SIMPLE AND MULTIPLE REGRESSION

SIMPLE AND MULTIPLE REGRESSION

Relació entre variables

•Entre variables discretes (exemple: veure Titanic) •Entre continues (Regressió !) •Entre discretes i continues (Regressió!)

> Rendiment en Matemàtiques,

> Nombre de llibres a casa

REGR factor score 1 for analysis 1

1200

1000

800

600

400

200

0

Desv. tνp. = 1,00

Media = 0,00

N = 10791,00

How many books at home Q19

6,05,04,03,02,01,0

4000

3000

2000

1000

0

Desv. tνp. = 1,30

Media = 3,8

N = 10670,00

Pisa 2003

> Rendiment en Matemàtiques,

> Nombre de llibres a casa

Pisa 2003

How many books at home Q19

76543210

RE

GR

fact

or s

core

1

for a

naly

sis

1

4

2

0

-2

-4

-6

Pisa 2003

1061512661927237235751155375N =

How many books at home Q19

Ren

dim

ent M

at

4

2

0

-2

-4

-6

Regressió Lineal ?

Pisa 2003 Regressió Lineal ?

Informe

REGR factor score 1 for analysis 1

-1,05056 375 ,99645420-,6346741 1155 ,93700815-,1713318 3575 ,91639840,1350648 2372 ,86959580,4411666 1927 ,86447655,5051664 1266 ,96605461,0066095 10670 ,99317157

How many books athome Q190-10 books11-25 books26-100 books101-200 books201-500 booksMore than 500 booksTotal

Media N Desv. típ.

Pisa 2003 Regressió Lineal ?

Regressió Lineal

ANOVAb

1628,360 1 1628,360 1952,838 ,000a

8895,433 10668 ,83410523,792 10669

RegresiónResidualTotal

Modelo1

Suma decuadrados gl

Mediacuadrática F Sig.

Variables predictoras: (Constante), How many books at home Q19a.

Variable dependiente: REGR factor score 1 for analysis 1b.

Coeficientesa

-1,126 ,027 -41,532 ,000

,301 ,007 ,393 44,191 ,000

(Constante)How many booksat home Q19

Modelo1

B Error típ.

Coeficientes noestandarizados

Beta

Coeficientesestandarizad

ost Sig.

Variable dependiente: REGR factor score 1 for analysis 1a.

* We first load the PISAespanya.sav file and then* this is the sintaxis file for SPSS analysis

*Q38 How often do these things happen in your math class*Student dont't listen to what the teacher says

CROSSTABS /TABLES=subnatio BY st38q02 /FORMAT= AVALUE TABLES /STATISTIC=CHISQ /CELLS= COUNT ROW .

FACTOR /VARIABLES pv1math pv2math pv3math pv4math pv5math pv1math1 pv2math1 pv3math1 pv4math1 pv5math1 pv1math2 pv2math2 pv3math2 pv4math2 pv5math2 pv1math3 pv2math3 pv3math3 pv4math3 pv5math3 pv1math4 pv2math4 pv3math4 pv4math4 pv5math4 /MISSING LISTWISE /ANALYSIS pv1math pv2math pv3math pv4math pv5math pv1math1 pv2math1 pv3math1 pv4math1 pv5math1 pv1math2 pv2math2 pv3math2 pv4math2 pv5math2 pv1math3 pv2math3 pv3math3 pv4math3 pv5math3 pv1math4 pv2math4 pv3math4 pv4math4 pv5math4 /PRINT INITIAL EXTRACTION FSCORE /PLOT EIGEN ROTATION /CRITERIA FACTORS(1) ITERATE(25) /EXTRACTION ML /ROTATION NOROTATE /SAVE REG(ALL) .

GRAPH /SCATTERPLOT(BIVAR)=st19q01 WITH fac1_1 /MISSING=LISTWISE .

