PDF created with FinePrint pdffactory Pro trial version Adott egy X folytonos változó, ami normális eloszlású

Adott egy X folytonos változó, ami normális eloszlású.

X Î N (m , s ) S Z Ó R Á S A N A L Í Z I S

Adottak ezen kívül az Y1,Y2,…,Yk diszkrét változók (faktorok)

H 0 : X - re nincs hatással Y1 Qtotal = Q1 + Q2 + ... + Qk + Q12 + Q13 + ... + Qhiba

A minta teljes szórásnégyzete

Qtotal =

å (x - x )

2

i





Az Y1 magyarázta rész





Az első két faktor interakciójához tartozó rész

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H 0 : X - re nincs hatással Y1 Qtotal = Q1 + Q2 + ... + Qk + Q12 + Q13 + ... + Qhiba A véletlen hiba okozta rész

Egyszerű csoportosítás

S Z Ó R Á S A N A L Í Z I S

X

A dolgozó fizetése

Y

A dolgozó beosztása (tisztviselő, őrző-védő, menedzser)

H 0 : A beosztás nincs hatással a fizetésre ì (t ) (t ) (t ) ü ì ( v ) (v ) (v ) ü ì ( m ) ( m ) ( m) ü í x1 , x2 ,..., xn ý, í x1 , x 2 ,..., xn ý, í x1 , x2 ,..., xn ý t þ î vþ î mþ î

Egyszerű csoportosítás Csoportátlagok:

S Z Ó R Á S A N A L Í Z I S

1 n t (t ) å x x (t ) = nt j = 1 j

x (v ) =

1 n v (v ) å x nv j = 1 j

x ( m) =

1 n m ( m) å x nm j = 1 j

Négyzetösszegek:

(

)

(

)

(

(

)2 + nv (x (v) - x )2 + nm (x (m) - x )2

)

nt 2 nv 2 nm 2 Qtotal = å x (jt ) - x + å x (jv ) - x + å x (jm) - x j =1 j =1 j =1 Q k = n t x (t ) - x

nt n v (v ) n m ( m) (t ) (t ) (v ) ( m) ) 2 Qb = å ( x j - x ) 2 + å ( x j - x )2 + å ( x j - x j =1 j =1 j =1


Egyszerű csoportosítás

Q total = Q k + Q b S Z Ó R Á S A N A L Í Z I S

H0

ð

Qk

3 -1 Qb

F-eloszlású (2, n-3)

n -3

H1

ð

Q m t x ( ) - x ( ) ± te × b n-3

n m + nt nm × nv

Student (n-3)


Descriptives Miles per Gallon

N American European Japanese Total

248 70 79 397

Mean 20,13 27,89 30,45 23,55

Std. Deviation 6,377 6,724 6,090 7,792

95% Confidence Interval for Mean Lower Bound Upper Bound 19,33 20,93 26,29 29,49 29,09 31,81 22,78 24,32

Std. Error ,405 ,804 ,685 ,391

Minimum 10 16 18 10

Maximum 39 44 47 47

Test of Homogeneity of Variances Miles per Gallon Levene Statistic ,106

df1

df2 394

2

Sig. ,900

ANOVA Miles per Gallon Sum of Squares Between Groups 7984,957 Within Groups 16056,415 Total 24041,372

df 2 394 396

Mean Square 3992,479 40,752

F 97,969

Sig. ,000

Report Miles per Gallon Country of Origin American European Japanese Total

Mean 20,13 27,89 30,45 23,55

N 248 70 79 397

Std. Deviation 6,377 6,724 6,090 7,792

Multiple Comparisons Dependent Variable: Miles per Gallon LSD

(I) Country of Origin American European Japanese

(J) Country of Origin European Japanese American Japanese American European

Mean Difference (I-J) -7,763* -10,322* 7,763* -2,559* 10,322* 2,559*

Std. Error ,864 ,825 ,864 1,048 ,825 1,048

Sig. ,000 ,000 ,000 ,015 ,000 ,015

95% Confidence Interval Lower Bound Upper Bound -9,46 -6,06 -11,94 -8,70 6,06 9,46 -4,62 -,50 8,70 11,94 ,50 4,62

*. The mean difference is significant at the .05 level.


