Preliminary hypotheses

Just by looking at the varibles above, one can make few assumptions regarding the inter-relationships between different variables. Let’s start with some assumptions:

Here, the main goal of the analysis is to study how crime rate is associated with other aspects in Boston. After going through the background information, it is a bit easier to identify interesting variables that could be related to criminal activities. Personally, I believe that the following are the four interesting variables that are associated with crime rates:

1. Accessibility to radial highways (rad) : In my opinion, accessibility to radial highways plays vital role when it comes to crime related activities. Criminals may prefer towns with rapid access to radial highways especially for escaping purpose. Therefore, I believe that accessibility to radial highways may contribute to higher crime rates.

2. Value of owner-occupied homes (medv) : Usually, the crime-related activities are common with individuals who do not have enough property. They say that one of the main reasons for crime is poverty. Individual with basic income may also have homes with lower value but I think we can also correlate the value of house one has with crime-rate. In other words, value of owner-occupied home should be negatively correlated with crime.

3. Distance to employment centers in Boston (dis) : I hypothesize that the farther the residential area from employment centers, the lower the crime rates.

4. Proportion of blacks(black) : There have been an impression that blacks commit more crime compared to whites. Therefore, I hypothesize that proportion of blacks is positively correlated with crime rate.

In the following section, we will see in details how my hypotheses are explained by the data.

Non-graphical data exploration

Non-graphical data exploration is the first step before we start analyzing the data. This step different aspects of the data here we will cover three of those:

1. measures of central tendency (mean, meadian)
2. measures of variability (interquartile range, standard deviation)
3. relationships between variables (correlation)

Now, let’s first look at the summary of the boston data in the form of table (instead of default layout) using pandoc.table function of pander package.

pandoc.table(summary(Boston), caption = "Summary of Boston data", split.table = 100)

## 
## --------------------------------------------------------------------------------------
##        crim               zn             indus            chas              nox       
## ------------------ ---------------- --------------- ----------------- ----------------
##  Min.  : 0.00632     Min.  : 0.00    Min.  : 0.46    Min.  :0.00000    Min.  :0.3850  
## 
##  1st Qu.: 0.08204   1st Qu.: 0.00    1st Qu.: 5.19   1st Qu.:0.00000   1st Qu.:0.4490 
## 
##  Median : 0.25651   Median : 0.00    Median : 9.69   Median :0.00000   Median :0.5380 
## 
##   Mean : 3.61352     Mean : 11.36     Mean :11.14     Mean :0.06917     Mean :0.5547  
## 
##  3rd Qu.: 3.67708   3rd Qu.: 12.50   3rd Qu.:18.10   3rd Qu.:0.00000   3rd Qu.:0.6240 
## 
##  Max.  :88.97620    Max.  :100.00    Max.  :27.74    Max.  :1.00000    Max.  :0.8710  
## --------------------------------------------------------------------------------------
## 
## Table: Summary of Boston data (continued below)
## 
##  
## --------------------------------------------------------------------------------------------------
##       rm              age              dis              rad              tax           ptratio    
## --------------- ---------------- ---------------- ---------------- --------------- ---------------
##  Min.  :3.561     Min.  : 2.90    Min.  : 1.130    Min.  : 1.000    Min.  :187.0    Min.  :12.60  
## 
##  1st Qu.:5.886   1st Qu.: 45.02   1st Qu.: 2.100   1st Qu.: 4.000   1st Qu.:279.0   1st Qu.:17.40 
## 
##  Median :6.208   Median : 77.50   Median : 3.207   Median : 5.000   Median :330.0   Median :19.05 
## 
##   Mean :6.285     Mean : 68.57     Mean : 3.795     Mean : 9.549     Mean :408.2     Mean :18.46  
## 
##  3rd Qu.:6.623   3rd Qu.: 94.08   3rd Qu.: 5.188   3rd Qu.:24.000   3rd Qu.:666.0   3rd Qu.:20.20 
## 
##  Max.  :8.780    Max.  :100.00    Max.  :12.127    Max.  :24.000    Max.  :711.0    Max.  :22.00  
## --------------------------------------------------------------------------------------------------
## 
## Table: Table continues below
## 
##  
## ------------------------------------------------
##      black            lstat           medv      
## ---------------- --------------- ---------------
##   Min.  : 0.32    Min.  : 1.73    Min.  : 5.00  
## 
##  1st Qu.:375.38   1st Qu.: 6.95   1st Qu.:17.02 
## 
##  Median :391.44   Median :11.36   Median :21.20 
## 
##   Mean :356.67     Mean :12.65     Mean :22.53  
## 
##  3rd Qu.:396.23   3rd Qu.:16.95   3rd Qu.:25.00 
## 
##  Max.  :396.90    Max.  :37.97    Max.  :50.00  
## ------------------------------------------------

