TEORIA BÀSICA I EXEMPLES: REGRESSSIÓ
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- Ῥόδη Φωτόπουλος
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1 TEORIA BÀSICA I EXEMPLES: REGRESSSIÓ SIMPLE, MÚLTIPLE I LOGÍSTICA Mètodes Estadístics, UPF, Hivern 2014
2 1 Descripció de dades i inferència estadística 2 3 Robust s.e. (Optional) 4 Case Influence statistics Multiple regression and multicolinearity 5
3 Fitxer de Dades Mostra aleatoria de mida n = 800 d una població Variables: despesa, renda, gènere (1/0, noi = 1), vot (1/0, partit A = 1) Fixer de dades és a la web (dues opcions.sav i el.txt): library(foreign) data=read.spss(" satorra/dades/m2014dadessim.sav") data= read.table(" satorra/dades/m2013regressiosamp.txt", header =T) names(data) "Lrenda" "Ldespeses" "Genere" "Vot" head(data) Lrenda Ldespeses Genere Vot
4 sis files del fitxer de dades Lrenda Ldespeses Genere Vot
5 Anàlisi Univariant (repliqueu amb SPSS) Descripció bàsica: attach(data) renda=exp(lrenda) summary(renda) Min. 1st Qu. Median Mean 3rd Qu. Max Mitjanes i desviacions estàndard: apply( data,2,mean) Lrenda Ldespeses Genere Vot apply( data,2,sd) Lrenda Ldespeses Genere Vot Destribució univariant (renda, Ldespeses): summary(renda) Min. 1st Qu. Median Mean 3rd Qu. Max sd(renda) = summary(ldespeses) Min. 1st Qu. Median Mean 3rd Qu. Max Albert5.421 Satorra 7.759
6 Renda: Mean, sd Mean: sd = > /sqrt(800) [1] > *( /sqrt(800)) [1] *( /sqrt(800)) [1] \% IC: ( , )
7 Histograma Histograma (freq.) de variable renda Frequency e+00 1e+05 2e+05 3e+05 4e+05 5e+05
8 Histograma de la variable log de renda Lrenda = log(renda) Histograma (freq.) de la variable log de renda Frequency
9 Comenteu, en aquesta base de dades: 1 Tipus de variables, tipus de distribució de les variables continues 2 Variable X estandarditzada x = x i x s x ( scale(lrenda) ) 3 Inferència sobre la renda mitjana µ de la població (estimació, intèrval de confiança,... ) 4 Mida de mostra per una determinada precisió: inferència sobre la mitjana de renda, vot =1,...
10 Relació bivariant: diagrama de dispersió 0e+00 1e+05 2e+05 3e+05 4e+05 5e renda despeses
11 Relació bivariant: diagrama de dispersió 0e+00 1e+05 2e+05 3e+05 4e+05 5e renda Ldespeses
12 Relació bivariant: diagrama de dispersió Ldespeses
13 Coeficient de correlació, r > cor(renda,despeses) [1] > cor(renda,ldespeses) [1] > cor(lrenda,ldespeses) [1] > round(cor(lrenda,ldespeses),2) [1] 0.44 > (cor(lrenda,ldespeses))ˆ2 [1] El coeficient de rcorrelació r entre log de despesa i el log de renda és: r = 0.44 El quadrat r 2 del coeficient de correlació, el , és el coeficient de determinació R 2 del tema següent, la regressió
14 Funció esperança condicionada: E(Y X) Regressió lineal: Regressió lineal simple: Y = α + βx + ɛ on ɛ és independent (incorrelacionada) amb X Regressió lineal múltiple: Y = α + β 1 X 1 + β 2 X β k X k + ɛ on ɛ és independent (incorrelacionada) amb X 1,..., X k Nomenclatura: α és el terme independent (la constant, el intercept ); els βs són coeficients de regressió. En la regressió múltiple, β 1,..., β k són coeficients de regressió parcial. El ɛ és el terme de perturbació del model.
