Chapter Two. An Introduction to Regression ( )

Size: px

Start display at page:

Download "Chapter Two. An Introduction to Regression ( )"

Vernon Stevenson
5 years ago
Views:

1 ubject: A Itroducto to Regresso Frst tage Chapter Two A Itroducto to Regresso ( ) 1 pg.

ubject: A Itroducto to Regresso Frst tage A Itroducto to Regresso Regresso aalss s a statstcal tool for the vestgato of relatoshps betwee varables.

2 ubject: A Itroducto to Regresso Frst tage A Itroducto to Regresso Regresso aalss s a statstcal tool for the vestgato of relatoshps betwee varables. Regresso s a statstcal techque to determe the lear relatoshp betwee two or more varables. Regresso s prmarl used for predcto. Ad Regresso ca be used for estmato, hpothess testg ad modelg causal relatoshps. A mportat task statstcs s to fd the relatoshps amog radom varables. I regresso problems tpcall oe of the varables s ofte called the Respose or Depedet varable whch s of partcular terest ad s deoted b. The other varables 1,,..., k, usuall called Eplaator, Regressors, or Idepedets varables, are prmarl used to predct or epla the behavors of. pg.

3 ubject: A Itroducto to Regresso Frst tage There are two kd of regresso: Lear Regresso These two graphs above represet Lear Regresso 3 pg.

4 ubject: A Itroducto to Regresso Frst tage Nolear Regresso These Three graphs above represet Nolear Regresso 4 pg.

5 ubject: A Itroducto to Regresso Frst tage What s Lear Regresso? Lear regresso s a approach for modelg the relatoshp betwee a scalar depedet varable ad oe or more eplaator varables (or depedet varables) deoted X. The case of oe eplaator varable s called mple Lear Regresso. For more tha oe eplaator varable, the process s called Multple Lear Regresso. 1. mple Lear Regresso mple lear regresso s defed as estmato of lear relatoshp betwee two varables ol oe of the varable s depedet or eplaator or predctor varable () ad the other s depedet or outcome or respose varable (). The formula of mple Lear Regresso s: Where: : s the depedet varable. : s the depedet varable. β 0 : s the tercept. β 1 : s the slope. ε : s the radom varable. 5 pg.

6 ubject: A Itroducto to Regresso Frst tage 6 pg. We ca estmate the parameters b ths wa: Ad ths s the estmated smple regresso equato: Y = β 0 + β 1 X Where, Y s the predcted respose whe the predctor varable s. The parameter β 0 ad β 1 are fed regresso parameters to be determed from the data. Ad β 0 s called the tercept ad β 1 s called the slope. β 0 s the value of whe ( = 0), ad β 1 s the chage whe creases b 1 ut. ce Y 1X ˆ ˆ 1 ) ( ) )( ( ˆ X X XY X Y X X Y Y X X ) ( ) ( ) )( (

7 ubject: A Itroducto to Regresso Frst tage Aalss Of Varace (ANOVA) ANOVA s a collecto of statstcal models used to aalze the dffereces amog group meas ad ther assocated procedures (such as "varato" amog ad betwee groups). The ANOVA table Costructg the ANOVA Table: A aalss of varace (ANOVA) table for regresso dsplas quattes that measure how much of the varablt the () varable s eplaed ad how much s ot eplaed b the regresso relatoshp wth the () varable(s). ce overall varato = error varato + regresso varato Total = Error + Regressor 7 pg.

8 ubject: A Itroducto to Regresso Frst tage Or.. Total =.. Regresso =.. Error = 8 pg.

9 ubject: A Itroducto to Regresso Frst tage Eample: 9 pg.

10 ubject: A Itroducto to Regresso Frst tage - Multple Lear Regresso: It s a relato betwee a depedet varable ad a set of eplaator varables ad the formula of multple lear regresso s : Or Y = β 0 + β 1 X 1 + β X + β 3 X β X + ε A multple lear regresso model s a lear model that descrbes how a -varable relates to two or more () varables (or trasformatos of -varables). For eample, suppose that a researcher s studg factors that mght affect sstolc blood pressures for wome aged 45 to 65 ears old. The respose varable s sstolc blood pressure (Y). uppose that two predctor varables of terest are age (X 1 ) ad bod mass de (X ). The geeral structure of a multple lear regresso model for ths stuato would be Y = β 0 + β 1 X 1 + β X + ε 10 pg.

11 ubject: A Itroducto to Regresso Frst tage Correlato Correlato aalss s to eame the relatoshp betwee two varables or more tha two varables. I other words, correlato aalss measures the stregth ad drecto of the relatoshp betwee two varables or more tha two varables. uch as the relatoshp betwee the cultural level of the parets ad the academc achevemet of ther chldre. The correlato ca be dvded to tpes as follows: 1- Correlato stregth a. Complete Correlato: Presece the full relatoshp betwee two varables such as the relatoshp betwee the area of a crcle ad ts dameter. b. Icomplete Correlato: There s a relatoshp betwee two varables, but dffcult or caot terpret the varable oe varable the other, for eample, the level of come affects the demad for a partcular good sze, but there are other varables that affect demad as the prce of ths tem ad the sze prces of alteratve goods as well as other varables. - Number of varables a. mple Correlato: It s the relatoshp betwee two varables ol. b. Partal Correlato: It s the relatoshp betwee two varables ol wth the presece of other varables c. Multple Correlato: It s the relatoshp betwee more tha two varables. 3- The shape of the relatoshp a. Lear Correlato: where represets the relatoshp betwee two varables (or more tha two varables) a straght le or a smple lear model, whch meas that a chage oe of the varables to be fed f the other varable s creased b a costat. b. No-Lear Correlato: where represets the relatoshp betwee two varables (or more tha two varables) a o-lear model, ad ths relatoshp meas that a chage oe of the two varables s ot fed f the other varable s creased b a specfc amout. 11 pg.

12 ubject: A Itroducto to Regresso Frst tage Perso Lear Correlato Coeffcet It s a statstcal dcator used to measure the stregth of lear correlato betwee two quatt varables (e, ca be measured quattatvel). If our smbol for Pearso lear correlato coeffcet s (r). The Pearso correlato formula s as follow: r ce: 1 ( )( ) 1 1 ( ) 1 1 ( ) 1 1 pg.

If r = +1 the the correlato betwee the two varables wll be etrusve ad complete. b. If r = 1 the the correlato betwee the two varables wll be reverse ad complete.

13 ubject: A Itroducto to Regresso Frst tage - mple Pearso correlato coeffcet propertes: 1. The value of smple Pearso correlato coeffcet s ragg betwee -1 ad +1 as: a. If r = +1 the the correlato betwee the two varables wll be etrusve ad complete. b. If r = 1 the the correlato betwee the two varables wll be reverse ad complete. c. If r = 0 the there s o correlato betwee the two varables. - The correlato coeffcet wll be affected b outlers (etreme) values. 13 pg.

14 ubject: A Itroducto to Regresso Frst tage Eample: Here are platg area of gree fodder per thousad hectares, ad the total meat producto thousad tos durg the perod from 1995 utl 00. Fd the correlato coeffcet betwee the amout ad the space Year pace Amout pg.

Objectives of Multiple Regression

Objectives of Multiple Regression Obectves of Multple Regresso Establsh the lear equato that best predcts values of a depedet varable Y usg more tha oe eplaator varable from a large set of potetal predctors {,,... k }. Fd that subset of