Chapter 1 ASPECTS OF MUTIVARIATE ANALYSIS

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1 Chapter ASPECTS OF MUTIVARIATE ANALYSIS. Itroductio Defiitio Wiipedia: Multivariate aalyi MVA i baed o the tatitical priciple of multivariate tatitic which ivolve obervatio ad aalyi of more tha oe tatitical variable at a time. The objective of cietific ivetigatio to which multivariate method mot aturally led themelve iclude the followig.. Data reductio or tructural implificatio. Sortig ad groupig 3. Ivetigatio of the depedece amog variable 4. Predictio 5. Hypothei cotructio ad Tetig Eample: I the real world mot data collectio cheme or deiged eperimet that provide data are multivariate i ature. Some eample of uch ituatio are give below. Durig a urvey of houehold everal meauremet o each houehold are tae. Thee meauremet beig tae o the ame houehold will be depedet. For eample the educatio level of the head of the houehold ad the aual icome of the family are related. Durig a productio proce a umber of differet meauremet uch a the teile tregth brittlee diameter etc. are tae o the ame uit. Collectively uch data are viewed a multivariate data. P a g e

2 Price of a car deped o everal factor ay year mileage warraty HP model amog may. Here year mileage warraty are correlated. Body fite deped o age height weight amout of eercie food habit etc. Here height ad weight are related. A ew drug i to be compared with a cotrol for it effectivee. Two differet group of patiet are aiged to each of the two treatmet ad they are oberved weely for et two moth. The periodic meauremet o the ame patiet will ehibit depedece ad thu the baic problem i multivariate i ature.. Applicatio of Multivariate Techique Some applicatio amog may are decribig below:. Data reductio or tructural implificatio. Sortig ad groupig 3. Ivetigatio of the depedece amog variable 4. Predictio 5. Hypothei cotructio ad Tetig Read page 3 ad 4 for applicatio for each of the above categorie. P a g e

3 . The Orgaizatio of Data Array Multivariate data arie wheever a ivetigator eeig to udertad a ocial or phyical pheomeo elect a umber p of variable or character to record. The value of thee variable are all recorded for each ditict item idividual or eperimetal uit. We will ue otatio j to idicate the particular value of the th variable that i oberved o the jth item or trial. That i j =meauremet of the th variable o the jth item Now meauremet o p variable ca be diplayed a follow Variable Variable. Variable Variable p Item. ip Item. p. Item j j j. j jp Item p Thee data ca be diplayed a a rectagular array Called X of row ad p colum. The array X cotai all of the obervatio o all of the variable. X......p......p... j j... j... jp p 3 P a g e

4 Eample. Page 6: Number of boo ad dollar ale A electio of four receipt from a uiverity bootore wa obtaied i order to ivetigate the ature of boo ale. Each receipt provided amog other thig the total amout of each ale ad the umber of boo old. The data are give below: Variable dollar ale Variable # of boo The the data array X i with 4 row ad colum X Here =4 =5 4 =3 Decriptive Statitic Decriptive tatitic decribe the data. For eample mea variace tadard deviatio correlatio ewe ad urtoi are decriptive tatitic. We will dicu motly dicu decriptive tatitic that meaure locatio variatio ad liear aociatio. The formal defiitio of thee quatitie are give below. Let be meauremet o variable. The the ample mea of thee meauremet i j j 4 P a g e

5 5 P a g e The ecod ample mea: j j The p ample mea:...p j j The ample variace which meaure the variability of the data alo called diperio OR pread of meauremet for variable i j j The ample variace of meauremet for p variable...p j j Note that ad the quare root of the ample variace i ow a the ample tadard deviatio SD. Note: Motly we will be ued SD to meaure the variability a it ha the ame uit of meauremet lie a mea or media. Sample Covariace: Coider pair of meauremet o each of variable &

6 A meaure of liear aociatio betwee the meauremet of variable ad i provided by the ample covariace. The ample covariace betwee variable ad i deoted by ad defied a j j j The ample covariace betwee ith ad th variable i deoted by i ad defied a i j ji i j ; i..p...p Thi i the average product of the deviatio from their repective mea. Sample correlatio coefficiet alo ow a Pearo product correlatio coefficiet The ample correlatio coefficiet betwee ith ad th variable i deoted by r i ad defied a r i ii i i i j j ji ji i i j j j ; i..p The ample correlatio coefficiet r ha the followig propertie:. The value of r lie betwee - ad + icluive.. r meaure the tregth of liear aociatio. Thu r=0 implie lac of liear aociatio betwee two variable. 6 P a g e

