Regression Diagnostics & Multiple Linear Regression

# Install and load libraries.
library(xlsx)
## Loading required package: rJava
## Loading required package: xlsxjars
library(ggplot2)
library(car)
## Warning: package 'car' was built under R version 3.4.4
## Loading required package: carData
## Warning: package 'carData' was built under R version 3.4.4
# Question 1
# Import the data from the Excel file containing data in Assignment 4.
occupation.prestige <- read.xlsx("homework4data.xlsx", sheetName = "Sheet1", header = TRUE)
colnames(occupation.prestige) <- c("Title", "Education", "Income", "Percent_of_Women", "Prestige_Score")
head(occupation.prestige) ; tail(occupation.prestige)
##                 Title Education Income Percent_of_Women Prestige_Score
## 1  GOV_ADMINISTRATORS     13.11  12351            11.16           68.8
## 2    GENERAL_MANAGERS     12.26  25879             4.02           69.1
## 3         ACCOUNTANTS     12.77   9271            15.70           63.4
## 4 PURCHASING_OFFICERS     11.42   8865             9.11           56.8
## 5            CHEMISTS     14.62   8403            11.68           73.5
## 6          PHYSICISTS     15.64  11030             5.13           77.6
##               Title Education Income Percent_of_Women Prestige_Score
## 97  TRAIN_ENGINEERS      8.49   8845             0.00           48.9
## 98      BUS_DRIVERS      7.58   5562             9.47           35.9
## 99     TAXI_DRIVERS      7.93   4224             3.59           25.1
## 100    LONGSHOREMEN      8.37   4753             0.00           26.1
## 101     TYPESETTERS     10.00   6462            13.58           42.2
## 102     BOOKBINDERS      8.55   3617            70.87           35.2
# Question 2
# Generate scatterplot for Prestige_Score and Education
png("scatterPlot.png", width = 1000, height = 750, unit = "px")     # Export plot as PNG file
plot(occupation.prestige$Education, occupation.prestige$Prestige_Score,
     main = "Scatter Plot of Education Level vs. Prestige Score",
     xlab = "Education Level (years)", ylab = "Prestige Score",
     type = "n", xlim = c(min(occupation.prestige$Education), max(occupation.prestige$Education)), 
     ylim = c(min(occupation.prestige$Prestige_Score), max(occupation.prestige$Prestige_Score)))
plot(occupation.prestige$Education, occupation.prestige$Prestige_Score,
     main = "Scatter Plot of Education Level vs. Prestige Score",
     xlab = "Education Level (years)", ylab = "Prestige Score",
     type = "n")
grid()
points(occupation.prestige$Education, occupation.prestige$Prestige_Score, col = "steelblue", pch = 20)
dev.off()
## quartz_off_screen 
##                 2
# Calculate the correlation.
cor(occupation.prestige$Education, occupation.prestige$Prestige_Score)
## [1] 0.8501769
# Question 3
# Perform a simple linear regression.
m <- lm(Prestige_Score ~ Education, data = occupation.prestige) ; m
## 
## Call:
## lm(formula = Prestige_Score ~ Education, data = occupation.prestige)
## 
## Coefficients:
## (Intercept)    Education  
##     -10.732        5.361
# Generate and save the residual plot.
png("residualPlot.png", width = 1000, height = 750, unit = "px")     # Export plot as PNG file
plot(m, which = 1, col = "steelblue", pch = 20)
dev.off()
## quartz_off_screen 
##                 2
# Generate a histogram of the residuals.
png("residualPlotHist.png", width = 1000, height = 750, unit = "px")     # Export plot as PNG file
hist(residuals(m), main = "Histogram of the Residuals for Prestige Score ~ Education", col = "steelblue")
dev.off()