REGRESSION /MISSING LISTWISE /STATISTICS COEFF OUTS R ANOVA /CRITERIA=PIN(.05) POUT(.10) /NOORIGIN /DEPENDENT fac1_1 /METHOD=ENTER st19q01 /PARTIALPLOT ALL /SCATTERPLOT=(*ZRESID ,*ZPRED ) /RESIDUALS HIST(ZRESID) NORM(ZRESID) .

library(foreign)help(read.spss)data=read.spss("G:/DATA/PISAdata2003/ReducedDataSpain.sav", use.value.labels=TRUE,to.data.fram=TRUE)names(data) [1] "SUBNATIO" "SCHOOLID" "ST03Q01" "ST19Q01" "ST26Q04" "ST26Q05" [7] "ST27Q01" "ST27Q02" "ST27Q03" "ST30Q02" "EC07Q01" "EC07Q02" [13] "EC07Q03" "EC08Q01" "IC01Q01" "IC01Q02" "IC01Q03" "IC02Q01" [19] "IC03Q01" "MISCED" "FISCED" "HISCED" "PARED" "PCMATH" [25] "RMHMWK" "CULTPOSS" "HEDRES" "HOMEPOS" "ATSCHL" "STUREL" [31] "BELONG" "INTMAT" "INSTMOT" "MATHEFF" "ANXMAT" "MEMOR" [37] "COMPLRN" "COOPLRN" "TEACHSUP" "ESCS" "W.FSTUWT" "OECD" [43] "UH" "FAC1.1" attach(data)mean(FAC1.1)[1] -8.95814e-16

tabulate(ST19Q01)[1] 106 0 15 1266 1927 2372 3575 1155 375> table(ST19Q01)ST19Q01 Miss Invalid N/A More than 500 books 106 0 15 1266 201-500 books 101-200 books 26-100 books 11-25 books 1927 2372 3575 1155 0-10 books 375

Efecto de Cultural Possession of the family

Gráfico de regresión parcial

Variable dependiente: REGR factor score 1 for analysis 1

Cultural possessions of the family (WLE)

3210-1-2-3

REG

R fa

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Data

Variables Y and X observedon a sample of size n:

yi , xi i =1,2, ..., n

Covariance and correlation

Scatterplot for various values of correlation

Coeficient de correlació r = 0 , tot i que hi ha una relació funcional exacta (no lineal!)

> cbind(x,y) x y [1,] -10 100 [2,] -9 81 [3,] -8 64 [4,] -7 49 [5,] -6 36 [6,] -5 25 [7,] -4 16 [8,] -3 9 [9,] -2 4[10,] -1 1[11,] 0 0[12,] 1 1[13,] 2 4[14,] 3 9[15,] 4 16[16,] 5 25[17,] 6 36[18,] 7 49[19,] 8 64[20,] 9 81[21,] 10 100>

Regressió Lineal Simple

Variables Y X

E Y | X = + X

Var (Y | X ) =

Linear relation: y = 1 + .6 X

Linear relation and sample data

Regression Model

Yi = + Xi + i

i : mean zero variance 2 nrmally distributed

Sample Data: Scatterplot

Fitted regression linea= 0.5789 b=0.6270

Fitted and true regression lines:a= 0.5789 b=0.6270 =1, =.6

Fitted and true regression lines in repeated (20) sampling

=1, =.6

OLS estimate of beta (under repeated sampling)

=1, =.6

Estimate of beta for different samples (100):

0.619 0.575 0.636 0.543 0.555 0.594 0.611 0.584 0.576 ......>

> mean(bs)

[1] 0.6042086

> sd(bs)

[1] 0.03599894

>

REGRESSION Analysis of the Simulated Data

(with R and other software )

Fitted regression line:a= 1.0232203, b= 0.6436286

=1, =.6

Regression Analysisregression = lm(Y ~X)summary(regression)

Call:lm(formula = Y ~ X)