After getting a statistical summary of, it’s worthwhile to see to what extent each variables are correlated. For that, we use corr() function on Boston data.

corr_boston<-cor(Boston) %>% round(2)
pandoc.table(corr_boston, split.table = 100)

## 
## ---------------------------------------------------------------------------------------------
##           crim     zn     indus   chas     nox     rm      age     dis     rad     tax  
## ------------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
##   **crim**       1     -0.2    0.41    -0.06   0.42    -0.22   0.35    -0.38   0.63    0.58  
## 
##    **zn**      -0.2      1     -0.53   -0.04   -0.52   0.31    -0.57   0.66    -0.31   -0.31 
## 
##   **indus**    0.41    -0.53     1     0.06    0.76    -0.39   0.64    -0.71    0.6    0.72  
## 
##   **chas**     -0.06   -0.04   0.06      1     0.09    0.09    0.09    -0.1    -0.01   -0.04 
## 
##    **nox**     0.42    -0.52   0.76    0.09      1     -0.3    0.73    -0.77   0.61    0.67  
## 
##    **rm**      -0.22   0.31    -0.39   0.09    -0.3      1     -0.24   0.21    -0.21   -0.29 
## 
##    **age**     0.35    -0.57   0.64    0.09    0.73    -0.24     1     -0.75   0.46    0.51  
## 
##    **dis**     -0.38   0.66    -0.71   -0.1    -0.77   0.21    -0.75     1     -0.49   -0.53 
## 
##    **rad**     0.63    -0.31    0.6    -0.01   0.61    -0.21   0.46    -0.49     1     0.91  
## 
##    **tax**     0.58    -0.31   0.72    -0.04   0.67    -0.29   0.51    -0.53   0.91      1   
## 
##  **ptratio**   0.29    -0.39   0.38    -0.12   0.19    -0.36   0.26    -0.23   0.46    0.46  
## 
##   **black**    -0.39   0.18    -0.36   0.05    -0.38   0.13    -0.27   0.29    -0.44   -0.44 
## 
##   **lstat**    0.46    -0.41    0.6    -0.05   0.59    -0.61    0.6    -0.5    0.49    0.54  
## 
##   **medv**     -0.39   0.36    -0.48   0.18    -0.43    0.7    -0.38   0.25    -0.38   -0.47 
## ---------------------------------------------------------------------------------------------
## 
## Table: Table continues below
## 
##  
## -----------------------------------------------
##           ptratio   black   lstat   medv  
## ------------- --------- ------- ------- -------
##   **crim**      0.29     -0.39   0.46    -0.39 
## 
##    **zn**       -0.39    0.18    -0.41   0.36  
## 
##   **indus**     0.38     -0.36    0.6    -0.48 
## 
##   **chas**      -0.12    0.05    -0.05   0.18  
## 
##    **nox**      0.19     -0.38   0.59    -0.43 
## 
##    **rm**       -0.36    0.13    -0.61    0.7  
## 
##    **age**      0.26     -0.27    0.6    -0.38 
## 
##    **dis**      -0.23    0.29    -0.5    0.25  
## 
##    **rad**      0.46     -0.44   0.49    -0.38 
## 
##    **tax**      0.46     -0.44   0.54    -0.47 
## 
##  **ptratio**      1      -0.18   0.37    -0.51 
## 
##   **black**     -0.18      1     -0.37   0.33  
## 
##   **lstat**     0.37     -0.37     1     -0.74 
## 
##   **medv**      -0.51    0.33    -0.74     1   
## -----------------------------------------------

The table above shows the correlation matrix of all variables. Bird’s eye view on the matrix shows that tax (full-value property-tax rate) and rad (index of accessibility to radial highways) are the most positively correlated variables, whereas dis (weighted mean of distances to five Boston employment centres) and age (proportion of owner-occupied units built prior to 1940) are the most negatively correlated variables. Moreover, chas (Charles river dummy variable) and rad are the two variables that are least correlated.

Multiple Linear Regression

Multiple linear regression is commonly used method for most of the the regression analyses. As a predictive analysis method, multiple linear regression describes given data and explain the relationship between one dependent variable and two or more independent variables. Our research question for the multiple linear regression is as follows: Can we explain the crime rate in Boston area based on the four variables that were hypothesized earlier in this report?

First, we need to check whether there is a linear relationship between the independent variables and the dependent variable in our multiple linear regression model. After that, we fit the model simply using lm() function and call output usig summary() function.

