This is a challenging dataset, in part because it’s real and messy. I will guide you through a simplified sensible analysis, but other models are possible.

Note that I needed to set cache=FALSE to assure all output was updated.

1 ANCOVA model: Albuquerque NM 87108, House and Apartment listing prices

Prof Erhardt constructed a dataset of listing prices for dwellings (homes and apartments) for sale from Zillow.com on Feb 26, 2016 at 1 PM for Albuquerque NM 87108. In this assignment we’ll develop a model to help understand which qualities that contribute to a typical dwelling’s listing price. We will then also predict the listing prices of new listings posted on the following day, Feb 27, 2016 by 2 PM.

Because we want to model a typical dwelling, it is completely reasonable to remove “unusual” dwellings from the dataset. Dwellings have a distribution with a long tail!

1.1 Unusual assignment, not top-down, but up-down-up-down

This is an unusual assignment because the workflow of this assignment isn’t top-down; instead, you’ll be scrolling up and down as you make decisions about the data and model you’re fitting. Yes, I have much of the code worked out for you. However, there are data decisions to make early in the code (such as excluding observations, transforming variables, etc.) that depend on the analysis (model checking) later. Think of it as a “choose your own adventure” that I’ve written for you.

1.1.1 Keep a record of your decisions

It is always desirable to make your work reproducible, either by someone else or by your future self. For each step you take, keep a diary of (a) what the next minor goal is, (b) what evidence/information you have, (c) what decision you make, and (d) what the outcome was.

For example, here’s the first couple steps of your diary:

  1. Include only “typical dwellings”. Based on scatterplot, remove extreme observations. Keep only HOUSE and APARTMENT.
  2. Exclude a few variables to reduce multicollinearity between predictor variables. Exclude Baths and LotSize.
  3. etc.

1.2 (2 p) (Step 1) Restrict data to “typical” dwellings

Step 1: After looking at the scatterplot below, identify what you consider to be a “typical dwelling” and exclude observations far from that range. For example, there are only a couple TypeSale that are common enough to model; remember to run factor() again to remove factor levels that no longer appear.

library(tidyverse)

# load ada functions
source("ada_functions.R")

# First, download the data to your computer,
#   save in the same folder as this Rmd file.

# read the data, skip the first two comment lines of the data file
dat_abq <-
  read_csv("ADA2_CL_14_HomePricesZillow_Abq87108.csv", skip=2) %>%
  mutate(
    id = 1:n()
  , TypeSale = factor(TypeSale)
    # To help scale the intercept to a more reasonable value
    #   Scaling the x-variables are sometimes done to the mean of each x.
    # center year at 1900 (negative values are older, -10 is built in 1890)
  , YearBuilt_1900 = YearBuilt - 1900
  ) %>%
  select(
    id, everything()
    , -Address, -YearBuilt
  )

head(dat_abq)
# A tibble: 6 x 9
     id TypeSale PriceList  Beds Baths Size_sqft LotSize DaysListed
  <int> <fct>        <dbl> <dbl> <dbl>     <dbl>   <dbl>      <dbl>
1     1 HOUSE       186900     3     2      1305    6969          0
2     2 APARTME~    305000     1     1      2523    6098          0
3     3 APARTME~    244000     1     1      2816    6098          0
4     4 CONDO       108000     3     2      1137      NA          0
5     5 CONDO        64900     2     1      1000      NA          1
6     6 HOUSE       275000     3     3      2022    6098          1
# ... with 1 more variable: YearBuilt_1900 <dbl>
## RETURN HERE TO SUBSET THE DATA

dat_abq <-
  dat_abq %>%
  filter(
    TRUE    # (X <= z)  # keep observations where variable X <= value z
  )
# note, if you remove a level from a categorical variable, then run factor() again

  # SOLUTION
  # these deletions are based only on the scatter plot in order to have
  #  "typical" dwellings




str(dat_abq)
tibble [143 x 9] (S3: tbl_df/tbl/data.frame)
 $ id            : int [1:143] 1 2 3 4 5 6 7 8 9 10 ...
 $ TypeSale      : Factor w/ 5 levels "APARTMENT","CONDO",..: 5 1 1 2 2 5 5 4 5 5 ...
 $ PriceList     : num [1:143] 186900 305000 244000 108000 64900 ...
 $ Beds          : num [1:143] 3 1 1 3 2 3 2 2 3 3 ...
 $ Baths         : num [1:143] 2 1 1 2 1 3 1 2 2 2 ...
 $ Size_sqft     : num [1:143] 1305 2523 2816 1137 1000 ...
 $ LotSize       : num [1:143] 6969 6098 6098 NA NA ...
 $ DaysListed    : num [1:143] 0 0 0 0 1 1 1 1 1 2 ...
 $ YearBuilt_1900: num [1:143] 54 48 89 96 85 52 52 65 58 52 ...