15 Efecte de Regressio scale(lrenda) scale(ldespeses) Y=X regressio Figure: Efecte de regressió
16 Figure: Dades de Francis Galton: ( ): Recta de regressió de Alçada de Fills vs. Alçada Pare
17 Exemple de regressió simple (dades estandarditzades) library(texreg) texreg(lm(scale(ldespeses) scale(lrenda)), scriptsize=t, stars=c(.05)) Model 1 (Intercept) 0.00 (0.03) scale(lrenda) 0.44 (0.03) R Adj. R Num. obs. 800 * p < 0.05 Table: Regressió amb dades estandarditzades
18 Exemple de regressió simple Model de regressió: Y = α + βx + ɛ, ɛ (0, σ 2 ɛ ), on Y = Ldespesa, X=Lrenda. Estimacions de α i β i R 2 a: 2.08 (0.21) b: 0.29 (0.02) R Adj. R Num. obs. 800 *** p < 0.001, ** p < 0.01, * p < 0.05, p < 0.1 Table: Taula de resultats 19% de la variació de Y ve explicada per la variació de X El coeficient de regressió de Y sobre X és positiu, 0.29, i altament significatiu (p < 0.001) Un increment de una unitat de X va associada a un increment de 0.29 del valor esperat de Y (variables expressades en logaritmes) Coeficients beta: de Lrenda =
19 El coeficient beta (coeficients de regressió estandarditzats) Són els coeficients de regressió quan les variables són estandarditzades; en aquest cas α = 0 coef.beta= *( )/ [1] > (coef.beta)ˆ2 [1] > cor(ldespeses,lrenda) [1] > (cor(ldespeses,lrenda))ˆ2 [1]
20 Regressió Múltiple Robust s.e. (Optional) re=lm(ldespeses Lrenda + texreg(re) Genere) Model de regressió: Y = α + β 1 X1 + β 2 X2 + ɛ, ɛ (0, σ 2 ɛ ), on Y = Ldespesa, X1=Lrenda, X2=Gènere Table: Multiple regression (Intercept) Lrenda Genere Estimates 2.98 (0.20) 0.23 (0.02) 0.55 (0.04) R AR n 800 this is OLS analysis
21 Robust s.e. (Optional) The linear multiple regression model (a bit of theory) It assumes, the regression function E(Y X) is lineal in its inputs X 1, X 2,..., X k ; i.e. E(Y) = α + β 1 X β k X k β 1 is the expected change in Y when we increase X 1 by one unit ceteris paribus all the other variables being constant. for prediction purposes, can sometimes outperform fancier more complicated models, specially in situations with small sample size it applies to transformed variables, so they encompass a large variety of functions for E(Y X) for the X s variables, it requires them to be continuous or binary variables we have Y = E(Y X) + ɛ, where the disturbance term ɛ is a random variable assumed to be independent of X, typically with variance that does not change with X (homoscedastic residuals) for the fitted model, we have Ŷ = a + b 1 X b k X k, where the bs are partial regression coefficients (obtained usually by OLS), and e = Y Ŷ define de residuals Note that E(Y X 1 ) is different than E(Y X 1, X 2 ) or E(Y X 1, X 2..., X k ). So, the regression coefficient b 1 for X 1 will typically change depending on which additional variables, besides X 1, we are conditioning. In causal analysis, researchers are interested in the change on Y 1 when we change X 1. This is a complicated issue that can only be answer properly with more context regarding the design of the data collection. So far we have been dealing only with a conditional expectation model (no elements have been introduced yet for proper causal analysis)
22 Robust s.e. (Optional) Regressió Múltiple 1 35% de variació de Y és explicada per la variació conjunta de Lrenda i Genere 2 Comparem el coeficients de regressió de Lrenda de la regressió simple i múltiple: 0.29 versus Interpretació dels coeficients de regressió: coeficients de regressió parcials. Variació de Y quan variem X1 ceteris paribus (control) les altres var. explicatives 4 La despesa difereix per gènere? 5...
