BIOSTATISTICS. Lecture 1 Data Presentation and Descriptive Statistics. dr. Petr Nazarov

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1 Microrry Center BIOSTATISTICS Lecture 1 Dt Presenttion Descriptive Sttistics dr. Petr Nzrov petr.nzrov@crp-snte.lu Lecture 1. Dt presenttion descriptive sttistics

2 COURSE OVERVIEW Orgniztion Orgniztion 10 sessions = 9 mes + 1 finlizing (30 hours in totl) 1 session = 1 hr lecture mixed with 2 hr prcticl work 3 intermedite tests + finlizing exm (solving tsks) Pln cn be found on moodle. However it my be corrected to fit level group (especilly, modeling prt will be moved furr) Gol: your FINAL knowledge skills in biologicl nlysis! not reltively fir mrk for your work Microst Excel Stwre with Dt Anlysis Add-In instlled Dt Lecture 1. Dt presenttion descriptive sttistics 2

3 COURSE OVERVIEW Recommended Literture Recommended Literture presenttion methodology Lecture 1. Dt presenttion descriptive sttistics 3

4 COURSE OVERVIEW Introduction Any biologicl study where numbers re mesured or reported Drug discovery BIOSTATISTICS: why where? Genomics systems biology Lecture 1. Dt presenttion descriptive sttistics Public helth 4

5 OUTLINE Lecture 1 Dt sttistic elements, vribles observtion types (qulittive quntittive) scles (nominl, ordinl, intervl, rtio) Descriptive sttistics: tbulr grphicl presenttion frequency distribution pie, br chrt histogrm representtion cumultive distributions crosstbultion sctter digrm Descriptive sttistics: numericl mesures mesures loction: men, mode, medin, quntiles/qurtiles/percentiles mesure vribility: vrince, strd devition, MAD, coefficient vrition or mesures: skewness distribution z-score. Chebyshev's orem. Detection outliers. Explortory nlysis. 5 number summry box plot Mesure ssocition between two vribles covrince correltion coefficient interprettion correltion coefficient Lecture 1. Dt presenttion descriptive sttistics 5

6 DATA AND STATISTICS Elements, vribles, observtions, scles types Lecture 1. Dt presenttion descriptive sttistics 6

7 DATA AND STATISTICS Dt: Elements, Vribles, Observtions Dt Dt The The fcts fcts figures figures collected, nlyzed, summrized for for presenttion interprettion. elements vribles observtion Person Plce Gender Net Worth ($BIL) Age Source Internet Fme Score Willim Gtes III 1 M Microst 9.5 Wrren Buffett 2 M Berkshire Hthwy 6.6 Crlos Slim Helu 3 M telecom 2.1 Lwrence Ellison 4 M Orcle 2.8 Ingvr Kmprd 5 M IKEA 2.4 Krl Albrecht 6 M Aldi 3.6 Mukesh Ambni 7 M petrochemicls 4.4 Lkshmi Mittl 8 M steel 5.4 Theo Albrecht 9 M Aldi 1.5 Amncio Orteg 10 M Zr 1.9 Jim Wlton 11 M Wl-Mrt 3.9 Alice Wlton 12 F Wl-Mrt 2.9 Cn we consider Plce s element? ( log 4.5) IFS = 3 10 N Lecture 1. Dt presenttion descriptive sttistics 7

8 DATA AND STATISTICS Dt Scles Types Dt Dt scles: Qulittive Nominl scle scle use use lbels lbels or or nmes nmes to to identify identify n n ttribute ttribute n n element. element. Ordinl scle scle exhibit exhibit properties properties nominl nominl order order or or rnk rnk is is meningful. meningful. Ex.1: Ex.2: Ex.1: Ex.2: Mle, Femle Rooms #: 101, 102, 103, Winners: The 1 st, 2 nd, 3 rd plces Mrks: A, B, C, Quntittive Intervl scle scle demonstrte demonstrte properties properties ordinl ordinl intervl intervl between between vlues vlues is is expressed expressed in in terms terms fixed fixed unit unit mesure mesure Rtio Rtio scle scle demonstrte demonstrte ll ll properties properties intervl intervl rtio rtio two two vlues vlues is is meningful. meningful. Ex.1: Exmintion score Ex.2: Internet fme score Ex.1: Weight Ex.2: Price Lecture 1. Dt presenttion descriptive sttistics 8

