Multivariate Statistical Analysis
Final Project: Housing Prices
Patrick Oster
Monday April 29, 2019; 19:33:06 PM
Project Goal
The goal of this project is to apply the material introduced in this class on a real world dataset and create a formal report to represent all your work.
Instructions
- Write a summary and description of the dataset you use. You may use summary statistics, descriptive statistics, figures, etc.
- Clearly state the research questions and your analysis plan of the project.
- State at least two possible multivariate analysis methods you want to use in the analysis. You may also use other multivariate analysis methods that’s not covered in the book to do your analysis.
- State whether your data satisfy the assumptions, how you fit the models, how you use the output from these models to answer your research questions (include numerical summaries, graphics and interpretations).
- Compare the models or further discussion of your models, provide suggestions to other people if they want to use similar methods in their analysis.
- Address any problems you meet while analyzing the dataset.
- Discuss further questions raised by your study (that might be investigated in future research or analysis).
Report
Your grade on this project will be based on a written report. You should write up your analysis and report in the form of a short technical/research article, preferably in the format of a high-quality statistics or mathematics journal.
- Your main part of report should be 4-7 pages, your writeup should briefly describe the points listed in the previous section.
- An abstract highlighting the results from your analysis should be included before the body of the work.
- Graphs, figures and tables can be helpful to understand your findings. An appendix should include your code and/or computer output. (extra figures/tables and appendix are not included in the 4-7 page limit)
- Interesting computer results can be included in the body of the report.
- A bibliography is necessary for any references you make.
Grading
- 20% Description of the data, summary of the possible research questions
- 40% Appropriate and correct analysis procedures
- 20% Complexity of your decisions, suggestions, extensions
- 10% Well-written and attractive report meeting the guidelines set above
- 10% Spelling, grammar and punctuation
Dataset: House Price Data
dat <- read.csv("/Users/Patrick/Dropbox/data/mvar/kc_house_data.csv", header = TRUE)
df.num <- dat %>% select(price, floors, bedrooms, bathrooms, sqft_living, sqft_living15,
sqft_lot, sqft_lot15, sqft_above, sqft_basement, lat, long, zipcode, yr_built,
yr_renovated, waterfront, view, condition, grade)
## Partitioning numeric data to explore more thorougly ## Home Attributes
df.home <- df.num %>% select(price, floors, bedrooms, bathrooms, yr_built, yr_renovated,
view)
# Square Footage Measures
df.sqft <- df.num %>% select(price, sqft_living, sqft_living15, sqft_lot, sqft_lot15,
sqft_above, sqft_basement)
# Location info
df.loc <- df.num %>% select(price, lat, long, zipcode, waterfront, condition,
grade)Variable Descriptions
- id: a notation for a house
- date: date the house was sold
- price: prediction target
- bedrooms: number of bedrooms per house
- bathrooms: number of bathrooms per house
- sqft_living: square footage of the home
- sqft_lot: square footage of the lot
- floors: total floors (levels) in house
- waterfront: house which has a view to a waterfront
- view: has been viewed
- condition: how good the condition is
- grade: overall grade given to the housing unit based on the King County grading system
- sqft_above: square footage of house apart from basement
- sqft_basement: square footage of the basement
- yr_built: built year
- yr_renovated: year when the house was renovated
- zipcode: zip
- lat: latitude coordinate
- long: longitude coordinate
- sqft_living15: living room area in 2015 (implies some renovations) which may have affected lotsize area
- sqft_lot15: lot size area in 2015 (implies some renovations)
Research Questions
Methodology of Analysis
Initial Exploratory Data Analysis
Missingness
Click in a column to sort by the corresponding variable.