Residuals: Min 1Q Median 3Q Max -6.0860 -2.1429 -0.1863 1.9695 9.4817

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 1.0232 0.3188 3.21 0.00180 ** X 0.6436 0.0377 17.07 < 2e-16 ***---Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1

Residual standard error: 3.182 on 98 degrees of freedomMultiple R-Squared: 0.7483, Adjusted R-squared: 0.7458 F-statistic: 291.4 on 1 and 98 DF, p-value: < 2.2e-16

>>

Regression Analysis with Stata

. use "E:\Albert\COURSES\cursDAS\AS2003\data\MONT.dta", clear

. regress y x

Source | SS df MS Number of obs = 100---------+------------------------------ F( 1, 98) = 291.42 Model | 2950.73479 1 2950.73479 Prob > F = 0.0000Residual | 992.280727 98 10.1253135 R-squared = 0.7483---------+------------------------------ Adj R-squared = 0.7458 Total | 3943.01551 99 39.8284395 Root MSE = 3.182

------------------------------------------------------------------------------ y | Coef. Std. Err. t P>|t| [95% Conf. Interval]---------+-------------------------------------------------------------------- x | .6436286 .0377029 17.071 0.000 .5688085 .7184488 _cons | 1.02322 .3187931 3.210 0.002 .3905858 1.655855------------------------------------------------------------------------------

. predict yh

. graph yh y x, c(s.) s(io)

Regression analysis with SPSS

Resumen del modelo

,865a ,748 ,746 3,1820Modelo1

R R cuadradoR cuadradocorregida

Error típ. de laestimación

Variables predictoras: (Constante), Xa.

ANOVAb

2950,735 1 2950,735 291,422 ,000a

992,281 98 10,1253943,016 99

RegresiónResidualTotal

Modelo1

Suma decuadrados gl

Mediacuadrática F Sig.

Variables predictoras: (Constante), Xa.

Variable dependiente: Yb.

Estimación

Coeficientesa

1,023 ,319 3,210 ,002,644 ,038 ,865 17,071 ,000

(Constante)X

Modelo1

B Error típ.

Coeficientes noestandarizados

Beta

Coeficientes

estandarizados

t Sig.

Variable dependiente: Ya.

Gráfico de dispersión

X

20100-10-20

Y20

10

0

-10

-20

Fitted Regression

FYi = 1.02 + .64 Xi , R2=.74s.e.: (.037) t-value: 17.07

Regression coeficient of X is significant (5% significance level),with the expected value of Y icreasing .64 for each unit increase of X. The 95% confidence interval for the regression coefficient is

[.64-1.96*.037, . .64+1.96*.037]=[.57, .71]

74% of the variation of Y is explained by the variation of X

Fitted regression line

Residual plot

OLS analysis

Variance decomposition

Properties of OLS estimation

Sampling distributions

Inferences

Student-t distribution

Significance tests

F-test

Prediction of Y

Multiple Regression:

t-test and CI

F-test

Confidence bounds for Y

Interpreting multiple regression by means of simple regression

Adjusted R2

Exemple de l’Anàlisi de Regressió

Dades de paisos.sav

Variables

Matrix Plot

Esperanηa de vida (1

Calories diΰries / h

PIB per cΰpita en $

Habitants per metge

Necessitat de transformar les variables !

Transformació de variables

Matrix Plot

Relacions (aproximadament) lineals entre variables transformades !

Esperanηa de vida (1

Calories diΰries / h

LOGPIB

LOGMETG

Anàlisi de RegressióResumen del modelob

,897a ,805 ,800 4,7680Modelo1

R R cuadradoR cuadradocorregida

Error típ. de laestimación

Variables predictoras: (Constante), LOGMETG, Caloriesdiàries / habitant, LOGPIB

a.

Variable dependiente: Esperança de vida (1992)b.