23 Robust s.e. (Optional) Residuals vs Fitted 164 Residuals
24 Robust s.e. (Optional) library(faraway); prplot(re,1) Lrenda beta*lrenda+res
25 Robust s.e. (Optional) library(faraway); prplot(re,2) Genere beta*genere+res
26 Robust s.e. (Optional) Exemple de regressió múltiple: dades Paisos.sav Pregunta: calories en la dieta afecta a l esperança de vida? Sintaxis de SPSS
27 Lectura de dades Robust s.e. (Optional) library(foreign) data= read.spss(" satorra/dades/paisos.sav") attach(data) names(data) CALORIES[(CALORIES == 9999)]=NA
28 Valors missing? Robust s.e. (Optional) > ESPVIDA [1] [27] [53] [79] [105] [131] [157] > > CALORIES [1] [27] [53] [79] [105] [131] [157] > CALORIES[(CALORIES == 9999)]=NA > CALORIES [1] NA [27] [53] [79] [105] NA NA [131] NA 3565 NA [157] NA
29 Robust s.e. (Optional) plot matricial ESPVIDA CALORIES SANITAT NIVELL ALFAB DIARIS TV AGRICULT
30 Robust s.e. (Optional) Regressió simple: ESPEV vs. CALORIES (Intercept) CALORIES Model (2.7159) (0.0010) R Adj. R Num. obs. 152 * p < 0.05 length(espvida)= 160 Table: Statistical models
31 Robust s.e. (Optional) Regressió simple: ESPEV vs. CALORIES, ALFAB (Intercept) CALORIES ALFAB Model (1.9458) (0.0009) (0.0227) R Adj. R Num. obs. 152 * p < 0.05 Table: Statistical models
32 Robust s.e. (Optional) ESPEV vs. CALORIES, ALFAB, NBAIX, NALT (Intercept) CALORIES ALFAB NBAIXTRUE NALTTRUE HABMETG Model (3.0704) ( ) (0.0268) (1.3683) (1.0609) ( ) R Adj. R Num. obs. 136 * p < 0.05 Table: Statistical models baix mitja alt : NALT = NIVELL == "alt"; NBAIX = NIVELL == "baix" ; HABMETG[HABMETG == ] =NA table(nivell); re=lm(espvida CALORIES + ALFAB + NBAIX +NALT + HABMETG)
33 Robust s.e. (Optional) CALORIES beta*calories+res Figure: Gràfic de regressió parcial: ESPVI versus CALORIES
34 Robust s.e. (Optional) ALFAB beta*alfab+res Figure: Gràfic de regressió parcial: ESPVI versus ALFAB
35 Robust s.e. (Optional) (Optional) Regression with regular s.e. r1=lm(ldespeses Lrenda + Genere) Estimate Std. Error t value Pr(> t ) (Intercept) <2e-16 *** Lrenda <2e-16 *** Genere <2e-16 *** # get X matrix/predictors X <- model.matrix(r1) # number of obs n <- dim(x)[1] # n of predictors k <- dim(x)[2] # calculate stan errs as in the above # sq root of diag elements in vcov se <- sqrt(diag(solve(crossprod(x)) * as.numeric(crossprod(resid(r1))/(n-k)))) > se (Intercept) Lrenda Genere
36 Robust s.e. (Optional) (Optional) Regression with heteroscedastic robust s.e. r1=lm(ldespeses Lrenda + Genere) X <- model.matrix(r1) n <- dim(x)[1] k <- dim(x)[2] # residual vector u <- matrix(resid(r1)) # meat part Sigma is a diagonal with uˆ2 as elements meat1 <- t(x) %*% diag(diag(crossprod(t(u)))) %*% X # degrees of freedom adjust dfc <- n/(n-k) # like before se <- sqrt(dfc*diag(solve(crossprod(x)) %*% meat1 %*% solve(crossprod(x)))) > se (Intercept) Lrenda Genere
37 Robust s.e. (Optional) (Optional) Regression with s.e. robust to clustering # clustered standard errors in regression #by : cl <- function(dat,fm, cluster){ require(sandwich, quietly = TRUE) require(lmtest, quietly = TRUE) M <- length(unique(cluster)) N <- length(cluster) K <- fm$rank dfc <- (M/(M-1))*((N-1)/(N-K)) uj <- apply(estfun(fm),2, function(x) tapply(x, cluster, sum)); vcovcl <- dfc*sandwich(fm, meat=crossprod(uj)/n) coeftest(fm, vcovcl) }
38 Robust s.e. (Optional) (Optional) Regression with s.e. robust to clustering r1=lm(ldespeses Lrenda + Genere) summary(r1) Estimate Std. Error t value Pr(> t ) (Intercept) <2e-16 *** Lrenda <2e-16 *** Genere <2e-16 *** clust= sample(1:40,800, replace=t) > tabulate(clust) [1] [26] cl(cbind(ldespesesa,genere, clust), fit, clust) Estimate Std. Error t value Pr(> t ) (Intercept) < 2.2e-16 *** Lrenda < 2.2e-16 *** Genere < 2.2e-16 ***
39 Robust s.e. (Optional) Material addicional de regressió simple i multiple 1 web del curs M2014 M2012Setmanes12: Detalls de la regressió lineal simple i multiple + sintaxis SPSS 2 Idra UCLA: SPSS Web Books Regression with SPSS
