In-Class Exercise 5

Author

KB

First published on 17-Dec-2022

7 Modeling the Spatial Variation of the Explanatory Factors of Water Point Status using Geographically Weighted Logistic Regression

7.1 Overview

In this exercise, we will build an explanatory model to discover factors affecting the water point status of Osun State in Nigeria. Osun is a state in southwestern Nigeria and is named after River Osun - a river which flows through the state. The state was established in Aug-1991 and is made up of 30 Local Government Areas (LGAs).

7.2 The Data

Two pre-processed data sets are used to build the explanatory model. They are:

  • Osun.rds - it contains LGA boundaries of Osun State. It is in sf polygon data frame, and

  • Osun_wp_sf.rds - it contains water points within the Osun State. It is in sf point data frame.

7.3 Model Variables

For the Logistic Regression Model that we are building, the following variables on water points are used:

  • Dependent variable: Water point status:

    • Class 0: Non-functional water points

    • Class 1: Functional water points.

      Water points with “Unknown” or “NA” status are excluded during pre-processing

  • Independent variables:

    • distance_to_primary_road

    • distance_to_secondary_road

    • distance_to_tertiary_road

    • distance_to_city

    • distance_to_town

    • water_point_population

    • local_population_1km

    • usage_capacity

    • is_urban

    • water_source_clean

      The first 7 variables are continuous variables while the remaining 3 are discrete variables.

7.4 Getting Started

The following packages are loaded into our R environment for the analysis:

  • R package for building and validating binary logistic regression models - blorr

  • R package for calibrating geographical weighted family of models - GWmodel

  • R package for multivariate data visualisation and analysis - corrplot

  • Spatial data handling - sf, spdep

  • Attribute data handling - tidyverse, especially readr, ggplot2 and dplyr

  • Rapid Exploratory Data Analysis - funModeling

  • Provide summary statistics about variables in data frames: Skimr, caret

  • Choropleth mapping - tmap, ggubr

We install and load the relevant packages using the following code chunk.

pacman::p_load(blorr,corrplot, ggpubr, sf, spdep, GWmodel, tmap, tidyverse,  funModeling, skimr, caret)

7.5 Import the data sets in R environment

The LGA boundaries of Osun State are imported and assigned to Osun with the following code chunk.

Osun <- read_rds("In-Class_Ex5/rds/Osun.rds")

The water points are imported and assigned to Osun_wp_sf with the following code chunk.

Osun_wp_sf <- read_rds("In-Class_Ex5/rds/Osun_wp_sf.rds")

7.6 Exploratory Data Analysis (EDA)

7.6.1 Check the proportion of functional and non-functional water points

We apply the following code to chart the status of water points in Osun

Osun_wp_sf %>% freq(input = 'status')
Warning: The `<scale>` argument of `guides()` cannot be `FALSE`. Use "none" instead as
of ggplot2 3.3.4.
ℹ The deprecated feature was likely used in the funModeling package.
  Please report the issue at <https://github.com/pablo14/funModeling/issues>.

  status frequency percentage cumulative_perc
1   TRUE      2642       55.5            55.5
2  FALSE      2118       44.5           100.0

We note that the % of non-functional water points is relatively high at 44.5%. At the same time, the proportion of both TRUE (functional) and FALSE (non-functional) classes are relatively balanced.

To visualise where these water points are located in Osun, we plot them by their status on a map using the following code chunk

tmap_mode("view")
tmap mode set to interactive viewing
actual_status <- tm_shape(Osun)+
  #tmap_options(check.and.fix=TRUE)+
  tm_polygons(alpha=0.4) +
  tm_shape(Osun_wp_sf) +
  tm_dots(col='status',
          alpha=0.8,
          palette = "RdBu") +
  tm_view(set.zoom.limits = c(9,12)) + 
  tm_layout(main.title = "Actual status of Water Points",
            main.title.position = "center",
            main.title.size = 1.0) 

actual_status

7.6.2 Inspect the variables for variable type and missing values

We use the skim() of skimr to get summary statistics of all the variables in the water point data frame, Osun_wp_sf .

