Correlação e Regressão Linear

1 Correlação

1.1 Importando os dados

library(foreign) ## Pacote Foreing é imcompatível com tidyverse - lê factor do spss ##    
dados<- read.spss("C:/Users/user/Desktop/Vida acadêmica/Disciplinas/Estatística Vitor/Aula 3/Dados/Regressao.sav")
attach(dados)
library(tibble) # Resolve o problema anterior #
dados <- as_tibble(dados)

1.2 Teste de normalidade das variáveis

shapiro.test(dados$Gastos_saude)

## 
##  Shapiro-Wilk normality test
## 
## data:  dados$Gastos_saude
## W = 0.99104, p-value = 0.6297

shapiro.test(dados$Gastos_lazer)

## 
##  Shapiro-Wilk normality test
## 
## data:  dados$Gastos_lazer
## W = 0.98349, p-value = 0.1499

shapiro.test(dados$Gastos_educacao)

## 
##  Shapiro-Wilk normality test
## 
## data:  dados$Gastos_educacao
## W = 0.98292, p-value = 0.1328

1.3 Correlação Simples

cor.test(dados$Gastos_educacao, dados$Gastos_saude)

## 
##  Pearson's product-moment correlation
## 
## data:  dados$Gastos_educacao and dados$Gastos_saude
## t = 13.463, df = 118, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.6961127 0.8402773
## sample estimates:
##     cor 
## 0.77825

cor.test(dados$Gastos_educacao, dados$Gastos_lazer)

## 
##  Pearson's product-moment correlation
## 
## data:  dados$Gastos_educacao and dados$Gastos_lazer
## t = 12.228, df = 118, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.6563268 0.8173331
## sample estimates:
##       cor 
## 0.7476177

cor.test(dados$Gastos_saude, dados$Gastos_lazer)

## 
##  Pearson's product-moment correlation
## 
## data:  dados$Gastos_saude and dados$Gastos_lazer
## t = 13.634, df = 118, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.7011641 0.8431539
## sample estimates:
##       cor 
## 0.7821115

cor(dados[,7:9])

##                 Gastos_saude Gastos_lazer Gastos_educacao
## Gastos_saude       1.0000000    0.7821115       0.7782500
## Gastos_lazer       0.7821115    1.0000000       0.7476177
## Gastos_educacao    0.7782500    0.7476177       1.0000000

1.4 Como determinar o método de correlação

cor.test(dados$Gastos_educacao, dados$Gastos_saude) # Pearson

## 
##  Pearson's product-moment correlation
## 
## data:  dados$Gastos_educacao and dados$Gastos_saude
## t = 13.463, df = 118, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.6961127 0.8402773
## sample estimates:
##     cor 
## 0.77825

cor.test(dados$Gastos_educacao, dados$Gastos_saude, exact = F, method = "spearman")

## 
##  Spearman's rank correlation rho
## 
## data:  dados$Gastos_educacao and dados$Gastos_saude
## S = 78681, p-value < 2.2e-16
## alternative hypothesis: true rho is not equal to 0
## sample estimates:
##       rho 
## 0.7267847

cor.test(dados$Gastos_educacao, dados$Gastos_saude, exact = F, method = "kendal")

## 
##  Kendall's rank correlation tau
## 
## data:  dados$Gastos_educacao and dados$Gastos_saude
## z = 8.8553, p-value < 2.2e-16
## alternative hypothesis: true tau is not equal to 0
## sample estimates:
##       tau 
## 0.5469319

1.5 Modelos de visualização das correlações

pairs(dados[,7:9])

1.5.1 Pelo pacote corrplot

library(corrplot)
Cor <- cor(dados[,7:9])
corrplot(Cor, tl.cex = 1)

corrplot(Cor, method = "color") # Cores

corrplot(Cor, method = "ellipse")

corrplot(Cor, method = "shade")

corrplot(Cor, method = "number")

corrplot(Cor, type = "upper")

corrplot(Cor, type = "lower", method = "number")

