Homework II
Refer to Real estate data set. Residential sales that occurred during the year 2002 were available from a city in the midwest of the US. Data on 522 arms-length transactions include sales price, style, finished square feet, number of bedrooms, pool, lot size, year built, air conditioning, and whether or not the lot in adjacent to a highway.
The city tax assessor was interested in predicting sales price based on the demographic variable information give above. Select a random sample of 320 observations to use in the model-building data set.
Read in the data and name the variables
real.est = read.table("E:/A 学习/1课程与书目/2014春/杨柳-高级应用数理统计/HW2 linear regression/RealEstate.txt",
header = F)
colnames(real.est) = c("id", "price", "area", "bedrooms", "bathrooms",
"aircon", "garage", "pool", "year", "quality", "style", "lot",
"highway")
o = ordered(real.est$quality)
real.est$quality = factor(o, levels = rev(levels(o)))
real.est$style = factor(real.est$style)
nrow(real.est)
## [1] 522
head(real.est, 10)
## id price area bedrooms bathrooms aircon garage pool year quality style lot highway
## 1 1 360000 3032 4 4 1 2 0 1972 2 1 22221 0
## 2 2 340000 2058 4 2 1 2 0 1976 2 1 22912 0
## 3 3 250000 1780 4 3 1 2 0 1980 2 1 21345 0
## 4 4 205500 1638 4 2 1 2 0 1963 2 1 17342 0
## 5 5 275500 2196 4 3 1 2 0 1968 2 7 21786 0
## 6 6 248000 1966 4 3 1 5 1 1972 2 1 18902 0
## 7 7 229900 2216 3 2 1 2 0 1972 2 7 18639 0
## 8 8 150000 1597 2 1 1 1 0 1955 2 1 22112 0
## 9 9 195000 1622 3 2 1 2 0 1975 3 1 14321 0
## 10 10 160000 1976 3 3 0 1 0 1918 3 1 32358 0
Randomly select a training sample of 320 out of 522 in total
The rest of the data is therefore split into the validation dataset
set.seed(1024)
indices = sample(nrow(real.est), 320)
training = real.est[indices, ]
nrow(training)
## [1] 320
validation = real.est[-indices, ]
nrow(validation)
## [1] 202
(i) Do any series multicollinearity problem exist here? Develop a best subset model for predicting sales price. Justify your choice of model.
# Compute the correlation matrix (excluding the factor variables 'id', 'quality' and 'style') and kappa to test multicollinearity
corr = cor(training[c(2:9, 12, 13)])
corr
## price area bedrooms bathrooms aircon garage pool year lot highway
## price 1.00000 0.80174 0.38361 0.64695 0.30115 0.55668 0.09867 0.579701 0.20131 -0.060195
## area 0.80174 1.00000 0.57249 0.73351 0.25785 0.49875 0.12362 0.434262 0.14992 -0.064564
## bedrooms 0.38361 0.57249 1.00000 0.58541 0.18748 0.31860 0.06488 0.262644 0.12980 -0.061422
## bathrooms 0.64695 0.73351 0.58541 1.00000 0.30711 0.43929 0.13995 0.490326 0.13419 -0.065155
## aircon 0.30115 0.25785 0.18748 0.30711 1.00000 0.28843 0.08978 0.417222 -0.11675 -0.126464
## garage 0.55668 0.49875 0.31860 0.43929 0.28843 1.00000 0.04161 0.465156 0.14422 -0.019479
## pool 0.09867 0.12362 0.06488 0.13995 0.08978 0.04161 1.00000 0.056705 -0.05613 -0.038468
## year 0.57970 0.43426 0.26264 0.49033 0.41722 0.46516 0.05670 1.000000 -0.08288 0.004762
## lot 0.20131 0.14992 0.12980 0.13419 -0.11675 0.14422 -0.05613 -0.082880 1.00000 0.191096
## highway -0.06020 -0.06456 -0.06142 -0.06515 -0.12646 -0.01948 -0.03847 0.004762 0.19110 1.000000
kappa(corr, exact = T)
## [1] 28.18
# Because kappa << 1000, no multicollinearity detected
# Do OLS regression
lm.sol = lm(price ~ . - id, data = training)
summary(lm.sol)
##
## Call:
## lm(formula = price ~ . - id, data = training)
##
## Residuals:
## Min 1Q Median 3Q Max
## -194940 -28433 -1988 27309 262060
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.05e+06 5.08e+05 -6.00 5.6e-09 ***
## area 9.90e+01 1.04e+01 9.49 < 2e-16 ***
## bedrooms -8.05e+03 4.42e+03 -1.82 0.06953 .
## bathrooms 9.62e+03 5.41e+03 1.78 0.07603 .
## aircon 5.39e+03 1.11e+04 0.49 0.62638
## garage 1.04e+04 7.13e+03 1.46 0.14640
## pool 9.82e+03 1.35e+04 0.73 0.46658
## year 1.57e+03 2.61e+02 6.02 4.9e-09 ***
## quality.L 1.06e+05 1.36e+04 7.79 1.1e-13 ***
## quality.Q 4.78e+04 6.80e+03 7.03 1.4e-11 ***
## style2 -2.76e+04 1.26e+04 -2.18 0.02983 *
## style3 -1.38e+04 1.23e+04 -1.12 0.26360
## style4 -5.35e+03 3.11e+04 -0.17 0.86341
## style5 -3.31e+04 2.04e+04 -1.62 0.10531
## style6 -7.99e+03 1.95e+04 -0.41 0.68275
## style7 -3.09e+04 1.15e+04 -2.69 0.00757 **
## style9 -9.33e+04 6.17e+04 -1.51 0.13130
## lot 1.24e+00 3.24e-01 3.84 0.00015 ***
## highway -3.78e+04 2.59e+04 -1.46 0.14574
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 60400 on 301 degrees of freedom
## Multiple R-squared: 0.82, Adjusted R-squared: 0.809
## F-statistic: 76.3 on 18 and 301 DF, p-value: <2e-16
# The model in whole is very significant, whereas some of the variables are not
# Do stepwise regression to testify the selection
# We can see the variables 'aircon', 'pool', 'style' and 'highway' already dropped
lm.step = step(lm.sol)
## Start: AIC=7064
## price ~ (id + area + bedrooms + bathrooms + aircon + garage +
## pool + year + quality + style + lot + highway) - id
##
## Df Sum of Sq RSS AIC
## - aircon 1 8.65e+08 1.10e+12 7062
## - pool 1 1.94e+09 1.10e+12 7062
## - style 7 4.52e+10 1.14e+12 7063
## <none> 1.10e+12 7064
## - garage 1 7.73e+09 1.10e+12 7064
## - highway 1 7.75e+09 1.10e+12 7064
## - bathrooms 1 1.15e+10 1.11e+12 7065
## - bedrooms 1 1.21e+10 1.11e+12 7065
## - lot 1 5.36e+10 1.15e+12 7077
## - year 1 1.32e+11 1.23e+12 7098
## - area 1 3.28e+11 1.42e+12 7145
## - quality 2 3.39e+11 1.44e+12 7146
##
## Step: AIC=7062
## price ~ area + bedrooms + bathrooms + garage + pool + year +
## quality + style + lot + highway
##
## Df Sum of Sq RSS AIC
## - pool 1 2.05e+09 1.10e+12 7060
## - style 7 4.54e+10 1.14e+12 7061
## <none> 1.10e+12 7062
## - garage 1 8.20e+09 1.11e+12 7062
## - highway 1 8.52e+09 1.11e+12 7062
## - bathrooms 1 1.15e+10 1.11e+12 7063
## - bedrooms 1 1.20e+10 1.11e+12 7063
## - lot 1 5.28e+10 1.15e+12 7075
## - year 1 1.40e+11 1.24e+12 7098
## - area 1 3.28e+11 1.43e+12 7143
## - quality 2 3.39e+11 1.44e+12 7144
##
## Step: AIC=7060
## price ~ area + bedrooms + bathrooms + garage + year + quality +
## style + lot + highway
##
## Df Sum of Sq RSS AIC
## - style 7 4.44e+10 1.14e+12 7059
## <none> 1.10e+12 7060
## - garage 1 8.10e+09 1.11e+12 7061
## - highway 1 8.66e+09 1.11e+12 7061
## - bathrooms 1 1.22e+10 1.11e+12 7062
## - bedrooms 1 1.25e+10 1.11e+12 7062
## - lot 1 5.17e+10 1.15e+12 7073
## - year 1 1.39e+11 1.24e+12 7097
## - quality 2 3.37e+11 1.44e+12 7142
## - area 1 3.36e+11 1.44e+12 7144
##
## Step: AIC=7059
## price ~ area + bedrooms + bathrooms + garage + year + quality +
## lot + highway
##
## Df Sum of Sq RSS AIC
## - highway 1 6.63e+09 1.15e+12 7059
## <none> 1.14e+12 7059
## - bathrooms 1 8.13e+09 1.15e+12 7059
## - garage 1 1.19e+10 1.16e+12 7060
## - bedrooms 1 2.07e+10 1.16e+12 7063
## - lot 1 6.57e+10 1.21e+12 7075
## - year 1 1.42e+11 1.29e+12 7094
## - quality 2 3.96e+11 1.54e+12 7150
## - area 1 4.15e+11 1.56e+12 7156
##
## Step: AIC=7059
## price ~ area + bedrooms + bathrooms + garage + year + quality +
## lot
##
## Df Sum of Sq RSS AIC
## <none> 1.15e+12 7059
## - bathrooms 1 8.90e+09 1.16e+12 7059
## - garage 1 1.22e+10 1.16e+12 7060
## - bedrooms 1 1.99e+10 1.17e+12 7062
## - lot 1 5.97e+10 1.21e+12 7073
## - year 1 1.37e+11 1.29e+12 7093
## - quality 2 4.03e+11 1.55e+12 7151
## - area 1 4.16e+11 1.57e+12 7156
summary(lm.step)
##
## Call:
## lm(formula = price ~ area + bedrooms + bathrooms + garage + year +
## quality + lot, data = training)
##
## Residuals:
## Min 1Q Median 3Q Max
## -191517 -32782 -1566 24054 287033
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.94e+06 4.90e+05 -6.01 5.1e-09 ***
## area 9.35e+01 8.82e+00 10.61 < 2e-16 ***
## bedrooms -1.01e+04 4.34e+03 -2.32 0.021 *
## bathrooms 8.29e+03 5.35e+03 1.55 0.122
## garage 1.29e+04 7.11e+03 1.82 0.070 .
## year 1.52e+03 2.50e+02 6.09 3.3e-09 ***
## quality.L 1.06e+05 1.28e+04 8.25 4.5e-15 ***
## quality.Q 5.15e+04 6.28e+03 8.21 5.9e-15 ***
## lot 1.24e+00 3.10e-01 4.02 7.4e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 60800 on 311 degrees of freedom
## Multiple R-squared: 0.811, Adjusted R-squared: 0.806
## F-statistic: 167 on 8 and 311 DF, p-value: <2e-16
# The variables 'bathrooms' and 'garage' still failed to pass the significance test, therefore we try to drop one more variable and evaluate which is the best to be dropped.
drop1(lm.step)
## Single term deletions
##
## Model:
## price ~ area + bedrooms + bathrooms + garage + year + quality +
## lot
## Df Sum of Sq RSS AIC
## <none> 1.15e+12 7059
## area 1 4.16e+11 1.57e+12 7156
## bedrooms 1 1.99e+10 1.17e+12 7062
## bathrooms 1 8.90e+09 1.16e+12 7059
## garage 1 1.22e+10 1.16e+12 7060
## year 1 1.37e+11 1.29e+12 7093
## quality 2 4.03e+11 1.55e+12 7151
## lot 1 5.97e+10 1.21e+12 7073
# We can see that the increase of AIC will be the least when dropping the variable 'bathrooms', so drop it and do regression again.
lm.drop = update(lm.step, . ~ . - bathrooms)
summary(lm.drop)
##
## Call:
## lm(formula = price ~ area + bedrooms + garage + year + quality +
## lot, data = training)
##
## Residuals:
## Min 1Q Median 3Q Max
## -213218 -31040 -2906 25497 293179
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.07e+06 4.84e+05 -6.34 7.9e-10 ***
## area 9.90e+01 8.10e+00 12.22 < 2e-16 ***
## bedrooms -8.16e+03 4.18e+03 -1.96 0.051 .
## garage 1.29e+04 7.13e+03 1.81 0.072 .
## year 1.59e+03 2.47e+02 6.43 4.9e-10 ***
## quality.L 1.09e+05 1.26e+04 8.64 3.1e-16 ***
## quality.Q 4.96e+04 6.16e+03 8.05 1.8e-14 ***
## lot 1.27e+00 3.10e-01 4.11 5.1e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 61000 on 312 degrees of freedom
## Multiple R-squared: 0.81, Adjusted R-squared: 0.806
## F-statistic: 190 on 7 and 312 DF, p-value: <2e-16
#'bedrooms' and 'garage' are no longer significant at 5% level. So make an attempt to drop them too.
drop1(lm.drop)
## Single term deletions
##
## Model:
## price ~ area + bedrooms + garage + year + quality + lot
## Df Sum of Sq RSS AIC
## <none> 1.16e+12 7059
## area 1 5.55e+11 1.71e+12 7183
## bedrooms 1 1.42e+10 1.17e+12 7061
## garage 1 1.21e+10 1.17e+12 7061
## year 1 1.54e+11 1.31e+12 7097
## quality 2 4.02e+11 1.56e+12 7151
## lot 1 6.27e+10 1.22e+12 7074
lm.drop2 = update(lm.drop, . ~ . - garage)
summary(lm.drop2)
##
## Call:
## lm(formula = price ~ area + bedrooms + year + quality + lot,
## data = training)
##
## Residuals:
## Min 1Q Median 3Q Max
## -215137 -30645 -1748 24441 293519
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.27e+06 4.73e+05 -6.90 2.9e-11 ***
## area 1.01e+02 8.05e+00 12.55 < 2e-16 ***
## bedrooms -7.67e+03 4.18e+03 -1.83 0.068 .
## year 1.70e+03 2.40e+02 7.06 1.1e-11 ***
## quality.L 1.13e+05 1.25e+04 9.09 < 2e-16 ***
## quality.Q 5.06e+04 6.16e+03 8.22 5.5e-15 ***
## lot 1.34e+00 3.09e-01 4.35 1.8e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 61200 on 313 degrees of freedom
## Multiple R-squared: 0.808, Adjusted R-squared: 0.804
## F-statistic: 219 on 6 and 313 DF, p-value: <2e-16
#'bedrooms' is still insignificant, so drop it anyway.
lm.opt = update(lm.drop2, . ~ . - bedrooms)
summary(lm.opt)
##
## Call:
## lm(formula = price ~ area + year + quality + lot, data = training)
##
## Residuals:
## Min 1Q Median 3Q Max
## -226104 -30361 -2895 21603 297610
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.23e+06 4.75e+05 -6.81 5.1e-11 ***
## area 9.38e+01 7.03e+00 13.34 < 2e-16 ***
## year 1.68e+03 2.41e+02 6.96 2.0e-11 ***
## quality.L 1.16e+05 1.24e+04 9.35 < 2e-16 ***
## quality.Q 5.31e+04 6.03e+03 8.80 < 2e-16 ***
## lot 1.30e+00 3.09e-01 4.22 3.3e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 61400 on 314 degrees of freedom
## Multiple R-squared: 0.806, Adjusted R-squared: 0.803
## F-statistic: 261 on 5 and 314 DF, p-value: <2e-16
All of the remaining variable are significant, so the best model should be:
price = -3232464.60 + 93.78 area + 1675.84 year + 116112.31 quality.L + 53066.14 quality.Q + 1.30 lot
(474861.58) (7.03) (240.89) (12416.79) (6028.18) (0.31)
(ii) Assess your model's ability to predict based on the validation data set.