9 DATA AND STATISTICS Tsk: Define Scles Person Plce Gender Net Worth ($BIL) Age Source Internet Fme Score Willim Gtes III 1 M Microst 9.5 Wrren Buffett 2 M Berkshire Hthwy 6.6 Crlos Slim Helu 3 M telecom 2.1 Lwrence Ellison 4 M Orcle 2.8 Ingvr Kmprd 5 M IKEA 2.4 Krl Albrecht 6 M Aldi 3.6 Mukesh Ambni 7 M petrochemicls 4.4 Lkshmi Mittl 8 M steel 5.4 Theo Albrecht 9 M Aldi 1.5 Amncio Orteg 10 M Zr 1.9 Jim Wlton 11 M Wl-Mrt 3.9 Alice Wlton 12 F Wl-Mrt 2.9 ( log 4.5) IFS = 3 10 N? Lecture 1. Dt presenttion descriptive sttistics 9

10 TABULAR AND GRAPHICAL PRESENTATION Frequency distribution, br pie chrts, histogrm, cumultive frequency distribution, sctter plot Lecture 1. Dt presenttion descriptive sttistics 10

11 TABULAR AND GRAPHICAL PRESENTATION Frequency Distribution Frequency Frequency distribution distribution A tbulr tbulr summry summry showing showing number number (frequency) (frequency) items items in in ech ech severl severl nonoverlpping nonoverlppingclsses. Mrks A B C B A B B A B C Frequency distribution: Mrk Frequency A 3 B 5 C 2 Totl 10 Reltive frequency distribution: Mrk Frequency A 0.3 B 0.5 C 0.2 Totl 1 In MS Excel use following functions: Percent frequency distribution: =COUNTIF(,element) to get number elements found in re =SUM() to get sum vlues in re Mrk Frequency A 30% B 50% C 20% Totl 100% Lecture 1. Dt presenttion descriptive sttistics 11

12 TABULAR AND GRAPHICAL PRESENTATION Exmple: Pncretitis Study The role smoking in etiology pncretitis hs been recognized for mny yers. To provide estimtes quntittive significnce se fctors, hospitl-bsed study ws crried out in estern Msschusetts Rhode Isl between ptients who hd hospitl dischrge dignosis pncretitis were included in this unmtched cse-control study. The control group consisted 217 ptients dmitted for diseses or thn those pncres biliry trct. Risk fctor informtion ws obtined from strdized interview with ech subject, conducted by trined interviewer. dpted from Chp T. Le, Introductory Biosttistics pncretitis.xls Pncretitis ptients: Smokers Ex-smokers Ex-smokers Smokers Smokers Smokers Ex-smokers Smokers Smokers Smokers Smokers Smokers Ex-smokers Smokers Smokers Ex-smokers Smokers Smokers Ex-smokers Ex-smokers Smokers Ex-smokers Smokers Smokers Never Smokers Ex-smokers Ex-smokers Smokers Ex-smokers Smokers Smokers Ex-smokers Smokers Smokers Smokers Smokers Smokers Ex-smokers Smokers Smokers Smokers Smokers Smokers Smokers Smokers Smokers Smokers Smokers Never Smokers Smokers Smokers Lecture 1. Dt presenttion descriptive sttistics 12

13 FREQUENCY DISTRIBUTION Reltive Frequency Distribution Frequency Frequency distribution distribution A tbulr tbulr summry summry showing showing number number (frequency) (frequency) items items in in ech ech severl severl nonoverlpping nonoverlppingclsses. Reltive Reltive frequency frequency distribution distribution A tbulr tbulr summry summry showing showing frction frction or or proportion proportion items items in in ech ech severl severl nonoverlpping nonoverlppingclsses. Sum Sum ll ll vlues vlues should should give give 1 Estimtion Estimtion probbility probbility distribution distribution When When number number experiments experiments n,, R.F.D. R.F.D. P.D. P.D. pncretitis.txt Frequency distribution: Smoking Cses Controls Never 2 56 Ex-smokers Smokers Totl Reltive frequency distribution: Smoking Cses Controls Never Ex-smokers Smokers Totl 1 1 In Excel use following functions: =COUNTIF(,element) to get number elements found in re =SUM() to get sum vlues in re Lecture 1. Dt presenttion descriptive sttistics 13

