CHAPTER 8 REGRESSION AND CORRELATION

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1 CHAPTER 8 REGREION AND CORRELATION

2 Determg for relatoshps ad makg predctos based from sample formato s oe of the ma dea of data aalss. Everoe wats to aswer questos lke Ca I predct how ma uts I ll sell f I sped amout of advertsg dollars? ; or Does drkg more det cola reall relate to more weght ga? ; or Do chldre s backpacks seem to be gettg heaver each ear school, or s t just me? Lear regresso tres to fd relatoshps betwee two or more varables ad comes up wth a model that tres to descrbe that relatoshp, much lke the wa the le = + 3 eplas the relatoshp betwee ad. But ulke math where fuctos lke = + 3 tell the etre stor about the two varables, statstcs, thgs do t come out that perfectl; some varablt ad error s volved (that s what makes t fu!). 8.1 Lear Correlato aalss It s a statstcal techque to determe the stregth or degree of lear relatoshp estg betwee two varables. A measure of the degree of lear relatoshp s called the correlato coeffcet, ρ. The followg are some eamples of scatter dagrams of two varables X ad Y. (1) ρ < 0 () ρ > 0 (3) ρ = 0 Tpes of Correlato

3 Correlato betwee varables ca be classfed terms of t drectos ad magtude. o A postve lear correlato ests wheever the depedet varable creases, the depedet varable also creases or whe the depedet varable decreases, the depedet varable also decreases. o A egatve lear correlato ests wheever the depedet varable creases, the depedet varable also decreases or whe the depedet varable creases, the depedet varable decreases. o A zero lear correlato ests, f o lear relatoshp ests betwee depedet ad depedet varables. (t ma be so, that the relatoshp s ot lear, the relatoshp mght be a epoetal or logarthmc relatoshp). Note that ths does ot mea that there s o assocato betwee the depedet varable wth the depedet varable. The Pearso product-momet correlato coeffcet or smpl the sample correlato coeffcet, r, estmates the stregth of lear relatoshp betwee two varables X ad Y, where r 1 1 1, 1 1,

4 1 1 Sample coeffcet of determato epresses the proporto of the total varato of the depedet varable that s eplaed or accouted for b the depedet varable. The sample coeffcet of determato s computed b just squarg the sample coeffcet ad s deoted b r. Eample From a radom sample of 9 studets from a class, scores o the mdterm eam () ad the fal eam () were as follows: X Y a) Determe ad terpret r. b) Determe ad terpret r. Soluto The followg are maual computatos for the requred formula to derve the sample correlato coeffcet ad the coeffcet of determato.

5 Total * , , r r Thus, the sample correlato coeffcet s appromatel Ths dcates a strog postve lear relatoshp betwee the mdterm eam ad the fal eam scores. Ad the sample coeffcet of determato s appromatel or % whch dcates that appromatel % of the varablt of the fal eam score s accouted for b the mdterm eam score.

6 Eercse 8.1 Do the followg. 1.. A stud was made o the amout of coverted sugar a certa process at dfferece temperatures. The followg observatos were recorded. Temp () Coverted Sugar () Determe the sample correlato coeffcet ad the sample coeffcet of determato.. The followg data gve the tmes of 0 swmmers who etered a 5 ard freestle swmmg competto. The values are the swmmers best tme for the seaso ad the values are ther tmes the competto Determe the sample correlato coeffcet ad the sample coeffcet of determato.

