SIMPLE AND MULTIPLE REGRESSION. Relació entre variables Entre variables discretes (exemple: veure...

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

Desv. tνp. = 1,00

Media = 0,00

N = 10791,00

How many books at home Q19

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

Desv. tνp. = 1,30

Media = 3,8

N = 10670,00

Pisa 2003

> Rendiment en Matemàtiques,

> Nombre de llibres a casa

Pisa 2003

76543210

Pisa 2003

1061512661927237235751155375N =

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 books

11-25 books

26-100 books

101-200 books

201-500 books

More than 500 books

Media N Desv. típ.

Pisa 2003 Regressió Lineal ?

Regressió Lineal

ANOVAb

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

8895,433 10668 ,834

10523,792 10669

Regresión

Residual

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

Coeficientesestandarizad

t 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

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 line

a= 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 10,125

3943,016 99

Regresión

Residual

Modelo1

Suma decuadrados gl

Variables predictoras: (Constante), Xa.

Variable dependiente: Yb.

Estimación

Coeficientesa

1,023 ,319 3,210 ,002

,644 ,038 ,865 17,071 ,000

(Constante)

Modelo1

B Error típ.

Coeficientes

estandarizados

t Sig.

Variable dependiente: Ya.

Gráfico de dispersión

20100-10-20

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 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

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

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 / habitant

LOGPIB

LOGMETG

Modelo1

B Error típ.

t Sig.

Variable dependiente: Esperança de vida (1992)a.

Diagnósticos por casoa

3,035 71,9Número de caso37

Residuo tip.Esperança de

vida (1992)

Residus vs y ajustada:

Grαfico de dispersiσn

Variable dependiente: Esperanηa de vida (1992)

Regresiσn Valor pronosticado tipificado

210-1-2

134133

131130129128

127126125

124123

119118

115114113112

555453

Regressió parcial: Grαfico de regresiσn parcial

Calories diΰries / habitant

Regressió parcial:

Grαfico de regresiσn parcial

LOGPIB

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

Regressió parcial Grαfico de regresiσn parcial

LOGMETG

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

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

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

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 / habitant

LOGPIB

LOGMETG

Modelo1

B Error típ.

t Sig.

Diagnósticos por casoa

3,040 71,9

-3,009 53,5

Número de caso37

Residuo tip.Esperança de

vida (1992)

Residus vs y ajustada Grαfico de dispersiσn

Regresiσn Valor pronosticado tipificado

210-1-2-3

Case statistics Case statistics

Case missfit

Potential for influence: Leverage

Influence

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, indicators

Indicator 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

DResid Difference residual and residual when datapoint deleted

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

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

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)

. 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

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 predictors

NB: VIF per coefficient

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 blocks

REGRESSION

/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

Modelo1

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 7,876

15101,350 1848

3718,200 8 464,775 75,127 ,000b

11383,150 1840 6,186

15101,350 1848

Regresión

Residual

Regresión

Residual

Modelo1

Suma decuadrados gl

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 ,000

11,733 ,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)

Gender

Modelo1

B Error típ.

t 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 7,877

15064,420 1844

1590,788 2 795,394 108,739 ,000b

13473,631 1842 7,315

15064,420 1844

4431,483 9 492,387 84,975 ,000c

10632,937 1835 5,795

15064,420 1844

Regresión

Residual

Regresión

Residual

Regresión

Residual

Modelo1

Suma decuadrados gl

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

Variable dependiente: Respondent's incomed.

Coeficientesa

11,700 ,210 55,778 ,000

-1,091 ,131 -,191 -8,340 ,000

7,869 ,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)

Gender

Highest year ofschool completed

(Constante)

Gender

Highest year ofschool completed

Modelo1

B Error típ.

t 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

Modelo1

Cambio enR cuadrado Cambio en F gl1 gl2

Sig. delcambio en F

Estadísticos de cambio

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

And so on…

Block 1

Categorical PredictorsBlock 2

Ask for R2 change

Model Summary

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

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

Model1

Adjusted RSquar

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

SIMPLE AND MULTIPLE REGRESSION. Relació entre variables Entre variables discretes (exemple: veure...

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