1.3 (2 p) (Step 3) Transform response, if necessary.

Step 3: Does the response variable require a transformation? If so, what transformation is recommended from the model diagnostic plots (Box-Cox)?

1.3.1 Solution

[answer]

dat_abq <-
  dat_abq %>%
  mutate(
    # Price in units of $1000
    PriceListK = PriceList / 1000

    # SOLUTION
  ) %>%
  select(
    -PriceList
  )

str(dat_abq)
tibble [143 x 9] (S3: tbl_df/tbl/data.frame)
 $ id            : int [1:143] 1 2 3 4 5 6 7 8 9 10 ...
 $ TypeSale      : Factor w/ 5 levels "APARTMENT","CONDO",..: 5 1 1 2 2 5 5 4 5 5 ...
 $ Beds          : num [1:143] 3 1 1 3 2 3 2 2 3 3 ...
 $ Baths         : num [1:143] 2 1 1 2 1 3 1 2 2 2 ...
 $ Size_sqft     : num [1:143] 1305 2523 2816 1137 1000 ...
 $ LotSize       : num [1:143] 6969 6098 6098 NA NA ...
 $ DaysListed    : num [1:143] 0 0 0 0 1 1 1 1 1 2 ...
 $ YearBuilt_1900: num [1:143] 54 48 89 96 85 52 52 65 58 52 ...
 $ PriceListK    : num [1:143] 186.9 305 244 108 64.9 ...

1.4 (2 p) (Step 4) Remove extremely influential observations.

Step 4: The goal is to develop a model that will work well for the typical dwellings. If an observation is highly influential, then it’s unusual.

## Remove influential observation
  dat_abq <-
    dat_abq %>%
    filter(
      TRUE   # !(id  %in% c( ... ))
    )

  # SOLUTION

1.5 Subset data for model building and prediction

Create a subset of the data for building the model, and another subset for prediction later on.

# remove observations with NAs
dat_abq <-
  dat_abq %>%
  na.omit()

# the data subset we will use to build our model
dat_sub <-
  dat_abq %>%
  filter(
    DaysListed > 0
  )

# the data subset we will predict from our model
dat_pred <-
  dat_abq %>%
  filter(
    DaysListed == 0
  ) %>%
  mutate(
    # the prices we hope to predict closely from our model
    PriceListK_true = PriceListK
    # set them to NA to predict them later
  , PriceListK = NA
  )

Scatterplot of the model-building subset.

# NOTE, this plot takes a long time if you're repeadly recompiling the document.
# comment the "print(p)" line so save some time when you're not evaluating this plot.
library(GGally)
library(ggplot2)
p <-
  ggpairs(
    dat_sub
  , mapping = ggplot2::aes(colour = TypeSale, alpha = 0.5)
  , lower = list(continuous = "points")
  , upper = list(continuous = "cor")
  , progress = FALSE
  )
#print(p)

There are clearly some unusual observations. Go back to the first code chunk and remove some observations that don’t represent a “typical” dwelling.

For example, remove these dwellings (in code above):

  • Super expensive dwelling
  • Dwellings with huge lots
  • Dwellings that were listed for years
  • Because most dwellings were APARTMENTs and HOUSEs, remove the others (there are only 1 or so of each).

Discuss the observed correlations or other outstanding features in the data.

1.5.1 Solution

[answer]

Features of data:

1.6 (2 p) (Step 2) Fit full two-way interaction model.

You’ll revisit this section after each modification of the data above.

Step 2: Let’s fit the full two-way interaction model and assess the assumptions. However, some of the predictor variables are highly correlated. Recall that the interpretation of a beta coefficient is “the expected increase in the response for a 1-unit increase in \(x\) with all other predictors held constant”. It’s hard to hold one variable constant if it’s correlated with another variable you’re increasing. Therefore, we’ll make a decision to retain some variables but not others depending on their correlation values. (In the PCA chapter, we’ll see another strategy.)

Somewhat arbitrarily, let’s exclude Baths (since highly correlated with Beds and Size_sqft). Let’s also exclude LotSize (since highly correlated with Size_sqft). Modify the code below. Notice that because APARTMENTs don’t have more than 1 Beds or Baths, those interaction terms need to be excluded from the model; I show you how to do this manually using the update() function.

Note that the formula below y ~ (x1 + x2 + x3)^2 expands into all main effects and two-way interactions.