REGRESSION /MISSING LISTWISE /STATISTICS COEFF OUTS R ANOVA /CRITERIA=PIN(.05) POUT(.10) /NOORIGIN /DEPENDENT espvida /METHOD=ENTER calories logpib logmetg /PARTIALPLOT ALL /SCATTERPLOT=(*ZRESID ,*ZPRED ) /SAVE RESID .

Coeficientesa

73,230 8,294 8,830 ,000,000 ,001 ,007 ,094 ,925

6,184 1,158 ,401 5,341 ,000-8,586 1,287 -,534 -6,671 ,000

(Constante)Calories diàries / habitantLOGPIBLOGMETG

Modelo1

B Error típ.

Coeficientes noestandarizados

Beta

Coeficientesestandarizad

ost Sig.

Variable dependiente: Esperança de vida (1992)a.

Diagnósticos por casoa

3,035 71,9Número de caso37

Residuo tip.Esperança de

vida (1992)

Variable dependiente: Esperança de vida (1992)a.

Residus vs y ajustada:

Grαfico de dispersiσn

Variable dependiente: Esperanηa de vida (1992)

Regresiσn Valor pronosticado tipificado

210-1-2

Reg

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σn R

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138

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134133

131130129128

127126125

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121119

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115114113112

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9695

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8382

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79

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7675

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7372

71

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61 6059

58

57

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555453

52

5150

49

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Regressió parcial: Grαfico de regresiσn parcial

Variable dependiente: Esperanηa de vida (1992)

Calories diΰries / habitant

800

600

400

200

0

-200

-400

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-800

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Esp

eran

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992)

20

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-20

Regressió parcial:

Grαfico de regresiσn parcial

Variable dependiente: Esperanηa de vida (1992)

LOGPIB

1,0,50,0-,5-1,0-1,5

Esp

eran

ηa d

e vi

da (1

992)

20

10

0

-10

-20

Regressió parcial Grαfico de regresiσn parcial

Variable dependiente: Esperanηa de vida (1992)

LOGMETG

,8,6,4,2,0-,2-,4-,6-,8-1,0

Esp

eran

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da (1

992)

20

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-10

-20

Anàlisi de RegressióREGRESSION /MISSING LISTWISE /STATISTICS COEFF OUTS R ANOVA /CRITERIA=PIN(.05) POUT(.10) /NOORIGIN /DEPENDENT espvida /METHOD=ENTER calories logpib logmetg cal2 /PARTIALPLOT ALL /SCATTERPLOT=(*ZRESID ,*ZPRED ) /SAVE RESID .

Resumen del modelob

,903a ,816 ,809 4,6550Modelo1

R R cuadradoR cuadradocorregida

Error típ. de laestimación

Variables predictoras: (Constante), CAL2, LOGPIB,LOGMETG, Calories diàries / habitant

a.

Variable dependiente: Esperança de vida (1992)b.

Coeficientesa

47,394 12,852 3,688 ,000,019 ,007 1,027 2,564 ,012

5,951 1,134 ,386 5,248 ,000-8,342 1,260 -,519 -6,620 ,000

-3,34E-06 ,000 -1,002 -2,589 ,011

(Constante)Calories diàries / habitantLOGPIBLOGMETGCAL2

Modelo1

B Error típ.

Coeficientes noestandarizados

Beta

Coeficientesestandarizad

ost Sig.

Variable dependiente: Esperança de vida (1992)a.

Diagnósticos por casoa

3,040 71,9-3,009 53,5

Número de caso3799

Residuo tip.Esperança de

vida (1992)

Variable dependiente: Esperança de vida (1992)a.

