Ethan Witkowski Spring 2019
homeprices <- read.csv("homeprices.csv")
# define variables
price <- homeprices[,"price"]
sqft <- homeprices[,"sqft"]
NE <- homeprices[,"NE"]
plot(price~sqft, col=NE+1)
#NE are light blue
# Linear model homes in NE and not in NE
fit1 <- abline(lm(price~sqft, subset=(NE==1)), col=1)
fit2 <- abline(lm(price~sqft, subset=(NE==0)), col=2)
The regression lines would have different slopes if NE and sqft had an interaction.
I do see evidence for an interaction, as the regression lines have different slopes.
##
## Call:
## lm(formula = price ~ sqft + NE)
##
## Residuals:
## Min 1Q Median 3Q Max
## -101.905 -9.114 1.353 7.282 78.072
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.108520 6.549015 0.475 0.636
## sqft 0.060879 0.003666 16.607 <2e-16 ***
## NE 3.720006 4.055259 0.917 0.361
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 20.46 on 114 degrees of freedom
## Multiple R-squared: 0.7158, Adjusted R-squared: 0.7108
## F-statistic: 143.5 on 2 and 114 DF, p-value: < 2.2e-16
There is no evidence for a statistically significant interaction term, as the p-value for NE is .361.
Yes, there is an outlier home with 3750 square feet.
homeprices <- homeprices[-79,]
# re-define variables
price <- homeprices[,"price"]
sqft <- homeprices[,"sqft"]
NE <- homeprices[,"NE"]
plot(price~sqft, col=NE+1)
# Linear model homes in NE and not in NE
fit1 <- abline(lm(price~sqft, subset=(NE==1)), col=1)
fit2 <- abline(lm(price~sqft, subset=(NE==0)), col=2)
##
## Call:
## lm(formula = price ~ sqft + NE)
##
## Residuals:
## Min 1Q Median 3Q Max
## -53.039 -9.151 0.327 10.096 69.868
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -7.320776 5.867117 -1.248 0.215
## sqft 0.069685 0.003442 20.245 <2e-16 ***
## NE -0.884970 3.563408 -0.248 0.804
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 17.61 on 113 degrees of freedom
## Multiple R-squared: 0.7907, Adjusted R-squared: 0.787
## F-statistic: 213.4 on 2 and 113 DF, p-value: < 2.2e-16
The slopes of the NE and non-NE regression lines are much closer.
The estimate of the NE interaction term is closer to 0 (now negative), and the p-value is much larger (.80).
congress <- read.csv("senate.csv")
# define variables
name <- congress[,"name"]
state <- congress[,"state"]
party <- congress[,"party"]
votes <- congress[,4:660]
pca <- prcomp(votes, center=T, scale=T)
PCscores <- pca$x
# set up plot with no points
plot(PCscores[,1], PCscores[,2], type="n")
# plot the names
text(PCscores[,1], PCscores[,2], label=name, cex=.4)
partycolor <- rep(NA, nrow(congress))
partycolor[party=="D"] <- "blue"
partycolor[party=="R"] <- "red"
pca <- prcomp(votes, center=T, scale=T)
PCscores <- pca$x
# set up plot with no points
plot(PCscores[,1], PCscores[,2], type="n")
# plot the names
text(PCscores[,1], PCscores[,2], label=name, cex=.4, col = partycolor)
The first principal compenent appears to represent the issue of immigration - as those on the left of the window oppose more immigration, while those on the right of the window support more immigration.
These congresspeople are outliers because they only served partial terms.
congress <- congress[-c(42,43,44,53,54,63,64,65),]
name <- congress[,"name"]
state <- congress[,"state"]
party <- congress[,"party"]
votes <- congress[,4:660]
partycolor <- rep(NA, nrow(congress))
partycolor[party=="D"] <- "blue"
partycolor[party=="R"] <- "red"The error occurs when attempting to perform the PCA because certain votes have 0 variance (all congresspeople voted the same way) – this is likely a result from a vote like “March 6th is National Tree Day,” where all the congresspeople would vote the same way.