To regain use of the VIM GUI and the R console, click outside the plot region.Summary Statistics
| price | floors | bedrooms | bathrooms | yr_built | yr_renovated | view |
|---|---|---|---|---|---|---|
| Min. : 75000 | Min. :1.000 | Min. : 0.000 | Min. :0.000 | Min. :1900 | Min. : 0.0 | Min. :0.0000 |
| 1st Qu.: 321950 | 1st Qu.:1.000 | 1st Qu.: 3.000 | 1st Qu.:1.750 | 1st Qu.:1951 | 1st Qu.: 0.0 | 1st Qu.:0.0000 |
| Median : 450000 | Median :1.500 | Median : 3.000 | Median :2.250 | Median :1975 | Median : 0.0 | Median :0.0000 |
| Mean : 540088 | Mean :1.494 | Mean : 3.371 | Mean :2.115 | Mean :1971 | Mean : 84.4 | Mean :0.2343 |
| 3rd Qu.: 645000 | 3rd Qu.:2.000 | 3rd Qu.: 4.000 | 3rd Qu.:2.500 | 3rd Qu.:1997 | 3rd Qu.: 0.0 | 3rd Qu.:0.0000 |
| Max. :7700000 | Max. :3.500 | Max. :33.000 | Max. :8.000 | Max. :2015 | Max. :2015.0 | Max. :4.0000 |
| price | sqft_living | sqft_living15 | sqft_lot | sqft_lot15 | sqft_above | sqft_basement |
|---|---|---|---|---|---|---|
| Min. : 75000 | Min. : 290 | Min. : 399 | Min. : 520 | Min. : 651 | Min. : 290 | Min. : 0.0 |
| 1st Qu.: 321950 | 1st Qu.: 1427 | 1st Qu.:1490 | 1st Qu.: 5040 | 1st Qu.: 5100 | 1st Qu.:1190 | 1st Qu.: 0.0 |
| Median : 450000 | Median : 1910 | Median :1840 | Median : 7618 | Median : 7620 | Median :1560 | Median : 0.0 |
| Mean : 540088 | Mean : 2080 | Mean :1987 | Mean : 15107 | Mean : 12768 | Mean :1788 | Mean : 291.5 |
| 3rd Qu.: 645000 | 3rd Qu.: 2550 | 3rd Qu.:2360 | 3rd Qu.: 10688 | 3rd Qu.: 10083 | 3rd Qu.:2210 | 3rd Qu.: 560.0 |
| Max. :7700000 | Max. :13540 | Max. :6210 | Max. :1651359 | Max. :871200 | Max. :9410 | Max. :4820.0 |
| price | lat | long | zipcode | waterfront | condition | grade |
|---|---|---|---|---|---|---|
| Min. : 75000 | Min. :47.16 | Min. :-122.5 | Min. :98001 | Min. :0.000000 | Min. :1.000 | Min. : 1.000 |
| 1st Qu.: 321950 | 1st Qu.:47.47 | 1st Qu.:-122.3 | 1st Qu.:98033 | 1st Qu.:0.000000 | 1st Qu.:3.000 | 1st Qu.: 7.000 |
| Median : 450000 | Median :47.57 | Median :-122.2 | Median :98065 | Median :0.000000 | Median :3.000 | Median : 7.000 |
| Mean : 540088 | Mean :47.56 | Mean :-122.2 | Mean :98078 | Mean :0.007542 | Mean :3.409 | Mean : 7.657 |
| 3rd Qu.: 645000 | 3rd Qu.:47.68 | 3rd Qu.:-122.1 | 3rd Qu.:98118 | 3rd Qu.:0.000000 | 3rd Qu.:4.000 | 3rd Qu.: 8.000 |
| Max. :7700000 | Max. :47.78 | Max. :-121.3 | Max. :98199 | Max. :1.000000 | Max. :5.000 | Max. :13.000 |
Visualizing Correlation
Geo-Spatial Visualization
king_county <- map_data(map = "county", region = "washington") %>% filter(subregion ==
"king")
king_base <- ggplot(data = king_county, mapping = aes(x = long, y = lat, group = group)) +
coord_fixed(1.3) + geom_polygon(color = "black", fill = NA)
king_base + geom_point(data = dat, mapping = aes(x = long, y = lat, group = zipcode,
color = price))
Principal Component Analysis
Assumptions for PCA
Performing PCA
Call:
princomp(x = df.num)
Standard deviations:
Comp.1 Comp.2 Comp.3 Comp.4
367145.9651446508 46451.9474231992 16856.7971618432 943.1351452520
Comp.5 Comp.6 Comp.7 Comp.8
513.4688851286 408.6524693460 383.6711990638 50.4180676714
Comp.9 Comp.10 Comp.11 Comp.12
23.8452181540 0.7576857097 0.6752474139 0.6251892634
Comp.13 Comp.14 Comp.15 Comp.16
0.5828901824 0.4600428983 0.3527710841 0.1220866594
Comp.17 Comp.18 Comp.19
0.1021258504 0.0765978564 0.0001628953
19 variables and 21613 observations.Factor Analysis
Assumptions for Factor Analysis
Performing Factor Analysis
equation <- formula(~price + floors + bedrooms + bathrooms + sqft_living + sqft_lot +
sqft_basement + zipcode + yr_built + yr_renovated + waterfront + view +
condition + grade)
FA.norot <- factanal(equation, factors = 3, data = df.num, rotation = "none")
FA.vmax <- factanal(equation, factors = 3, data = df.num, rotation = "varimax")
FA.pmax <- factanal(equation, factors = 3, data = df.num, rotation = "varimax")Rotation Sums of Squared Loadings: the values of the table represent the distribution of the variance after the varimax rotation. Factor Matrix: this table contains the unrotated factor loadings, which are the correlations between the variable and the factor.
Rotated Factor Matrix: this table contains the rotated factor loadings (factor pattern matrix), which represent both how the variables are weighted for each factor but also the correlation between the variables and the factor.