  ## SOLUTION
  lm_full <-
    lm(
      PriceListK ~ (TypeSale + Beds + Size_sqft + DaysListed + YearBuilt_1900)^2
    , data = dat_sub
    )
  #lm_full <-
  #  lm(
  #    PriceListK ~ (Beds + Baths + Size_sqft + LotSize + DaysListed + YearBuilt_1900)^2
  #  , data = dat_sub
  #  )
  lm_full

Call:
lm(formula = PriceListK ~ (TypeSale + Beds + Size_sqft + DaysListed + 
    YearBuilt_1900)^2, data = dat_sub)

Coefficients:
                             (Intercept)  
                               2.183e+00  
               TypeSaleFOR SALE BY OWNER  
                               1.447e+02  
                     TypeSaleFORECLOSURE  
                               2.084e+03  
                           TypeSaleHOUSE  
                               2.859e+02  
                                    Beds  
                              -1.299e+02  
                               Size_sqft  
                               1.445e-01  
                              DaysListed  
                              -2.441e-01  
                          YearBuilt_1900  
                               3.642e-01  
          TypeSaleFOR SALE BY OWNER:Beds  
                                      NA  
                TypeSaleFORECLOSURE:Beds  
                              -1.706e+02  
                      TypeSaleHOUSE:Beds  
                                      NA  
     TypeSaleFOR SALE BY OWNER:Size_sqft  
                                      NA  
           TypeSaleFORECLOSURE:Size_sqft  
                               3.828e-01  
                 TypeSaleHOUSE:Size_sqft  
                               3.209e-02  
    TypeSaleFOR SALE BY OWNER:DaysListed  
                                      NA  
          TypeSaleFORECLOSURE:DaysListed  
                              -1.066e+00  
                TypeSaleHOUSE:DaysListed  
                               3.068e-01  
TypeSaleFOR SALE BY OWNER:YearBuilt_1900  
                                      NA  
      TypeSaleFORECLOSURE:YearBuilt_1900  
                              -3.516e+01  
            TypeSaleHOUSE:YearBuilt_1900  
                              -5.218e+00  
                          Beds:Size_sqft  
                              -4.801e-04  
                         Beds:DaysListed  
                              -1.260e-01  
                     Beds:YearBuilt_1900  
                               2.629e+00  
                    Size_sqft:DaysListed  
                               2.270e-04  
                Size_sqft:YearBuilt_1900  
                              -1.584e-03  
               DaysListed:YearBuilt_1900  
                              -2.272e-03  
  library(car)
  try(Anova(lm_full, type=3))
Error in Anova.III.lm(mod, error, singular.ok = singular.ok, ...) : 
  there are aliased coefficients in the model
  ## Note that this doesn't work because APARTMENTs only have 1 bed and 1 bath.
  ## There isn't a second level of bed or bath to estimate the interaction.
  ## Therefore, remove those two terms
  lm_full <-
    update(
      lm_full
    , . ~ . - TypeSale:Beds
    )
  library(car)
  try(Anova(lm_full, type=3))
Error in Anova.III.lm(mod, error, singular.ok = singular.ok, ...) : 
  there are aliased coefficients in the model
## Uncomment this line when you're ready to assess the model assumptions
# plot diagnostics
#lm_diag_plots(lm_full)

# List the row numbers with id numbers
#   The row numbers appear in the residual plots.
#   The id number can be used to exclude values in code above.
dat_sub %>% select(id) %>% print(n = Inf)
# A tibble: 114 x 1
       id
    <int>
  1     6
  2     7
  3     8
  4     9
  5    10
  6    12
  7    13
  8    14
  9    15
 10    17
 11    18
 12    19
 13    20
 14    21
 15    22
 16    24
 17    27
 18    29
 19    30
 20    31
 21    32
 22    33
 23    35
 24    36
 25    37
 26    38
 27    39
 28    40
 29    41
 30    42
 31    45
 32    46
 33    47
 34    48
 35    49
 36    50
 37    51
 38    52
 39    53
 40    54
 41    55
 42    56
 43    57
 44    58
 45    59
 46    60
 47    61
 48    62
 49    64
 50    65
 51    66
 52    67
 53    68
 54    69
 55    70
 56    71
 57    72
 58    77
 59    78
 60    79
 61    80
 62    81
 63    83
 64    84
 65    85
 66    86
 67    87
 68    91
 69    92
 70    93
 71    94
 72    95
 73    96
 74    99
 75   100
 76   101
 77   102
 78   103
 79   104
 80   105
 81   106
 82   108
 83   109
 84   110
 85   111
 86   113
 87   114
 88   115
 89   116
 90   117
 91   119
 92   120
 93   121
 94   122
 95   123
 96   124
 97   125
 98   126
 99   127
100   128
101   129
102   130
103   131
104   132
105   133
106   134
107   135
108   136
109   137
110   138
111   140
112   141
113   142
114   143

After Step 2, interpret the residual plots. What are the primary issues in the original model?

1.6.1 Solution

[answer]

1.7 (2 p) (Step 5) Model selection, check model assumptions.

Using step(..., direction="both") with the BIC criterion, perform model selection.