#Find votes with 0 variance
#sort(apply(congress, 2, sd),
# decreasing=T)
#Removing votes with 0 variance
congress <- congress[,-c(19,55,73,136,152,168,208,348,354,369,378,531,532,533,575)]
# define variables
name <- congress[,"name"]
state <- congress[,"state"]
party <- congress[,"party"]
votes <- congress[,4:645]
partycolor <- rep(NA, nrow(congress))
partycolor[party=="D"] <- "blue"
partycolor[party=="R"] <- "red"
#Run PCA without 0 variance columns/votes
pca <- prcomp(votes, center=T, scale=T)
PCscores <- pca$x
# set up plot with no points
plot(PCscores[,1], PCscores[,2], type="n")
# plot the names
text(PCscores[,1], PCscores[,2], label=name, cex=.4, col = partycolor)
## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6 PC7
## Standard deviation 19.3090 4.85082 3.75175 3.54452 3.27306 3.07030 2.83982
## Proportion of Variance 0.5807 0.03665 0.02192 0.01957 0.01669 0.01468 0.01256
## Cumulative Proportion 0.5807 0.61740 0.63932 0.65889 0.67558 0.69026 0.70282
## PC8 PC9 PC10 PC11 PC12 PC13 PC14
## Standard deviation 2.7873 2.72667 2.59993 2.51917 2.48951 2.47893 2.36630
## Proportion of Variance 0.0121 0.01158 0.01053 0.00989 0.00965 0.00957 0.00872
## Cumulative Proportion 0.7149 0.72650 0.73703 0.74692 0.75657 0.76614 0.77486
## PC15 PC16 PC17 PC18 PC19 PC20 PC21
## Standard deviation 2.34425 2.27075 2.18388 2.13768 2.12731 2.06938 2.04662
## Proportion of Variance 0.00856 0.00803 0.00743 0.00712 0.00705 0.00667 0.00652
## Cumulative Proportion 0.78342 0.79146 0.79888 0.80600 0.81305 0.81972 0.82625
## PC22 PC23 PC24 PC25 PC26 PC27 PC28
## Standard deviation 2.02094 1.97710 1.96364 1.92130 1.85368 1.84701 1.79356
## Proportion of Variance 0.00636 0.00609 0.00601 0.00575 0.00535 0.00531 0.00501
## Cumulative Proportion 0.83261 0.83870 0.84470 0.85045 0.85580 0.86112 0.86613
## PC29 PC30 PC31 PC32 PC33 PC34 PC35
## Standard deviation 1.76072 1.74406 1.7362 1.70562 1.68979 1.65832 1.6025
## Proportion of Variance 0.00483 0.00474 0.0047 0.00453 0.00445 0.00428 0.0040
## Cumulative Proportion 0.87096 0.87570 0.8804 0.88492 0.88937 0.89365 0.8976
## PC36 PC37 PC38 PC39 PC40 PC41 PC42
## Standard deviation 1.57693 1.57490 1.54750 1.54280 1.52666 1.48846 1.44246
## Proportion of Variance 0.00387 0.00386 0.00373 0.00371 0.00363 0.00345 0.00324
## Cumulative Proportion 0.90153 0.90539 0.90912 0.91283 0.91646 0.91991 0.92315
## PC43 PC44 PC45 PC46 PC47 PC48 PC49
## Standard deviation 1.43197 1.41240 1.40638 1.37297 1.35808 1.33494 1.32301
## Proportion of Variance 0.00319 0.00311 0.00308 0.00294 0.00287 0.00278 0.00273
## Cumulative Proportion 0.92634 0.92945 0.93253 0.93547 0.93834 0.94112 0.94384
## PC50 PC51 PC52 PC53 PC54 PC55 PC56
## Standard deviation 1.28762 1.27373 1.25304 1.23111 1.22412 1.19898 1.17983
## Proportion of Variance 0.00258 0.00253 0.00245 0.00236 0.00233 0.00224 0.00217
## Cumulative Proportion 0.94643 0.94895 0.95140 0.95376 0.95609 0.95833 0.96050
## PC57 PC58 PC59 PC60 PC61 PC62 PC63
## Standard deviation 1.17186 1.12923 1.12008 1.08965 1.08360 1.0761 1.04978
## Proportion of Variance 0.00214 0.00199 0.00195 0.00185 0.00183 0.0018 0.00172
## Cumulative Proportion 0.96264 0.96463 0.96658 0.96843 0.97026 0.9721 0.97378
## PC64 PC65 PC66 PC67 PC68 PC69 PC70
## Standard deviation 1.03804 1.01841 0.98616 0.97879 0.95521 0.93119 0.92716
## Proportion of Variance 0.00168 0.00162 0.00151 0.00149 0.00142 0.00135 0.00134
## Cumulative Proportion 0.97546 0.97707 0.97859 0.98008 0.98150 0.98285 0.98419
## PC71 PC72 PC73 PC74 PC75 PC76 PC77
## Standard deviation 0.86058 0.84344 0.82900 0.79851 0.79280 0.76406 0.75222
## Proportion of Variance 0.00115 0.00111 0.00107 0.00099 0.00098 0.00091 0.00088
## Cumulative Proportion 0.98535 0.98645 0.98752 0.98852 0.98950 0.99041 0.99129
## PC78 PC79 PC80 PC81 PC82 PC83 PC84
## Standard deviation 0.7171 0.71142 0.69528 0.66705 0.64853 0.61729 0.59207
## Proportion of Variance 0.0008 0.00079 0.00075 0.00069 0.00066 0.00059 0.00055
## Cumulative Proportion 0.9921 0.99288 0.99363 0.99432 0.99498 0.99557 0.99612
## PC85 PC86 PC87 PC88 PC89 PC90 PC91
## Standard deviation 0.57318 0.54770 0.53620 0.51079 0.49581 0.48150 0.44311
## Proportion of Variance 0.00051 0.00047 0.00045 0.00041 0.00038 0.00036 0.00031
## Cumulative Proportion 0.99663 0.99710 0.99754 0.99795 0.99833 0.99869 0.99900
## PC92 PC93 PC94 PC95 PC96 PC97
## Standard deviation 0.40029 0.38510 0.36650 0.33527 0.29436 9.132e-15
## Proportion of Variance 0.00025 0.00023 0.00021 0.00018 0.00013 0.000e+00
## Cumulative Proportion 0.99925 0.99948 0.99969 0.99987 1.00000 1.000e+00
The first PCA accounts for 58.07% of the variance.
The second PCA accounts for 3.66% of variance by itself, 69.93% in cumulative with PCA1.
36 PC’s are necessary to account for 90% of the variance in the dataset.
The first PC appears to represent if the congress person supports pro-choice.
The second PC appears to represent voting on national debt, as there are many tea party members on the bottom left of the graph.
Murkowski and Collins are outliers, as they are moderate republicans as opposed to the more staunch republicans; for example, they are both pro-choice.
Yes, King and Sanders, and they more closely resemble democrats.
The democrats are more coherent in their voting.
It seems republicans are more spread on fiscal issues, as the tea party republicans are on the bottom left.
There also seems to be a trend in republicans where those who are more fiscally lenient are also more likely to vote pro-choice.