# Compute the predictions of validation dataset using lm.opt model, and validate the accuracy with the actual price data.
pred = predict(lm.opt, validation[, c(3, 9, 10, 12)])
# Average absolute prediction error
mean(abs(pred - validation[, 2])/validation[, 2])
## [1] 0.147
# Test whether there is a significant difference between the true and the predicted.
t.test(pred, validation[, 2])
##
## Welch Two Sample t-test
##
## data: pred and validation[, 2]
## t = 0.0139, df = 401.7, p-value = 0.9889
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -26340 26715
## sample estimates:
## mean of x mean of y
## 271942 271754
chisq.test(pred, validation[, 2], correct = T)
## Warning: Chi-squared approximation may be incorrect
##
## Pearson's Chi-squared test
##
## data: pred and validation[, 2]
## X-squared = 31714, df = 31557, p-value = 0.2655
No significant difference between them. So the model's predictive validity could be accepted.
(iii) Perform appropriate diagnostic checks to evaluate outliers and assess their influence.
In order to perform a diagnosis of outliers against the linear regression model lm.opt, we refer a comprehensive diagnostic function given in the Xue Yi's book (p. 354)
Remember, it is according to the model lm.opt based on the training data, rather than on the whole data or validation data, that we consider the influence of the outliers. therefore, the outliers we find below are only from training data.
source("D:\\Rworkspace\\Reg_Diag.R")
diag = Reg_Diag(lm.opt)