## quartz_off_screen 
##                 2
# Identify outliers by ID (Title).
outlierTest(m)
## No Studentized residuals with Bonferonni p < 0.05
## Largest |rstudent|:
##    rstudent unadjusted p-value Bonferonni p
## 53 -2.98896          0.0035306      0.36012
influence.measures(m)
## Influence measures of
##   lm(formula = Prestige_Score ~ Education, data = occupation.prestige) :
## 
##        dfb.1_  dfb.Edct    dffit cov.r   cook.d     hat inf
## 1   -0.061640  0.089475  0.13600 1.017 9.24e-03 0.01729    
## 2   -0.046715  0.087777  0.17951 0.984 1.59e-02 0.01288    
## 3   -0.030017  0.046759  0.07804 1.028 3.06e-03 0.01530    
## 4    0.000112  0.017378  0.07133 1.021 2.56e-03 0.01042    
## 5   -0.074698  0.093601  0.11422 1.043 6.56e-03 0.02985    
## 6   -0.076393  0.091628  0.10474 1.059 5.53e-03 0.04176    
## 7   -0.035776  0.043812  0.05164 1.056 1.35e-03 0.03499    
## 8   -0.098101  0.118487  0.13682 1.052 9.41e-03 0.03921    
## 9   -0.073971  0.093220  0.11475 1.041 6.62e-03 0.02883    
## 10  -0.013436  0.016818  0.02049 1.052 2.12e-04 0.03005    
## 11  -0.023925  0.042224  0.08123 1.024 3.32e-03 0.01343    
## 12  -0.016454  0.030280  0.06072 1.028 1.86e-03 0.01305    
## 13   0.091907 -0.121853 -0.16216 1.020 1.31e-02 0.02252    
## 14   0.053768 -0.068085 -0.08443 1.045 3.59e-03 0.02803    
## 15  -0.101343  0.128978  0.16119 1.030 1.30e-02 0.02725    
## 16   0.115088 -0.147966 -0.18784 1.020 1.76e-02 0.02584    
## 17  -0.149751  0.178865  0.20324 1.047 2.07e-02 0.04348    
## 18   0.076137 -0.098313 -0.12564 1.034 7.92e-03 0.02529    
## 19   0.193132 -0.235241 -0.27504 1.018 3.74e-02 0.03652    
## 20  -0.071104  0.089712  0.11064 1.042 6.16e-03 0.02863    
## 21  -0.180068  0.213785  0.24086 1.044 2.89e-02 0.04621    
## 22   0.023260 -0.031494 -0.04327 1.040 9.45e-04 0.02085    
## 23   0.058754 -0.071984 -0.08490 1.053 3.63e-03 0.03488    
## 24  -0.229559  0.272622  0.30727 1.029 4.68e-02 0.04607    
## 25   0.147272 -0.175000 -0.19740 1.052 1.95e-02 0.04579    
## 26  -0.003582  0.004467  0.00541 1.053 1.48e-05 0.03079    
## 27  -0.035198  0.060375  0.11272 1.016 6.36e-03 0.01375    
## 28  -0.038895  0.026171 -0.06106 1.026 1.88e-03 0.01201    
## 29  -0.085572  0.115864  0.15918 1.017 1.26e-02 0.02085    
## 30   0.022881 -0.027881 -0.03262 1.058 5.37e-04 0.03640    
## 31  -0.052105  0.080764  0.13395 1.013 8.96e-03 0.01540    
## 32   0.011121  0.012069  0.09388 1.013 4.41e-03 0.00997    
## 33  -0.000985  0.001558  0.00265 1.036 3.55e-06 0.01498    
## 34  -0.000212  0.009914  0.03961 1.028 7.91e-04 0.01046    
## 35   0.003498 -0.018573 -0.06203 1.024 1.94e-03 0.01077    
## 36   0.002304 -0.027291 -0.10224 1.011 5.23e-03 0.01056    
## 37  -0.000226 -0.001296 -0.00619 1.031 1.93e-05 0.01025    
## 38  -0.012283  0.001583 -0.04387 1.026 9.70e-04 0.00982    
## 39  -0.000622 -0.006184 -0.02769 1.029 3.87e-04 0.01032    
## 40  -0.066681  0.047837 -0.09565 1.020 4.59e-03 0.01307    
## 41   0.057290 -0.120050 -0.26932 0.921 3.46e-02 0.01223   *
## 42  -0.014722 -0.011925 -0.10788 1.007 5.81e-03 0.00993    
## 43  -0.022408  0.015915 -0.03261 1.032 5.37e-04 0.01287    
## 44  -0.032084  0.016321 -0.06831 1.022 2.35e-03 0.01040    
## 45  -0.026901  0.006919 -0.08267 1.016 3.43e-03 0.00987    
## 46  -0.017888 -0.037964 -0.22633 0.933 2.46e-02 0.01009   *
## 47   0.002490  0.003418  0.02392 1.029 2.89e-04 0.01001    
## 48   0.000322 -0.041939 -0.16981 0.976 1.42e-02 0.01044    
## 49  -0.021146 -0.013460 -0.14015 0.991 9.73e-03 0.00990    
## 50  -0.003213  0.001878 -0.00598 1.031 1.81e-05 0.01088    
## 51  -0.010089 -0.013849 -0.09693 1.012 4.70e-03 0.01001    
## 52  -0.090894  0.046930 -0.19104 0.963 1.78e-02 0.01043    
## 53  -0.191802  0.122576 -0.32191 0.868 4.80e-02 0.01147   *
## 54  -0.114703  0.063968 -0.22425 0.941 2.43e-02 0.01067    
## 55   0.002851 -0.014444 -0.04773 1.027 1.15e-03 0.01079    
## 56  -0.002116 -0.002296 -0.01786 1.030 1.61e-04 0.00997    
## 57   0.004162  0.003177  0.02971 1.029 4.45e-04 0.00992    
## 58   0.026503 -0.017736  0.04191 1.030 8.86e-04 0.01194    
## 59   0.007220  0.002891  0.04099 1.027 8.47e-04 0.00985    
## 60  -0.015447  0.012965 -0.01749 1.043 1.54e-04 0.02176    
## 61  -0.170428  0.134528 -0.21150 0.984 2.20e-02 0.01647    
## 62   0.030055 -0.006096  0.09869 1.010 4.87e-03 0.00984    
## 63  -0.109537  0.073503 -0.17258 0.983 1.47e-02 0.01198    
## 64  -0.126493  0.108599 -0.13885 1.031 9.66e-03 0.02525    
## 65  -0.173952  0.150877 -0.18845 1.023 1.77e-02 0.02731    
## 66  -0.149981  0.127078 -0.16758 1.020 1.40e-02 0.02307    
## 67   0.338779 -0.297116  0.36208 0.967 6.35e-02 0.03001    
## 68  -0.155702  0.121569 -0.19650 0.988 1.90e-02 0.01588    
## 69  -0.015692  0.011819 -0.02093 1.035 2.21e-04 0.01440    
## 70   0.142332 -0.120867  0.15855 1.023 1.26e-02 0.02341    
## 71  -0.075307  0.063587 -0.08455 1.037 3.60e-03 0.02257    
## 72   0.068541 -0.057874  0.07695 1.038 2.98e-03 0.02257    
## 73  -0.092928  0.079420 -0.10262 1.037 5.30e-03 0.02445    
## 74   0.154960 -0.136661  0.16455 1.036 1.36e-02 0.03160    
## 75   0.063563 -0.055956  0.06764 1.050 2.31e-03 0.03106    
## 76  -0.004478  0.002243 -0.00966 1.031 4.71e-05 0.01036    
## 77   0.079278 -0.060303  0.10415 1.021 5.44e-03 0.01475    
## 78   0.019036 -0.015178  0.02326 1.037 2.73e-04 0.01707    
## 79   0.140145 -0.116239  0.16141 1.016 1.30e-02 0.02037    
## 80   0.016950 -0.013475  0.02081 1.037 2.19e-04 0.01689    
## 81   0.076645 -0.058535  0.10007 1.022 5.02e-03 0.01490    
## 82   0.154339 -0.118181  0.20070 0.983 1.98e-02 0.01501    
## 83  -0.032130  0.013088 -0.08038 1.018 3.24e-03 0.01007    
## 84   0.095684 -0.085264  0.10046 1.051 5.08e-03 0.03506    
## 85   0.070968 -0.058086  0.08335 1.033 3.50e-03 0.01906    
## 86   0.035724 -0.017750  0.07759 1.019 3.02e-03 0.01035    
## 87   0.043186 -0.038113  0.04582 1.052 1.06e-03 0.03181    
## 88   0.029005 -0.021268  0.04028 1.032 8.18e-04 0.01359    
## 89   0.045105 -0.025154  0.08818 1.016 3.90e-03 0.01067    
## 90   0.225683 -0.182605  0.26969 0.961 3.53e-02 0.01810    
## 91   0.226986 -0.198448  0.24351 1.010 2.94e-02 0.02919    
## 92   0.224470 -0.198589  0.23752 1.020 2.80e-02 0.03258    
## 93  -0.017647  0.014745 -0.02011 1.042 2.04e-04 0.02121    
## 94   0.109797 -0.088129  0.13282 1.018 8.82e-03 0.01752    