R E G R E S S Z I Ó A N A L Í Z I S

Y

függőváltozó

X1, X2, ... Xp

független változók

Y» f(X1, X2, ... Xp )

fÎF

becslés

E(Y- f*(X1, X2, ... Xp ))2 = min E(Y- f(X1, X2, ... Xp ))2 fÎF A legkisebb négyzetek módszere h(a,b,c,...) =

n

S (Y - f(X

X2i, ... Xpi,a,b,c,... ))2 ® min

1i,

i

a,b,c,...

i=1

I.


• Lineáris regresszió

f(X) = B0 + B1 X

• Többváltozós lineáris regresszió f(X1 , X2 ,...,Xp ) = B0 + B1 X1 + B2 X2+...+ Bp Xp • Polinomiális regresszió f(X1 , X2 ,...,Xp ) = B0 + B1 X + B2 X2+...+ BpXp X1=X, X2=X2, ... , Xp=Xp • Kétparaméteres (lineárisra visszavezethető) regresszió pl. Y=f(X) = Bo·

e

B1 X

Þ lnY = B1 X + ln Bo

II.

Kétparaméteres (lineárisra visszavezethető) regresszió

y = b0 + b1 x + b2 x 2

y = b0 × b1

x

quadratic compound

y = exp(b0 + b1 × x)

growth

y = b0 + b1 × ln x

logarithmic

y = b0 + b1 × x + b2 × x + b3 × x 2

b ö æ y = expç b0 + 1 ÷ xø è

y = b0 +

b1 x

exponential inverse

y = b0 + x b1

power

3

y=

cubic

S

y = b0 + exp(b1 × x )

1 1 / u + b0 + b1

x

logistic



• Nemlineáris regresszió f(X) = B1 + B2 exp(B3 X)

aszimptotikus I.

f(X) = B1 - B2 · (B3 )X

aszimptotikus II.

f(X) = (B1 + B2 X)-1/B3

sűrűség

f(X) = B1 · (1- B3 · exp(B2 X2)) f(X) = B1 · exp( - B2 exp( - B3 X2))) f(X) = B1 · exp( - B2 /(X + B3 ))

Gauss Gompertz Johnson-Schumacher

III.


• Nemlineáris regresszió f(X) = (B1 + B3 X)B2

log-módosított

f(X) = B1 - ln(1 + B2 exp( - B3 X )

log-logisztikus

f(X) = B1 + B2 exp( - B3 X )

Metcherlich

f(X) = B1 · X / (X + B2 )

Michaelis Menten

f(X) = (B1 B2 +B3 XB4)/(B2 + XB4 )

Morgan-Merczer-Florin

f(X) = B1 /(1+B2 exp( - B3 X +B4X2 + B5X3 )) Peal-Reed

IV.


• Nemlineáris regresszió f(X) = (B1 + B2 X +B3X2 + B4X3)/ B5X3 f(X) = (B1 + B2 X +B3X2 )/ B4X2 f(X) = B1/((1+B3 · exp(B2

X))(1/B4)

f(X) = B1/((1+B3 · exp(B2 X)) f(X) = (B1

(1-B4) ·

B2 exp( - B3

X))1/(1-B4)

köbök aránya négyzetek aránya Richards Verhulst Von Bertalanffy

f(X) = B1 - B2 exp( -B3 X B4)

Weibull

f(X) = 1/(B1 + B2 X +B3X2 )

Yield sűrűség

V.



• Szakaszonkénti lineáris regresszió

VI.


• Poligoniális regresszió

VII.


• Többváltozós lineárisis regresszió kategória-változóval

VIII.



• Logisztikus regresszió

Y dichotóm A

Y=

bekövetkezik { 1,0, haha azaz AA esemény esemény nem következik be

• A választó fog szavazni • A páciensnek szívinfarktusa lesz • Az üzletet meg fogják kötni

X1 , X2 ,...,Xp

ordinális szintű független változók

• eddig hányszor ment el, kor, iskola, jövedelem • napi cigi, napi pohár, kor, stressz • ár, mennyiség, piaci forgalom, raktárkészlet

IX.