No Rotation
Call:
factanal(x = equation, factors = 3, data = df.num, rotation = "none")
Uniquenesses:
price floors bedrooms bathrooms sqft_living
0.192 0.506 0.561 0.282 0.089
sqft_lot sqft_basement zipcode yr_built yr_renovated
0.976 0.514 0.873 0.315 0.942
waterfront view condition grade
0.871 0.762 0.823 0.245
Loadings:
Factor1 Factor2 Factor3
price 0.769 0.331 0.327
floors 0.439 -0.484 0.259
bedrooms 0.551 -0.368
bathrooms 0.811 -0.224
sqft_living 0.943 -0.131
sqft_lot 0.149
sqft_basement 0.380 0.460 -0.361
zipcode -0.204 0.266 0.121
yr_built 0.396 -0.727
yr_renovated 0.218
waterfront 0.142 0.199 0.264
view 0.317 0.308 0.205
condition 0.393 -0.118
grade 0.835 -0.141 0.194
Factor1 Factor2 Factor3
SS loadings 3.830 1.565 0.654
Proportion Var 0.274 0.112 0.047
Cumulative Var 0.274 0.385 0.432
Test of the hypothesis that 3 factors are sufficient.
The chi square statistic is 12931.95 on 52 degrees of freedom.
The p-value is 0
Length Class Mode
converged 1 -none- logical
loadings 42 loadings numeric
uniquenesses 14 -none- numeric
correlation 196 -none- numeric
criteria 3 -none- numeric
factors 1 -none- numeric
dof 1 -none- numeric
method 1 -none- character
STATISTIC 1 -none- numeric
PVAL 1 -none- numeric
n.obs 1 -none- numeric
call 5 -none- call Varimax Rotation
Call:
factanal(x = equation, factors = 3, data = df.num, rotation = "varimax")
Uniquenesses:
price floors bedrooms bathrooms sqft_living
0.192 0.506 0.561 0.282 0.089
sqft_lot sqft_basement zipcode yr_built yr_renovated
0.976 0.514 0.873 0.315 0.942
waterfront view condition grade
0.871 0.762 0.823 0.245
Loadings:
Factor1 Factor2 Factor3
price 0.482 0.130 0.748
floors 0.157 0.672 0.133
bedrooms 0.661
bathrooms 0.693 0.458 0.166
sqft_living 0.852 0.251 0.349
sqft_lot 0.145
sqft_basement 0.567 -0.382 0.137
zipcode -0.205 -0.257 0.137
yr_built 0.243 0.764 -0.206
yr_renovated -0.132 0.201
waterfront 0.359
view 0.179 0.449
condition -0.412
grade 0.557 0.506 0.435
Factor1 Factor2 Factor3
SS loadings 2.688 1.987 1.374
Proportion Var 0.192 0.142 0.098
Cumulative Var 0.192 0.334 0.432
Test of the hypothesis that 3 factors are sufficient.
The chi square statistic is 12931.95 on 52 degrees of freedom.
The p-value is 0
Length Class Mode
converged 1 -none- logical
loadings 42 loadings numeric
uniquenesses 14 -none- numeric
correlation 196 -none- numeric
criteria 3 -none- numeric
factors 1 -none- numeric
dof 1 -none- numeric
method 1 -none- character
rotmat 9 -none- numeric
STATISTIC 1 -none- numeric
PVAL 1 -none- numeric
n.obs 1 -none- numeric
call 5 -none- call Promax Rotation
Call:
factanal(x = equation, factors = 3, data = df.num, rotation = "varimax")
Uniquenesses:
price floors bedrooms bathrooms sqft_living
0.192 0.506 0.561 0.282 0.089
sqft_lot sqft_basement zipcode yr_built yr_renovated
0.976 0.514 0.873 0.315 0.942
waterfront view condition grade
0.871 0.762 0.823 0.245
Loadings:
Factor1 Factor2 Factor3
price 0.482 0.130 0.748
floors 0.157 0.672 0.133
bedrooms 0.661
bathrooms 0.693 0.458 0.166
sqft_living 0.852 0.251 0.349
sqft_lot 0.145
sqft_basement 0.567 -0.382 0.137
zipcode -0.205 -0.257 0.137
yr_built 0.243 0.764 -0.206
yr_renovated -0.132 0.201
waterfront 0.359
view 0.179 0.449
condition -0.412
grade 0.557 0.506 0.435
Factor1 Factor2 Factor3
SS loadings 2.688 1.987 1.374
Proportion Var 0.192 0.142 0.098
Cumulative Var 0.192 0.334 0.432
Test of the hypothesis that 3 factors are sufficient.
The chi square statistic is 12931.95 on 52 degrees of freedom.
The p-value is 0
Length Class Mode
converged 1 -none- logical
loadings 42 loadings numeric
uniquenesses 14 -none- numeric
correlation 196 -none- numeric
criteria 3 -none- numeric
factors 1 -none- numeric
dof 1 -none- numeric
method 1 -none- character
rotmat 9 -none- numeric
STATISTIC 1 -none- numeric
PVAL 1 -none- numeric
n.obs 1 -none- numeric
call 5 -none- call