1.7.1 Solution

## BIC
# option: test="F" includes additional information
#           for parameter estimate tests that we're familiar with
# option: for BIC, include k=log(nrow( [data.frame name] ))
lm_red_BIC <-
  step(
    lm_full
  , direction = "both"
  , test = "F"
  , trace = 0
  , k = log(nrow(dat_sub))
  )
lm_final <- lm_red_BIC
## Uncomment this line when you're ready to assess the model assumptions
# plot diagnostics
#lm_diag_plots(lm_final)

Model assumptions appear to be reasonably met. A few influential observations exist.

1.8 (4 p) (Step 6) Plot final model, interpret coefficients.

If you arrived at the same model I did, then the code below will plot it. Eventually (after Step 7), the fitted model equations will describe the each dwelling TypeSale and interpret the coefficients.

library(car)
#Anova(lm_final, type=3)
summary(lm_final)

Call:
lm(formula = PriceListK ~ TypeSale + Size_sqft + YearBuilt_1900 + 
    Size_sqft:YearBuilt_1900, data = dat_sub)

Residuals:
    Min      1Q  Median      3Q     Max 
-184.16  -41.71   -1.96   28.74  417.46 

Coefficients:
                            Estimate Std. Error t value Pr(>|t|)    
(Intercept)               -3.416e+02  9.358e+01  -3.650 0.000407 ***
TypeSaleFOR SALE BY OWNER  1.160e+02  7.686e+01   1.509 0.134125    
TypeSaleFORECLOSURE        9.160e+00  3.669e+01   0.250 0.803342    
TypeSaleHOUSE              1.045e+02  1.783e+01   5.862 5.13e-08 ***
Size_sqft                  2.592e-01  3.855e-02   6.724 8.93e-10 ***
YearBuilt_1900             4.883e+00  1.630e+00   2.995 0.003409 ** 
Size_sqft:YearBuilt_1900  -2.750e-03  6.432e-04  -4.276 4.14e-05 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 75.1 on 107 degrees of freedom
Multiple R-squared:  0.9377,    Adjusted R-squared:  0.9342 
F-statistic: 268.6 on 6 and 107 DF,  p-value: < 2.2e-16

Fitted model equation is \[ \widehat{\log_{10}(\textrm{PriceList})} = -342 + 116 I(\textrm{TypeSale} = \textrm{HOUSE}) + 9.16 \log_{10}(\textrm{Size_sqft}) \]

1.8.1 Solution

After Step 7, return and intepret the model coefficients above.

[answer]

1.9 (2 p) (Step 7) Transform predictors.

We now have enough information to see that a transformation of a predictor can be useful. See the curvature with Size_sqft? This is one of the headaches of regression modelling, everything depends on everything else and you learn as you go. Return to the top and transform Size_sqft and LotSize.

A nice feature of this transformation is that the model interaction goes away. Our interpretation is now on the log scale, but it’s a simpler model.

1.10 (4 p) (Step 8) Predict new observations, interpret model’s predictive ability.

Using the predict() function, we’ll input the data we held out to predict earlier, and use our final model to predict the PriceListK response. Note that 10^lm_pred is the table of values on the scale of “thousands of dollars”.

Interpret the predictions below the output.

How well do you expect this model to predict? Justify your answer.

# predict new observations, convert to data frame
lm_pred <-
  as.data.frame(
    predict(
      lm_final
    , newdata = dat_pred
    , interval = "prediction"
    )
  ) %>%
  mutate(
    # add column of actual list prices
    PriceListK = dat_pred$PriceListK_true
  )
lm_pred
       fit       lwr      upr PriceListK
1 171.0553  21.13390 320.9766      186.9
2 213.6503  60.32045 366.9801      305.0
3 133.5463 -21.88382 288.9764      244.0
# on "thousands of dollars" scale
10^lm_pred
            fit          lwr           upr    PriceListK
1 1.135702e+171 1.361129e+21           Inf 7.943282e+186
2 4.469914e+213 2.091480e+60           Inf 1.000000e+305
3 3.517992e+133 1.306721e-22 9.471238e+288 1.000000e+244
# attributes of the three predicted observations
dat_pred %>% print(n = Inf, width = Inf)
# A tibble: 3 x 10
     id TypeSale   Beds Baths Size_sqft LotSize DaysListed YearBuilt_1900
  <int> <fct>     <dbl> <dbl>     <dbl>   <dbl>      <dbl>          <dbl>
1     1 HOUSE         3     2      1305    6969          0             54
2     2 APARTMENT     1     1      2523    6098          0             48
3     3 APARTMENT     1     1      2816    6098          0             89
  PriceListK PriceListK_true
  <lgl>                <dbl>
1 NA                    187.
2 NA                    305 
3 NA                    244 

1.10.1 Solution

[answer]