## 95  -0.047652  0.040510 -0.05300 1.042 1.42e-03 0.02358    
## 96  -0.036866  0.068902  0.14021 1.003 9.78e-03 0.01293    
## 97   0.164822 -0.130238  0.20421 0.987 2.05e-02 0.01653    
## 98   0.091397 -0.077440  0.10212 1.035 5.24e-03 0.02307    
## 99  -0.092328  0.076526 -0.10645 1.030 5.69e-03 0.02029    
## 100 -0.096939  0.077518 -0.11793 1.022 6.97e-03 0.01726    
## 101 -0.003765  0.002012 -0.00767 1.031 2.97e-05 0.01053    
## 102  0.001089 -0.000855  0.00136 1.037 9.38e-07 0.01617
# Question 4:
# Calculate muitple linear regression for prestige score ~ education + income + women.
n <- lm(Prestige_Score ~ Education + Income + Percent_of_Women, data = occupation.prestige) ; n
## 
## Call:
## lm(formula = Prestige_Score ~ Education + Income + Percent_of_Women, 
##     data = occupation.prestige)
## 
## Coefficients:
##      (Intercept)         Education            Income  Percent_of_Women  
##        -6.794334          4.186637          0.001314         -0.008905
# Display summary of n.
summary(n)
## 
## Call:
## lm(formula = Prestige_Score ~ Education + Income + Percent_of_Women, 
##     data = occupation.prestige)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -19.8246  -5.3332  -0.1364   5.1587  17.5045 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      -6.7943342  3.2390886  -2.098   0.0385 *  
## Education         4.1866373  0.3887013  10.771  < 2e-16 ***
## Income            0.0013136  0.0002778   4.729 7.58e-06 ***
## Percent_of_Women -0.0089052  0.0304071  -0.293   0.7702    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 7.846 on 98 degrees of freedom
## Multiple R-squared:  0.7982, Adjusted R-squared:  0.792 
## F-statistic: 129.2 on 3 and 98 DF,  p-value: < 2.2e-16
# Display ANOVA table of n.
anova(n)
## Analysis of Variance Table
## 
## Response: Prestige_Score
##                  Df  Sum Sq Mean Sq  F value    Pr(>F)    
## Education         1 21608.4 21608.4 350.9741 < 2.2e-16 ***
## Income            1  2248.1  2248.1  36.5153 2.739e-08 ***
## Percent_of_Women  1     5.3     5.3   0.0858    0.7702    
## Residuals        98  6033.6    61.6                       
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Calculate the F-statistic for df = 3, 98 and alpha = 0.025.
qf(0.975, df1 = 3, df2 = 98)
## [1] 3.252415
# Question 5.
# Calculate t-statistic to test with.
qt(0.0975, df = 98)
## [1] -1.304849
# Determine confidence intervals for regression variables.
confint(n, conf.level = 0.95)
##                          2.5 %       97.5 %
## (Intercept)      -1.322220e+01 -0.366468202
## Education         3.415272e+00  4.958002277
## Income            7.623127e-04  0.001864808
## Percent_of_Women -6.924697e-02  0.051436660
# Generate residual plots with fitted values against residuals.
par(mfrow = c(1,2))
plot(fitted(n), resid(n), axes = TRUE, frame.plot = TRUE,
     col = "steelblue", pch = 20, ann = FALSE)
abline(h = 0, lty = 2)
title(main = "Residuals vs. Fitted (Without Curve)", xlab = "Fitted values", ylab = "Residuals")
mtext("Residuals vs Fitted", side = 3, line = 0.25)
plot(n, which = 1, col = "steelblue", pch = 20, ann = TRUE)
title(main = "Residuals vs. Fitted (With Curve)")

