Appendices
Ben Abraham
Figure 1: Scatterplot Matrix of Data
Figure 2: Histogram of Median Income
Figure 3: Histogram of the Proportion of Residents with a Bachelor’s Degree
Figure 4: Histogram of Unemployment Rate
Figure 5: Histogram of Average Household Size
Figure 6: Boxplot of Region
Figure 7: Initial Model Summary
##
## Call:
## lm(formula = medianIncome ~ pctBach + pctUnemployed + avgHouse +
## region, data = income)
##
## Residuals:
## Min 1Q Median 3Q Max
## -34.902 -3.997 -0.219 3.692 25.481
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.33032 3.12608 1.065 0.2872
## pctBach 1.19672 0.06105 19.603 < 2e-16 ***
## pctUnemployed -1.07963 0.10388 -10.394 < 2e-16 ***
## avgHouse 14.65802 1.28533 11.404 < 2e-16 ***
## regionNortheast 6.48713 1.08507 5.979 4.00e-09 ***
## regionSoutheast -1.90928 0.76426 -2.498 0.0128 *
## regionSouthwest -4.08575 1.01331 -4.032 6.29e-05 ***
## regionWest -2.00312 1.03756 -1.931 0.0540 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.74 on 562 degrees of freedom
## Multiple R-squared: 0.6583, Adjusted R-squared: 0.6541
## F-statistic: 154.7 on 7 and 562 DF, p-value: < 2.2e-16Figure 8: Distribution of Population Variable Before Log Transformation
Figure 9: Distribution of Population Variable After Logarithmic Transformation
Figure 10: Scatterplot of Poverty Rate and Median Income
Figure 11: Scatterplot of Median Income and log(population) by Region
Figure 12: Summary Output with Interaction Term on Average House Size
##
## Call:
## lm(formula = medianIncome ~ pctBach + pctUnemployed + avgHouse +
## log(population) * region, data = income)
##
## Residuals:
## Min 1Q Median 3Q Max
## -32.211 -3.923 0.005 3.659 24.364
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.29143 4.29127 -0.301 0.76357
## pctBach 1.05553 0.07264 14.532 < 2e-16 ***
## pctUnemployed -1.18570 0.10556 -11.233 < 2e-16 ***
## avgHouse 13.21591 1.32590 9.968 < 2e-16 ***
## log(population) 1.07142 0.39107 2.740 0.00635 **
## regionNortheast -34.13597 10.12528 -3.371 0.00080 ***
## regionSoutheast 4.35310 6.22527 0.699 0.48468
## regionSouthwest 6.13410 6.75428 0.908 0.36418
## regionWest 9.05138 5.92776 1.527 0.12734
## log(population):regionNortheast 3.34152 0.87525 3.818 0.00015 ***
## log(population):regionSoutheast -0.62392 0.59803 -1.043 0.29726
## log(population):regionSouthwest -0.98777 0.65297 -1.513 0.13091
## log(population):regionWest -1.02977 0.57415 -1.794 0.07343 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.577 on 557 degrees of freedom
## Multiple R-squared: 0.6775, Adjusted R-squared: 0.6705
## F-statistic: 97.51 on 12 and 557 DF, p-value: < 2.2e-16Figure 13: Scatterplot Matrix of Strong Linear Relationships (pctHS, pctEmployed, pctMarried, and pctMarriedHouse)
Figure 14: Scatterplot Matrix of Strong Linear Relationships (pctPrivateHC, pctPublicHC, and pctEmployerHC)
Figure 15: Scatterplot of Median Income vs Population with a logarithmic transformation
Figure 16: Model Output for Initial Model with pctHS as additional predictor