diag
## residual s1 standard s2 student s3 hat_matrix s4 DFFITS s5 cooks_distance s6 COVRATIO s7
## 114 -70451.6 -1.1673594 -1.1680364 0.034279 -2.201e-01 8.062e-03 1.0283
## 515 -2973.5 -0.0487508 -0.0486733 0.013618 -5.719e-03 5.469e-06 1.0333
## 182 -22638.2 -0.3703335 -0.3698241 0.009219 -3.567e-02 2.127e-04 1.0261
## 198 -1798.6 -0.0294050 -0.0293582 0.008047 -2.644e-03 1.169e-06 1.0276
## 11 -149312.2 -2.5016735 * -2.5229559 * 0.055484 * -6.115e-01 * 6.127e-02 0.9564
## 388 14434.7 0.2365357 0.2361797 0.012574 2.665e-02 1.187e-04 1.0312
## 85 -5864.0 -0.0968391 -0.0966863 0.027771 -1.634e-02 4.465e-05 1.0483
## 165 -78070.7 -1.2837426 -1.2850735 0.019382 -1.807e-01 5.429e-03 1.0072
## 308 -13791.1 -0.2258654 -0.2255238 0.011502 -2.433e-02 9.893e-05 1.0302
## 302 -28953.9 -0.4741392 -0.4735532 0.011258 -5.053e-02 4.266e-04 1.0265
## 10 11167.2 0.1846746 0.1843903 0.030491 3.270e-02 1.788e-04 1.0507
## 490 -9971.7 -0.1634503 -0.1631968 0.013170 -1.885e-02 5.942e-05 1.0324
## 180 -13729.6 -0.2245631 -0.2242232 0.008899 -2.125e-02 7.547e-05 1.0275
## 232 -71945.2 -1.1782372 -1.1789686 0.011408 -1.266e-01 2.670e-03 1.0040
## 499 -13331.9 -0.2183272 -0.2179958 0.011335 -2.334e-02 9.108e-05 1.0301
## 92 -131230.3 -2.1695238 * -2.1824858 * 0.029895 -3.831e-01 * 2.417e-02 0.9597
## 429 18847.0 0.3090166 0.3085711 0.013719 3.639e-02 2.214e-04 1.0316
## 432 -1095.3 -0.0179507 -0.0179221 0.012904 -2.049e-03 7.020e-07 1.0326
## 344 11867.8 0.1956086 0.1953088 0.024006 3.063e-02 1.569e-04 1.0436
## 248 -14260.0 -0.2331806 -0.2328292 0.008412 -2.144e-02 7.688e-05 1.0269
## 16 146890.3 2.4133749 * 2.4321918 * 0.017763 3.271e-01 * 1.755e-02 0.9275
## 194 40790.2 0.6690842 0.6684947 0.014560 8.126e-02 1.102e-03 1.0256
## 54 -7615.1 -0.1251510 -0.1249547 0.018328 -1.707e-02 4.874e-05 1.0380
## 243 -31119.3 -0.5088549 -0.5082536 0.008368 -4.669e-02 3.642e-04 1.0228
## 437 15703.2 0.2573522 0.2569692 0.012808 2.927e-02 1.432e-04 1.0312
## 382 -36389.3 -0.5948100 -0.5941970 0.007637 -5.213e-02 4.538e-04 1.0202
## 249 -22675.3 -0.3712811 -0.3707708 0.011038 -3.917e-02 2.564e-04 1.0280
## 147 100843.7 1.6552381 1.6598577 0.015861 2.107e-01 7.360e-03 0.9827
## 229 25251.8 0.4137329 0.4131862 0.012300 4.611e-02 3.553e-04 1.0286
## 286 36694.9 0.6116363 0.6110256 0.045659 * 1.337e-01 2.983e-03 1.0605
## 475 -2503.6 -0.0410231 -0.0409579 0.012437 -4.596e-03 3.532e-06 1.0321
## 319 13055.4 0.2139868 0.2136614 0.013068 2.459e-02 1.011e-04 1.0319
## 46 -13726.4 -0.2248707 -0.2245305 0.012070 -2.482e-02 1.030e-04 1.0308
## 392 31085.3 0.5108376 0.5102356 0.018198 6.947e-02 8.061e-04 1.0331
## 191 15408.3 0.2542799 0.2539008 0.026431 4.183e-02 2.926e-04 1.0457
## 351 -38227.4 -0.6245224 -0.6239148 0.006582 -5.078e-02 4.307e-04 1.0185
## 112 -47684.4 -0.7869944 -0.7865163 0.026607 -1.300e-01 2.822e-03 1.0349
## 450 21572.3 0.3535606 0.3530675 0.012938 4.042e-02 2.731e-04 1.0302
## 69 116885.6 1.9336224 1.9421383 0.031146 3.482e-01 * 2.003e-02 0.9791
## 189 27584.7 0.4526240 0.4520502 0.015213 5.619e-02 5.275e-04 1.0310
## 176 7028.2 0.1151586 0.1149775 0.012412 1.289e-02 2.778e-05 1.0319
## 419 61717.5 1.0110094 1.0110452 0.011933 1.111e-01 2.057e-03 1.0116
## 380 -43502.1 -0.7116219 -0.7110614 0.009164 -6.838e-02 7.806e-04 1.0188
## 435 -14491.4 -0.2373306 -0.2369736 0.011460 -2.552e-02 1.088e-04 1.0300
## 453 -16571.3 -0.2722881 -0.2718862 0.017947 -3.676e-02 2.258e-04 1.0365
## 38 -61181.8 -1.0098951 -1.0099272 0.026868 -1.678e-01 4.693e-03 1.0272
## 441 852.1 0.0139678 0.0139455 0.013305 1.619e-03 4.385e-07 1.0331
## 83 -2317.5 -0.0383026 -0.0382416 0.029376 -6.653e-03 7.400e-06 1.0501
## 213 30446.1 0.4978545 0.4972574 0.008396 4.576e-02 3.498e-04 1.0231
## 468 -5464.2 -0.0901905 -0.0900479 0.026765 -1.493e-02 3.728e-05 1.0472
## 140 92679.0 1.5218728 1.5250825 0.016702 1.988e-01 6.557e-03 0.9916
## 328 -8206.3 -0.1341317 -0.1339218 0.007538 -1.167e-02 2.277e-05 1.0267
## 307 -26211.6 -0.4300714 -0.4295125 0.015113 -5.321e-02 4.730e-04 1.0313
## 482 -30124.4 -0.4937255 -0.4931301 0.012933 -5.645e-02 5.323e-04 1.0279
## 480 -5023.6 -0.0823438 -0.0822134 0.013142 -9.488e-03 1.505e-05 1.0328
## 177 -89933.1 -1.5284518 -1.5317246 0.082059 * -4.580e-01 * 3.481e-02 1.0618
## 96 170808.2 3.0258173 * 3.0660254 * 0.155087 * 1.314e+00 * 2.801e-01 * 1.0102
## 374 -34799.5 -0.5699256 -0.5693119 0.011469 -6.132e-02 6.281e-04 1.0248
## 357 -5298.8 -0.0867153 -0.0865782 0.010001 -8.702e-03 1.266e-05 1.0295
## 45 -15021.1 -0.2466445 -0.2462753 0.016581 -3.198e-02 1.710e-04 1.0353
## 117 4493.7 0.0741132 0.0739957 0.025236 1.191e-02 2.370e-05 1.0456
## 154 3869.8 0.0636607 0.0635597 0.020243 9.136e-03 1.396e-05 1.0403
## 295 -4192.9 -0.0685333 -0.0684246 0.007557 -5.971e-03 5.961e-06 1.0270
## 26 -48552.4 -0.7945501 -0.7940826 0.009949 -7.960e-02 1.057e-03 1.0172
## 387 31788.7 0.5210895 0.5204842 0.013268 6.035e-02 6.085e-04 1.0277
## 311 19039.0 0.3113352 0.3108871 0.008454 2.871e-02 1.377e-04 1.0261
## 186 81458.1 1.3309357 1.3325788 0.006806 1.103e-01 2.023e-03 0.9921
## 274 -13814.4 -0.2261596 -0.2258176 0.010730 -2.352e-02 9.246e-05 1.0294
## 415 31554.1 0.5169474 0.5163433 0.012131 5.722e-02 5.469e-04 1.0266
## 193 -28400.2 -0.4684417 -0.4678587 0.025434 -7.558e-02 9.545e-04 1.0415
## 179 51110.2 0.8408282 0.8404349 0.020331 1.211e-01 2.445e-03 1.0265
## 211 81112.6 1.3581479 1.3599840 0.054282 * 3.258e-01 * 1.765e-02 1.0404
## 381 -42092.4 -0.6905964 -0.6900201 0.014995 -8.514e-02 1.210e-03 1.0254
## 135 186273.2 3.0492204 * 3.0904601 * 0.010530 3.188e-01 * 1.649e-02 0.8602
## 175 74398.9 1.2168882 1.2178240 0.008915 1.155e-01 2.220e-03 0.9997
## 517 -8934.5 -0.1464422 -0.1462138 0.013072 -1.683e-02 4.734e-05 1.0324
## 120 -140254.2 -2.3146912 * -2.3309748 * 0.026525 -3.848e-01 * 2.433e-02 0.9444
## 354 -83639.5 -1.3662394 -1.3681347 0.006315 -1.091e-01 1.977e-03 0.9898
## 138 145787.9 2.3864815 * 2.4045852 * 0.010522 2.480e-01 1.009e-02 0.9230
## 50 39811.4 0.6595346 0.6589402 0.033907 1.234e-01 2.544e-03 1.0464
## 195 -5360.0 -0.0875690 -0.0874305 0.006632 -7.144e-03 8.532e-06 1.0260
## 76 -84382.3 -1.3951965 -1.3973110 0.030134 -2.463e-01 1.008e-02 1.0125
## 489 6781.6 0.1111776 0.1110026 0.013458 1.296e-02 2.810e-05 1.0330
## 133 100159.5 1.6404284 1.6448777 0.011562 1.779e-01 5.246e-03 0.9793
## 111 -44027.9 -0.7259866 -0.7254387 0.024834 -1.158e-01 2.237e-03 1.0348
## 149 57738.7 0.9470552 0.9468993 0.014485 1.148e-01 2.197e-03 1.0167
## 443 -14770.9 -0.2421221 -0.2417588 0.013212 -2.797e-02 1.308e-04 1.0318
## 340 -19492.9 -0.3204277 -0.3199694 0.018765 -4.425e-02 3.273e-04 1.0368
## 368 -28372.4 -0.4644266 -0.4638458 0.010449 -4.767e-02 3.796e-04 1.0258
## 301 -10801.2 -0.1772285 -0.1769549 0.015184 -2.197e-02 8.072e-05 1.0344
## 315 -29350.6 -0.4803594 -0.4797702 0.010124 -4.852e-02 3.933e-04 1.0252
## 153 2184.8 0.0360480 0.0359907 0.026034 5.884e-03 5.789e-06 1.0465
## 123 -38543.9 -0.6518239 -0.6512258 0.072890 * -1.826e-01 5.567e-03 1.0906
## 72 220417.8 3.6431630 * 3.7167581 * 0.029455 6.475e-01 * 6.713e-02 0.8105
## 80 165038.4 2.7356931 * 2.7644770 * 0.035027 5.267e-01 * 4.528e-02 0.9140
## 227 -71917.5 -1.1789939 -1.1797291 0.013434 -1.377e-01 3.155e-03 1.0061
## 3 12885.5 0.2110092 0.2106879 0.011259 2.248e-02 8.450e-05 1.0301
## 105 -103447.0 -1.7160777 -1.7214344 0.036523 -3.352e-01 * 1.861e-02 0.9998
## 470 9742.9 0.1595657 0.1593179 0.011493 1.718e-02 4.934e-05 1.0307
## 156 37150.2 0.6101399 0.6095290 0.017026 8.022e-02 1.075e-03 1.0296
## 109 -127949.6 -2.1139270 * -2.1257386 * 0.028646 -3.650e-01 * 2.196e-02 0.9629
## 246 74615.6 1.2192267 1.2201754 0.006954 1.021e-01 1.735e-03 0.9976
## 107 -37902.4 -0.6260616 -0.6254544 0.028193 -1.065e-01 1.895e-03 1.0411
## 438 11491.3 0.1882318 0.1879424 0.011835 2.057e-02 7.073e-05 1.0308
## 389 15525.7 0.2544079 0.2540287 0.012538 2.862e-02 1.370e-04 1.0310
## 183 51970.6 0.8520499 0.8516772 0.013575 9.991e-02 1.665e-03 1.0191
## 296 6966.6 0.1138276 0.1136486 0.006820 9.417e-03 1.483e-05 1.0261
## 342 -2461.8 -0.0403339 -0.0402697 0.012225 -4.480e-03 3.356e-06 1.0319
## 142 60020.6 0.9897429 0.9897107 0.024930 1.583e-01 4.174e-03 1.0260
## 141 55503.0 0.9114462 0.9111998 0.016782 1.190e-01 2.363e-03 1.0204
## 414 5121.3 0.0840404 0.0839074 0.015408 1.050e-02 1.842e-05 1.0351
## 90 17558.7 0.2897738 0.2893507 0.026475 4.772e-02 3.806e-04 1.0454
## 161 104619.8 1.7178335 1.7232124 0.016566 2.237e-01 8.285e-03 0.9794
## 240 -82826.1 -1.3572789 -1.3591086 0.012640 -1.538e-01 3.931e-03 0.9966
## 170 21694.2 0.3562068 0.3557110 0.016523 4.611e-02 3.553e-04 1.0339
## 322 10230.5 0.1677311 0.1674713 0.013616 1.968e-02 6.473e-05 1.0328
## 55 -4483.8 -0.0766227 -0.0765013 0.092064 * -2.436e-02 9.922e-05 1.1226
## 201 -47249.6 -0.7828589 -0.7823752 0.034152 -1.471e-01 3.612e-03 1.0431
## 157 79330.0 1.2975946 1.2990142 0.008993 1.237e-01 2.547e-03 0.9959
## 108 70760.6 1.1803047 1.1810466 0.047041 * 2.624e-01 1.146e-02 1.0415
## 181 19698.9 0.3218236 0.3213638 0.006589 2.617e-02 1.145e-04 1.0241
## 272 -52708.1 -0.8623377 -0.8619848 0.009442 -8.416e-02 1.181e-03 1.0145
## 137 24851.5 0.4074538 0.4069120 0.013660 4.789e-02 3.832e-04 1.0302
## 395 -11070.9 -0.1843561 -0.1840723 0.043848 * -3.942e-02 2.598e-04 1.0654
## 25 34588.3 0.5871035 0.5864899 0.079743 * 1.726e-01 4.978e-03 1.1004
## 290 5539.7 0.0909607 0.0908170 0.016566 1.179e-02 2.323e-05 1.0363
## 474 20388.9 0.3341532 0.3336800 0.012864 3.809e-02 2.425e-04 1.0304
## 200 -2833.8 -0.0463498 -0.0462761 0.008884 -4.381e-03 3.209e-06 1.0284
## 332 -50230.8 -0.8203521 -0.8199239 0.005926 -6.331e-02 6.687e-04 1.0123
## 456 40446.8 0.6629786 0.6623858 0.013155 7.648e-02 9.765e-04 1.0243
## 110 -29734.8 -0.4904974 -0.4899035 0.025603 -7.941e-02 1.054e-03 1.0413
## 128 -115706.8 -1.9205012 -1.9288022 0.037572 * -3.811e-01 * 2.400e-02 0.9866
## 116 8552.6 0.1410823 0.1408620 0.025626 2.284e-02 8.725e-05 1.0457
## 127 -144731.2 -2.3943861 * -2.4126976 * 0.031242 -4.333e-01 * 3.081e-02 0.9421
## 386 19867.3 0.3259623 0.3254979 0.015029 4.021e-02 2.702e-04 1.0328
## 257 -92184.2 -1.5109625 -1.5140689 0.013072 -1.743e-01 5.040e-03 0.9886
## 271 -83527.8 -1.3714333 -1.3733671 0.016460 -1.777e-01 5.246e-03 0.9997
## 518 -39272.5 -0.6515769 -0.6509788 0.036782 -1.272e-01 2.702e-03 1.0497
## 451 -381.4 -0.0062624 -0.0062524 0.016517 -8.103e-04 1.098e-07 1.0364
## 14 -31560.1 -0.5208939 -0.5202886 0.026676 -8.613e-02 1.239e-03 1.0418
## 169 21526.8 0.3518701 0.3513787 0.007633 3.082e-02 1.587e-04 1.0247
## 51 35287.8 0.5818907 0.5812769 0.024909 9.291e-02 1.442e-03 1.0386
## 188 70276.1 1.1484485 1.1490341 0.007176 9.769e-02 1.589e-03 1.0011
## 210 -32504.5 -0.5420191 -0.5414086 0.046463 * -1.195e-01 2.386e-03 1.0630
## 369 -21385.3 -0.3499932 -0.3495037 0.010101 -3.531e-02 2.083e-04 1.0273
## 291 11211.8 0.1837372 0.1834543 0.012723 2.083e-02 7.251e-05 1.0318
## 224 49651.0 0.8114958 0.8110535 0.007424 7.014e-02 8.209e-04 1.0141
## 331 -33818.6 -0.5542196 -0.5536072 0.012752 -6.292e-02 6.612e-04 1.0264
## 159 55588.1 0.9093182 0.9090668 0.009142 8.732e-02 1.271e-03 1.0126
## 457 9292.4 0.1540917 0.1538519 0.035768 2.963e-02 1.468e-04 1.0567
## 280 -35659.9 -0.5885329 -0.5879194 0.026583 -9.716e-02 1.577e-03 1.0402
## 465 13367.6 0.2190123 0.2186800 0.012250 2.435e-02 9.915e-05 1.0310
## 89 38753.6 0.6396147 0.6390118 0.026653 1.057e-01 1.867e-03 1.0391
## 59 40622.6 0.6703629 0.6697741 0.026370 1.102e-01 2.029e-03 1.0380
## 427 15165.4 0.2488295 0.2484574 0.015117 3.078e-02 1.584e-04 1.0337
## 478 37175.3 0.6093076 0.6086965 0.013006 6.987e-02 8.154e-04 1.0255
## 204 1325.3 