40 Case Influence statistics Multiple regression and multicolinearity Variable depenent Y binaria Fins ara Y era una variable continua Regressió logística (i la regressió probit ) Y és binaria Com en la regressió habitual, les variables explicatives poden ser continues o binaries
41 Case Influence statistics Multiple regression and multicolinearity No serveix la regressió lineal? La relació es no-lineal Els terme d error és heteroscedastics El terme d error no té distribució normal Exemple: Y = Vot, X = Lrenda Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) <2e-16 *** Lrenda <2e-16 *** Multiple R-squared: Ŷ = Lrenda + ɛ, R 2 =.22
42 Case Influence statistics Multiple regression and multicolinearity Vot Lrenda Figure: Vot vs. Lrenda
43 Regressió logística (el model) Case Influence statistics Multiple regression and multicolinearity Suposem que Y i Bernoulli (π i ) π i = P(Y i = 1), i = 1,..., n probabilitats odds (probabilitats en contra) logit π o(odds) = π/(1 π) L(logit) = ln (o) L i = ln π i 1 π i π i = 1 1+e L i Model lineal logit: L i = β 0 + β 1 X 1 + β 2 X 2 + β 3 X 3 Model no-lineal de probabilitat: π i = e (β 0+β 1 X 1 +β 2 X 2 +β 3 X 3 )
44 Ajust de la regressió logística Case Influence statistics Multiple regression and multicolinearity Exemple: Y = Vot, X = Lrenda π i = 1 1+e L i π i = 1 1+e Lrenda i glm(vot Lrenda, family = "binomial") oefficients: Estimate Std. Error z value Pr(> z ) (Intercept) <2e-16 *** Lrenda <2e-16 *** Number of Fisher Scoring iterations: 4 ˆL = Lrenda
45 Case Influence statistics Multiple regression and multicolinearity Vot linear model logistic model Lrenda Figure: Vot vs. Lrenda: linear versus logistic fits
46 Interpretació dels paràmetres Case Influence statistics Multiple regression and multicolinearity ˆL = Lrenda exp(-1.208)= = Odds disminueixen en un 70% quan X X + 1 % d augment/decreixement odds (exp(β) 1) 100
47 Vot versus Lrenda + Genere Case Influence statistics Multiple regression and multicolinearity glm(formula = Vot Lrenda + Genere, family = "binomial" (Intercept) <2e-16 *** Lrenda <2e-16 *** Genere <2e-16 *** ˆL = Lrenda Genere > 100*(exp( )-1) [1] > 100*(exp( )-1) [1]
48 Case Influence statistics Multiple regression and multicolinearity Lrenda Lrenda+1 disminueix un 73% els odds de Vot = 1, controlant per gènere Els odds de Vot = 1 dels nois (Gènere = 1) són un 1266% superiors que els de les noies (Gènere = 0), controlant per Renda
49 Case Influence statistics Multiple regression and multicolinearity Vot Simple Mult. Homes Mult. Dones Lrenda Figure: Corbes logístiques, marginals (reg. simple) i condicionals (regr. múltiple)
50 Case Influence statistics Multiple regression and multicolinearity (Optional) More on logistic regression: lrm lrm( Vot Lrenda + Genere, y =T, x=t) Logistic Regression Model Model Likelihood Discrimination Rank Discrim. Ratio Test Indexes Indexes Obs 800 LR chi R C d.f. 2 g Dxy Pr(> chi2) < gr gamma max deriv 5e-07 gp tau-a Brier Coef S.E. Wald Z Pr(> Z ) Intercept < Lrenda < Genere <0.0001
51 Case Influence statistics Multiple regression and multicolinearity (Optional) More on logistic regression: e b and the % of increment of odds Suppose the fitted logistic regression where L = x Logit2 = (x + 1)(e b 1) 100 = (e 2 1) 100 = 639% and an unit increase of x = 1, x x + 1. ### when p is around 0.5 x= 1 Logit1 = *x Logit2 = *(x+1) prob1= 1/(1+ exp( -Logit1)) prob2= 1/(1+ exp(-logit2 )) ((prob2-prob1)/prob1)*100 > prob1 [1] 0.5 > prob2 [1] ### p = molt baixa x= 1
52 Optional: Case influence Case Influence statistics Multiple regression and multicolinearity Y all exclude 2 and 11 exclude 1 exclude 1 and X
53 (Optional): multicolinearity Case Influence statistics Multiple regression and multicolinearity datafile=read.table("/users/albertsatorra/rstudio/datasets/regressiomulticol.dat") reg=lm(y X1+factor(X2)+X3) summary(reg) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) X factor(x2) factor(x2) ** factor(x2) ** X Signif. codes: 0 *** ** 0.01 * Residual standard error: on 94 degrees of freedom Multiple R-squared: 0.433, Adjusted R-squared: F-statistic: on 5 and 94 DF, p-value: 1.961e-10