# Determine influential points.
influence.measures(n)
## Influence measures of
##   lm(formula = Prestige_Score ~ Education + Income + Percent_of_Women,      data = occupation.prestige) :
## 
##        dfb.1_  dfb.Edct  dfb.Incm  dfb.P__W     dffit cov.r   cook.d
## 1   -2.31e-02  8.73e-03  4.60e-02 -2.87e-03  9.82e-02 1.056 2.43e-03
## 2   -1.04e-01  6.21e-01 -1.04e+00 -4.87e-01 -1.07e+00 1.448 2.83e-01
## 3   -2.45e-02  3.87e-02 -6.39e-03 -2.65e-02  8.04e-02 1.045 1.63e-03
## 4    6.08e-03  9.59e-03  2.89e-03 -2.65e-02  6.56e-02 1.045 1.08e-03
## 5   -1.25e-01  1.99e-01 -1.22e-01 -1.27e-01  2.35e-01 1.043 1.38e-02
## 6   -8.44e-02  1.09e-01 -4.10e-02 -6.59e-02  1.39e-01 1.086 4.88e-03
## 7   -1.02e-01  1.39e-01 -7.57e-02 -5.66e-02  1.59e-01 1.068 6.39e-03
## 8   -2.64e-02  2.25e-02  9.79e-03 -1.12e-02  4.98e-02 1.094 6.27e-04
## 9   -5.24e-02  6.87e-02 -1.53e-02 -5.20e-02  1.09e-01 1.071 2.97e-03
## 10   2.26e-03 -3.11e-03  9.96e-04  2.34e-03 -4.55e-03 1.087 5.23e-06
## 11  -4.66e-02  1.82e-01 -1.82e-01 -1.95e-01  2.55e-01 1.027 1.62e-02
## 12  -2.44e-02  8.48e-02 -7.24e-02 -8.76e-02  1.29e-01 1.043 4.17e-03
## 13   9.21e-02 -1.49e-01  8.32e-02  9.25e-02 -1.91e-01 1.026 9.07e-03
## 14   2.39e-02 -2.12e-02  3.06e-03 -1.13e-02 -3.73e-02 1.078 3.52e-04
## 15  -1.91e-01  2.16e-01 -9.12e-02  1.60e-02  2.97e-01 0.970 2.17e-02
## 16   8.44e-02 -1.00e-01  5.31e-02 -1.15e-02 -1.37e-01 1.060 4.74e-03
## 17   3.33e-02  2.70e-03 -6.74e-02 -1.79e-02 -9.68e-02 1.152 2.36e-03
## 18   2.72e-02 -2.36e-02  6.58e-03 -2.23e-02 -4.86e-02 1.091 5.95e-04
## 19   2.00e-01 -2.11e-01  5.46e-02  2.20e-02 -2.83e-01 0.997 1.99e-02
## 20  -2.29e-01  5.10e-01 -4.70e-01 -4.04e-01  5.85e-01 1.027 8.39e-02
## 21  -1.67e-01  1.55e-01  5.41e-03 -3.42e-02  2.44e-01 1.043 1.48e-02
## 22  -3.76e-02  2.82e-02 -4.62e-03  4.57e-02  7.92e-02 1.089 1.58e-03
## 23   7.06e-03 -7.75e-03  3.04e-03 -8.56e-05 -1.00e-02 1.086 2.55e-05
## 24   8.99e-02  1.54e-01 -4.41e-01 -2.01e-01 -4.96e-01 1.335 6.17e-02
## 25   2.28e-01 -1.90e-01 -7.51e-02  7.74e-02 -3.96e-01 0.990 3.86e-02
## 26   9.77e-02  2.84e-02 -2.52e-01 -6.58e-02 -3.53e-01 1.058 3.10e-02
## 27  -1.30e-01  4.85e-02  3.35e-02  3.44e-01  4.61e-01 0.958 5.19e-02
## 28  -1.12e-02  1.48e-02 -7.62e-03 -3.30e-02 -4.41e-02 1.078 4.91e-04
## 29  -2.31e-01  2.03e-01 -7.61e-02  2.41e-01  4.77e-01 0.914 5.50e-02
## 30   