##
## Call:
## lm(formula = medianIncome ~ pctBach + pctUnemployed + avgHouse +
## pctHS + region, data = income)
##
## Residuals:
## Min 1Q Median 3Q Max
## -34.725 -3.981 -0.219 3.608 25.313
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.60444 5.37226 -0.113 0.910458
## pctBach 1.26982 0.10156 12.503 < 2e-16 ***
## pctUnemployed -1.05365 0.10782 -9.772 < 2e-16 ***
## avgHouse 14.78915 1.29377 11.431 < 2e-16 ***
## pctHS 0.06704 0.07443 0.901 0.368149
## regionNortheast 6.33726 1.09793 5.772 1.3e-08 ***
## regionSoutheast -1.81973 0.77082 -2.361 0.018579 *
## regionSouthwest -3.83806 1.05013 -3.655 0.000282 ***
## regionWest -1.74486 1.07663 -1.621 0.105650
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.741 on 561 degrees of freedom
## Multiple R-squared: 0.6588, Adjusted R-squared: 0.6539
## F-statistic: 135.4 on 8 and 561 DF, p-value: < 2.2e-16Figure 17: Model Output for Initial Model with pctPublicHC as additional predictor
##
## Call:
## lm(formula = medianIncome ~ pctBach + pctUnemployed + avgHouse +
## pctPublicHC + region, data = income)
##
## Residuals:
## Min 1Q Median 3Q Max
## -34.139 -3.554 0.172 3.277 23.704
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 44.46475 4.84029 9.186 < 2e-16 ***
## pctBach 0.73993 0.07070 10.466 < 2e-16 ***
## pctUnemployed -0.44499 0.11250 -3.956 8.61e-05 ***
## avgHouse 7.34271 1.36538 5.378 1.11e-07 ***
## pctPublicHC -0.60738 0.05767 -10.532 < 2e-16 ***
## regionNortheast 7.02737 0.99368 7.072 4.56e-12 ***
## regionSoutheast -1.72441 0.69917 -2.466 0.01395 *
## regionSouthwest -2.90111 0.93353 -3.108 0.00198 **
## regionWest -0.27622 0.96297 -0.287 0.77434
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.164 on 561 degrees of freedom
## Multiple R-squared: 0.7147, Adjusted R-squared: 0.7107
## F-statistic: 175.7 on 8 and 561 DF, p-value: < 2.2e-16Figure 18: Model Output for Initial Model with pctPrivateHC as additional predictor
##
## Call:
## lm(formula = medianIncome ~ pctBach + pctUnemployed + avgHouse +
## pctPrivateHC + region, data = income)
##
## Residuals:
## Min 1Q Median 3Q Max
## -40.184 -2.946 -0.205 2.719 24.931
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -34.56687 3.77524 -9.156 < 2e-16 ***
## pctBach 0.71386 0.06235 11.449 < 2e-16 ***
## pctUnemployed -0.28900 0.10493 -2.754 0.00607 **
## avgHouse 14.18280 1.10272 12.862 < 2e-16 ***
## pctPrivateHC 0.57884 0.04060 14.257 < 2e-16 ***
## regionNortheast 5.97321 0.93118 6.415 3e-10 ***
## regionSoutheast 0.62774 0.67911 0.924 0.35569
## regionSouthwest 1.67395 0.95827 1.747 0.08121 .
## regionWest 2.08065 0.93472 2.226 0.02641 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.779 on 561 degrees of freedom
## Multiple R-squared: 0.7492, Adjusted R-squared: 0.7456
## F-statistic: 209.5 on 8 and 561 DF, p-value: < 2.2e-16Figure 19: Model Output for Initial Model with pctEmployerHC as additional predictor
##
## Call:
## lm(formula = medianIncome ~ pctBach + pctUnemployed + avgHouse +
## pctEmployerHC + region, data = income)
##
## Residuals:
## Min 1Q Median 3Q Max
## -30.8301 -3.3400 -0.1718 2.9261 23.9453
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -5.25492 2.75695 -1.906 0.0572 .