1.5.2 Pelo Hmisc

library(Hmisc)
m <- rcorr(as.matrix(dados[,7:9]))
m

##                 Gastos_saude Gastos_lazer Gastos_educacao
## Gastos_saude            1.00         0.78            0.78
## Gastos_lazer            0.78         1.00            0.75
## Gastos_educacao         0.78         0.75            1.00
## 
## n= 120 
## 
## 
## P
##                 Gastos_saude Gastos_lazer Gastos_educacao
## Gastos_saude                  0            0             
## Gastos_lazer     0                         0             
## Gastos_educacao  0            0

m$r # matriz de coeficiente de correlação

##                 Gastos_saude Gastos_lazer Gastos_educacao
## Gastos_saude       1.0000000    0.7821115       0.7782500
## Gastos_lazer       0.7821115    1.0000000       0.7476177
## Gastos_educacao    0.7782500    0.7476177       1.0000000

m$P # matriz de valor p

##                 Gastos_saude Gastos_lazer Gastos_educacao
## Gastos_saude              NA            0               0
## Gastos_lazer               0           NA               0
## Gastos_educacao            0            0              NA

m$n # matriz de n

##                 Gastos_saude Gastos_lazer Gastos_educacao
## Gastos_saude             120          120             120
## Gastos_lazer             120          120             120
## Gastos_educacao          120          120             120

corrplot(m$r, p.mat=m$P, sig.level = 0.05) # Gráfico de bolinha eliminando as correlações não significativas

corrplot(m$r, p.mat=m$P, sig.level = 0.05, method = "number", type = "upper") # Gráfico números eliminando as correlações não significativas

1.5.3 Pelo GGally

library(ggplot2)
library(GGally)

ggcorr(dados, digits = 2, label = T)

ggpairs(dados[,7:9],lower = list(continuous="smooth"))

dados2 <- dados[,c(3,2,6,8,9,10,11)]
ggpairs(dados2 ,columns=2:7,aes(colour=Profissao))

1.5.4 Pelo corrgram

library(corrgram)
corrgram(dados)

corrgram(dados,lower.panel = panel.pts,upper.panel = panel.conf,diag.panel = panel.density)

1.5.5 Pelo car

library(car)
spm(~ `Gastos_saude` + `Gastos_lazer` + `Gastos_educacao`, data = dados, by.groups = F, legend = F)

spm(~ `Gastos_saude` + `Gastos_lazer` + `Gastos_educacao` | Profissao, data = dados, by.groups = TRUE, legend = F)

2 Regressão Linear

2.1 Importando os dados

library(foreign) ## Pacote Foreing é imcompatível com tidyverse - lê factor do spss ##    
dados<- read.spss("C:/Users/user/Desktop/Vida acadêmica/Disciplinas/Estatística Vitor/Aula 3/Dados/Regressao.sav")
attach(dados)
library(tibble) # Resolve o problema anterior #
dados <- as_tibble(dados)
dados2 <- dados[,7:9] # Selecionar apenas as variáveis que temos interesse em trabalhar

2.2 Criando o modelo de regressão

reg <- lm(Gastos_educacao ~., data = dados2)
summary(reg) # Resumo do modelo

## 
## Call:
## lm(formula = Gastos_educacao ~ ., data = dados2)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -115.661  -27.284   -3.401   21.809  140.070 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   6.77161   12.67814   0.534    0.594    
## Gastos_saude  0.48772    0.08523   5.723 8.22e-08 ***
## Gastos_lazer  0.31038    0.07555   4.108 7.42e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 44.42 on 117 degrees of freedom
## Multiple R-squared:  0.6554, Adjusted R-squared:  0.6495 
## F-statistic: 111.3 on 2 and 117 DF,  p-value: < 2.2e-16

2.3 Avaliação do modelo

library(QuantPsyc)
lm.beta(reg) # valores padronizados

## Gastos_saude Gastos_lazer 
##    0.4984003    0.3578131

reg$coefficients # Coeficientes

##  (Intercept) Gastos_saude Gastos_lazer 
##    6.7716131    0.4877232    0.3103845

confint(reg) # intervalo de confiança dos coenficientes  

##                    2.5 %     97.5 %
## (Intercept)  -18.3367719 31.8799982
## Gastos_saude   0.3189333  0.6565131
## Gastos_lazer   0.1607626  0.4600065

attributes(reg)