7 8. Smple lear Regresso Aalss Regresso Aalss s a statstcal techque used for determg the fuctoal form of the relatoshp betwee two or more varables, where oe varable s called the depedet varable or the respose varable Y ad the rest are called the depedet or cocomtat varables X s. Is objectve s usuall to be able to predct or estmate the value of the respose varable gve the values of the depedet varable(s). I a math class, we have leared that relatoshps are dsplaed b meas of graphs. If we are gve the value of, we ca compute for the value of, as f predctg what should be the value of. But statstcs, relatoshps are ot that eas to predct. For stace, we kow that the heght of a perso has a fluece o hs weght. However, t s ot the ol reaso. Stll ma other factors affect the weght of a perso such as age, se, bod structure, other evrometal factors, ad geetc ssues. Oe wa of eplorg relatoshps s va scatter plots. The followg data shows 10 emploed dvduals from large compaes Makat, who were tervewed, the umber of ears the compa ad the umber of pad leaves s determed for each. No. of rs Servce No. of Leaves

8 No. ofpad Leaves Plottg the values a Cartesa coordate sstem we have the followg No. of Years Servce NOTE Two major codtos must be satsfed before applg the smple lear regresso model to a data. The depedet varable should have a ormal dstrbuto for each value of the depedet varable The depedet varable should have a costat amout of dsperso (stadard devato) for each value of the depedet varable.

9 The followg llustrato shows what the assumptos of lear regresso s. Smple lear regresso allows us to descrbe the relatoshp betwee a depedet varable Y ad the depedet varable X usg a lear equato kow as the smple lear regresso equato, Y o 1 X where β o ad β 1 are called the regresso coeffcets ad From a sample, a estmate of the regresso equato s of the form ŷ bo b1

10 where b1 ad bo b , 1 1, Eample A stud was made b a facal aalst of Jeas ad Shrts Ic. to determe the relatoshp betwee ther advertsg epedtures (X) ad sales (Y) (both hudred thousads of pesos). The followg data were recorded: Epedtures Sales Determe the regresso equato. We have the followg computatos:

11 Total From the above computatos we ote that Usg the formulas to derve the estmated regresso equato, we have

12 , , b Ad b o b 17 ( ) Thus the estmated regresso equato s gve b Note that ths procedure s ver tedous ad legth. However, ma statstcal packages ca easl compute the eeded formato to costruct the estmated regresso equato. Such statstcal tool s a plug- from Mcrosoft Ecel called PHStat.

13 The followg are computer geerated outputs from Ecel: Regresso Aalss Regresso Statstcs Multple R R Square Adjusted R Square Stadard Error Observatos 10 Coeffcets Stadard Error t Stat Itercept Epedtures Based from the computer output the regresso equato s gve b: Ths s what we epected ad the same wth our maual computato.,

14 Eercse Do the followg. 1. The followg are the test scores of 10 studets a pre test math ad ther fal eam calculus. Pretest Calculus Grade Determe the estmated regresso equato where depedet varable s the fal grade calculus.. Determe the estmated regresso equato for eercse 8.1 No. 1. Graph the estmated regresso le ad costruct a scatter plot of the gve data o the same graph. 3. Determe the estmated regresso equato for eercse 8.1 No.. Graph the estmated regresso le ad costruct a scatter plot of the gve data o the same graph. 8.3 Resduals A resdual s the dfferece betwee the observed value of ad the predcted value of ). Specfcall, for a data pot, ou take ts observed -value ; ths comes from the data set,

15 ad subtract the epected -value; ths s the value of, o the estmated le. If the resdual s large, the le does t ft well that spot. If the resdual s small, the le fts well that spot. I geeral, a postve resdual dcates that the estmated regresso equato uderestmated at that pot, ad a egatve resdual meas t overestmated at that pot. Resduals helps us determe whether the -values come from a ormal dstrbuto. Ths would allow us to measure how far off our predctos were from the actual observatos. Resduals also help detfg problems that occurred the process of trg to ft a straght le to the observatos. Ths s better vewed o a graph.

16 Based o the computed regresso equato, the followg resdual values are computed.

17 Observato Predcted Sales Resduals Stadardzed Resduals Stadardzed resduals just mea covertg the resduals to a Z-score, so ou see where t falls o the stadard ormal dstrbuto. Ths meas that we eed to use the formula, z where, ad are the mea ad stadard devato of the samplg dstrbuto of the resduals. The followg graphs shows the scatter plot of the data, resdual plot ad ormal probablt plot.