• Logisztikus regresszió 1 P(Y=1) = P(A) » ————— 1 - e-Z Z = B0 + B1 X1 + B2 X2+...+ Bp Xp P(A) ODDS = ————— » e Z 1 - P(A)

Þ

log (ODDS) = Z = B0 + B1 X1 + B2 X2+...+ Bp Xp

X.

• Logisztikus regresszió R E G A legnagyobb valószínűség elve R E S L(e1,e2,...,en) = P(Y1= e1, Y2= e2, ... , Yn= en) = S Z = P(Y1= e1) P(Y2= e2) L P(Yn= en) » I Ó A 1 1 1 N » ———— · ———— · L · ———— -Z -Z 1 2 A 1 e-Zn 1-e 1-e L Í Z I ln L(e ,e ,...,e ) = 1 ln —————————————— 1 2 n S

S (

XI.

1 - exp (B0 + B1 X1 + B2 X2+...+ Bp Xp)

)



• Lineáris regresszió A lineáris kapcsolat kitüntetett: (1) a legegyszerűbb és leggyakoribb. (2) két dimenziós normális eloszlás esetén a kapcsolat nem is lehet más (vagy lineáris vagy egyáltalán nincs).

XII.


• Lineáris regresszió A teljes négyzetösszeg

A maradékösszeg

A regressziós összeg

XIII.

A lineáris regresszió Q = Qres + Qreg

(xi, yi )

y

res

(xi, yˆ i )

reg

( x, y ) 0

yˆi = B0 + B1 xi x


A lineáris regresszió A teljes négyzetösszeg felbontása: Q = Qres + Qreg freg szabadsági foka n-2, mert n tagú az összeg, de ezek között két összefüggés van. Ha nincs lineáris regresszió, a varianciák hányadosa (1, n-2) szabadsági fokú F eloszlást követ.

fres szabadsági foka mindössze 1, mert az átlag konstans

Q reg F =

s s

2 reg 2 res

=

f reg Q reg ( n - 2 ) = Q res Q res f res

A lineáris regresszió A legkisebb négyzetek módszere alapelve: y

yˆi

= B0 + B1 xi (x3, y3) e3

(x1, y1) e1 0

e2

(x5, y5) e5 e4 (x4, y4)

(x2, y2) x

Regressziós kapcsolat keresése változók között



M o de l Summa ry R Cou ntry o f Origin = Am erican Mo del (Se lected ) R S q uare 1 ,92 0 a ,84 6

Adj usted R S quare ,84 5

Std . Erro r of the Estimate 38,866

a. Pre dictor s: (Co nstan t), Veh icle W eight (lbs.)

ANO VAb,c Mod el 1

Sum of Squ ares Reg ressio n 207 9737 Res idual 379 148,4 Tot al 245 8885

df

Mea n Squ are F 1 207 9737,0 24 137 6,806 251 151 0,552 252

Sig. ,000 a

a. Pre dictors : (Con stant), Vehic le W eig ht (lbs .) b. Dep enden t Varia ble: E ngine Displac emen t (cu. in ches) c. Sele cting only ca ses fo r which Coun try of O rigin = Ame rican

Coefficientsa,b

Model 1

(Constant) Vehicle Weight (lbs.)

Unstandardized Coefficients B Std. Error -140,192 10,736 ,115 ,003

Standardized Coefficients Beta

t -13,058 37,105

,920

Sig. ,000 ,000

a. Dependent Variable: Engine Displacement (cu. inches) b. Selecting only cases for which Country of Origin = American

Model Summary R Country of Origin = European (Selected) R Square ,895a ,801

Adjusted Std. Error of R Square the Estimate ,798 10,045 a. Predictors: (Constant), Vehicle Weight (lbs.)

Model 1

ANO VAb,c Mod el 1

Sum of Squ ares Reg ressio n 288 72,390 Res idual 716 3,774 Tot al 360 36,164

df 1 71 72

Mea n Squ are 288 72,390 100 ,898

F 286 ,154

Sig. ,000 a

a. Pre dictors : (Con stant), Vehic le W eig ht (lbs .) b. Dep enden t Varia ble: E ngine Displac emen t (cu. in ches) c. Sele cting only ca ses fo r which Coun try of O rigin = Euro pean

Coefficientsa,b

Model 1


Unstandardized Coefficients B Std. Error 10,275 5,980 ,041 ,002

Standardized Coefficients Beta ,895

t 1,718 16,916

Sig. ,090 ,000

a. Dependent Variable: Engine Displacement (cu. inches) b. Selecting only cases for which Country of Origin = European


Model Summary

Model 1

R Country of Origin = Japanese (Selected) ,841a

R Square ,708

Adjusted R Square ,704

Std. Error of the Estimate 12,585

a. Predictors: (Constant), Vehicle Weight (lbs.)