#let's first create subset of Boston data (my_Bos) using five variables kept in (my_var)
myvar<-c("crim", "black", "medv", "dis", "rad")
my_Bos<-select(Boston, one_of(myvar))
ggpairs(my_Bos)

Here, the crime variable is positively correlated with rad (0.626) and negatively correlated with black (-0.385), medv (-0.388) and dis.(-0.38)

#fit linear model
lm_crim<-lm(crim~. , data=my_Bos)
#summarize the model
summary(lm_crim)

## 
## Call:
## lm(formula = crim ~ ., data = my_Bos)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -10.816  -1.668  -0.468   0.738  76.323 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  6.529163   1.730608   3.773 0.000181 ***
## black       -0.009053   0.003633  -2.492 0.013021 *  
## medv        -0.141722   0.034902  -4.061 5.68e-05 ***
## dis         -0.292778   0.159593  -1.835 0.067168 .  
## rad          0.483580   0.042263  11.442  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6.524 on 501 degrees of freedom
## Multiple R-squared:  0.4293, Adjusted R-squared:  0.4247 
## F-statistic:  94.2 on 4 and 501 DF,  p-value: < 2.2e-16

From the output, residuals represent the difference between the actual values i.e the crime rate (dependent variable) in Boston and each of the four explanatory variables. The next section shows the coefficients of the model. From the output, we saw that the intercept is 6.529163 with coefficients for dependant variables being -0.009053(black), -0.141722(medv), -0.292778(dis) and 0.483580(rad). Next, we can also see Standard errors for the coefficients which measure the average amount that the coefficient estimates differ from the dependent variables. Moreover, the t-value measures how far the coefficient estimation is away from zero with the assumption that the farther the better. In other words, we reject the null hypothesis when it’s far from zero.

In the above model, the t-values are farthest in rad followed by medv, black and dis. These values are also larger compared to standard error except dis, which indiciate that these variables do have some relationship with the crime. The last column indicates the probability for getting any value >= t. In general, a p-value of 0.05 is good cut-off for significance. Thus, we can see that three (black, medv and rad) out of four choosen variables were significant.

The residual standard error indicates the quality of the regression model. This value some how gives how much we can be confident of our model. Here, the residual error is based on 501 degrees of freedom, where the degrees of freedom are data points used for estimating the parameters.

Finally, I would like to highlight to what extent my assumptions have been addressed by the model. Although I hypothesized that all of my four chosen variables have impact on crime, it turned out that the distance to employment centers (dis) is not significant variable to predict crime rate. Besides that, all three variables were significant with rad being the most significant one followed by medv and black.

Interpretation (model validation)

The three different diagnostic plots are generated above.The assumptions behind all three models is linearity and normality. Based on the above plots, we can conclude that the errors are normally distributed (clearly observed in q-q plot). Similarly, residual versus fitted model showed that the errors are not dependent on the crime variable. Moreover, we can see that few outliers (381, 406 and 410) have influences to the assumption that is clearly represented in case of residual vs leverage model and can be seen in other two plots as well. All the models have adressed the outliers nicely. Thus, assumptions in all models are more or less valid.

Session information:

sessionInfo()

## R version 3.4.2 (2017-09-28)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 17.10
## 
## Matrix products: default
## BLAS: /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.7.1
## LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.7.1
## 
## locale:
##  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
##  [3] LC_TIME=fi_FI.UTF-8        LC_COLLATE=en_US.UTF-8    
##  [5] LC_MONETARY=fi_FI.UTF-8    LC_MESSAGES=en_US.UTF-8   
##  [7] LC_PAPER=fi_FI.UTF-8       LC_NAME=C                 
##  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
## [11] LC_MEASUREMENT=fi_FI.UTF-8 LC_IDENTIFICATION=C       
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
## [1] GGally_1.3.2  ggplot2_2.2.1 dplyr_0.7.4   corrplot_0.84 pander_0.6.1 
## [6] MASS_7.3-47  
## 
## loaded via a namespace (and not attached):
##  [1] Rcpp_0.12.12       knitr_1.16         bindr_0.1         
##  [4] magrittr_1.5       munsell_0.4.3      colorspace_1.3-2  
##  [7] R6_2.2.2           rlang_0.1.4        plyr_1.8.4        
## [10] stringr_1.2.0      tools_3.4.2        grid_3.4.2        
## [13] gtable_0.2.0       htmltools_0.3.6    lazyeval_0.2.0    
## [16] yaml_2.1.14        rprojroot_1.2      digest_0.6.12     
## [19] assertthat_0.2.0   tibble_1.3.4       bindrcpp_0.2      
## [22] reshape2_1.4.2     RColorBrewer_1.1-2 glue_1.2.0        
## [25] evaluate_0.10.1    rmarkdown_1.8      labeling_0.3      
## [28] stringi_1.1.5      compiler_3.4.2     scales_0.4.1      
## [31] backports_1.1.0    reshape_0.8.7      pkgconfig_2.0.1