Note: the plot above s graphed usg PHStat of Mcrosoft Ecel. Checkg for ormalt s oe of the assumptos of the lear regresso model ad the resduals helps us determe f ths codto s met.

19 Note: the plot above s graphed usg PHStat of Mcrosoft Ecel. Checkg for ormalt s oe of the assumptos of the lear regresso model ad the resduals helps us determe f ths codto s met. Ad ths codto of ormalt s met, f we ca see o the resdual plot lots of(stadardzed) resduals close to zero; ad fewer resduals as we go farther from zero. The followg descrbes some gudeles o makg a correct cocluso ad terpretato of the results a data aalss. It s a usual mstake to terpret the result of a regresso aalss to a cause ad effect coclusos. However, ths ca ol be doe ad asserted f ad ol f the method or

20 procedure of collectg formato s doe b a well desged epermet that cotrols possble varable that ma affect the depedet as well as the depedet varable. If ths s ot the case the what we ca ol assert s that there s a relatoshp betwee the varables cocered. We caot clude the terpretato as to what s causg the relatoshp. Etrapolatg o the result of the regresso aalss. Ths s aother commo mstake whe dog coclusos a lear regresso aalss. For stace, the depedet varable s the lot area of a house. We caot assume the value of the depedet varable to be just 1 square feet or a value of 1, 000, 000 square feet. Or f the depedet varable s the heght of a tree, t makes o sese to use a egatve value of the depedet varable. Kow the lmtato of a smple lear regresso. Smple lear regresso does ot mea t s smple to compute or eas to work o. But t s smple a sese that we do t actuall cosder other factors ad just focus o a specfc oe. However, most varable that s usuall cosdered real world stuatos are affected b ot just oe factor or varable. Most lkel, a combato of ma factors would be approprate to aalze the behavor of oe varable. For stace, the heght of a perso s ot just affected b ts age, t ca also be affected b the evromet at whch the perso lves, aother factor would volve geetc structure of the perso, socal status of the perso, ad ma other factors.

21 Eercse Do the followg 1. OP compouds are used largel as pestcdes. However t s mportat to stud ther effects o speces that are eposed to them. A epermet was coducted whch dfferet doses of a partcular pestcde was admstered to 5 groups of 5 rabbts. The respose varable was a measure of bra actvt. It was postulated that bra actvt would decrease wth crease OP dosage. Dose Actvt Dose Actvt Dose Actvt Dose Actvt a) Obta the estmated lear regresso equato for bra actvt usg dosage as the depedet varable. b) Use the obtaed lear regresso equato obtaed (A) to predct bra actvt whe dosage s c) Compute the correlato coeffcet ad descrbe the relatoshp betwee bra actvt ad OP dosage.

22 . Determe resduals ad stadardzed resdual for eercse 8.1 No. 1. Plot the stadardzed resdual agast the estmated values of the depedet varable. (Use the stadardzed resduals as the as ad the estmated values of the depedet varable as the -as) Iterpret the result. 3. Determe resduals ad stadardzed resdual for eercse 8.1 No.. Plot the stadardzed resdual agast the estmated values of the depedet varable. (Use the stadardzed resduals as the as ad the estmated values of the depedet varable as the -as) Iterpret the graph. 4. A facal aalst wats to determe the relatoshp betwee the umbers of tem sold of a partcular product at dfferet stores metro mala. The followg sample data shows the umber of tems sold ad the regular prce of a certa product amog dfferet stores. Prce () Php Items Sold () a. Determe the regresso equato b. Compute the correlato coeffcet c. Determe the coeffcet of determato d. What s the epected umber of tems sold f the prce s Php0. e. Costruct a scatter plot for the sample data f. Sketch the estmated regresso equato the scatter plot g. Costruct a table for the resduals ad stadardzed resduals