ANOVAb,c Model 1

Sum of Squares 29570,727 12195,577 41766,304

Regression Residual Total

df 1 77 78

Mean Square 29570,727 158,384

F 186,703

Sig. ,000a

a. Predictors: (Constant), Vehicle Weight (lbs.) b. Dependent Variable: Engine Displacement (cu. inches) c. Selecting only cases for which Country of Origin = Japanese

Coefficientsa,b

Model 1


Unstandardized Coefficients B Std. Error -32,235 9,977 ,061 ,004

Standardized Coefficients Beta ,841

t -3,231 13,664

Sig. ,002 ,000

a. Dependent Variable: Engine Displacement (cu. inches) b. Selecting only cases for which Country of Origin = Japanese

Variables Entered/Removedb,c Model 1

Variables Entered Engine Displacem ent (cu. inches), Time to Accelerate from 0 to 60 mph (sec), Horsepow er, Vehicle Weight a (lbs.)

Variables Removed

Method

.

Enter

a. All requested variables entered. b. Dependent Variable: Miles per Gallon c. Models are based only on cases for which Country of Origin = American

Model Summary

Model 1

R Country of Origin = American (Selected) ,865a

R Square ,748

Adjusted R Square ,744

Std. Error of the Estimate 3,244

a. Predictors: (Constant), Engine Displacement (cu. inches), Time to Accelerate from 0 to 60 mph (sec), Horsepower, Vehicle Weight (lbs.)


ANOVAb,c Model 1

Sum of Squares 7482,899 2515,631 9998,529


df 4 239 243

Mean Square 1870,725 10,526

F 177,730

Sig. ,000a

a. Predictors: (Constant), Engine Displacement (cu. inches), Time to Accelerate from 0 to 60 mph (sec), Horsepower, Vehicle Weight (lbs.) b. Dependent Variable: Miles per Gallon c. Selecting only cases for which Country of Origin = American

Coefficientsa,b

Model 1

Unstandardized Coefficients B Std. Error 46,620 2,498 -,019 ,015 -,003 ,001

(Constant) Horsepower Vehicle Weight (lbs.) Time to Accelerate from 0 to 60 mph (sec) Engine Displacement (cu. inches)

Standardized Coefficients Beta -,117 -,342

t 18,661 -1,259 -3,642

Sig. ,000 ,209 ,000

-,429

,129

-,183

-3,315

,001

-,034

,007

-,529

-4,802

,000

a. Dependent Variable: Miles per Gallon b. Selecting only cases for which Country of Origin = American

Variables Entered/Removeda,b Model 1

Variables Entered

Variables Removed

Vehicle Weight (lbs.)

.

Engine Displacem ent (cu. inches)

.

Time to Accelerate from 0 to 60 mph (sec)

.

2

3

Method Stepwise (Criteria: Probabilit y-ofF-to-enter <= ,050, Probabilit y-ofF-to-remo ve >= ,100). Stepwise (Criteria: Probabilit y-ofF-to-enter <= ,050, Probabilit y-ofF-to-remo ve >= ,100). Stepwise (Criteria: Probabilit y-ofF-to-enter <= ,050, Probabilit y-ofF-to-remo ve >= ,100).

a. Dependent Variable: Miles per Gallon b. Models are based only on cases for which Country of Origin = American


Model Summary

Model 1 2 3

R Country of Origin = American (Selected) ,845a ,858b ,864c

Adjusted R Square ,712 ,734 ,744

R Square ,713 ,736 ,747

Std. Error of the Estimate 3,442 3,308 3,248

a. Predictors: (Constant), Vehicle Weight (lbs.) b. Predictors: (Constant), Vehicle Weight (lbs.), Engine Displacement (cu. inches) c. Predictors: (Constant), Vehicle Weight (lbs.), Engine Displacement (cu. inches), Time to Accelerate from 0 to 60 mph (sec)