0.0216643 0.0216298 0.007830 1.921e-03 6.173e-07 1.0274
## 391 -25877.2 -0.4253948 -0.4248393 0.018862 -5.890e-02 5.798e-04 1.0353
## 428 32207.7 0.5279566 0.5273494 0.013263 6.114e-02 6.244e-04 1.0275
## 53 -41614.9 -0.6816866 -0.6811044 0.011886 -7.470e-02 9.316e-04 1.0225
## 266 -81964.2 -1.3409586 -1.3426717 0.009404 -1.308e-01 2.845e-03 0.9941
## 277 -24220.0 -0.3955315 -0.3949996 0.005822 -3.023e-02 1.527e-04 1.0222
## 373 -3986.5 -0.0651759 -0.0650725 0.008048 -5.861e-03 5.744e-06 1.0275
## 219 -87413.1 -1.4437513 -1.4462588 0.028042 -2.457e-01 1.002e-02 1.0076
## 511 82930.9 1.3786091 1.3805966 0.040533 * 2.838e-01 * 1.338e-02 1.0244
## 497 -42300.1 -0.6963889 -0.6958167 0.021729 -1.037e-01 1.795e-03 1.0323
## 486 -46565.7 -0.7635104 -0.7630022 0.013763 -9.013e-02 1.356e-03 1.0221
## 214 -2200.9 -0.0360243 -0.0359670 0.010320 -3.673e-03 2.255e-06 1.0299
## 19 -640.0 -0.0105171 -0.0105003 0.018140 -1.427e-03 3.406e-07 1.0382
## 318 -66242.8 -1.0827313 -1.0830295 0.007535 -9.437e-02 1.483e-03 1.0043
## 253 -84588.5 -1.3877129 -1.3897696 0.014849 -1.706e-01 4.838e-03 0.9972
## 268 -46276.1 -0.7611085 -0.7605975 0.019834 -1.082e-01 1.954e-03 1.0285
## 190 -21816.5 -0.3564795 -0.3559835 0.006933 -2.974e-02 1.479e-04 1.0239
## 396 -29838.2 -0.4922637 -0.4916690 0.025844 -8.008e-02 1.071e-03 1.0415
## 412 46142.2 0.7567405 0.7562245 0.014217 9.082e-02 1.376e-03 1.0228
## 35 -45023.8 -0.7365125 -0.7359747 0.009162 -7.077e-02 8.360e-04 1.0181
## 226 -72769.1 -1.1934936 -1.1943036 0.014325 -1.440e-01 3.450e-03 1.0063
## 202 -21875.2 -0.3607160 -0.3602158 0.024888 -5.755e-02 5.535e-04 1.0427
## 288 -14450.2 -0.2366942 -0.2363381 0.011788 -2.581e-02 1.114e-04 1.0304
## 477 -5965.8 -0.0978068 -0.0976524 0.013553 -1.145e-02 2.190e-05 1.0331
## 196 -31071.1 -0.5082934 -0.5076922 0.009252 -4.906e-02 4.021e-04 1.0238
## 309 -6793.6 -0.1111356 -0.1109607 0.009221 -1.070e-02 1.916e-05 1.0286
## 28 -2962.3 -0.0485005 -0.0484234 0.010876 -5.078e-03 4.311e-06 1.0305
## 263 -28222.4 -0.4622708 -0.4616912 0.011729 -5.030e-02 4.227e-04 1.0272
## 34 -52460.2 -0.8585693 -0.8582091 0.010107 -8.672e-02 1.254e-03 1.0153
## 261 14722.4 0.2410795 0.2407176 0.011181 2.560e-02 1.095e-04 1.0297
## 327 -18631.8 -0.3051085 -0.3046674 0.011262 -3.252e-02 1.767e-04 1.0291
## 221 1757.4 0.0287360 0.0286902 0.008317 2.628e-03 1.154e-06 1.0279
## 121 -50264.6 -0.8317056 -0.8312963 0.031579 -1.501e-01 3.759e-03 1.0387
## 37 -84613.4 -1.4141556 -1.4164197 0.050787 * -3.276e-01 * 1.783e-02 1.0335
## 251 -73908.3 -1.2073728 -1.2082566 0.006465 -9.746e-02 1.581e-03 0.9977
## 171 -2503.6 -0.0410516 -0.0409863 0.013797 -4.848e-03 3.929e-06 1.0335
## 218 44551.2 0.7315896 0.7310471 0.016752 9.542e-02 1.520e-03 1.0261
## 144 197592.3 3.2931908 * 3.3462376 * 0.045479 * 7.304e-01 * 8.612e-02 0.8648
## 304 15408.5 0.2519182 0.2515422 0.008067 2.268e-02 8.602e-05 1.0264
## 136 127998.4 2.0905835 * 2.1019315 * 0.006075 1.643e-01 4.452e-03 0.9428
## 160 90160.0 1.4761586 1.4789467 0.010898 1.552e-01 4.001e-03 0.9884
## 431 15418.1 0.2524908 0.2521140 0.011335 2.699e-02 1.218e-04 1.0298
## 403 40063.7 0.6566219 0.6560260 0.012921 7.506e-02 9.406e-04 1.0242
## 73 297609.9 * 4.9227472 * 5.1162949 * 0.030921 9.139e-01 * 1.289e-01 0.6496 *
## 336 -10742.6 -0.1759674 -0.1756957 0.011823 -1.922e-02 6.175e-05 1.0309
## 56 13379.4 0.2249509 0.2246105 0.062061 * 5.778e-02 5.580e-04 1.0857
## 103 -226103.7 -3.8105642 * -3.8956312 * 0.066495 * -1.040e+00 * 1.724e-01 0.8219
## 15 18491.8 0.3027413 0.3023029 0.010778 3.156e-02 1.664e-04 1.0286
## 98 55182.2 0.9133770 0.9131353 0.032218 1.666e-01 4.629e-03 1.0366
## 408 -6555.7 -0.1074753 -0.1073060 0.013493 -1.255e-02 2.633e-05 1.0330
## 416 -40735.9 -0.6705327 -0.6699439 0.021427 -9.913e-02 1.641e-03 1.0327
## 436 13531.3 0.2220424 0.2217059 0.015332 2.767e-02 1.280e-04 1.0342
## 206 45184.8 0.7411782 0.7406452 0.014584 9.010e-02 1.355e-03 1.0236
## 247 143555.6 2.3507655 * 2.3679486 * 0.011217 2.522e-01 1.045e-02 0.9267
## 454 29152.1 0.4804790 0.4798898 0.023953 7.518e-02 9.443e-04 1.0397
## 71 74449.2 1.2326084 1.2336322 0.032727 2.269e-01 8.567e-03 1.0236
## 5 18910.5 0.3088272 0.3083819 0.005844 2.364e-02 9.345e-05 1.0234
## 346 -9373.7 -0.1531829 -0.1529445 0.007159 -1.299e-02 2.820e-05 1.0262
## 93 94231.0 1.5548137 1.5583462 0.026109 2.552e-01 1.080e-02 0.9992
## 439 -13552.2 -0.2222680 -0.2219313 0.014295 -2.673e-02 1.194e-04 1.0331
## 24 -117203.1 -1.9201099 -1.9284046 0.012116 -2.136e-01 7.537e-03 0.9612
## 2 81477.3 1.3313813 1.3330274 0.007002 1.119e-01 2.083e-03 0.9922
## 178 116918.8 1.9130527 1.9212331 0.009637 1.895e-01 5.935e-03 0.9593
## 293 -43068.6 -0.7031849 -0.7026177 0.005369 -5.162e-02 4.449e-04 1.0152
## 335 -25902.0 -0.4235647 -0.4230105 0.008466 -3.909e-02 2.553e-04 1.0245
## 350 -4319.7 -0.0708124 -0.0707001 0.013344 -8.222e-03 1.130e-05 1.0330
## 151 58138.6 0.9602561 0.9601366 0.028071 1.632e-01 4.439e-03 1.0304
## 501 2471.6 0.0405863 0.0405217 0.016701 5.281e-03 4.663e-06 1.0366
## 1 17744.2 0.2906264 0.2902023 0.011624 3.147e-02 1.656e-04 1.0296
## 469 -644.7 -0.0105739 -0.0105570 0.014417 -1.277e-03 2.726e-07 1.0342
## 285 51638.7 0.8454108 0.8450258 0.010777 8.820e-02 1.298e-03 1.0164
## 292 -2716.4 -0.0443812 -0.0443106 0.006706 -3.641e-03 2.216e-06 1.0262
## 487 -2130.9 -0.0349802 -0.0349245 0.016055 -4.461e-03 3.328e-06 1.0359
## 62 -47222.4 -0.7736070 -0.7731112 0.012051 -8.539e-02 1.217e-03 1.0200
## 82 52239.9 0.8635852 0.8632347 0.029772 1.512e-01 3.814e-03 1.0357
## 68 -71974.4 -1.2116320 -1.2125389 0.064392 * -3.181e-01 * 1.684e-02 1.0593
## 63 -49063.9 -0.8241471 -0.8237251 0.060290 * -2.086e-01 7.263e-03 1.0707
## 325 -50723.1 -0.8286194 -0.8282049 0.006471 -6.684e-02 7.453e-04 1.0126
## 323 20970.4 0.3426821 0.3422000 0.007088 