Residus vs y ajustada Grαfico de dispersiσn

Variable dependiente: Esperanηa de vida (1992)

Regresiσn Valor pronosticado tipificado

210-1-2-3

Reg

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4

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Case statistics

Case missfit Potential for influence: LeverageInfluence

Residuals

Hat matrix

Influence Analysis

Diagnostic case statistics After fitting regression, use the instructionFpredict namevar predicted value of y

, cooksd Cook’s D influence measure, dfbeta(x1) DFBETAS for regression coefficient on var x1, dfits DFFITS influence measures, hat Diagonal elements of hat matrix (leverage)

, leverage (same as hat) , residuals

, rstandard standardized residuals , .rstudent Studentized residuals , stdf standard erros of predicte individual y, standard error of forecast, stdp standard errors of predicted mean y, stdr standard errors of residuals , welsch Welsch’s distance influence measure

.... display tprob(47, 2.337)Sort namevar list x1 x2 –5/1

Leverages after fit …

. fpredict lev, leverage

. gsort -lev . list nombre lev in 1/5 nombre lev 1. RECAGUREN SOCIEDAD LIMITADA, .0803928 2. EL MARQUES DEL AMPURDAN S,L, .0767692 3. ADOBOS Y DERIVADOS S,L, .0572497 4. CONSERVAS GARAVILLA SA .0549707 5. PLANTA DE ELABORADOS ALIMENTICIOS MA .0531497 .

Box plot of leverage values . predict lev, leverage. graph lev, box s([_n])

Cases with extreme leverages

Leverage versus residual square plot. lvr2plot, s([_n])

Dfbeta’s: . fpredict beta, dfbeta(nt_paau). graph beta, box s([_n])

Regression: Outliers, basic ideaOutlier

Regression: Outliers, indicatorsIndicator Description Rule of thumb

(when “wrong”)

Resid Residual: actual – predicted -

ZResid Standardized residual: residual divided by standarddeviation residual

> 3 (in absolute value)

SResid Studentized Residu: residual divided by standarddeviation residual at that particular datapoint of X values

> 3 (in absolute value)

DResid Difference residual and residual when datapoint deleted

-

SDResid See DResid, standardized by standard deviation at that particular datapoint of X values

> 3 (in absolute value)

Regression: Outliers, in SPSS

Regression: Influential Points, Basic Idea

Influential Point (no outlier!)

Regression: Influential Points, Indicators

Indicator Description Rule of thumb

Lever Afstand tussen punt en overige puntenNB: potentieel invloedrijk

> 2 (p+1) / n

Cook Verandering in residuen van overige cases als een bepaalde case niet in de regressie meedoet

> 1

DfBeta Verschil tussen beta’s wanneer de case wel meedoet en wanneer die niet meedoet in het modelNB: voor elke beta krijgen we deze

-

SdfBeta DfBeta / Standaardfout DfBetaNB: voor elke beta

> 2/√n

DfFit Verschil tussen voorspelde waarde als case wel versus niet meedoet in model

-

SDfFit DfFit / standaarddeviatie SdFit > 2 /√(p/n)

CovRatio Verandering in Varianties/Covarianties als punt niet meedoet

Abs (CovRatio – 1)> 3 p / n

Regression: Influential points, in SPSS

Case 2 is an influential point

Regression: Influential Points, what to do?

• Nothing at all..• Check data• Delete a-typical datapoints, then repeat

regression without these datapoints

“to delete a point or not is an issue statisticians disagree on”

MULTICOLLINEARITY

Diagnostic tools

Regression: Multicollinearity

• If predictors correlate “high”, then we speak of multicollinearity

• Is this a problem? If you want to asess the influence of each predictor, yes it is, because:– Standarderrors blow up, making coefficients not-

significant

Analyzing math data. use "G:\Albert\COURSES\cursDAS\AS2003b\data\mat.dta", clear

. save "G:\Albert\COURSES\CursMetEstad\Curs2004\Metodes\mathdata.dta"file G:\Albert\COURSES\CursMetEstad\Curs2004\Metodes\mathdata.dta saved

. edit- preserve

. gen perform = (nt_m1+ nt_m2+ nt_m3)/3(110 missing values generated)

. corr perform nt_paau nt_acces nt_exp(obs=189)

| perform nt_paau nt_acces nt_exp---------+------------------------------------ perform | 1.0000 nt_paau | 0.3535 1.0000nt_acces | 0.5057 0.8637 1.0000 nt_exp | 0.5002 0.3533 0.7760 1.0000

. outfile nt_exp nt_paau nt_acces perform using "G:\Albert\COURSES\CursMetEsta> d\Curs2004\Metodes\mathdata.dat"

.