1.84e-02 -2.00e-02  4.31e-03  5.20e-03 -2.74e-02 1.082 1.89e-04
## 31  -1.43e-01  1.21e-01 -3.76e-02  2.06e-01  3.73e-01 0.931 3.39e-02
## 32   1.15e-02  5.62e-02 -6.36e-02 -6.32e-02  1.43e-01 0.992 5.10e-03
## 33  -7.86e-03  1.92e-02 -1.39e-02 -1.67e-02  2.77e-02 1.067 1.94e-04
## 34   2.26e-03  1.06e-02 -6.18e-03 -1.97e-02  3.67e-02 1.054 3.39e-04
## 35   7.96e-04  1.42e-04 -6.35e-04 -4.14e-03 -5.21e-03 1.105 6.85e-06
## 36   1.31e-02 -4.45e-03  8.85e-04 -6.06e-02 -8.40e-02 1.096 1.78e-03
## 37  -8.83e-03  1.04e-02 -8.27e-03  4.28e-02  7.68e-02 1.059 1.49e-03
## 38  -1.69e-03 -1.72e-03 -2.39e-03  4.45e-02  6.34e-02 1.092 1.01e-03
## 39  -5.45e-03  2.82e-03 -4.35e-04  2.96e-02  4.48e-02 1.073 5.07e-04
## 40  -7.00e-02 -4.47e-04  5.30e-02  6.83e-02 -1.24e-01 1.029 3.82e-03
## 41   1.11e-01 -1.29e-01  1.09e-01 -1.99e-01 -4.03e-01 0.942 3.96e-02
## 42   9.66e-03 -5.64e-04  1.58e-03 -7.95e-02 -1.11e-01 1.086 3.09e-03
## 43  -3.02e-02  3.23e-03  1.61e-02  2.86e-02 -5.10e-02 1.056 6.56e-04
## 44  -1.03e-02 -3.46e-04  1.12e-02 -1.39e-02 -4.40e-02 1.055 4.89e-04
## 45  -4.50e-04  1.19e-02 -1.01e-02 -6.13e-02 -7.61e-02 1.099 1.46e-03
## 46   1.59e-02 -9.09e-02  1.03e-01 -3.82e-02 -2.70e-01 0.877 1.76e-02
## 47  -3.45e-03  1.05e-02 -8.95e-03  4.04e-02  8.78e-02 1.041 1.94e-03
## 48   1.30e-02 -5.19e-02  3.29e-02 -2.33e-02 -1.87e-01 0.936 8.54e-03
## 49   6.78e-03 -2.28e-02  3.03e-02 -7.79e-02 -1.64e-01 1.015 6.70e-03
## 50  -2.12e-02  1.35e-02 -9.01e-03  4.26e-03 -3.76e-02 1.051 3.57e-04
## 51  -3.37e-02 -2.21e-02  2.16e-03  9.75e-02 -1.87e-01 0.974 8.67e-03
## 52  -3.47e-02 -2.09e-03  5.76e-02 -1.19e-01 -2.52e-01 0.976 1.56e-02
## 53  -1.25e-01 -3.40e-01  6.15e-01  5.10e-01 -7.08e-01 0.834 1.18e-01
## 54  -8.00e-02 -2.15e-01  3.71e-01  3.42e-01 -4.51e-01 0.945 4.95e-02
## 55  -4.06e-05 -2.43e-02  1.15e-02  3.40e-02 -7.62e-02 1.039 1.46e-03
## 56  -2.32e-03 -3.93e-03  2.93e-03  4.06e-03 -1.97e-02 1.051 9.76e-05
## 57   3.57e-03 -8.87e-04  3.73e-03 -3.82e-04  1.95e-02 1.051 9.58e-05
## 58  -1.19e-02  8.67e-03 -6.85e-03  5.61e-03 -2.01e-02 1.065 1.02e-04
## 59   3.98e-03  6.56e-04  7.14e-04 -8.35e-03  1.65e-02 1.060 6.86e-05
## 60   7.05e-03 -5.94e-03  1.33e-03  4.63e-03  1.01e-02 1.072 2.59e-05
## 61  -1.79e-01  3.54e-02  1.03e-01  1.14e-01 -2.60e-01 0.937 1.66e-02
## 62   3.71e-02  4.20e-03 -7.91e-03 -5.85e-02  1.14e-01 1.022 3.23e-03