54 Optional: Case influence Case Influence statistics Multiple regression and multicolinearity Y all exclude 2 and 11 exclude 1 exclude 1 and X
55 Case Influence statistics Multiple regression and multicolinearity (Optional) s.e. robust to cluster effects: library rms library(rms ) fit =lrm( Vot Lrenda + Genere, y =T, x=t) library(rms ) length(vot) [1] 800 # assume we have a variable clust clust= sample(1:40,800, replace=t) robcov(fit,cluster=clust) Logistic Regression Model Model Likelihood Discrimination Rank Discrim. Ratio Test Indexes Indexes Obs 800 LR chi R C d.f. 2 g Dxy Pr(> chi2) < gr gamma max deriv 5e-07 gp tau-a Brier Coef S.E. Wald Z Pr(> Z ) Intercept < Lrenda < Genere <0.0001
56 Case Influence statistics Multiple regression and multicolinearity (Optional) s.e. robust to cluster effects: bootstrap > bootcov(fit, cluster=clust) Logistic Regression Model lrm(formula = Vot Lrenda + Genere, x = T, y = T) Coef S.E. Wald Z Pr(> Z ) Intercept < Lrenda < Genere <0.0001
57 Case Influence statistics Multiple regression and multicolinearity Material addicional de regressió logística 1 web del curs, M2014: Slides Logit Regression, més detalls sobre la regressió logistica + altre material en la secció de regressió logística. 2 Idra UCLA: SPSS Data Analysis Examples Logit Regression R Data Analysis Examples: Logit Regression
58 Fitxer de Dades Mostra aleatoria de mida n = 1000 d una població Variables: data= read.table(" satorra/m/dadesregressio2014.txt", header =T) #data= read.spss(" satorra/m/dadesme2014.sav") #data=as.data.frame(data) names(data) "Y1" "Y2" "X1" "X2" "X3" "X4" "X5" "X6" head(data) > head(data) Y1 Y2 X1 X2 X3 X4 X5 X Y1 is logexpenses Y2 is voting X5 is home = 1 X6 is categorial X1 to X4 are indicators related with income (latent)
59 Y Y2 X X2 X X X5 Figure: Matrix plot of the new data set
60 Multiple regression fit1 = lm(y1 X1+ X2+X3+X4+X5+factor(X6)) summary(fit1) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) <2e-16 *** X X X X X factor(x6) <2e-16 *** factor(x6) factor(x6) Residual standard error: on 991 degrees of freedom Multiple R-squared: 0.454, Adjusted R-squared: F-statistic: 103 on 8 and 991 DF, p-value: < 2.2e-16
61 Multiple regression (excluding X2) fit1 = lm(y1 X1+X3+X4+X5+factor(X6)) summary(fit1) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) < 2e-16 *** X e-12 *** X X X factor(x6) < 2e-16 *** factor(x6) factor(x6) Residual standard error: on 992 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: on 7 and 992 DF, p-value: < 2.2e-16
62 Diagnostic in linear multiple regression (Fox s library(car)) library(car) fit1 = lm(y1 X1+X2+X3+X4+X5+factor(X6)) # Evaluate Collinearity vif(fit) # variance inflation factors sqrt(vif(fit)) > 2 # problem? VIF > 4? ncvtest(fit1) Non-constant Variance Score Test Variance formula: fitted.values Chisquare = Df = 1 p = > durbinwatsontest(fit1) lag Autocorrelation D-W Statistic p-value Alternative hypothesis: rho!= 0 ## Multicolineality > vif(fit1) GVIF Df GVIFˆ(1/(2*Df)) X X X X X factor(x6)
63 X1 Component+Residual(Y1) X2 Component+Residual(Y1) X3 Component+Residual(Y1) X4 Component+Residual(Y1) X5 Component+Residual(Y1) factor(x6) Component+Residual(Y1) Component + Residual Plots Figure: component + residual plot
64 Regression with a principal component > F1 = princomp(cbind(x1,x2,x3,x4))$score[,1] > fit1 = lm(y1 F1+X5+factor(X6)) > summary(fit1) Call: lm(formula = Y1 F1 + X5 + factor(x6)) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) < 2e-16 *** F e-16 *** X factor(x6) < 2e-16 *** factor(x6) factor(x6) Signif. codes: 0 *** ** 0.01 * Residual standard error: on 994 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: on 5 and 994 DF, p-value: < 2.2e-16
65 Logistic Regression library(rms) lrm(formula = Y2 X1 + X2 + X3 + X4 + X5 + factor(x6)) Model Likelihood Discrimination Rank Discrim. Ratio Test Indexes Indexes Obs 1000 LR chi R C d.f. 8 g Dxy Pr(> chi2) < gr gamma max deriv 1e-07 gp tau-a Brier Coef S.E. Wald Z Pr(> Z ) Intercept X X < X X X X6= < X6= X6=
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