## $names
##  [1] "coefficients"  "residuals"     "effects"       "rank"         
##  [5] "fitted.values" "assign"        "qr"            "df.residual"  
##  [9] "xlevels"       "call"          "terms"         "model"        
## 
## $class
## [1] "lm"

reg$residuals # Resíduos

##            1            2            3            4            5            6 
##    8.3634527   -2.5703931   11.7618970  -16.7265027  -12.8437423  -54.0174274 
##            7            8            9           10           11           12 
##   -7.5372200   29.8925726  -26.6492687    7.4220346  -24.0804711   -7.0976214 
##           13           14           15           16           17           18 
##    2.5705041  -16.9886448   12.6911752   33.5125748  -44.4135455   -9.3066095 
##           19           20           21           22           23           24 
##  -13.3562795   10.8909361   13.3002035   61.5254364   58.2245308   -2.4799487 
##           25           26           27           28           29           30 
##   -9.2399304   27.3418608   43.3285293   73.7075034  -18.8691957   -4.0203202 
##           31           32           33           34           35           36 
##  -54.8592480    9.4448288  -31.1430678   -1.0130422    9.9823699  -57.1175370 
##           37           38           39           40           41           42 
##   13.8614342  -70.3352527    3.3935962  -16.3435324  -12.9843367   11.7386997 
##           43           44           45           46           47           48 
##   -7.2231398  -29.1899776  -33.0587438   -2.7815943   22.0636254   11.5579561 
##           49           50           51           52           53           54 
##   54.6050464   -4.8696808   69.2931615  140.0702050   23.5576743  -69.6930647 
##           55           56           57           58           59           60 
##  -51.3396623   -0.6097038   54.3110781  -23.6953159   -8.9216996  -56.3831082 
##           61           62           63           64           65           66 
##    0.9936549   80.9650165  -24.5465472  -18.7598765   24.8446653   38.8933283 
##           67           68           69           70           71           72 
##  138.6587158  -18.7280427   13.2172556    3.8136439   39.1603427  -25.8308848 
##           73           74           75           76           77           78 
##   75.8071875  -23.7410423  -21.5347860   -5.1079370  -34.3178134  -15.7308293 
##           79           80           81           82           83           84 
##  -57.6546518   61.6634769   57.0592270   -2.7568010   44.3644819  -30.2491707 
##           85           86           87           88           89           90 
##    2.5128170   21.7241753  -33.7349861   69.2257583  -46.5802218   11.5779443 
##           91           92           93           94           95           96 
##  -57.3143296  -15.3792346   71.7846987  -32.8310974  -52.2430165  -31.3418051 
##           97           98           99          100          101          102 
##   35.9005007  131.1325011  -16.2699197  -33.7607054   14.4566575    8.5567068 
##          103          104          105          106          107          108 
##   17.4956001   11.8462054   -6.6427302  -98.9584315   32.0056719  -30.5492327 
##          109          110          111          112          113          114 
##   13.9947866  -50.2903500 -115.6611205  -71.5015894   30.7914873  -45.0076109 
##          115          116          117          118          119          120 
##   62.3673767   54.0897106  -59.6192608  -42.0659612  -53.8936725  -12.9539942

par(mfrow = c(2,2), mar = c(0,0,0,0))
plot(reg) # Linearidade, normalidade, homocedasticidade, casos influentes 

dev.off()

## null device 
##           1

anova(reg)

## Analysis of Variance Table
## 
## Response: Gastos_educacao
##               Df Sum Sq Mean Sq F value    Pr(>F)    
## Gastos_saude   1 405737  405737 205.633 < 2.2e-16 ***
## Gastos_lazer   1  33303   33303  16.879 7.421e-05 ***
## Residuals    117 230854    1973                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

2.4 Distância d Cook

plot(reg, pch = 18, col = "red", which=c(4))

cooks.distance(reg)