## 63  -2.80e-02  2.65e-02  2.14e-02 -1.49e-01 -2.25e-01 1.071 1.27e-02
## 64  -1.11e-01  1.25e-01 -6.09e-02 -1.29e-01 -1.89e-01 1.064 8.93e-03
## 65  -1.93e-01  1.41e-01 -1.83e-02 -3.47e-02 -2.19e-01 1.002 1.19e-02
## 66  -1.58e-01  9.88e-02  6.93e-03 -6.37e-03 -1.85e-01 1.006 8.50e-03
## 67   3.91e-01 -1.78e-01 -9.20e-02 -1.90e-01  4.53e-01 0.871 4.90e-02
## 68  -1.05e-01 -2.42e-02  1.44e-01  8.31e-02 -2.15e-01 1.006 1.15e-02
## 69  -5.42e-02  2.39e-02  7.25e-04  3.79e-02 -7.87e-02 1.053 1.56e-03
## 70   1.56e-01 -1.21e-01  2.89e-02  3.66e-02  1.82e-01 1.012 8.28e-03
## 71  -1.18e-01  8.07e-02 -1.56e-02  1.16e-02 -1.32e-01 1.036 4.39e-03
## 72   5.42e-02 -3.69e-02  7.12e-03 -5.32e-03  6.05e-02 1.061 9.23e-04
## 73  -4.64e-02  4.63e-02 -1.37e-02 -5.30e-02 -8.33e-02 1.086 1.75e-03
## 74   1.45e-01 -1.28e-01  4.95e-02  4.06e-02  1.62e-01 1.049 6.54e-03
## 75   6.64e-02 -5.65e-02  1.67e-02  2.39e-02  7.68e-02 1.072 1.49e-03
## 76  -2.73e-02  7.17e-03 -1.56e-03  3.02e-02 -6.00e-02 1.052 9.06e-04
## 77   7.16e-02 -3.58e-02  3.15e-03 -3.95e-02  9.88e-02 1.043 2.45e-03
## 78  -1.66e-02  9.17e-03 -1.32e-03  8.13e-03 -2.13e-02 1.066 1.15e-04
## 79   1.24e-01 -8.39e-02  2.64e-02 -3.52e-02  1.47e-01 1.035 5.39e-03
## 80  -1.60e-03  9.31e-04 -1.07e-04  4.38e-04 -1.96e-03 1.062 9.74e-07
## 81   6.89e-02 -3.52e-02  3.29e-03 -3.52e-02  9.36e-02 1.044 2.20e-03
## 82   1.71e-01 -2.36e-01  1.52e-01  3.55e-01  4.53e-01 0.898 4.93e-02
## 83  -3.29e-02 -4.36e-02  7.45e-02  9.68e-02 -1.28e-01 1.041 4.14e-03
## 84   1.21e-01 -1.69e-01  1.10e-01  1.89e-01  2.36e-01 1.131 1.40e-02
## 85   5.59e-02 -2.62e-02 -4.90e-03 -3.23e-02  7.10e-02 1.061 1.27e-03
## 86   3.55e-02 -3.83e-03 -6.12e-03 -4.64e-02  8.13e-02 1.045 1.67e-03
## 87   2.30e-05 -1.33e-05 -5.25e-07 -8.92e-06  2.55e-05 1.084 1.64e-10
## 88  -1.46e-02  1.06e-02 -6.82e-03  5.67e-03 -2.17e-02 1.065 1.18e-04
## 89   4.90e-02 -1.18e-03 -1.89e-02 -6.57e-02  1.07e-01 1.036 2.87e-03
## 90   2.11e-01 -1.97e-01  1.45e-01 -2.05e-02  2.86e-01 0.980 2.02e-02
## 91   2.20e-01 -1.37e-01  1.32e-02 -7.70e-02  2.45e-01 1.011 1.49e-02
## 92   1.86e-01 -1.38e-01  4.43e-02 -3.91e-02  2.03e-01 1.044 1.03e-02
## 93  -3.38e-02  1.20e-02  1.07e-02  2.22e-02 -4.34e-02 1.070 4.75e-04
## 94   9.20e-02 -5.59e-02  1.55e-02 -4.16e-02  1.18e-01 1.045 3.49e-03
## 95  -6.12e-02  1.90e-02  2.52e-02  4.31e-02 -7.95e-02 1.070 