Regression. Albert Satorra. Mètodes Estadístics, UPF, hivern 2013

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1 Regression Albert Satorra Mètodes Estadístics, UPF, hivern 2013 Albert Satorra ( Mètodes Estadístics, UPF, hivern 2013 ) GRAU en CP, hivern / 24

2 Continguts 1 Summary statistics Standardization Transforming non-normal data Inference: estimation of the mean ANOVA: variation across groups 2 Regression analysis More data sets, for examples Albert Satorra ( Mètodes Estadístics, UPF, hivern 2013 ) GRAU en CP, hivern / 24

3 Summary statistics Summary of distribution of Nota summary(nota) Min. 1st Qu. Median Mean 3rd Qu. Max length(nota) = 365 mean(nota) = sd(nota) = Albert Satorra ( Mètodes Estadístics, UPF, hivern 2013 ) GRAU en CP, hivern / 24

4 Summary statistics Histogram of Nota Density Albert Satorra ( Mètodes Estadístics, UPF, hivern 2013 ) GRAU en CP, hivern / 24 Nota

5 Standardization Percentiles, under normal distribution mean(nota) =6.48; sd(nota) =1.56 Standardized variable: Znota = (Nota- 6.48)/ 1.56 How many people above 5? 1 - pnorm( (5-6.48)/ 1.56) = i.e. 83% Compare with true value, sum(nota > 5)/length(Nota) = % PERCENTILES: > 5 is the percentile since pnorm( (5-6.48)/ 1.56)= > 7 is the percentile, since pnorm( (7-6.48)/ 1.56) [1] In general, e.g. el 30 percentile is: mean(nota) + qnorm(0.32)*sd(nota) = Albert Satorra ( Mètodes Estadístics, UPF, hivern 2013 ) GRAU en CP, hivern / 24

6 Standardization Valor immobles, non-normality data=read.table(" = T) summary(valor) Min. 1st Qu. Median Mean 3rd Qu. Max We normalize by taking log of Valor logvalor = log(valor) summary(logvalor) Min. 1st Qu. Median Mean 3rd Qu. Max mean(logvalor)=13.54 sd(logvalor)= 0.78 for the 3rd Qu , we have that pnorm((log( ) - mean(logvalor))/sd(logvalor)) [1] which is close to the third quartile The 60 percentile is to be found as qnorm(0.6)= So: exp( qnorm(0.6)* ) = Note that the true percentile of is sum(valor < )/length(Valor) = 0.585; that is, 58.5 This accuracy would not be obtained if we did not use the log transformation of the variable Valor The normal approximation of the 60 percentile is: mean(valor) + qnorm(0.60)*sd(valor) = Note that is the 71 percentile, since sum(valor < )/length(Valor) = Albert Satorra ( Mètodes Estadístics, UPF, hivern 2013 ) GRAU en CP, hivern / 24

7 Standardization Histogram of Valor Density 0e+00 1e 07 2e 07 3e 07 4e 07 5e 07 6e 07 7e 07 0e+00 1e+06 2e+06 3e+06 4e+06 5e+06 Albert Satorra ( Mètodes Estadístics, UPF, hivern 2013 ) GRAU en CP, hivern / 24 Valor

8 Standardization Histogram of logvalor Density logvalor Albert Satorra ( Mètodes Estadístics, UPF, hivern 2013 ) GRAU en CP, hivern / 24

9 Inference: estimation of the mean Estimation of the population mean using a sample sample1=sample(round(nota,2),20) sample1 [1] summary(sample1) Min. 1st Qu. Median Mean 3rd Qu. Max sd(sample1)= t.test(sample1) data: sample1 t = , df = 19, p-value = 3.314e-13 alternative hypothesis: true mean is not equal to 0 95 percent confidence interval: (we know that the true value is 6.48) t.test(sample1, conf.level = 0.9)... alternative hypothesis: true mean is not equal to 0 90 percent confidence interval: Albert Satorra ( Mètodes Estadístics, UPF, hivern 2013 ) GRAU en CP, hivern / 24