14 TABULAR AND GRAPHICAL PRESENTATION Br Pie Chrts pncretitis.xls 80% 80% 70% 70% 60% 60% 50% 50% 40% 40% 30% 30% 20% 20% 10% 10% 0% 0% Never Ex-smokers Smokers Never Ex-smokers Smokers Pncretitis Pncretitis Control Control Pncretitis Pncretitis Never Never Ex-smokers Ex-smokers Smokers Smokers Control Control Try to void using in scientific reports. For public/business presenttions only! Never Never Ex-smokers Ex-smokers Smokers Smokers In MS Excel use following steps: Chrt Wizrd Columns Set rnge (both columns Percent freq. distribution) Chrt Wizrd Pie Set rnge (one columns Percent freq. distribution) Lecture 1. Dt presenttion descriptive sttistics 14

15 TABULAR AND GRAPHICAL PRESENTATION Crosstbultion pncretitis.xls Disese Smoking or pncretitis Totl Ex-smokers Never Smokers Totl In Excel use following steps: Dt Pivot Tble PivotChrt MS Office list + Pivot Tble Set rnge, including heders Select output set lyout by drg--dropping nmes into tble Lecture 1. Dt presenttion descriptive sttistics 15

16 TABULAR AND GRAPHICAL PRESENTATION Exmple: Mice Dt Series Tordf Tordf MG, MG, Bchmnov Bchmnov AA AA Survey clcium & sodium intke Survey clcium & sodium intke metbolism metbolism with with bone bone body body composition composition Project Project symbol: symbol: Tordf3 Tordf3 Accession Accession number: number: MPD:103 MPD:103 mice.xls 790 mice from different strins prmeter Strting ge Ending ge Strting weight Ending weight Weight chnge Bleeding time Ionized C in blood Blood ph Bone minerl density Len tissues weight Ft weight Lecture 1. Dt presenttion descriptive sttistics 16

17 The following re weights in grms for 970 mice: TABULAR AND GRAPHICAL PRESENTATION Histogrm mice.xls Sorted weights show tht vlues re in grms. Let us divide weight into bins bins bins Weight,g Frequency >= More 0 Lecture 1. Dt presenttion descriptive sttistics 17

18 TABULAR AND GRAPHICAL PRESENTATION Histogrm Now, let us use bin-size = 1 grm Bin Frequency More 0 In Excel use following steps: Frequency Weight, g Specify column bins (intervl) upper-limits Tools Dt Anlysis Histrogrm select input, bins, output (Anlysis ToolPk should be instlled) use Chrt Wizrd Columns to visulize results Lecture 1. Dt presenttion descriptive sttistics 18

19 TABULAR AND GRAPHICAL PRESENTATION Cumultive Frequency Distribution Cumultive Cumultive frequency frequency distribution distribution A tbulr tbulr summry summry quntittive quntittive showing showing number number items items with with vlues vlues less less thn thn or or equl equl to to upper upper clss clss limit limit ech ech clss. clss. Frequency Weight, g Cumultive reltive frequency Ogive Weight, g Lecture 1. Dt presenttion descriptive sttistics 19

20 TABULAR AND GRAPHICAL PRESENTATION Sctter Plot mice.xls Let us look on mutul dependency Strting Ending weights Ending weight In Excel use following steps: Select region Use Chrt Wizrd XY (Sctter) Strting weight Lecture 1. Dt presenttion descriptive sttistics 20

21 NUMERICAL MEASURES Popultion smple, mesures loction, quntiles, qurtiles percentiles, mesures vribility, z-score, detection outliers, explortion nlysis, box plot, covrition, correltion Lecture 1. Dt presenttion descriptive sttistics 21