##            1            2            3            4            5            6 
## 3.626821e-04 3.567185e-05 2.979681e-04 1.843818e-03 7.679278e-04 4.580775e-03 
##            7            8            9           10           11           12 
## 6.495522e-04 6.428010e-03 4.970680e-03 1.395884e-04 3.494716e-03 7.032325e-04 
##           13           14           15           16           17           18 
## 2.082471e-05 2.793338e-03 5.198148e-04 7.453809e-03 7.441787e-03 1.545638e-04 
##           19           20           21           22           23           24 
## 3.971420e-04 6.505589e-04 6.929172e-04 1.077426e-02 7.853514e-03 3.912614e-05 
##           25           26           27           28           29           30 
## 2.788476e-04 1.845123e-03 1.306472e-02 1.099294e-02 7.926165e-04 1.444299e-04 
##           31           32           33           34           35           36 
## 5.375211e-03 3.542030e-04 6.180320e-03 1.505719e-06 3.018508e-04 1.678888e-02 
##           37           38           39           40           41           42 
## 8.068665e-04 1.424050e-02 3.234163e-05 3.922071e-04 5.446103e-04 7.242952e-04 
##           43           44           45           46           47           48 
## 9.592285e-05 2.745744e-03 1.157433e-02 1.740064e-05 1.378161e-03 2.126748e-04 
##           49           50           51           52           53           54 
## 8.958884e-03 1.349452e-04 1.203245e-02 3.829091e-02 1.612302e-03 1.633712e-02 
##           55           56           57           58           59           60 
## 3.914979e-03 1.213124e-06 1.770253e-02 1.567729e-03 3.882119e-04 7.803860e-03 
##           61           62           63           64           65           66 
## 3.150686e-06 9.624244e-03 1.922356e-03 6.169198e-04 2.544011e-03 4.343879e-03 
##           67           68           69           70           71           72 
## 1.580055e-01 1.682506e-03 5.662678e-03 5.787266e-05 6.681944e-03 5.921682e-03 
##           73           74           75           76           77           78 
## 1.319510e-02 3.191760e-03 1.577229e-03 2.239681e-04 2.255262e-03 6.573966e-04 
##           79           80           81           82           83           84 
## 5.113247e-03 1.382224e-02 5.882502e-03 8.606703e-05 1.445028e-02 2.942380e-03 
##           85           86           87           88           89           90 
## 1.107930e-05 8.876767e-04 9.936581e-03 2.067455e-02 4.953311e-03 8.997315e-04 
##           91           92           93           94           95           96 
## 1.920048e-02 1.826554e-03 6.358865e-02 5.523618e-03 2.076753e-02 3.090727e-03 
##           97           98           99          100          101          102 
## 2.000111e-03 5.747722e-02 8.525485e-04 3.265669e-03 3.755280e-04 1.374947e-04 
##          103          104          105          106          107          108 
## 1.424988e-03 6.143592e-04 1.787136e-04 2.486814e-02 4.126715e-03 3.698866e-03 
##          109          110          111          112          113          114 
## 1.559297e-03 8.568435e-03 5.785654e-02 1.865775e-02 4.549437e-03 4.514693e-03 
##          115          116          117          118          119          120 
## 1.023297e-02 5.271212e-03 1.600172e-02 1.415505e-02 9.505620e-03 1.266218e-03

meuCD <- cooks.distance(reg)
mecdr <- round(meuCD,5)
sort(mecdr)

