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PHN TCH HI QUI TUYN TNH N GIN 17.1 Phng trnh hi qui tuyn tnh

Phn tch hi qui tuyn tch n gin (Simple Linear Regression Analysis) l

tm s lin h gia 2 bin s lin tc: bin c lp (bin d on) trn trc honh x

vi bin ph thuc (bin kt cc) trn trc tung y. Sau v mt ng thng hi

qui v t phng trnh ng thng ny ta c th d on c bin y (v d: cn

nng) khi c x (v d: tui)

V d 1: Ta c 1 mu gm 6 tr t 1-6 tui, c cn nng nh bng sau:

Tui Cn nng (kg)

1 10 2 12 3 14 4 16 5 18 6 20

Ni cc cp (x,y) ny ta thy c dng 1 phng trnh bc nht: y=2x+8

(trong 2 l dc v 8 l im ct trn trc tung y khi x=0). Trong thng k

phng trnh ng thng (bc nht) ny c vit di dng:

y= x + [1]


y l phng trnh hi qui tuyn tnh, trong gi l dc (slope) v l chn

(intercept), im ct trn trc tung khi x=0.

Thc ra phng trnh hi qui tuyn tnh ny ch c trn l thuyt, ngha l cc tr s

ca xi (i=1,2,3,4,5,6) v yi tng ng, lin h vi nhau 100% (hoc h s tng

quan R=1)

Trong thc t him khi c s lin h 100% ny m thng c s sai lch gia tr

s quan st yi v tr s yi c on nm trn ng hi qui.

17.1.1 M hnh hi qui tuyn tnh

V d 2: Ta c 1 mu gm 6 tr em khc c cn nng theo bng sau:

Tui Cn nng (kg)

1 11 2 11 3 14 4 16 5 18 6 20

Khi v ng thng hi qui, ta thy cc tr s quan st y3, y4, y5, y6 nm trn ng

thng, cn y1 v y2 khng nm trn ng thng ny v s lin h gia xi v yi


khng cn l 100% m ch cn 97% v c s sai lch ti y1 v y2. S sai lch ny

trong thng k gi l phn d (residual) hoc errors.

Gi y1, y2, y3, y4, y5, y6 l tr s quan st v y1, y2, y3, y4, y5, y6 l tr s c on

nm trn ng hi qui, 1, 2, 3, 4, 5, 6 l phn d.

Nh vy 1= y1 y1

2 = y2 y2

3 = y3 y3

4 = y4 y4

5 = y5 y5

6 = y6 y6

Khi phng trnh hi qui tuyn tnh c vit di dng tng qut nh sau:

y= xi + i+ i [2]

Nh vy nu phn d i cng nh s lin h gia x,y cng ln v ngc li. Phn

lin h cn i gi l phn hi qui. M hnh hi qui tuyn tch c m t nh sau:

D liu= Hi qui (Regression) + Phn d (Residual)

17.1.2 c tnh h s tng quan v chn

Mun v c phng trnh hi qui tuyn tnh cn phi c tnh c dc

v chn trn trc tung.

V d 3: Nu chng ta chn mt mu thc t gm 30 em t 1-6 tui v kt qu cn

nng tng ng ca 30 em c v trong biu sau:


Lc ny ta khng th ni 30 im trn biu m phi v 1 ng thng i cng

gn vi tt c cc im cng tt. Nh vy 3 ng thng biu ta chn ng

thng no?. Nguyn tc chn ng thng no i gn c 30 im, c ngha lm sao

tng cc phn d i nh nht:

i= (yi- x )

v tng bnh phng ca phn d:

(i)2= (yi- x )

2

y l phng trnh bc 2 theo x. Trong ton hc, mun tm tr cc tiu ca 1

phng trnh bc 2, ngi ta ly o hm v cho o hm trit tiu (bng 0) s tm

c tr cc tiu ca x. Gii phng trnh ny, ta s tnh c 2 thng s v v

t 2 thng s ny ta s v c ng thng hi qui. Phng php ny trong ton

hc gi l phng php bnh phng nh nht (least square method).

Gii phng trnh trn ta c:

= r SySx

(r l h s tng quan; Sy l lch chun ca y v Sx l lch chun ca x)

r = 1

n-1 (

xi- x

Sx ) (

yi- y

Sy )

= y - x

v phng trnh hi qui tuyn tnh ca y theo x (bnh phng nh nht) l:

y = xi +

17.2 Phn tch hi qui tuyn tnh trong SPSS

Nhp s liu tui v cn nng cn c ca 30 tr 1-6 tui vo SPSS:

Ct 1: tui; ct 2: cn nng


Vo menu: >Analyze> Regression> Linear


Bng 17.1 Tm tt m hnh

H s tng quan R=0,918 v R2=0,843

Bng 17. 2 Phn tch ANOVA vi bin ph thuc l cn nng

Tng bnh phng phn hi qui (Regression)=336,14


Tng bnh phng phn d (Residual)=62,8

Trung bnh bnh phng hi qui: 336,14/ 1 (bc t do)=336,14

Trung bnh bnh phng phn d: 62,8/ 28(bc t do=n-2)=2,24

F= 336,14

2,24 = 149,8 v p


T phng trnh ny ta c th c on c cn nng theo tui ca tr, tuy nhin

nm trong mt gii hn no chng hn nh t 1-12 tui, v sau tui ny cn nng

tr s tng vt trong thi k dy th v khng cn lin h tuyn tnh vi tui na.