2.891e-02 1.397e-04 1.0243
## 222 -49842.2 -0.8209119 -0.8204846 0.022582 -1.247e-01 2.595e-03 1.0295
## 7 -31167.2 -0.5091102 -0.5085088 0.006307 -4.051e-02 2.742e-04 1.0207
## 131 -182421.7 -3.0133399 * -3.0530048 * 0.028291 -5.209e-01 * 4.406e-02 0.8797
## 338 -7012.5 -0.1146402 -0.1144599 0.007897 -1.021e-02 1.744e-05 1.0272
## 447 17525.5 0.2876578 0.2872372 0.015840 3.644e-02 2.220e-04 1.0341
## 75 48542.0 0.8003697 0.7999106 0.024709 1.273e-01 2.705e-03 1.0324
## 88 -42745.9 -0.7051634 -0.7045978 0.025710 -1.145e-01 2.187e-03 1.0363
## 313 -2957.0 -0.0484119 -0.0483350 0.010845 -5.061e-03 4.283e-06 1.0305
## 411 34670.3 0.5681234 0.5675098 0.012564 6.401e-02 6.845e-04 1.0259
## 264 21079.6 0.3456609 0.3451757 0.013938 4.104e-02 2.815e-04 1.0314
## 184 53274.4 0.8702388 0.8699016 0.006334 6.945e-02 8.045e-04 1.0111
## 418 -15135.2 -0.2483041 -0.2479327 0.014879 -3.047e-02 1.552e-04 1.0335
## 378 -57436.4 -0.9384451 -0.9382663 0.006802 -7.765e-02 1.005e-03 1.0092
## 283 -24139.1 -0.3951318 -0.3946002 0.010452 -4.055e-02 2.748e-04 1.0270
## 242 -52921.3 -0.8670784 -0.8667348 0.012301 -9.673e-02 1.561e-03 1.0173
## 520 -41021.0 -0.6727098 -0.6721223 0.014092 -8.036e-02 1.078e-03 1.0250
## 300 27417.7 0.4496494 0.4490774 0.014187 5.387e-02 4.849e-04 1.0300
## 152 32278.1 0.5291353 0.5285278 0.013352 6.148e-02 6.315e-04 1.0276
## 95 144476.4 2.3887945 * 2.4069588 * 0.030125 4.242e-01 * 2.954e-02 0.9415
## 48 -8576.5 -0.1413368 -0.1411160 0.023691 -2.198e-02 8.079e-05 1.0437
## 522 -11466.0 -0.1882186 -0.1879292 0.016036 -2.399e-02 9.622e-05 1.0352
## 330 -13056.7 -0.2137521 -0.2134270 0.010707 -2.220e-02 8.242e-05 1.0295
## 343 -7404.8 -0.1216358 -0.1214448 0.017377 -1.615e-02 4.361e-05 1.0371
## 41 7525.7 0.1233383 0.1231447 0.012855 1.405e-02 3.302e-05 1.0323
## 370 -26961.1 -0.4414432 -0.4408765 0.010980 -4.645e-02 3.606e-04 1.0268
## 279 15762.2 0.2590553 0.2586701 0.018416 3.543e-02 2.098e-04 1.0371
## 326 -11686.6 -0.1908784 -0.1905853 0.006104 -1.494e-02 3.730e-05 1.0249
## 377 -54207.5 -0.8865326 -0.8862296 0.008691 -8.298e-02 1.148e-03 1.0129
## 287 11432.1 0.1868913 0.1866038 0.007910 1.666e-02 4.641e-05 1.0268
## 462 -60501.1 -0.9945150 -0.9944977 0.018740 -1.374e-01 3.148e-03 1.0193
## 173 39277.9 0.6423293 0.6417274 0.008573 5.967e-02 5.946e-04 1.0201
## 491 -29564.5 -0.4851321 -0.4845406 0.015308 -6.041e-02 6.098e-04 1.0305
## 345 -26549.8 -0.4340209 -0.4334593 0.007844 -3.854e-02 2.482e-04 1.0237
## 270 -13102.0 -0.2147040 -0.2143775 0.012648 -2.426e-02 9.842e-05 1.0315
## 424 8560.5 0.1405138 0.1402943 0.015887 1.783e-02 5.312e-05 1.0354
## 8 -29057.1 -0.4768929 -0.4763055 0.015670 -6.010e-02 6.034e-04 1.0311
## 379 -51392.0 -0.8395400 -0.8391444 0.006456 -6.764e-02 7.633e-04 1.0122
## 349 -34180.9 -0.5608473 -0.5602342 0.015181 -6.956e-02 8.081e-04 1.0288
## 79 181012.4 2.9843513 * 3.0227730 * 0.024570 4.797e-01 * 3.739e-02 0.8793
## 9 7346.2 0.1208087 0.1206190 0.019588 1.705e-02 4.860e-05 1.0394
## 97 -64526.9 -1.0713382 -1.0715912 0.038148 * -2.134e-01 7.587e-03 1.0367
## 479 -406.2 -0.0066544 -0.0066438 0.012158 -7.371e-04 9.083e-08 1.0319
## 506 7087.8 0.1163522 0.1161693 0.016085 1.485e-02 3.689e-05 1.0357
## 132 -12287.1 -0.2036829 -0.2033718 0.035126 -3.880e-02 2.517e-04 1.0556
## 355 -12957.0 -0.2124016 -0.2120783 0.013331 -2.465e-02 1.016e-04 1.0322
## 17 -10845.9 -0.1781720 -0.1778970 0.017497 -2.374e-02 9.422e-05 1.0368
## 260 17669.8 0.2888327 0.2884107 0.007678 2.537e-02 1.076e-04 1.0256
## 168 5041.5 0.0823884 0.0822580 0.007195 7.002e-03 8.198e-06 1.0266
## 461 -12852.7 -0.2107453 -0.2104243 0.013821 -2.491e-02 1.037e-04 1.0327
## 167 44026.5 0.7246631 0.7241140 0.021333 1.069e-01 1.908e-03 1.0311
## 455 5338.2 0.0874224 0.0872841 0.011402 9.374e-03 1.469e-05 1.0309
## 86 125448.9 2.0796173 * 2.0907514 * 0.035180 3.992e-01 * 2.628e-02 0.9721
## 265 -60335.1 -0.9878014 -0.9877632 0.010809 -1.033e-01 1.777e-03 1.0114
## 324 3288.0 0.0537288 0.0536434 0.007066 4.525e-03 3.424e-06 1.0265
## 65 16629.5 0.2726727 0.2722704 0.013818 3.223e-02 1.736e-04 1.0321
## 466 32378.9 0.5590164 0.5584035 0.110484 * 1.968e-01 6.469e-03 1.1391
## 393 -15376.6 -0.2536302 -0.2532520 0.025462 -4.094e-02 2.801e-04 1.0447
## 33 -112604.1 -1.8473137 -1.8544745 0.014840 -2.276e-01 8.568e-03 0.9690
## 360 -7528.6 -0.1232585 -0.1230650 0.010809 -1.286e-02 2.767e-05 1.0302
## 459 -4818.8 -0.0789428 -0.0788178 0.012048 -8.704e-03 1.267e-05 1.0316
## 276 -79637.7 -1.3014389 -1.3028835 0.007183 -1.108e-01 2.042e-03 0.9939
## 44 -40.5 -0.0006654 -0.0006643 0.017519 -8.871e-05 1.316e-09 1.0375
## 148 105617.5 1.7616293 1.7675783 0.046935 * 3.923e-01 * 2.547e-02 1.0076
## 333 -3996.6 -0.0654262 -0.0653224 0.010644 -6.775e-03 7.675e-06 1.0302
## 361 -37370.7 -0.6140910 -0.6134809 0.018079 -8.324e-02 1.157e-03 1.0306
## 250 -65167.0 -1.0664508 -1.0666848 0.009959 -1.070e-01 1.907e-03 1.0074
## 471 24623.2 0.4036747 0.4031361 0.013480 4.712e-02 3.711e-04 1.0300
## 60 -81897.2 -1.3378401 -1.3395312 0.006404 -1.075e-01 1.923e-03 0.9913
## 347 -1984.8 -0.0324240 -0.0323724 0.006455 -2.609e-03 1.138e-06 1.0259
## 317 -23686.3 -0.3878976 -0.3873722 0.011354 -4.151e-02 2.880e-04 1.0281
## 143 71657.8 1.1758315 1.1765508 0.015271 1.465e-01 3.574e-03 1.0081
## 256 -20503.9 -0.3348403 -0.3343664 0.005792 -2.552e-02 1.089e-04 1.0231
## 22 7266.3 0.1189929 0.1188059 0.011288 1.269e-02 2.694e-05 1.0307
## 217 31094.5 0.5086419 0.5080406 0.009114 4.872e-02 3.966e-04 1.0236
## 215 -86553.3 -1.4373878 -1.4398420 0.038612 * -2.886e-01 * 1.383e-02 1.0191
## 225 19519.4 0.3189863 0.3185295 0.007189 2.711e-02 1.228e-04 1.0247
## 254 15508.9 0.2567628 0.2563805 0.032669 4.712e-02 3.711e-04 1.0524
## 504 273.2 0.0044763 0.0044692 0.012637 5.056e-04 4.274e-08 1.0324
## 413 14350.8 0.2365177 0.2361618 0.023879 3.694e-02 2.281e-04 1.0431
## 383 -3145.9 -0.0514506 -0.0513688 0.008740 -4.823e-03 3.890e-06 1.0283
## 510 -5813.9 -0.0952135 -0.0950632 0.011418 -1.022e-02 1.745e-05 1.0309
## 129 -152323.3 -2.5478716 * -2.5705216 * 0.052330 * -6.040e-01 * 5.974e-02 0.9489
## 278 -11675.7 -0.1907189 -0.1904260 0.006287 -1.515e-02 3.836e-05 1.0251
## 205 51507.3 0.8419843 0.8415930 0.007778 7.451e-02 9.262e-04 1.0135
## 484 4234.9 0.0693644 0.0692544 0.011679 7.528e-03 9.476e-06 1.0313
# We also could use the built-in function of R-base below.
# influence.measures(lm.opt)
# Sum up the number of stars an observation got in Reg_Diag, indicating the degree of its outlying
starred <- function(row) {