Multiple regression: perform vs nt_acces nt_paau

. regress perform nt_acces nt_paau Source | SS df MS Number of obs = 245 ---------+------------------------------ F( 2, 242) = 31.07 Model | 71.1787647 2 35.5893823 Prob > F = 0.0000 Residual | 277.237348 242 1.14560888 R-squared = 0.2043 ---------+------------------------------ Adj R-squared = 0.1977 Total | 348.416112 244 1.42793489 Root MSE = 1.0703 ------------------------------------------------------------------------------ perform | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- nt_acces | 1.272819 .2427707 5.243 0.000 .7946054 1.751032 nt_paau | -.2755092 .1835091 -1.501 0.135 -.6369882 .0859697 _cons | -1.513124 .9729676 -1.555 0.121 -3.42969 .4034425 ------------------------------------------------------------------------------

.Perform = rendiment a mates I a III

Collinearity

Diagnostics for multicollinearity . corre nt_paau nt_exp nt_acces (obs=276) | nt_paau nt_exp nt_acces --------+--------------------------- nt_paau| 1.0000 nt_exp| 0.3435 1.0000 nt_acces| 0.8473 0.7890 1.0000

. fit perform nt_paau nt_exp nt_access . vif Variable | VIF 1/VIF ---------+---------------------- nt_acces | 1201.85 0.000832 nt_paau | 514.27 0.001945 nt_exp | 384.26 0.002602 ---------+---------------------- Mean VIF | 700.13

.

Any explanatory variable with a VIF greater than 5 (or tolerance less than .2) show a degree of collinearity that may be Problematic

This ratio is called Tolerance

In the case of just nt_paau an nt_exp we Get

. vifVariable | VIF 1/VIF ---------+---------------------- nt_exp | 1.14 0.875191 nt_paau | 1.14 0.875191---------+----------------------Mean VIF | 1.14

.

VIF = 1/(1 – Rj2)

Multiple regression: perform vs nt_paau nt_exp

. regress perform nt_paau nt_exp Source | SS df MS Number of obs = 189 ---------+------------------------------ F( 2, 186) = 37.24 Model | 75.2441994 2 37.6220997 Prob > F = 0.0000 Residual | 187.897174 186 1.01019986 R-squared = 0.2859 ---------+------------------------------ Adj R-squared = 0.2783 Total | 263.141373 188 1.39968815 Root MSE = 1.0051 ------------------------------------------------------------------------------ perform | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- nt_paau | .3382551 .1109104 3.050 0.003 .119451 .5570593 nt_exp | .9040681 .1396126 6.476 0.000 .6286403 1.179496 _cons | -3.295308 1.104543 -2.983 0.003 -5.474351 -1.116266 ------------------------------------------------------------------------------

. predict yh(option xb assumed; fitted values)(82 missing values generated)

. predict e, resid(169 missing values generated)

.