##      34      56      61      85      13      46      39       2      24      70 
## 0.00000 0.00000 0.00000 0.00001 0.00002 0.00002 0.00003 0.00004 0.00004 0.00006 
##      82      43      50      10      30     102      18     105      48      76 
## 0.00009 0.00010 0.00013 0.00014 0.00014 0.00014 0.00015 0.00018 0.00021 0.00022 
##      25       3      35      32       1     101      40      59      19      15 
## 0.00028 0.00030 0.00030 0.00035 0.00036 0.00038 0.00039 0.00039 0.00040 0.00052 
##      41     104      64       7      20      78      21      12      42       5 
## 0.00054 0.00061 0.00062 0.00065 0.00065 0.00066 0.00069 0.00070 0.00072 0.00077 
##      29      37      99      86      90     120      47     103     109      58 
## 0.00079 0.00081 0.00085 0.00089 0.00090 0.00127 0.00138 0.00142 0.00156 0.00157 
##      75      53      68      92       4      26      63      97      77      65 
## 0.00158 0.00161 0.00168 0.00183 0.00184 0.00185 0.00192 0.00200 0.00226 0.00254 
##      44      14      84      96      74     100      11     108      55     107 
## 0.00275 0.00279 0.00294 0.00309 0.00319 0.00327 0.00349 0.00370 0.00391 0.00413 
##      66     114     113       6      89       9      79     116      31      94 
## 0.00434 0.00451 0.00455 0.00458 0.00495 0.00497 0.00511 0.00527 0.00538 0.00552 
##      69      81      72      33       8      71      17      16      60      23 
## 0.00566 0.00588 0.00592 0.00618 0.00643 0.00668 0.00744 0.00745 0.00780 0.00785 
##     110      49     119      62      87     115      22      28      45      51 
## 0.00857 0.00896 0.00951 0.00962 0.00994 0.01023 0.01077 0.01099 0.01157 0.01203 
##      27      73      80     118      38      83     117      54      36      57 
## 0.01306 0.01320 0.01382 0.01416 0.01424 0.01445 0.01600 0.01634 0.01679 0.01770 
##     112      91      88      95     106      52      98     111      93      67 
## 0.01866 0.01920 0.02067 0.02077 0.02487 0.03829 0.05748 0.05786 0.06359 0.15801

dados$cooks.distance <- cooks.distance(reg)
library(plotly)
library(magrittr)
plot_ly(data = dados, y = dados$cooks.distance, type = "scatter") %>% layout(title = "Distância de Cook") # O valor mostrado é sempre +1 na base de dados

2.5 Calcular a correlação entre as VI’s (multicolineariedade)

round(cor(dados2),2)

##                 Gastos_saude Gastos_lazer Gastos_educacao
## Gastos_saude            1.00         0.78            0.78
## Gastos_lazer            0.78         1.00            0.75
## Gastos_educacao         0.78         0.75            1.00

library(car)
vif(reg) # Ponto de corte: VIF até 10

## Gastos_saude Gastos_lazer 
##     2.575318     2.575318

mean(vif(reg))

## [1] 2.575318

1/vif(reg) # Tolerância: Ponto de corte: valores menores do que 0.2

## Gastos_saude Gastos_lazer 
##    0.3883015    0.3883015

2.6 Durbin-Watson (autocorrelação dos resíduos)

library(lmtest)
dwtest(reg) # Valores próximos a 2

## 
##  Durbin-Watson test
## 
## data:  reg
## DW = 1.8241, p-value = 0.1552
## alternative hypothesis: true autocorrelation is greater than 0

library(car)
durbinWatsonTest(reg) #H0 resíduos são aleatórios

##  lag Autocorrelation D-W Statistic p-value
##    1      0.08742109      1.824128   0.296
##  Alternative hypothesis: rho != 0

2.7 Resíduos

res <- rstandard(lm(Gastos_educacao ~., data = dados2))
res <- as.data.frame(res)