22 NUMERICAL MEASURES Popultion Smple Popultion Popultion prmeter prmeter A numericl numericl vlue vlue used used s s summry summry mesure mesure for for popultion popultion (e.g., (e.g., popultion popultion men men µ, µ, vrince vrince σ 2 σ, 2, strd strd devition devition σ) σ) POPULATION µ men σ 2 vrince N number elements (usully N= ) SAMPLE x m, men s 2 vrince n number elements Smple Smple sttistic sttistic A numericl numericl vlue vlue used used s s summry summry mesure mesure for for smple smple (e.g., (e.g., smple smple men men m, m, smple smple vrince vrince s s 2, 2, smple smple strd strd devition devition s) s) mice.xls 790 mice from different strins All existing lbortory Mus musculus ID Strin Sex Strting ge Ending ge Strting weight Ending weight Weight chnge Bleeding time Ionized C in blood Blood ph Bone minerl density Len tissues weight Ft weight 1 129S1/SvImJ f S1/SvImJ f S1/SvImJ f S1/SvImJ f S1/SvImJ f S1/SvImJ f S1/SvImJ f S1/SvImJ f S1/SvImJ f S1/SvImJ f S1/SvImJ m S1/SvImJ m S1/SvImJ m S1/SvImJ m S1/SvImJ m S1/SvImJ m S1/SvImJ m S1/SvImJ m S1/SvImJ m Lecture 1. Dt presenttion descriptive sttistics 22

23 NUMERICAL MEASURES Mesures Loction Men Men A mesure mesure centrl centrl loction loction computed computed by by summing summing vlues vlues dividing dividing by by number number observtions. observtions. x = µ = p = m = x i N n x ( x true) = i n i Medin Medin A mesure mesure centrl centrl loction loction provided provided by by vlue vlue in in middle middle when when re re rrnged rrnged in in scending scending order. order. Weight Mode Mode A mesure mesure loction, loction, defined defined s s vlue vlue tht tht occurs occurs with with gretest gretest frequency. frequency. Mode = 23 Medin = 23.5 Men = 31.7 Lecture 1. Dt presenttion descriptive sttistics 23

24 NUMERICAL MEASURES Mesures Loction mice.xls Histogrm p.d.f. pproximtion medinmen mode Femle proportion p f = Density weight, g Bleeding time In Excel use following functions: = AVERAGE() = MEDIAN() = MODE() Density medin = 55 men = 61 mode = N = 760 Bwidth = Lecture 1. Dt presenttion descriptive sttistics 24

25 NUMERICAL MEASURES Quntiles, Qurtiles Percentiles Percentile Percentile A vlue vlue such such tht tht t t lest lest p% p% observtions observtions re re less less thn thn or or equl equl to to this this vlue, vlue, t t lest lest (100-p)% (100-p)% observtions observtions re re greter greter thn thn or or equl equl to to this this vlue. vlue. The The 50-th 50-th percentile percentile is is medin. medin. Qurtiles Qurtiles The The 25th, 25th, 50th, 50th, 75th 75th percentiles, percentiles, referred referred to to s s first first qurtile, qurtile, second second qurtile qurtile (medin), (medin), third third qurtile, qurtile, respectively. respectively. In Excel use following functions: =PERCENTILE(,p) Weight Q 1 = 21 Q 2 = 23.5 Q 3 = 39 Lecture 1. Dt presenttion descriptive sttistics 25

26 NUMERICAL MEASURES Mesures Vribility Interqurtile Interqurtile rnge rnge (IQR) (IQR) A mesure mesure vribility, vribility, defined defined to to be be difference difference between between third third first first qurtiles. qurtiles. Vrince Vrince A mesure mesure vribility vribility bsed bsed on on squred squred devitions devitions vlues vlues bout bout men. men. Strd Strd devition devition A mesure mesure vribility vribility computed computed by by tking tking positive positive squre squre root root vrince. vrince. IQR = Q 3 Q 1 popultion smple s σ 2 x i = N ( ) 2 µ 2 ( x m) i = n 1 2 Smple strd devition = s = Popultion strd devition = σ = 2 s 2 σ Weight IQR = 18 Vrince = St. dev. = 17.9 In Excel use following functions: =VAR(), =STDEV() Lecture 1. Dt presenttion descriptive sttistics 26