## pctBach 0.82407 0.05879 14.017 < 2e-16 ***
## pctUnemployed -0.50938 0.09808 -5.194 2.89e-07 ***
## avgHouse 9.23565 1.17057 7.890 1.59e-14 ***
## pctEmployerHC 0.52071 0.03694 14.094 < 2e-16 ***
## regionNortheast 4.91738 0.93992 5.232 2.38e-07 ***
## regionSoutheast -0.60407 0.66385 -0.910 0.3632
## regionSouthwest 0.03900 0.91939 0.042 0.9662
## regionWest 1.77718 0.93187 1.907 0.0570 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.797 on 561 degrees of freedom
## Multiple R-squared: 0.7477, Adjusted R-squared: 0.7441
## F-statistic: 207.8 on 8 and 561 DF, p-value: < 2.2e-16Figure 20: Model Output for Initial Model with pctMarried as additional predictor
##
## Call:
## lm(formula = medianIncome ~ pctBach + pctUnemployed + avgHouse +
## pctMarried + region, data = income)
##
## Residuals:
## Min 1Q Median 3Q Max
## -37.074 -3.936 -0.210 3.247 21.820
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -28.04716 4.28148 -6.551 1.30e-10 ***
## pctBach 1.28153 0.05700 22.482 < 2e-16 ***
## pctUnemployed -0.56114 0.10921 -5.138 3.84e-07 ***
## avgHouse 14.85696 1.18676 12.519 < 2e-16 ***
## pctMarried 0.48622 0.04901 9.921 < 2e-16 ***
## regionNortheast 7.52896 1.00720 7.475 2.98e-13 ***
## regionSoutheast -1.05613 0.71076 -1.486 0.13786
## regionSouthwest -2.92412 0.94276 -3.102 0.00202 **
## regionWest -2.49602 0.95914 -2.602 0.00950 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.222 on 561 degrees of freedom
## Multiple R-squared: 0.7093, Adjusted R-squared: 0.7052
## F-statistic: 171.1 on 8 and 561 DF, p-value: < 2.2e-16Figure 21: Model Output for Initial Model with pctPoverty as additional predictor
##
## Call:
## lm(formula = medianIncome ~ pctBach + pctUnemployed + avgHouse +
## pctPoverty + I(pctPoverty^2) + region, data = income)
##
## Residuals:
## Min 1Q Median 3Q Max
## -14.7751 -2.4285 -0.1139 2.2769 18.9909
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 44.952435 2.612758 17.205 < 2e-16 ***
## pctBach 0.701727 0.040951 17.136 < 2e-16 ***
## pctUnemployed 0.117395 0.075872 1.547 0.1224
## avgHouse 10.184353 0.807770 12.608 < 2e-16 ***
## pctPoverty -2.845072 0.135196 -21.044 < 2e-16 ***
## I(pctPoverty^2) 0.039348 0.003021 13.025 < 2e-16 ***
## regionNortheast 4.289551 0.669844 6.404 3.21e-10 ***
## regionSoutheast 1.926483 0.485602 3.967 8.22e-05 ***
## regionSouthwest 1.144354 0.647474 1.767 0.0777 .
## regionWest -0.119917 0.641727 -0.187 0.8518
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.136 on 560 degrees of freedom
## Multiple R-squared: 0.8718, Adjusted R-squared: 0.8697
## F-statistic: 423 on 9 and 560 DF, p-value: < 2.2e-16Figure 22: Model Output for Initial Model with pctMarriedHouse as additional predictor
##
## Call:
## lm(formula = medianIncome ~ pctBach + pctUnemployed + avgHouse +
## pctMarriedHouse + region, data = income)
##
## Residuals:
## Min 1Q Median 3Q Max
## -41.694 -3.719 -0.212 3.161 23.365
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -16.75544 3.39452 -4.936 1.05e-06 ***