10 Inference: estimation of the mean Estimation of the population mean using a sample sample2=sample(round(nota,2),20) Sample 2: summary(sample2) Min. 1st Qu. Median Mean 3rd Qu. Max sd(sample2) [1] t.test(sample2) data: sample2 t = , df = 19, p-value = 1.702e-15 alternative hypothesis: true mean is not equal to 0 95 percent confidence interval: Albert Satorra ( Mètodes Estadístics, UPF, hivern 2013 ) GRAU en CP, hivern / 24

11 ANOVA: variation across groups Group variation? (groups are classrooms) ANOVA: aov2= aov(testexamenfinal ~ as.factor(grupteoria)) summary(aov2) Df Sum Sq Mean Sq F value Pr(>F) as.factor(grupteoria) e-07 *** Residuals Signif. codes: 0 *** ** 0.01 * YES! Albert Satorra ( Mètodes Estadístics, UPF, hivern 2013 ) GRAU en CP, hivern / 24

12 ANOVA: variation across groups Class group variation of Nota Albert Satorra ( Mètodes Estadístics, UPF, 1hivern ) 3 4 GRAU 5 en CP, hivern / 24

13 ANOVA: variation across groups Group variation? (groups are types of exam) aggregate(nota, list(tipus),mean) Group.1 x boxplot(nota ~ Tipus, col=2:5, main="per tipus (1 a 4) d examen") dev.copy2pdf(file="/albertnou/courses/metodes05_07_10/boxplot2.pdf") ANOVA aov1= aov(nota ~ as.factor(tipus)) summary(aov1) Df Sum Sq Mean Sq F value Pr(>F) #as.factor(tipus) # Residuals Albert Satorra ( Mètodes Estadístics, UPF, hivern 2013 ) GRAU en CP, hivern / 24

14 ANOVA: variation across groups per tipus (1 a 4) d'examen Albert Satorra ( Mètodes Estadístics, UPF, hivern 2013 ) GRAU en CP, hivern / 24

15 ANOVA: variation across groups On categorial data relations Examples of categorical variables include marital status (never married, married, divorced, or widowed) or race (black, white, Asian, native American, and Inuit). We are interested in the study of associations between variables that may have more than two categories. More details on relation among categorical variables are to be found here: Slides Albert Satorra ( Mètodes Estadístics, UPF, hivern 2013 ) GRAU en CP, hivern / 24

16 Regression analysis The regression model relates a single outcome to one or a number of covariates. The model embodies basic ideas about explaining variation, statistical control, and the functional form of social and behavioral relationships. Albert Satorra ( Mètodes Estadístics, UPF, hivern 2013 ) GRAU en CP, hivern / 24

17 Regression analysis Variation of grade of final exam (Y) across different levels of midterm exam (X) scatterplot of Y vs X TestExamenFinal Albert Satorra ( Mètodes Estadístics, UPF, hivern 2013 ) GRAU en CP, hivern / 24

18 Regression analysis Simple Linear Regression > out=lm(testexamenfinal ~ MitjanaControls) > summary(out) Call: lm(formula = TestExamenFinal ~ MitjanaControls) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) * MitjanaControls e-14 *** --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 363 degrees of freedom Multiple R-squared: 0.143, Adjusted R-squared: Albert Satorra ( Mètodes Estadístics, UPF, hivern 2013 ) GRAU en CP, hivern / 24

19 Regression analysis Simple Linear Regression library(foreign) help(read.spss) data = read.spss(" use.value.labels = TRUE, to.data.frame = TRUE, max.value.labels = Inf, trim.factor.names = FALSE, trim_values = TRUE, reencode = NA, us attach(data); names(data) plot(gnpcap, LIFE_EXPECTANCY, type ="n", axes=f) axis(1) axis(2) text(gnpcap[-27], LIFE_EXPECTANCY[-27], COUNTRY[-27], col="blue", cex=0.8) abline(lm(life_expectancy ~ GNPCAP), lty=4, col="red", lwd=3) text(gnpcap[27], LIFE_EXPECTANCY[27], COUNTRY[27], col="red", cex=1.2) dev.copy2pdf(file="/albertnou/courses/metodes05_07_10/yvsx2.pdf") Albert Satorra ( Mètodes Estadístics, UPF, hivern 2013 ) GRAU en CP, hivern / 24