27 NUMERICAL MEASURES Mesures Vribility Coefficient Coefficient vrition vrition A mesure mesure reltive reltive vribility vribility computed computed by by dividing dividing strd strd Strd devition devition devition by by men. 100 % men. Weight Men CV = 57% Medin Medin bsolute bsolute devition devition (MAD) (MAD) MAD MAD is is robust robust mesure mesure vribility vribility univrite univrite smple smple quntittive quntittive.. MAD = medin ( x medin( x) ) i Set 1 Set Set 1 Set 2 Men Medin St.dev MAD Lecture 1. Dt presenttion descriptive sttistics 27

28 NUMERICAL MEASURES Mesures Vribility Skewness Skewness A mesure mesure shpe shpe distribution. distribution. Dt Dt skewed skewed to to left left result result in in negtive negtive skewness; skewness; symmetric symmetric distribution distribution results results in in zero zero skewness; skewness; skewed skewed to to right right result result in in positive positive skewness. skewness. Skewness = n x m ( )( ) i n 1 n 2 s i 3 dpted from Anderson et l Sttistics for Business Economics Lecture 1. Dt presenttion descriptive sttistics 28

29 NUMERICAL MEASURES z-score Detection Outliers z-score z-score A vlue vlue computed computed by by dividing dividing devition devition bout bout men men (x (x i - i - x) x) by by strd strd devition devition s. s. A z-score z-score is is referred referred to to s s strdized strdized vlue vlue denotes denotes number number strd strd devitions devitions x x i is i is from from men. men. Chebyshev s orem For For ny ny set, set, t t lest lest (1 (1 1/z 1/z 2 2 )) vlues vlues must must be be within within z strd devitions from from men, men, where where z ny ny vlue vlue > z i = x i m s Weight z-score For ANY distribution: At lest 75 % vlues re within z = 2 strd devitions from men At lest 89 % vlues re within z = 3 strd devitions from men At lest 94 % vlues re within z = 4 strd devitions from men At lest 96% vlues re within z = 5 strd devitions from men Lecture 1. Dt presenttion descriptive sttistics 29

30 NUMERICAL MEASURES Detection Outliers For bell-shped distributions: Approximtely 68 % vlues re within 1 st.dev. from men Approximtely 95 % vlues re within 2 st.dev. from men Almost ll points re inside 3 st.dev. from men Outlier Outlier An An unusully unusully smll smll or or unusully unusully lrge lrge vlue. vlue. For For bell-shped bell-shped distributions distributions points points with with z >3 z >3 cn cn be be considered considered s s outliers. outliers. Exmple: Gussin distribution Weight z-score Lecture 1. Dt presenttion descriptive sttistics 30

31 NUMERICAL MEASURES Tsk: Detection Outliers mice.xls Using Excel, try to identify outlier mice on bsis Weight chnge vrible z i = x i s m For For bell-shped bell-shped distributions distributions points points with with z >3 z >3 cn cn be be considered considered s s outliers. outliers. In Excel use following functions: = AVERAGE() - men, m = STDEV() - strd devition, s = bs() - bsolute vlue sort by z-scle to identify outliers Lecture 1. Dt presenttion descriptive sttistics 31

32 NUMERICAL MEASURES Explortion Dt Anlysis Five-number Five-number summry summry An An explortory explortory nlysis nlysis technique technique tht tht uses uses five five numbers numbers to to summrize summrize : : smllest smllest vlue, vlue, first first qurtile, qurtile, medin, medin, third third qurtile, qurtile, lrgest lrgest vlue vlue children.xls Min. Min. : : Q 1 : 1 : Medin: Medin: Q 3 : 3 : Mx. Mx. : : In Excel use: Tool Dt Anlysis Descriptive Sttistics Box Box plot plot A grphicl grphicl summry summry bsed bsed on on five-number five-number summry summry Min Q 2 Q 1 Box Qplot 3 Mx In Excel use (indirect): Chrt Wizrd Stock Open-high-low-close open Q3 high Q3+1.5*IQR low Q1-1.5*IQR close Q1 1.5 IQR Lecture 1. Dt presenttion descriptive sttistics 32