23 Chapter Summar 1. The sample correlato coeffcet determes the drecto ad magtude of the lear relatoshp of the varables volved.. The coeffcet of determato epresses the accoutablt of the depedet varable to the varablt of the depedet varable. That s, the gve percetage measure the wa the depedet varable eplas the varablt of the depedet varable. 3. The scatter plot presets a vsual terpretato of the sample data. Ad mplctl dcates f there s a possblt of a lear relatoshp betwee the varables volved 4. Regresso aalss provdes a tool that helps statstcas predct values of a certa varable usg ts relatoshp wth aother varable. 5. I terpretg the result of a regresso aalss, the followg should be cosdered: Kow the lmtatos of the smple lear regresso Use values for the depedet varable that makes sese. Avod etrapolato. Do ot assume a cause ad effect coclusos whe the procedure of obtag the formato s ot eplctl dcated. Ths s to avod makg such assumptos that all other factors that ma affect the values of both the depedet ad depedet varable.

24 STATISTICAL TOOLS The followg are just some of the avalable statstcal tools that ca be used to ad the computato of comple formulas. PHStat for Mcrosoft Ecel Data Aalss Tool Pack for Mcrosoft Ecel. Statstca SAS SP MTab

25 Chapter Revew Wrte the letter correspodg to the correct aswer. 1. It determes f there s a lear relatoshp betwee two varables. a. regresso b. Mea c. coeffcet of varato d. correlato coeffcet. Whch correlato coeffcet represets the strogest lear relatoshp betwee two varables. a. 0.0 b. 0.5 c d Whch of the followg correlato coeffcet dcates a perfect lear relatoshp. a. 1 b. -1 c. both a ad b d. oe 4. Whch of the followg scatter plot dcates a perfect postve lear relatoshp? a. b. c. d. 5. The value of r = -0.5 meas that the relatoshp betwee the two varable s a. egatve lear b. postve lear c. o lear relatoshp d. caot be determed For os A mathematcs placemet test s gve to all eterg freshme at the Wsto Uverst. The followg are the score of 0 studets the placemet test ad ther respectve fourth ear fal grade math. Placemet Score 4th Yr Math Grade Placemet Score 4th Yr Math Grade

26 If the estmated regresso equato s costructed the equato would be a. ˆ b. ˆ c. ˆ d. ˆ Computg for the sample correlato coeffcet we would obta the value a b c d The sample coeffcet of determato s evaluated to be a % b % c % d % 9. The resdual s, whe the 4 th ear math grade s 80. a b c d

27 10. The estmated placemet test score s, where the 4 th ear math grade s 75. a. 7. b. 79 c 74. d If aother freshma applcat took the test, what would be hs epected grade f hs fal grade 4 th r math s 99. a. 99. b. 90 c 95 d Based from the data provded, the relatoshp betwee the math placemet scores ad the 4 th ear grade math shows, a. a postve lear relatoshp b. egatve lear relatoshp c o lear relatoshp d. o relatoshp 13. Whch of the followg s true about the relatoshp of the correlato coeffcet ad the slope of the estmated regresso equato? a. the have dfferet sgs b. the have the same sg c. the the same d. o cocluso ca be made For umber 14-18

28 The followg data shows the mleage of 10 trucks of Joh Lee Truckg Compa ad the umber of das for the delver. Mles () Das () If the estmated regresso equato s costructed the equato would be a. ˆ b. ˆ c. ˆ d. ˆ Computg for the sample correlato coeffcet we would obta the value a b c d The sample coeffcet of determato s evaluated to be a % b % c % d %

29 17. The resdual s, where the umber of mleage s 85. a b c d The estmated umber of das for delver s, where the umber of mleage s 750. a b c d What ca we sa about the relatoshp betwee the mleage of the trucks of the compa agast the umber of das for delver. a. whe the mleage creases the umber of das for delver also creases b. whe the mleage creases the umber of das for delver decreases c. the mleage has a effect wth the umber of das for delver d. the mleage does ot have a effect o the umber of das for delver 0. Whe small (large) value of the depedet varable are assocated wth small(large) values of a depedet varable, we ca sa that the relatoshp betwee the two varables s, a. egatve lear relatoshp b. postve lear relatoshp c. o lear relatoshp d. o relatoshp

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