7 3. r=± a perfect liear aociatio. 4. r> 0 implie a tedecy for oe value of the pair to be large whe other value i large ad alo both value to be mall together. 5. r < 0 implie a tedecy for oe value i the pair to be large tha it average whe other value i maller tha it average 6. The value of r i remai uchaged if the meauremet of the ith variable are chaged to variable chaged to y j y ji = a ji + b ad the value of the th = c j + b provided that the cotat a ad c have ame ig. That mea r i ivariat i both locatio ad cale of meauremet. Array of Baic Decriptive Statitic The decriptive tatitic computed from meauremet o p variable ca be orgaized ito array. Sample mea X p Sample variace ad covariace S p p p p pp 7 P a g e

8 Sample correlatio R r r p r r p r r p p Eample. page 0 The array X S ad R for bivariate data i Eample. are give below X 50 4 S R r =-0.36 wea egative liear relatiohip betwee two variable X ad X. Graphical Techique: Scatter Plot: Uig SPSS we obtai the followig catter plot betwee variable &. Variable : Variable : P a g e

9 Figure.: A catter plot betwee variable ad Uig SPSS we obtai r =0.96. Strog correlatio betwee variable &. The catter diagram Figure. gave the ame impreio about the trog liear relatiohip betwee variable &. Eample.4 A catter plot for baeball data Table.: 977 Salary ad Fial Record for the Natioal League Eat Team Player payroll Wo-lot percetage Philadelphia Pittburg St. Loui Chicago Motreal New Yor P a g e

10 The catter plot uig SPSS Figure.4: Salarie ad wo-lot percetage Table - page 4 Eample.6 page 7: A zoologit obtaied meauremet o =5 lizard. The weight or ma i give i gram while the out-vet legth SVL ad hid limb pa HLS are give i millimeter. The data are diplayed i Table.3. Table.3: Lizard Size Data Lizard Ma SVL HLS P a g e

11 Uig Miitab 3D Scatterplot of Ma v HLS v SVL 5 Ma HLS SVL P a g e

12 From Figure.6 ad.7 we ca ee that mot of the variatio i catter about a oe-dimeioal traight lie..4 Data Diplay ad Pictorial Repreetatio Coider data i Eample.6 ad do a matri plot which i a liig multiple two-dimeioal plot. Correlatio Matri Ma Pearo Correlatio SVL HLS Correlatio Ma SVL HLS.96 **.96 ** Sig. -tailed N Pearo.96 **.938 ** Correlatio Sig. -tailed N Pearo.96 **.938 ** Correlatio Sig. -tailed N *. Correlatio i igificat at the 0.0 level -tailed. P a g e

13 .5 Ditace If we coider the poit P= i the plae the traight lie ditace do P from P to the origi O=00 i accordig to the Pythagorea theorem i give by d O P The ituatio i illutrated i Figure.9 page P a g e

14 4 P a g e I geeral if the poit P ha p coordiate o that P=... p the traight lie ditace from P to origi O=00..0 i P O d p ' All poit p that lie a cotat quared ditace uch a c from the origi atify the followig equatio c O P d p The traight lie ditace betwee two arbitrary poit P ad Q with coordiate P=.. p ad Q=y y...y p i give by y y y y Q P d p p Stadardized Ditace: * * P O d -3 Uig -3 we ee that all poit which have coordiate ad are a cotat quared ditace c from the origi mut atify c -4 Equatio -4 i the equatio of a ellipe cetered at the origi whoe major ad mior ae coicide with the coordiate ae. Thi geeral cae i how i Figure. page 3.

15 Eample.4 page 3: Calculatig a tatitical ditace d O P 4 All poit that are a cotat ditace from the origi atify the equatio 4 The coordiate of ome poit a uit ditace from the origi are preeted i the followig Table Coordiate: / Ditace: / P a g e

16 A plot of the equatio 4 i give below The epreio i -3 ca be geeralized to accommodate the calculatio of tatitical ditace from a arbitrary poit P= to ay fied poit Q=y y. If we aume that the coordiate variable vary idepedetly o oe aother the ditace from P to Q i give by d P Q y y Let the poit P ad Q have p coordiated uch that P=... p ad Q=y y y p. Suppoe Q i a fied poit [it could be O=0 0 0] ad the coordiate variable vary idepedetly of oe aother. Let. pp be ample variace cotructed from meauremet o p repectively. The the tatitical ditace from P to Q i y y p y p d P Q pp 6 P a g e