33 NUMERICAL MEASURES Exmple: Mice Weight Exmple Exmple Build Build box box plot plot for for weights weights mle mle femle femle mice mice mice.xls 1. Build 5 number summries for mles femles Femle Mle Min Q Q Q Mx Combine numbers into following order open Q3 high Q3+min(1.5*(Q3-Q1),Mx) low Q1-mx(1.5*(Q3-Q1),Min) close Q1 Mouse weight In Excel use: Chrt Wizrd Stock Open-high-low-close Put series-in-rows Adjust colors, etc Weight, g Femle Mle Lecture 1. Dt presenttion descriptive sttistics 33

34 NUMERICAL MEASURES Mesure Assocition between 2 Vribles Covrince Covrince A mesure mesure liner liner ssocition ssocition between between two two vribles. vribles. Positive Positive vlues vlues indicte indicte positive positive reltionship; reltionship; negtive negtive vlues vlues indicte indicte negtive negtive reltionship. reltionship. σ xy = popultion ( xi µ x )( yi µ y ) N s xy smple ( x x)( y y) = n 1 i i mice.xls Ending weight vs. Strting weight Ending weight Strting weight In Excel use function: =COVAR() s xy = 39.8 hrd to interpret Lecture 1. Dt presenttion descriptive sttistics 34

35 NUMERICAL MEASURES Mesure Assocition between 2 Vribles Correltion Correltion (Person (Person product product moment moment correltion correltion coefficient) coefficient) A mesure mesure liner liner ssocition ssocition between between two two vribles vribles tht tht tkes tkeson on vlues vlues between between Vlues Vlues ner ner indicte indicte strong strong positive positive liner liner reltionship, reltionship, vlues vlues ner ner -1-1 indicte indicte strong strong negtive negtive liner liner reltionship; reltionship; vlues vlues ner ner zero zero indicte indicte lck lck liner liner reltionship. reltionship. popultion ( x x)( y y) σ xy i i ρxy = = σ σ σ σ N x y x y r xy smple sxy = = s s x y ( xi x)( yi y) s s ( n 1) x y 60 Ending weight In Excel use function: =CORREL() r xy = Strting weight mice.xls Lecture 1. Dt presenttion descriptive sttistics 35

36 NUMERICAL MEASURES Correltion Coefficient If If we we hve hve only only 2 points points in in x x y y sets, sets, wht wht vlues vlues would would you you expect expect for for correltion correltion b/w b/w xx y y? Wikipedi Lecture 1. Dt presenttion descriptive sttistics 36

37 NUMERICAL MEASURES Weighted Men Weighted Weighted men men The The men men obtined obtined by by ssigning ssigning ech ech observtion observtion weight weight tht tht reflects reflects its its importnce importnce m = w x i w i i As n exmple need weighted men, consider following smple five purchses rw mteril over severl months Note tht cost per pound vries from $2.80 to $3.40, quntity purchsed hs vried from 500 to Suppose tht mnger sked for informtion bout men cost per pound rw mteril. If we would use simple men cost p.p.: we overestimte verge cost! Anderson et l Sttistics for Business Economics Lecture 1. Dt presenttion descriptive sttistics 37

38 NUMERICAL MEASURES Grouped Men Grouped Grouped Dt Dt vilble vilble in in clss clss intervls intervls s s summrized summrized by by frequency frequency distribution. distribution. Individul Individul vlues vlues originl originl re re not not vilble. vilble. not vilble children.xls Bin Frequency More 0 Men for grouped m = k i f i n M i Vrince for grouped s 2 = k i f i ( M m) i n 1 2 Lecture 1. Dt presenttion descriptive sttistics 38

39 QUESTIONS? Thnk you for your ttention to be continued Lecture 1. Dt presenttion descriptive sttistics 39