ANOVAd,e Model 1

2

3

Regression Residual Total Regression Residual Total Regression Residual Total

Sum of Squares 7131,610 2866,919 9998,529 7360,992 2637,538 9998,529 7466,203 2532,326 9998,529

df 1 242 243 2

Mean Square 7131,610 11,847

F 601,987

Sig. ,000a

3680,496

336,298

,000b

235,869

,000c

241 243 3 240 243

10,944 2488,734 10,551

a. Predictors: (Constant), Vehicle Weight (lbs.) b. Predictors: (Constant), Vehicle Weight (lbs.), Engine Displacement (cu. inches) c. Predictors: (Constant), Vehicle Weight (lbs.), Engine Displacement (cu. inches), Time to Accelerate from 0 to 60 mph (sec) d. Dependent Variable: Miles per Gallon e. Selecting only cases for which Country of Origin = American

Coefficientsa,b

Model 1 2

3

(Constant) Vehicle Weight (lbs.) (Constant) Vehicle Weight (lbs.) Engine Displacement (cu. inches) (Constant) Vehicle Weight (lbs.) Engine Displacement (cu. inches) Time to Accelerate from 0 to 60 mph (sec)

Unstandardized Coefficients B Std. Error 43,104 ,964 -,007 ,000 39,642 1,196 -,004 ,001

Standardized Coefficients Beta -,845 -,490

t 44,715 -24,535 33,148 -5,811

Sig. ,000 ,000 ,000 ,000

-,025

,005

-,386

-4,578

,000

44,713 -,003

1,989 ,001

-,377

22,476 -4,176

,000 ,000

-,038

,007

-,580

-5,626

,000

-,336

,107

-,143

-3,158

,002

a. Dependent Variable: Miles per Gallon b. Selecting only cases for which Country of Origin = American

Excluded Variablesd

Model 1

2

3

Horsepower Time to Accelerate from 0 to 60 mph (sec) Engine Displacement (cu. inches) Horsepower Time to Accelerate from 0 to 60 mph (sec) Horsepower

Beta In -,140a ,009 -,386

a

a

,059b -,143

b

-,117c

t -2,243

Sig. ,026

Partial Correlation -,143

Collinearity Statistics Tolerance ,301

,226

,822

,015

,794

-4,578

,000

-,283

,154

,752

,453

,049

,180

-3,158

,002

-,200

,513

-1,259

,209

-,081

,122

a. Predictors in the Model: (Constant), Vehicle Weight (lbs.) b. Predictors in the Model: (Constant), Vehicle Weight (lbs.), Engine Displacement (cu. inches) c. Predictors in the Model: (Constant), Vehicle Weight (lbs.), Engine Displacement (cu. inches), Time to Accelerate from 0 to 60 mph (sec) d. Dependent Variable: Miles per Gallon


Model Summary and Parameter Estimates Dependent Variable: Miles per Gallon Equation Linear Logarithmic Inverse Power Exponential Logistic

R Square ,595 ,658 ,659 ,705 ,669 ,669

Model Summary F df1 572,709 1 751,882 1 754,263 1 933,576 1 788,834 1 788,834 1

df2 390 390 390 390 390 390

Sig. ,000 ,000 ,000 ,000 ,000 ,000

Parameter Estimates Constant b1 39,855 -,157 108,452 -18,536 3,963 1808,017 1023,877 -,836 47,300 -,007 ,021 1,007

The independent variable is Horsepower.


Model Summary and Parameter Estimates Dependent Variable: Miles per Gallon Equation Power

R Square ,705

Model Summary F df1 933,576 1

df2 390

Sig. ,000

Parameter Estimates Constant b1 1023,877 -,836


ANOVA


Sum of Squares 31,889 13,321 45,210

df 1 390 391

Mean Square 31,889 ,034

F 933,576

Sig. ,000


Coefficients

ln(Horsepower) (Constant)

Unstandardized Coefficients B Std. Error -,836 ,027 1023,877 128,800

Standardized Coefficients Beta -,840

t -30,554 7,949

Sig. ,000 ,000

The dependent variable is ln(Miles per Gallon).


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