1.59e-03
## 96   1.53e-03 -2.03e-02  5.76e-02  2.88e-03  8.33e-02 1.080 1.75e-03
## 97   1.41e-01 -1.26e-01  9.20e-02 -2.44e-02  1.97e-01 1.022 9.64e-03
## 98   7.02e-02 -4.59e-02  8.60e-03 -1.64e-02  7.91e-02 1.060 1.57e-03
## 99  -1.12e-01  3.13e-02  4.90e-02  8.19e-02 -1.51e-01 1.039 5.75e-03
## 100 -1.23e-01  2.19e-02  6.49e-02  1.15e-01 -1.85e-01 1.021 8.50e-03
## 101 -8.60e-03  4.37e-04  3.52e-03  8.10e-03 -1.83e-02 1.055 8.46e-05
## 102  2.31e-02 -2.81e-02  1.52e-02  3.93e-02  5.34e-02 1.080 7.19e-04
##        hat inf
## 1   0.0270    
## 2   0.3422   *
## 3   0.0176    
## 4   0.0145    
## 5   0.0464    
## 6   0.0538    
## 7   0.0457    
## 8   0.0498    
## 9   0.0389    
## 10  0.0414    
## 11  0.0433    
## 12  0.0262    
## 13  0.0309    
## 14  0.0348    
## 15  0.0331    
## 16  0.0367    
## 17  0.0988   *
## 18  0.0467    
## 19  0.0380    
## 20  0.1021    
## 21  0.0479    
## 22  0.0485    
## 23  0.0407    
## 24  0.2435   *
## 25  0.0558    
## 26  0.0753    
## 27  0.0569    
## 28  0.0359    
## 29  0.0484    
## 30  0.0379    
## 31  0.0363    
## 32  0.0132    
## 33  0.0249    
## 34  0.0154    
## 35  0.0572    
## 36  0.0542    
## 37  0.0254    
## 38  0.0489    
## 39  0.0317    
## 40  0.0193    
## 41  0.0434    
## 42  0.0500    
## 43  0.0188    
## 44  0.0171    
## 45  0.0556    
## 46  0.0154   *
## 47  0.0173    
## 48  0.0117    
## 49  0.0221    
## 50  0.0129    
## 51  0.0167    
## 52  0.0271    
## 53  0.0646   *
## 54  0.0517    
## 55  0.0138    
## 56  0.0104    
## 57  0.0106    
## 58  0.0223    
## 59  0.0173    
## 60  0.0285    
## 61  0.0211    
## 62  0.0151    
## 63  0.0588    
## 64  0.0484    
## 65  0.0280    
## 66  0.0232    
## 67  0.0366   *
## 68  0.0287    
## 69  0.0219    
## 70  0.0245    
## 71  0.0238    
## 72  0.0238    
## 73  0.0463    
## 74  0.0352    
## 75  0.0345    
## 76  0.0176    
## 77  0.0201    
## 78  0.0233    
## 79  0.0259    
## 80  0.0188    
## 81  0.0196    
## 82  0.0415    
## 83  0.0251    
## 84  0.0978   *
## 85  0.0260    
## 86  0.0180    
## 87  0.0385    
## 88  0.0227    
## 89  0.0187    
## 90  0.0337    
## 91  0.0358    
## 92  0.0409    
## 93  0.0288    
## 94  0.0250    
## 95  0.0334    
## 96  0.0412    
## 97  0.0306    
## 98  0.0263    
## 99  0.0287    
## 100 0.0281    
## 101 0.0132    
## 102 0.0379