping, andneighbouringintervalsofattributevalues, whicharedefinedby a lowerand an upperbound. Neighbouringvaluesimpliesthateachvaluebelongsto a classanddoes not lieoutiside(completenessoftheclassification).

17. 150 200 height [cm] classlimits: • (160; 170] contains all values, that are > 160 but  170. 150 160 170 180 190 200 height [cm] ] ( ] ( ( ] ] ( ( ] ( Descriptivestatistics height • complete • disjoint (eachvaluebelongstoonlyoneclass) classification:

18. Class number i Class limits (ai-1; ai] Tally sheet frequency Cumulative frequency absolute ni relative hi absolute Ni relative Hi 1  150 0 0.00 0 0.00 2 (150; 160] 5 0.05 5 0.05 3 (160; 170] 30 0.30 35 0.35 4 (170; 180] 35 0.35 70 0.70 5 (180; 190] 25 0.25 95 0.95 6 (190; 200] 5 0.05 100 1.00 7 > 200 0 0.00 100 1.00 N=100 1,00 Descriptivestatistics height [cm]

19. Descriptivestatistics Graphicalrepresentation

20. Descriptivestatistics • Piechart (R function: pie() ) • Shows absolute frequencies • Example: bloodgroups

21. Descriptivestatistics • Bar chart (R function: barplot() ) • Shows relative frequencies • Example: bloodgroups

22. Frequencies Cumulativefrequencies Number of children Tally sheet absoluteni relative hi 5 0.10 1 0 5 0.10 0.50 25 2 1 20 0.40 40 0.80 3 2 15 0.30 45 4 3 0.90 5 0.10 48 0.96 5 4 3 0.06 50 1.00 6 >4 2 0.04 1.00 N = 50 Descriptivestatistics • Representationofcumulativefrequencieswithempiricaldistributionfunction F • Discretetrait: NumberofChildren relative Hi absoluteNi

23. F H h i i 1.0 1.0 0.8 0.8 hi 0.6 0.6 0.4 0.4 0.2 0.2 hi 0.0 0.0 0 4 >4 2 1 3 0 1 2 3 4 >4 F: Empiricaldistributionfunction Sincetheattributeis quantitative discrete, weobtain a stepfunction Descriptivestatistics Number of children Bar chart

24. Descriptivestatistics • Histogramms (R function: hist() ) • Construction: • Data issubdevidedintoclasses • Surfacearea ofcolumnsisproportional totherespectivefrequencies • Columns areneighbouringsinceclassesareneighbouring

25. 1 0,8 0,6 0,4 0,2 0 height[cm] 150 160 170 180 190 200 Descriptivestatistics Example: Height [cm] hi Histogram

26. f empiricaldensityfunction f 0,8 0,6 F • • 0,4 0,8 • empiricaldistributionfunction F (forcontinuoustrait) 0,6 0,2 0,4 • height[cm] 0 0,2 200 200 150 150 160 160 170 170 180 180 190 190 • • height[cm] 0 Descriptivestatistics

27. f empirical density function f 0,8 hi 0,6 F • • 0,4 hi 0,8 • empirical distribution function F 0,6 0,2 0,4 • height[cm] 0 0,2 200 200 150 150 160 160 170 170 180 180 190 190 • • height[cm] 0 Descriptivestatistics

28. DescriptiveStatistics • Note: Slides 23 and 26 bothshowempiricaldistributionfunctions. In thefirstcase, weobtain a stepfunctionsincethetraitunderinvestigationisdiscrete.

29. Descriptivestatistics Measuresofcentraltendency, dispersionandspread

30. Descriptivestatistics • Measuresofcentraltendency: • A numbertocharacterizethe „center“ ofthedata • Most important: • Mean • Median

31. sample sample ranks ranks x(1)=3 x(1)=3 x1=5 x1=5 x(2)=4 x(2)=4 x2 =9 x2 =9 x(3)=5 x(3)=5 x3=3 x3=3 x(4)=6 x(4)=6 x4=8 x4=8 x(5)=8 x(5)=7 x5=19 x5=19 x(6)=9 x(6)=8 x6=4 x6=4 x(7)=19 x(7)=9 x7=6 x7=6 x(8)=19 x8=7 Descriptivestatistics • Median (R function: median() ) • Sample: Order accordingto: Ordered sample: • Median n = 8 even: n = 7 odd:

32. Descriptivestatistics • Mean (R function: mean() ) • Sample: • Sample size: n • Mean

33. Descriptivestatistics • Comparison of median andmean: • Bothsampleshave median 2500 • and arethemeanvalues • Meancanstronglybeinfluencedby a singlevalue • Median ismore robust against extreme values („outliers“) Nevertheless, themeanismoreoftenused in practicesinceithasotherdesirableproperties (seelater).

34. outlier x Descriptivestatistics How to treat outliers? 1) Discard No! 2) Check value and correct Yes!

35. sample A sample B x x Descriptivestatistics • Measuretheamountofvariationofthedata!  The mean (or median) is not sufficenttodescribe a sample

36. Descriptivestatistics • Measuresofdispersionandspread: • Numbers tocharacterizetheamountvariationaroundthecenter (= mean) • Most important: • Minimum, maximum, range (dispersion) • Empiricalvariance (spread) • Empiricalstandarddeviation (spread)

37. ranks sample x(1)=3 x1=5 x(2)=4 x2 =9 x(3)=5 x3=3 x(4)=6 x4=8 x(5)=8 x5=19 x(6)=9 x6=4 x(7)=19 x7=6 n=7 n=7 Descriptivestatistics • range: • minimum: min = x(1) • maximum: max = x(n) • range:R = x(n) – x(1)

38. Descriptivestatistics • Variance (R function: var() ): • A measureto express thespreadaroundthecenter(mean) by a singlevalue • The squareddeviationofeachattributevaluefromthemeanisconsidered. • Formulafortheempiricalvariancefrom a sample ofelements: • The empircalstandarddeviationis just thesquarerootofthevariance, . (R function: sd() ).

39. x1 = 75 x2 = 2 x3 = 270 x4 = n = 4 = 100 x4=53 isnot free,but givenbyothervalueswhenthemeanisknown. s2has (n-1) degreesoffreedom (f) Whydevidebyinsteadof? Example:

40. Data • Ifyouhavedatayouwanttoanalyse, please bring italong!