2.3 Least-Square regressions

Transcription

1 .3 Leat-Square regreon Eample.10 How do chldren grow? The pattern of growth vare from chld to chld, o we can bet undertandng the general pattern b followng the average heght of a number of chldren. Here the mean heght of a group of chldren n Kalama, an Egptan vllage that wa the te of a tud of nutrton n developng countre. The data were obtaned b meaurng the heght of 161 chldren from the vllage each month from 18 to 9 month of age. Age n month Data Heght n centmeter Scatterplot Correlaton r Regreon lne: ŷ lecture Leat-quare Regreon Suppoe we have n obervaton on two varable and : ( 1, 1 ), (, ),, ( n, n ). Aume there a trong lnear aocaton between and. Then we ma ft a traght lne Y a + bx to the data. Where a---ntercept, b---lope We ue leat quare method to calculate a and b:.e. mnmze We have b a r b ( )( ( r ) The equaton ŷ a + b called the leat-quare (LS) regreon lne. ) ( a b ). Predcton wth LS regreon lne We can ue a regreon lne to predct the repone for a pecfc value of the eplanator varable. But be aware of etrapolaton whch the ue of a regreon lne for predcton far outde the range of value of the eplanator varable that ou ued to obtan the lne. Such predcton are often not relable. lecture 5 1

2 3. Interpretng the regreon lne For LS regreon lne ŷ a + b, b----- the amount b whch change when ncreae b one unt. a----- the value of when 0. Note: the LS regreon lne alwa pae through the pont on the graph of agant. o the regreon lne ma be wrtten n an alternatve wa: (, ) r ( ) 4. Correlaton and regreon r coeffcent of determnaton r It the fracton of the varaton n the value of that eplaned b the LS regreon of on. Varaton of ha two ource: Value of var a change. Value of are cattered above and below ŷ. ˆ lecture Cauton about Regreon and Correlaton 1. Redual A redual the dfference between an oberved value of the repone varable and the value predcted b the regreon lne. That, redual oberved predcted ŷ. Redual plot A redual plot a catter plot of the regreon redual agant the eplanator varable. The tpcal pattern of redual plot: Untructured horzontal band centered at zero. no pattern mean the fttng good. Curved pattern mean relatonhp between and not lnear, o the fttng no good. Fan-haped pattern mean varaton of ncreae a ncreae, and the regreon fttng not good. lecture 5 4

3 3. Outler and nfluental obervaton Eample: Doe the age at whch a chld begn to talk predct a later core on a tet of mental ablt? A tud of cogntve development n oung chldren recorded the age n month at whch each of 1 chldren poke ther frt word and ther Geell Adaptve Score, the reult of an ablt tet taken much later. Chld Age Score Data Chld Age Score The ftted regreon lne : ŷ , r 0.41 Scatterplot lecture 5 5 Eample (contnue) Redual plot Re-ft a lne wth Chld 18 removed An outler an obervaton that le outde the overall pattern of the other obervaton. Pont that are outler n the drecton of a catter plot have large regreon redual, but other outler need not have large redual. An obervaton nfluental for a tattcal calculaton f removng t would markedl change the reult of the calculaton. Pont that are outler n the drecton of a catterplot are often nfluental for the leat quare regreon lne. lecture 5 6 3

4 4. Lurkng varable It a varable that not among the eplanator or repone varable n a tud and et ma nfluence the nterpretaton of relatonhp among thoe varable. Eample: The Math. Department of a large tate unvert mut plan the number of ecton and ntructor requred for t elementar coure. The department hope that number of tudent n thoe coure can be predcted from number of frt-ear tudent. Here the data for everal ear. X----no. of frt-ear tudent, Y----no. of tudent who enroll n elementar Math. coure. Year X Y Regreon lne: Ŷ r lecture 5 7 Comment: The nd redual plot ugget that a change took place between 1997 and 1998 that caued a hgher proporton of tudent to take math coure begnnng n In fact, one of the chool n the unvert changed t program to requre that enterng tudent take another math coure. Th change the lurkng varable that eplan the pattern we oberved. lecture 5 8 4

5 5. Cauaton, Common repone, Confoundng Cauaton The aocaton between two varable X and Y eplaned b a drect caue-and-effect lnk. One varable X caue the other Y. Eample: X mother bod ma nde Y daughter bod ma nde Common repone The oberved aocaton between the varable X and Y eplaned b a lurkng varable Z. Both X and Y change n repone to change n Z. We a that X and Y are common repone of Z. Th common repone create an aocaton even though there ma be no drect caual lnk between X and Y. Eample: X a tudent SAT core a a hgh chool enor Y the tudent frt-ear college grade pont average. Confoundng Two varable are confounded when ther effect on a repone varable can t be dtnguhed from each other. The confoundng varable ma be ether eplanator varable or lurkng varable. Eample: X number of ear of educaton a worker ha Y the worker ncome lecture 5 9 Summar of Correlaton and Regreon: Correlaton and Regreon are powerful tool for meaurng the aocaton between two varable and for epreng the dependence of one varable on the other. But the mut be ued wth an awarene of ther lmtaton. Below are ome bac rule: Correlaton meaure onl lnear aocaton, and fttng a traght lne make ene onl when the overall pattern of the relatonhp lnear. Alwa plot our data before calculatng. Etrapolaton (ung a ftted model far outde the range of the data that we ued to ft t) often produce unrelable predcton. Correlaton and leat-quare regreon are not retant. Alwa plot our data and look for potentall nfluental pont. Lurkng varable can make a correlaton or regreon mleadng. Plot the redual agant tme and agant other varable that ma nfluence the relatonhp between X and Y. When ou calculate and nterpret correlaton and regreon, be aware of poble common repone and confoundng effect. Even a ver trong aocaton between two varable not b telf good evdence that there a caue-and-effect lnk between the varable. How to etablh cauaton? The bet method to etablh a drect caual lnk between X and Y to conduct a carefull degned eperment n whch the effect of poble lurkng varable are controlled. We wll tud th n net chapter. lecture

6 Eample: We have een that nvetment theor ue the tandard devaton of return to decrbe the volatlt or rk of an nvetment. To decrbe how the rk of a pecfc ecurt related to that of the market a a whole, we ue leat-quare regreon. Fgure.4 plot the monthl percent total return on Phlp Morr common tock agant the monthl return on the Standard & Poor 500-tock nde, whch repreent the market, for the perod between Augut 1990 and June The one clear outler turn out not to be ver nfluental. Here are the bac decrptve meaure: bar r0.551 bar (a) Fnd the equaton of the leat-quare lne from th nformaton. What percent of the volatlt n Phlp Morr tock eplaned b the lnear relatonhp wth the market a a whole? (b) Eplan carefull what the lope of the lne tell u about how Phlp Morr tock repond to change n the market. Th lope called beta n nvetment theor. lecture