2.7.1 Outliers dos resíudos

dados$Zscoreresiduos <- res
dados$Zscoreresiduos

##             res
## 1    0.19106773
## 2   -0.05875611
## 3    0.26645191
## 4   -0.38356939
## 5   -0.29299923
## 6   -1.22165535
## 7   -0.17499727
## 8    0.68658446
## 9   -0.61177766
## 10   0.16831918
## 11  -0.55138069
## 12  -0.16580330
## 13   0.05839625
## 14  -0.39271095
## 15   0.28837696
## 16   0.76859963
## 17  -1.01072808
## 18  -0.21060769
## 19  -0.30263319
## 20   0.24901132
## 21   0.30279665
## 22   1.39652446
## 23   1.31961946
## 24  -0.05683525
## 25  -0.20997836
## 26   0.61995141
## 27   0.99457236
## 28   1.66913520
## 29  -0.42754746
## 30  -0.09275853
## 31  -1.24146502
## 32   0.21505584
## 33  -0.71375376
## 34  -0.02290413
## 35   0.22669973
## 36  -1.30474411
## 37   0.31581995
## 38  -1.59663785
## 39   0.07702071
## 40  -0.36951630
## 41  -0.29504083
## 42   0.26822887
## 43  -0.16348428
## 44  -0.66326426
## 45  -0.76594406
## 46  -0.06303081
## 47   0.50078587
## 48   0.26141080
## 49   1.23999559
## 50  -0.11140259
## 51   1.57132682
## 52   3.17129358
## 53   0.53480859
## 54  -1.58421356
## 55  -1.16081224
## 56  -0.01385548
## 57   1.24349956
## 58  -0.53776202
## 59  -0.20365066
## 60  -1.27838639
## 61   0.02257616
## 62   1.83056377
## 63  -0.55770465
## 64  -0.42449575
## 65   0.56594111
## 66   0.88287651
## 67   3.19328916
## 68  -0.42740106
## 69   0.32112879
## 70   0.08683754
## 71   0.89261986
## 72  -0.59588817
## 73   1.71801828
## 74  -0.54307788
## 75  -0.48956485
## 76  -0.11774613
## 77  -0.77689919
## 78  -0.35687238
## 79  -1.30379579
## 80   1.40275232
## 81   1.29132739
## 82  -0.06398945
## 83   1.01937752
## 84  -0.68731845
## 85   0.05685993
## 86   0.49175188
## 87  -0.77793966
## 88   1.57774353
## 89  -1.05560799
## 90   0.26558911
## 91  -1.31171132
## 92  -0.35372578
## 93   1.67038748
## 94  -0.74992143
## 95  -1.20124481
## 96  -0.71200693
## 97   0.81188170
## 98   2.98063647
## 99  -0.36968858
## 100 -0.76635229
## 101  0.32716433
## 102  0.19368930
## 103  0.39912042
## 104  0.27003744
## 105 -0.15128613
## 106 -2.24424441
## 107  0.72887595
## 108 -0.69558263
## 109  0.32208338
## 110 -1.14324258
## 111 -2.63614247
## 112 -1.62661945
## 113  0.70270898
## 114 -1.01981190
## 115  1.41477417
## 116  1.22410529
## 117 -1.35949972
## 118 -0.96822305
## 119 -1.22476179
## 120 -0.29780718

round(dados$Zscoreresiduos, digits = 3)

##        res
## 1    0.191
## 2   -0.059
## 3    0.266
## 4   -0.384
## 5   -0.293
## 6   -1.222
## 7   -0.175
## 8    0.687
## 9   -0.612
## 10   0.168
## 11  -0.551
## 12  -0.166
## 13   0.058
## 14  -0.393
## 15   0.288
## 16   0.769
## 17  -1.011
## 18  -0.211
## 19  -0.303
## 20   0.249
## 21   0.303
## 22   1.397
## 23   1.320
## 24  -0.057
## 25  -0.210
## 26   0.620
## 27   0.995
## 28   1.669
## 29  -0.428
## 30  -0.093
## 31  -1.241
## 32   0.215
## 33  -0.714
## 34  -0.023
## 35   0.227
## 36  -1.305
## 37   0.316
## 38  -1.597
## 39   0.077
## 40  -0.370
## 41  -0.295
## 42   0.268
## 43  -0.163
## 44  -0.663
## 45  -0.766
## 46  -0.063
## 47   0.501
## 48   0.261
## 49   1.240
## 50  -0.111
## 51   1.571
## 52   3.171
## 53   0.535
## 54  -1.584
## 55  -1.161
## 56  -0.014
## 57   1.243
## 58  -0.538
## 59  -0.204
## 60  -1.278
## 61   0.023
## 62   1.831
## 63  -0.558
## 64  -0.424
## 65   0.566
## 66   0.883
## 67   3.193
## 68  -0.427
## 69   0.321
## 70   0.087
## 71   0.893
## 72  -0.596
## 73   1.718
## 74  -0.543
## 75  -0.490
## 76  -0.118
## 77  -0.777
## 78  -0.357
## 79  -1.304
## 80   1.403
## 81   1.291
## 82  -0.064
## 83   1.019
## 84  -0.687
## 85   0.057
## 86   0.492
## 87  -0.778
## 88   1.578
## 89  -1.056
## 90   0.266
## 91  -1.312
## 92  -0.354
## 93   1.670
## 94  -0.750
## 95  -1.201
## 96  -0.712
## 97   0.812
## 98   2.981
## 99  -0.370
## 100 -0.766
## 101  0.327
## 102  0.194
## 103  0.399
## 104  0.270
## 105 -0.151
## 106 -2.244
## 107  0.729
## 108 -0.696
## 109  0.322
## 110 -1.143
## 111 -2.636
## 112 -1.627
## 113  0.703
## 114 -1.020
## 115  1.415
## 116  1.224
## 117 -1.359
## 118 -0.968
## 119 -1.225
## 120 -0.298