20 Regression analysis Variation of life expectation (Y) against GNP (X) LIFE_EXPECTANCY Greece Bosnia Portugal Croatia Poland Czech Rep. Georgia Lithuania mania Bulgaria Hungary Estonia Belarus ine Latvia Russia Spain Ireland Iceland Sweden Netherlands France Norway Italy United Kingdom Finland Germany Belgium Denmark Austria Switzerland GNPCAP Albert Satorra ( Mètodes Estadístics, UPF, hivern 2013 ) GRAU en CP, hivern / 24

21 Regression analysis Simple Linear Regression > summary(lm(life_expectancy ~ GNPCAP)) Call: lm(formula = LIFE_EXPECTANCY ~ GNPCAP) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) 7.039e e < 2e-16 *** GNPCAP 3.804e e e-08 *** --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 29 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: on 1 and 29 DF, p-value: 2.142e-08 Albert Satorra ( Mètodes Estadístics, UPF, hivern 2013 ) GRAU en CP, hivern / 24

22 Regression analysis Simple Linear Regression using IBM SPSS Statistics These are the steps to produce the chart by using syntax : 1 Open the data file (in the IBM SPSS Statistics main menu, select FILE... OPEN... DATA, browse to the folder where the data file resides, select the data file and click Open) 2 From the main menu, select FILE... NEW... SYNTAX (not FILE... NEW... SCRIPT; here is where the difference between syntax and script comes in...). This will open a Syntax window. 3 copy and paste the lines below into the new Syntax window 4 from the main menu in the Syntax window, select RUN... ALL Having the data file opened, copy and paste the next lines in the Syntax window and then run it by selecting R GGRAPH /GRAPHDATASET NAME="graphdataset" VARIABLES=GNPCAP LIFE_EXPECTANCY COUNTRY MISSING=LISTWISE REPORTMISSING=NO /GRAPHSPEC SOURCE=INLINE. BEGIN GPL SOURCE: s=usersource(id("graphdataset")) DATA: GNPCAP=col(source(s), name("gnpcap")) DATA: LIFE_EXPECTANCY=col(source(s), name("life_expectancy")) DATA: COUNTRY=col(source(s), name("country"), unit.category()) GUIDE: axis(dim(1), label("gnp per Capita (US Dollars)")) GUIDE: axis(dim(2), label("life expectancy")) ELEMENT: point(position(gnpcap*life_expectancy), label(country)) END GPL. Albert Satorra ( Mètodes Estadístics, UPF, hivern 2013 ) GRAU en CP, hivern / 24

23 Regression analysis Dades immobles: Valor versus Superficie Construida data=read.table(" [1] "TipologiaInm" "Calidad" "TamanoPoblacion" [8] "NumGarajes" "Ascensor" plot(valor, SuperficieConstruida) plot( SuperficieConstruida, Valor) abline(lm(valor ~ SuperficieConstruida), col="red", lty=3, lwd=2) out1= lm(valor ~ SuperficieConstruida) summary(out1) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) Albert (Intercept) Satorra ( Mètodes Estadístics, UPF, hivern ) GRAU en CP, <2e-16 hivern 2013*** 22 / 24

24 Regression analysis More data sets, for examples Multiple Regression (CPS data) data=read.table(" names(data) [1] "female" "age" "mstat" "edyr" "hgc" "reth" "hw" qqnorm(hw) qqnorm(log(hw)) hist(hw) out1 <- lm(log(hw) ~ edyr + age + female+ mstat, data = data) summary(out1) library(car) avplots(out1) Albert Satorra ( Mètodes Estadístics, UPF, hivern 2013 ) GRAU en CP, hivern / 24

25 Regression analysis More data sets, for examples Other data sets: fatality data (Iraq), welfare data, ) ### Fatality data library(foreign) fata=read.dta(" ### iraq data fata=read.table(" ## welfare data dades=read.table(" Albert Satorra ( Mètodes Estadístics, UPF, hivern 2013 ) GRAU en CP, hivern / 24