. corr nt_exp nt_paau nt_acces(obs=276)

| nt_exp nt_paau nt_acces---------+--------------------------- nt_exp | 1.0000 nt_paau | 0.3435 1.0000nt_acces | 0.7890 0.8473 1.0000

Regression: Multicollinearity, Indicators

Indicator description Rule of thumb (when “wrong”)

Overall F_Test versus test coefficients

Overall F-Test is significant, but individual coefficients are not

-

Beta Standardized coefficient Outside [-1, +1]

Tolerance Tolerance = unique variance of a predictor (not shared/explained by other predictors) …NB: Tolerance per coefficient

< 0.01

Variantie Inflation Factor

√ VIF indicates how much the standard error of a particular coefficient is inflated due to correlatation between this particular predictor and the other predictorsNB: VIF per coefficient

>10

Eigenvalues …rather technical… +/- 0

Condition Index …rather technical… > 30

Variance Proportion …rather technical…look tot “loadings” on the dimensions

Loadings around 1

Regression: Multicollinearity, in SPSS

diagnostics

Regression: Multicollineariteit, in SPSS

Beta > 1 Tolerance, VIF in orde

Regressie: Multicollineariteit, in SPSS

2 eigenwaarden rond 0 Deze variabelen zorgen voor multicoll.CI in orde

Regression: Multicollinearity, what to do?

• Nothing… (if there is no interest in the individual coefficients, only in good prediction)

• Leave one (or more) predictor(s) out• Use PCA to reduce high correlated variables to

smaller number of uncorrelated variables

Variables Categòriques

Use: Survey_sample.sav, in i/.../data

Salari vs gènere | anys d’educacióstatus de treball

Creació de variables dicotòmiques

GET FILE='G:\Albert\Web\Metodes2005\Dades\survey_sample.sav'.

COMPUTE D1 = wrkstat=1.EXECUTE .COMPUTE D2 = wrkstat=2.EXECUTE .COMPUTE D3 = wrkstat=3.EXECUTE .COMPUTE D4 = wrkstat=4. EXECUTE .COMPUTE D5 = wrkstat=5.EXECUTE .COMPUTE D6 = wrkstat=6.EXECUTE .COMPUTE D7 = wrkstat=7.EXECUTE .COMPUTE D8 = wrkstat=8.EXECUTE .

Regressió en blocksREGRESSION

/MISSING LISTWISE

/STATISTICS COEFF OUTS R ANOVA CHANGE

/CRITERIA=PIN(.05) POUT(.10)

/NOORIGIN

/DEPENDENT rincome

/METHOD=ENTER sex

/METHOD=ENTER d2 d3 d4 d5 d6 d7 d8 .

Resumen del modelo

,192a ,037 ,036 2,806 ,037 70,384 1 1847 ,000,496b ,246 ,243 2,487 ,210 73,059 7 1840 ,000

Modelo12

R R cuadradoR cuadradocorregida

Error típ. de laestimación

Cambio enR cuadrado Cambio en F gl1 gl2

Sig. delcambio en F

Estadísticos de cambio

Variables predictoras: (Constante), Gendera.

Variables predictoras: (Constante), Gender, D8, D3, D6, D5, D4, D7, D2b.

ANOVAc

554,345 1 554,345 70,384 ,000a

14547,005 1847 7,87615101,350 18483718,200 8 464,775 75,127 ,000b

11383,150 1840 6,18615101,350 1848

RegresiónResidualTotalRegresiónResidualTotal

Modelo1

2

Suma decuadrados gl

Mediacuadrática F Sig.

Variables predictoras: (Constante), Gendera.

Variables predictoras: (Constante), Gender, D8, D3, D6, D5, D4, D7, D2b.

Variable dependiente: Respondent's incomec. Coeficientesa

11,708 ,210 55,882 ,000-1,096 ,131 -,192 -8,390 ,00011,733 ,188 62,433 ,000

-,629 ,119 -,110 -5,299 ,000-3,016 ,171 -,366 -17,681 ,000-,955 ,432 -,045 -2,211 ,027

-2,977 ,377 -,161 -7,898 ,000-2,860 ,452 -,128 -6,330 ,000-3,309 ,513 -,131 -6,456 ,000-4,046 ,337 -,247 -11,994 ,000-1,414 ,882 -,032 -1,603 ,109

(Constante)Gender(Constante)GenderD2D3D4D5D6D7D8

Modelo1

2

B Error típ.