dados$large.residuals <- dados$Zscoreresiduos > 2.9 | dados$Zscoreresiduos < - 2.9
summary(dados$large.residuals)

##     res         
##  Mode :logical  
##  FALSE:117      
##  TRUE :3

2.7.2 Normalidade para os resíduos (Shapiro-Wilk)

res <- rstandard(lm(Gastos_educacao ~., data = dados2))
shapiro.test(res)

## 
##  Shapiro-Wilk normality test
## 
## data:  res
## W = 0.96922, p-value = 0.007498

hist(res, breaks = 15, freq = T, col = "lightgray", main = "Distribuição dos resíduos", xlab = "Escore")

qq <- qplot(sample = res)
ggplotly(qq)

2.7.3 Teste de variância - homocedasticidade (Breush-Pagan)

library(ggplot2)
dados$fitted <- reg$fitted.values
dados$standr <- reg$residuals
dados[,c("Gastos_educacao", "fitted")]

## # A tibble: 120 x 2
##    Gastos_educacao fitted
##              <dbl>  <dbl>
##  1            97.5   89.1
##  2           186.   189. 
##  3           154.   142. 
##  4            57.1   73.8
##  5            81.8   94.6
##  6           126.   180. 
##  7           134.   142. 
##  8           103.    73.5
##  9            43.2   69.8
## 10           159.   151. 
## # ... with 110 more rows

ggplot(data=dados, aes(fitted, standr)) + geom_point() + geom_smooth(color = "blue") + labs(x= "Valores ajustados", y = "Resíduos padronizados")

ggplot(data=dados, aes(fitted,standr)) + geom_point() + geom_smooth(method = "lm", color = "blue") + labs(x= "Valores ajustados", y = "Resíduos padronizados")

library(car)
ncvTest(lm(Gastos_educacao ~., data = dados2)) #p = 0.19005

## Non-constant Variance Score Test 
## Variance formula: ~ fitted.values 
## Chisquare = 1.717224, Df = 1, p = 0.19005

2.8 Regressaõ sem outliers

dados <- subset(dados, dados$large.residuals == "FALSE")
reg2 <- lm(Gastos_educacao ~ Gastos_lazer + Gastos_saude, data = dados) 
summary(reg2)

## 
## Call:
## lm(formula = Gastos_educacao ~ Gastos_lazer + Gastos_saude, data = dados)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -113.271  -25.631   -2.201   18.593   84.620 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   1.87481   11.19846   0.167    0.867    
## Gastos_lazer  0.32816    0.06701   4.897 3.23e-06 ***
## Gastos_saude  0.47246    0.07661   6.167 1.09e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 38.99 on 114 degrees of freedom
## Multiple R-squared:  0.7158, Adjusted R-squared:  0.7108 
## F-statistic: 143.6 on 2 and 114 DF,  p-value: < 2.2e-16

2.9 Regressão com variável categórica

reg3 <- lm(Gastos_educacao ~  Gastos_lazer + Gastos_saude + Relacionamento, data = dados)
summary(reg3)

## 
## Call:
## lm(formula = Gastos_educacao ~ Gastos_lazer + Gastos_saude + 
##     Relacionamento, data = dados)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -114.731  -24.498   -1.114   22.141   83.173 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)             3.42971   11.97227   0.286    0.775    
## Gastos_lazer            0.32914    0.07026   4.684 8.01e-06 ***
## Gastos_saude            0.47073    0.07976   5.902 3.98e-08 ***
## RelacionamentoSolteiro -0.93265    8.16267  -0.114    0.909    
## RelacionamentoCasado   -7.15388   10.80257  -0.662    0.509    
## RelacionamentoViuvo    11.77865   40.60580   0.290    0.772    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 39.41 on 111 degrees of freedom
## Multiple R-squared:  0.7173, Adjusted R-squared:  0.7045 
## F-statistic: 56.32 on 5 and 111 DF,  p-value: < 2.2e-16