## pctBach 1.25832 0.05584 22.533 < 2e-16 ***
## pctUnemployed -0.50621 0.10831 -4.674 3.71e-06 ***
## avgHouse 9.45179 1.26435 7.476 2.97e-13 ***
## pctMarriedHouse 0.53750 0.04956 10.846 < 2e-16 ***
## regionNortheast 6.81804 0.98791 6.901 1.40e-11 ***
## regionSoutheast -1.34442 0.69744 -1.928 0.05440 .
## regionSouthwest -2.60536 0.93218 -2.795 0.00537 **
## regionWest -2.20778 0.94440 -2.338 0.01975 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.133 on 561 degrees of freedom
## Multiple R-squared: 0.7175, Adjusted R-squared: 0.7135
## F-statistic: 178.1 on 8 and 561 DF, p-value: < 2.2e-16Figure 23: Model Output for Initial Model with log(population) as additional predictor
##
## Call:
## lm(formula = medianIncome ~ pctBach + pctUnemployed + avgHouse +
## log(population) + region, data = income)
##
## Residuals:
## Min 1Q Median 3Q Max
## -33.241 -4.038 -0.246 3.489 26.403
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.75753 3.26507 0.232 0.81661
## pctBach 1.10030 0.07124 15.445 < 2e-16 ***
## pctUnemployed -1.14677 0.10655 -10.762 < 2e-16 ***
## avgHouse 13.59488 1.34310 10.122 < 2e-16 ***
## log(population) 0.68906 0.26602 2.590 0.00984 **
## regionNortheast 5.74146 1.11732 5.139 3.83e-07 ***
## regionSoutheast -2.03266 0.76189 -2.668 0.00785 **
## regionSouthwest -4.01150 1.00861 -3.977 7.88e-05 ***
## regionWest -1.60416 1.04376 -1.537 0.12488
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.706 on 561 degrees of freedom
## Multiple R-squared: 0.6624, Adjusted R-squared: 0.6575
## F-statistic: 137.6 on 8 and 561 DF, p-value: < 2.2e-16Figure 24: Model Output of Model with Added Variables
##
## Call:
## lm(formula = medianIncome ~ pctBach + pctUnemployed + pctPublicHC +
## pctPrivateHC + pctEmployerHC + pctMarried + pctMarriedHouse +
## avgHouse + pctPoverty + I(pctPoverty^2) + log(population) +
## region, data = income)
##
## Residuals:
## Min 1Q Median 3Q Max
## -13.9888 -2.2627 -0.0614 2.0926 19.1450
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 64.508155 6.915550 9.328 < 2e-16 ***
## pctBach 0.483897 0.051860 9.331 < 2e-16 ***
## pctUnemployed 0.252294 0.078867 3.199 0.001458 **
## pctPublicHC -0.242579 0.048105 -5.043 6.23e-07 ***
## pctPrivateHC -0.145378 0.045023 -3.229 0.001316 **
## pctEmployerHC 0.147528 0.040336 3.657 0.000279 ***
## pctMarried -0.105996 0.073333 -1.445 0.148911
## pctMarriedHouse 0.192539 0.073325 2.626 0.008882 **
## avgHouse 3.160867 1.208195 2.616 0.009134 **
## pctPoverty -2.792358 0.144430 -19.334 < 2e-16 ***
## I(pctPoverty^2) 0.040955 0.002981 13.737 < 2e-16 ***
## log(population) 0.623204 0.173646 3.589 0.000361 ***
## regionNortheast 3.590693 0.656014 5.473 6.70e-08 ***
## regionSoutheast 1.331682 0.460446 2.892 0.003976 **
## regionSouthwest 1.376873 0.647401 2.127 0.033880 *
## regionWest 0.951050 0.630222 1.509 0.131851
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.779 on 554 degrees of freedom
## Multiple R-squared: 0.8941, Adjusted R-squared: 0.8913
## F-statistic: 311.9 on 15 and 554 DF, p-value: < 2.2e-16Figure 25: Comparing models with a quadratic term and one without
Figure 26: Plot of Residuals for Model of the Second Power
Figure 27: Model with third power exponent
##
## Call:
## lm(formula = medianIncome ~ pctBach + pctUnemployed + pctPublicHC +
## pctPrivateHC + pctEmployerHC + pctMarried + pctMarriedHouse +
## avgHouse + pctPoverty + I(pctPoverty^2) + I(pctPoverty^3) +
## log(population) + region, data = income)
##
## Residuals:
## Min 1Q Median 3Q Max
## -14.7294 -2.2431 -0.0443 1.8356 17.5094
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 75.4935674 6.9495323 10.863 < 2e-16 ***