V d mun c on cn nng ca tr t qun th nghin cu ny:

7 tui Cn nng= 7,77 + 1,96 x7 = 21,49 kg

8 tui Cn nng= 7,77 + 1,96 x8 = 23,45 kg

17. 3 Cc gi nh trong phn tch hi qui tuyn tnh

Phn tch hi qui tuyn tnh khng ch l vic m t cc d liu quan st

c trong mu (sample) nghin cu m cn phi suy rng cho mi lin h

trong dn s (population). V vy, trc khi trnh by v din dch m hnh hi

qui tuyn tnh cn phi d tm vi phm cc gi nh. Nu cc gi nh b vi

phm th cc kt qu c lng khng ng tin cy c.

Cc gi nh cn thit trong hi qui tuyn tnh:

1. xi l bin s c nh, khng c sai st ngu nhin trong o lng.

2. Phn d (tr s quan st tr cho tr s c on) phn phi theo lut

phn phi chun

3. Phn d c tr trung bnh bng 0 v phng sai khng thay i cho mi

tr xi

4. Khng c tng quan gia cc phn d

V d: Mt nghin cu tm s tng quan gia cholesterol mu vi b dy

lp ni trung mc (NTM) ca ng mch cnh o c trn siu m vi

d liu ghi nhn 100 BN nh sau:


Biu phn tn (scatter) l mt phng tin tt nh gi mc

ng thng ph hp vi d liu quan st.

Vo menu: Analyze> Curve Estimation


Vo mn hnh Curve Estimation

Nhp chuyn BEDAYNTM (B dy ni trung mc) vo Dependent (s) v

CHOLESTEROL vo Variable. nh du nhy vo cc Include

constant in equation, Plot models v Linear (nu mun c lng s

lin h gia 2 bin theo dng phng trnh bc 2 th nh thm du nhy

vo Quadratic). Nhn OK, ta c biu sau:


y l phng trnh hi qui tuyn tnh vi y= 0,748 + 0,062x

Gi nh x ( cholesterol mu) l mt bin c nh, khng c sai st trong o

lng. Gi nh ny khng c vn nu bnh nhn c o mt phng

th nghim chun.

Cc gi nh cn li thc hin trong SPSS nh sau: Vo menu: Analyze> Regression> Linear...


Vo mn hnh Linear, Nhp chuyn BEDAYNTM qua Dependent v CHOLESTEROL qua Independent(s)

Nhn nt Plots, m hp thoi Plots:

Nhp chuyn phn d *ZRESID vo X (trc honh) v gi tr d on

vo Y (trc tung) xem phn d c phn b ngu nhin v phng sai

c c nh cho mi tr ca xi. Nhn du nhy vo Histogram v Normal

probability plot xem phn d c phn phi chun.


Nhn Continue, sau nhn OK cho kt qu sau:

Nh vy phn d c trung bnh (mean)=0 v lch chun (SD)=0,394

Biu phn b phn d c dng hnh chung u 2 bn, tr trung bnh

gn bng zero v SD gn bng 1. Nh vy gi nh phn d c phn phi

chun khng b vi phm.

Hoc xem biu P-P plot so snh gia phn phi tch ly ca phn d

quan st (Observed Cum Prob) trn trc honh v phn phi tch ly k


vng (Expected Cum Prob) trn trc tung. Nu cc im u nm gn

ng cho th phn phi phn d c coi nh gn chun.

Cui cng xem gi nh cc phng sai khng i vi mi gi tr ca x

(cholesterol mu) hoc gi l homoscedasticity. Nu cc tr phn d

phn tn ngu nhin quanh gi tr zero (ng ngang) th coi nh phng

sai khng thay i, v gi nh v homoscedasticity khng b vi phm.


Nu phng sai thay i (ln dn hoc nh dn theo gi tr ca x) th gi l

Heteroscedascity (gi nh v phng sai c nh b vi phm). V d nh hnh

di y:

Tm li, vi v d trn cc gi nh ca phn tch hi qui tuyn tnh u tha

mn v ta c th kt lun l b dy ni trung mc ng mch cnh c lin h

tuyn tnh vi nng cholesterol mu theo phng trnh :

Y (B dy ni trung mc)= 0,062 X cholesterol + 0,748.

Nh vy c nng cholesterol tng ln 1 mmol/L th b dy ni trung mc

ng mch cnh tng ln 0,062mm.

Ti liu tham kho:

1. McClave J T and Sincich T. 2000. Simple linear regression in Statistics, 8th

edition, Prentice-Hall, USA, pp. 505-557.

2. Moore D. S. and McCabe G. P. 1999. Looking at Data-Relationships (Chapter

2), in Introduction to the Practice of Statistics, W.H. Freeman and Company,

New York, pp. 102-145.