Coeficientes noestandarizados

Beta

Coeficientesestandarizad

ost Sig.

Variable dependiente: Respondent's incomea.

Regressió en blocks REGRESSION

/MISSING LISTWISE

/STATISTICS COEFF OUTS R ANOVA CHANGE

/CRITERIA=PIN(.05) POUT(.10)

/NOORIGIN

/DEPENDENT rincome

/METHOD=ENTER sex

/METHOD=ENTER educ

/METHOD=ENTER d2 d3 d4 d5 d6 d7 d8 . ANOVAd

547,844 1 547,844 69,553 ,000a

14516,575 1843 7,87715064,420 18441590,788 2 795,394 108,739 ,000b

13473,631 1842 7,31515064,420 18444431,483 9 492,387 84,975 ,000c

10632,937 1835 5,79515064,420 1844

RegresiónResidualTotalRegresiónResidualTotalRegresiónResidualTotal

Modelo1

2

3

Suma decuadrados gl

Mediacuadrática F Sig.

Variables predictoras: (Constante), Gendera.

Variables predictoras: (Constante), Gender, Highest year of school completedb.

Variables predictoras: (Constante), Gender, Highest year of school completed, D3,D8, D5, D6, D4, D7, D2

c.

Variable dependiente: Respondent's incomed.

Coeficientesa

11,700 ,210 55,778 ,000-1,091 ,131 -,191 -8,340 ,0007,869 ,379 20,754 ,000

-1,135 ,126 -,198 -8,996 ,000

,283 ,024 ,263 11,941 ,000

8,455 ,342 24,715 ,000-,679 ,115 -,119 -5,896 ,000

,241 ,021 ,224 11,302 ,000

-2,891 ,166 -,351 -17,451 ,000-,801 ,418 -,038 -1,915 ,056

-2,625 ,366 -,142 -7,169 ,000-2,647 ,438 -,119 -6,047 ,000-3,554 ,497 -,141 -7,157 ,000-3,802 ,327 -,232 -11,621 ,000-1,044 ,854 -,024 -1,223 ,222

(Constante)Gender(Constante)GenderHighest year ofschool completed(Constante)GenderHighest year ofschool completedD2D3D4D5D6D7D8

Modelo1

2

3

B Error típ.

Coeficientes noestandarizados

Beta

Coeficientesestandarizad

ost Sig.

Variable dependiente: Respondent's incomea.

Resumen del modelo

,191a ,036 ,036 2,807 ,036 69,553 1 1843 ,000,325b ,106 ,105 2,705 ,069 142,582 1 1842 ,000,542c ,294 ,291 2,407 ,189 70,034 7 1835 ,000

Modelo123

R R cuadradoR cuadradocorregida

Error típ. de laestimación

Cambio enR cuadrado Cambio en F gl1 gl2

Sig. delcambio en F

Estadísticos de cambio

Variables predictoras: (Constante), Gendera.

Variables predictoras: (Constante), Gender, Highest year of school completedb.

Variables predictoras: (Constante), Gender, Highest year of school completed, D3, D8, D5, D6, D4, D7, D2c.

Categorical Predictors

Is income dependent on years of age and religion ?

Categorical PredictorsCompute dummy variable for each category, except last

Categorical Predictors

And so on…

Categorical PredictorsBlock 1

Categorical PredictorsBlock 2

Categorical PredictorsAsk for R2 change

Categorical Predictors

Model Summary

.101a .010 .010 5.424 .010 14.688 1 1421 .000

.172b .030 .026 5.379 .019 7.064 4 1417 .000

Model12

R

RSqua

re

Adjusted RSquar

e

Std. Errorof the

EstimateR SquareChange F Change df1 df2

Sig. FChange

Change Statistics

Predictors: (Constant), Age of Respondenta.

Predictors: (Constant), Age of Respondent, Jewish, Cath, None, Protb.

Look at R Square change for

importance of categorical

variable