## pctBach 0.4298655 0.0510807 8.415 3.36e-16 ***
## pctUnemployed 0.1970514 0.0770211 2.558 0.01078 *
## pctPublicHC -0.2134773 0.0468948 -4.552 6.53e-06 ***
## pctPrivateHC -0.1114624 0.0440187 -2.532 0.01161 *
## pctEmployerHC 0.1268039 0.0392631 3.230 0.00131 **
## pctMarried -0.1012288 0.0711116 -1.424 0.15515
## pctMarriedHouse 0.1756318 0.0711542 2.468 0.01388 *
## avgHouse 2.9738844 1.1719297 2.538 0.01144 *
## pctPoverty -4.8812225 0.3742212 -13.044 < 2e-16 ***
## I(pctPoverty^2) 0.1433403 0.0172535 8.308 7.55e-16 ***
## I(pctPoverty^3) -0.0014901 0.0002476 -6.019 3.19e-09 ***
## log(population) 0.7619289 0.1699441 4.483 8.94e-06 ***
## regionNortheast 3.4398760 0.6365926 5.404 9.72e-08 ***
## regionSoutheast 1.1640698 0.4473361 2.602 0.00951 **
## regionSouthwest 1.6153092 0.6289959 2.568 0.01049 *
## regionWest 1.4815323 0.6174127 2.400 0.01674 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.664 on 553 degrees of freedom
## Multiple R-squared: 0.9006, Adjusted R-squared: 0.8978
## F-statistic: 313.3 on 16 and 553 DF, p-value: < 2.2e-16Figure 28: Residual Plot for Final Model
Figure 29: QQ Plot for Final Model
Figure 30: VIF Output for Final Model
## GVIF Df GVIF^(1/(2*Df))
## pctBach 3.071871 1 1.752676
## pctUnemployed 2.757156 1 1.660468
## pctPublicHC 5.397817 1 2.323320
## pctPrivateHC 9.032486 1 3.005410
## pctEmployerHC 5.992370 1 2.447932
## pctMarried 9.081810 1 3.013604
## pctMarriedHouse 8.399578 1 2.898203
## avgHouse 3.433620 1 1.853003
## pctPoverty 230.999949 1 15.198682
## I(pctPoverty^2) 833.986076 1 28.878817
## I(pctPoverty^3) 229.031705 1 15.133793
## log(population) 2.422183 1 1.556336
## region 3.045808 4 1.149378Figure 31: Expected Change in Response for a one Standard Deviation Increase in each variable
## pctBach
## 2.265543## pctUnemployed
## 0.652522## pctPublicHC
## -1.624543## pctPrivateHC
## -1.168933## pctEmployerHC
## 1.214346## pctMarried
## -0.6589385## pctMarriedHouse
## 1.098819## avgHouse
## 0.7222598## log(population)
## 1.071782Figure 32: Model Output without pctMarried
##
## Call:
## lm(formula = medianIncome ~ pctBach + pctUnemployed + pctPublicHC +
## pctPrivateHC + pctEmployerHC + pctMarriedHouse + avgHouse +
## pctPoverty + I(pctPoverty^2) + +I(pctPoverty^3) + log(population) +
## region, data = income)
##
## Residuals:
## Min 1Q Median 3Q Max
## -14.6110 -2.2452 -0.1178 1.9644 17.3239
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 70.6485643 6.0646488 11.649 < 2e-16 ***
## pctBach 0.4331119 0.0510770 8.480 < 2e-16 ***
## pctUnemployed 0.2111851 0.0764492 2.762 0.005928 **
## pctPublicHC -0.2143074 0.0469346 -4.566 6.13e-06 ***
## pctPrivateHC -0.1006821 0.0434025 -2.320 0.020718 *
## pctEmployerHC 0.1375627 0.0385645 3.567 0.000392 ***
## pctMarriedHouse 0.0893411 0.0372958 2.395 0.016930 *
## avgHouse 3.8280148 1.0076119 3.799 0.000161 ***
## pctPoverty -4.8356300 0.3731933 -12.957 < 2e-16 ***
## I(pctPoverty^2) 0.1431291 0.0172688 8.288 8.72e-16 ***
## I(pctPoverty^3) -0.0014941 0.0002478 -6.030 3.00e-09 ***
## log(population) 0.7493768 0.1698724 4.411 1.23e-05 ***
## regionNortheast 3.6292745 0.6231100 5.824 9.72e-09 ***
## regionSoutheast 1.2355500 0.4449205 2.777 0.005672 **
## regionSouthwest 1.6984049 0.6268613 2.709 0.006950 **
## regionWest 1.5260297 0.6171919 2.473 0.013715 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.667 on 554 degrees of freedom
## Multiple R-squared: 0.9003, Adjusted R-squared: 0.8976
## F-statistic: 333.4 on 15 and 554 DF, p-value: < 2.2e-16