2.10 Regressão sem o intercepto

reg2 <- lm(Gastos_educacao ~ Gastos_lazer + Gastos_saude, data = dados)
summary(reg2) # Intercepto não é significativo no modelo

## 
## Call:
## lm(formula = Gastos_educacao ~ Gastos_lazer + Gastos_saude, data = dados)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -113.271  -25.631   -2.201   18.593   84.620 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   1.87481   11.19846   0.167    0.867    
## Gastos_lazer  0.32816    0.06701   4.897 3.23e-06 ***
## Gastos_saude  0.47246    0.07661   6.167 1.09e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 38.99 on 114 degrees of freedom
## Multiple R-squared:  0.7158, Adjusted R-squared:  0.7108 
## F-statistic: 143.6 on 2 and 114 DF,  p-value: < 2.2e-16

reg4 <- lm(Gastos_educacao ~ -1 + Gastos_lazer + Gastos_saude, data = dados)
summary(reg4)

## 
## Call:
## lm(formula = Gastos_educacao ~ -1 + Gastos_lazer + Gastos_saude, 
##     data = dados)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -113.761  -24.874   -1.214   18.909   84.879 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## Gastos_lazer  0.33288    0.06054   5.498 2.35e-07 ***
## Gastos_saude  0.47494    0.07485   6.346 4.54e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 38.82 on 115 degrees of freedom
## Multiple R-squared:  0.9602, Adjusted R-squared:  0.9596 
## F-statistic:  1389 on 2 and 115 DF,  p-value: < 2.2e-16

2.11 Regressão Polinomial

reg5 <- lm(Gastos_educacao ~ poly(Gastos_lazer,2) + poly(Gastos_saude,2), data = dados)
summary(reg5)

## 
## Call:
## lm(formula = Gastos_educacao ~ poly(Gastos_lazer, 2) + poly(Gastos_saude, 
##     2), data = dados)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -111.116  -26.763   -2.869   20.454   86.201 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)             179.037      3.626  49.381  < 2e-16 ***
## poly(Gastos_lazer, 2)1  309.224     63.612   4.861 3.83e-06 ***
## poly(Gastos_lazer, 2)2  -36.191     60.138  -0.602    0.549    
## poly(Gastos_saude, 2)1  387.513     63.723   6.081 1.69e-08 ***
## poly(Gastos_saude, 2)2   48.797     60.020   0.813    0.418    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 39.22 on 112 degrees of freedom
## Multiple R-squared:  0.7175, Adjusted R-squared:  0.7074 
## F-statistic:  71.1 on 4 and 112 DF,  p-value: < 2.2e-16

reg6 <- lm(Gastos_educacao ~ poly(Gastos_lazer,3) + poly(Gastos_saude,3), data = dados)
summary(reg6)

## 
## Call:
## lm(formula = Gastos_educacao ~ poly(Gastos_lazer, 3) + poly(Gastos_saude, 
##     3), data = dados)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -104.489  -24.529   -1.938   21.496   84.159 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)             179.037      3.613  49.556  < 2e-16 ***
## poly(Gastos_lazer, 3)1  271.136     69.090   3.924 0.000152 ***
## poly(Gastos_lazer, 3)2  -45.928     60.416  -0.760 0.448766    
## poly(Gastos_lazer, 3)3  -18.563     53.547  -0.347 0.729508    
## poly(Gastos_saude, 3)1  417.398     67.317   6.200 1.01e-08 ***
## poly(Gastos_saude, 3)2   55.299     60.558   0.913 0.363154    
## poly(Gastos_saude, 3)3   80.197     55.605   1.442 0.152072    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 39.08 on 110 degrees of freedom
## Multiple R-squared:  0.7245, Adjusted R-squared:  0.7094 
## F-statistic:  48.2 on 6 and 110 DF,  p-value: < 2.2e-16