sum(as.numeric(gsub("\\*", "1", row)), na.rm = T)
}
diag.count = apply(diag[, seq(2, 14, by = 2)], 1, starred)
table(diag.count)
## diag.count
## 0 1 2 3 4 5
## 278 13 11 12 4 2
By filtering out the observations with the condition that diag.count >= 5, 4, 3, we can rerun the regression and see how much it is improved.
real.est[names(which(diag.count >= 5)), ]
## id price area bedrooms bathrooms aircon garage pool year quality style lot highway
## 96 96 600000 2344 4 3 1 2 0 1925 1 1 86004 0
## 73 73 920000 3857 4 5 1 3 0 1997 1 1 32793 0
training1 = training[!training$id %in% names(which(diag.count >=
5)), ]
lm.opt1 = update(lm.opt, data = training1)
summary(lm.opt1)
##
## Call:
## lm(formula = price ~ area + year + quality + lot, data = training1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -216872 -29292 -3575 23381 232311
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.41e+06 4.58e+05 -7.43 1.0e-12 ***
## area 9.45e+01 6.73e+00 14.04 < 2e-16 ***
## year 1.77e+03 2.33e+02 7.59 3.7e-13 ***
## quality.L 1.05e+05 1.21e+04 8.70 < 2e-16 ***
## quality.Q 4.82e+04 5.77e+03 8.34 2.4e-15 ***
## lot 9.91e-01 3.04e-01 3.25 0.0013 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 58200 on 312 degrees of freedom
## Multiple R-squared: 0.811, Adjusted R-squared: 0.808
## F-statistic: 268 on 5 and 312 DF, p-value: <2e-16
pred1 = predict(lm.opt1, validation[, c(3, 9, 10, 12)])
mean(abs(pred1 - validation[, 2])/validation[, 2])
## [1] 0.1478
Disappointedly, R-squared decreased, so did the significance of the coefficient “lot”, whereas the significance of “area” and “year” increased. What is more, the average absolute prediction error even increased.
Stricter filtering of outliers shows a little improvements of R-squared and prediction accuracy, but not very remarkable.
real.est[names(which(diag.count >= 4)), ]
## id price area bedrooms bathrooms aircon garage pool year quality style lot highway
## 11 11 190000 2812 7 5 0 2 1 1966 3 7 56639 0
## 96 96 600000 2344 4 3 1 2 0 1925 1 1 86004 0
## 144 144 675000 3855 4 4 1 3 0 1996 2 7 35845 0
## 73 73 920000 3857 4 5 1 3 0 1997 1 1 32793 0
## 103 103 479000 5032 7 3 1 3 0 1989 1 7 22000 0
## 129 129 465000 4453 7 5 1 2 0 1974 1 7 15595 0
training2 = training[!training$id %in% names(which(diag.count >=
4)), ]
lm.opt2 = update(lm.opt, data = training2)
summary(lm.opt2)
##
## Call:
## lm(formula = price ~ area + year + quality + lot, data = training2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -183491 -28349 -3497 22294 218535
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.25e+06 4.36e+05 -7.45 9.4e-13 ***
## area 1.01e+02 6.68e+00 15.12 < 2e-16 ***
## year 1.68e+03 2.21e+02 7.60 3.5e-13 ***
## quality.L 1.06e+05 1.17e+04 9.11 < 2e-16 ***
## quality.Q 5.31e+04 5.48e+03 9.68 < 2e-16 ***
## lot 9.37e-01 2.92e-01 3.21 0.0015 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 54700 on 308 degrees of freedom
## Multiple R-squared: 0.827, Adjusted R-squared: 0.825
## F-statistic: 295 on 5 and 308 DF, p-value: <2e-16
pred2 = predict(lm.opt2, validation[, c(3, 9, 10, 12)])
mean(abs(pred2 - validation[, 2])/validation[, 2])
## [1] 0.1454
real.est[names(which(diag.count >= 3)), ]
## id price area bedrooms bathrooms aircon garage pool year quality style lot highway
## 11 11 190000 2812 7 5 0 2 1 1966 3 7 56639 0
## 92 92 389900 2817 4 3 1 3 0 1996 1 7 31214 0
## 16 16 527000 3232 5 5 1 2 0 1984 2 6 21445 0
## 96 96 600000 2344 4 3 1 2 0 1925 1 1 86004 0
## 135 135 515000 2950 5 3 1 2 0 1969 2 1 21598 0
## 120 120 367000 2940 4 7 1 2 0 1988 1 7 22003 0
## 72 72 830000 3889 4 4 1 3 0 1991 1 7 28378 0
## 80 80 647000 2464 3 3 1 3 0 1992 1 1 31703 0
## 109 109 370000 2936 4 4 1 3 0 1987 1 7 16437 0
## 127 127 380000 3460 5 4 1 2 0 1972 1 1 18571 0
## 144 144 675000 3855 4 4 1 3 0 1996 2 7 35845 0
## 73 73 920000 3857 4 5 1 3 0 1997 1 1 32793 0
## 103 103 479000 5032 7 3 1 3 0 1989 1 7 22000 0
## 131 131 336000 3301 3 4 1 2 0 1977 1 3 18741 0
## 95 95 640000 2705 3 3 1 3 0 1994 1 1 22196 0
## 79 79 725000 3242 3 3 1 3 0 1989 1 1 27173 0
## 86 86 609000 2654 5 3 1 3 0 1997 1 1 12821 0
## 129 129 465000 4453 7 5 1 2 0 1974 1 7 15595 0
training3 = training[!training$id %in% names(which(diag.count >=
3)), ]
lm.opt3 = update(lm.opt, data = training3)
summary(lm.opt3)
##
## Call:
## lm(formula = price ~ area + year + quality + lot, data = training3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -114611 -25600 -3137 22629 146903
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.95e+06 3.64e+05 -8.08 1.6e-14 ***
## area 9.87e+01 5.68e+00 17.36 < 2e-16 ***
## year 1.53e+03 1.85e+02 8.28 4.4e-15 ***
## quality.L 1.09e+05 1.04e+04 10.50 < 2e-16 ***
## quality.Q 5.28e+04 4.95e+03 10.67 < 2e-16 ***
## lot 8.71e-01 2.45e-01 3.56 0.00043 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 45400 on 296 degrees of freedom
## Multiple R-squared: 0.857, Adjusted R-squared: 0.854
## F-statistic: 354 on 5 and 296 DF, p-value: <2e-16
pred3 = predict(lm.opt3, validation[, c(3, 9, 10, 12)])
mean(abs(pred3 - validation[, 2])/validation[, 2])
## [1] 0.1412