Osun_wp_sf %>%
  skim()
Warning: Couldn't find skimmers for class: sfc_POINT, sfc; No user-defined `sfl`
provided. Falling back to `character`.
Data summary
Name Piped data
Number of rows 4760
Number of columns 75
_______________________
Column type frequency:
character 47
logical 5
numeric 23
________________________
Group variables None

Variable type: character

skim_variable n_missing complete_rate min max empty n_unique whitespace
source 0 1.00 5 44 0 2 0
report_date 0 1.00 22 22 0 42 0
status_id 0 1.00 2 7 0 3 0
water_source_clean 0 1.00 8 22 0 3 0
water_source_category 0 1.00 4 6 0 2 0
water_tech_clean 24 0.99 9 23 0 3 0
water_tech_category 24 0.99 9 15 0 2 0
facility_type 0 1.00 8 8 0 1 0
clean_country_name 0 1.00 7 7 0 1 0
clean_adm1 0 1.00 3 5 0 5 0
clean_adm2 0 1.00 3 14 0 35 0
clean_adm3 4760 0.00 NA NA 0 0 0
clean_adm4 4760 0.00 NA NA 0 0 0
installer 4760 0.00 NA NA 0 0 0
management_clean 1573 0.67 5 37 0 7 0
status_clean 0 1.00 9 32 0 7 0
pay 0 1.00 2 39 0 7 0
fecal_coliform_presence 4760 0.00 NA NA 0 0 0
subjective_quality 0 1.00 18 20 0 4 0
activity_id 4757 0.00 36 36 0 3 0
scheme_id 4760 0.00 NA NA 0 0 0
wpdx_id 0 1.00 12 12 0 4760 0
notes 0 1.00 2 96 0 3502 0
orig_lnk 4757 0.00 84 84 0 1 0
photo_lnk 41 0.99 84 84 0 4719 0
country_id 0 1.00 2 2 0 1 0
data_lnk 0 1.00 79 96 0 2 0
water_point_history 0 1.00 142 834 0 4750 0
clean_country_id 0 1.00 3 3 0 1 0
country_name 0 1.00 7 7 0 1 0
water_source 0 1.00 8 30 0 4 0
water_tech 0 1.00 5 37 0 20 0
adm2 0 1.00 3 14 0 33 0
adm3 4760 0.00 NA NA 0 0 0
management 1573 0.67 5 47 0 7 0
adm1 0 1.00 4 5 0 4 0
New Georeferenced Column 0 1.00 16 35 0 4760 0
lat_lon_deg 0 1.00 13 32 0 4760 0
public_data_source 0 1.00 84 102 0 2 0
converted 0 1.00 53 53 0 1 0
created_timestamp 0 1.00 22 22 0 2 0
updated_timestamp 0 1.00 22 22 0 2 0
Geometry 0 1.00 33 37 0 4760 0
ADM2_EN 0 1.00 3 14 0 30 0
ADM2_PCODE 0 1.00 8 8 0 30 0
ADM1_EN 0 1.00 4 4 0 1 0
ADM1_PCODE 0 1.00 5 5 0 1 0

Variable type: logical

skim_variable n_missing complete_rate mean count
rehab_year 4760 0 NaN :
rehabilitator 4760 0 NaN :
is_urban 0 1 0.39 FAL: 2884, TRU: 1876
latest_record 0 1 1.00 TRU: 4760
status 0 1 0.56 TRU: 2642, FAL: 2118

Variable type: numeric

skim_variable n_missing complete_rate mean sd p0 p25 p50 p75 p100 hist
row_id 0 1.00 68550.48 10216.94 49601.00 66874.75 68244.50 69562.25 471319.00 ▇▁▁▁▁
lat_deg 0 1.00 7.68 0.22 7.06 7.51 7.71 7.88 8.06 ▁▂▇▇▇
lon_deg 0 1.00 4.54 0.21 4.08 4.36 4.56 4.71 5.06 ▃▆▇▇▂
install_year 1144 0.76 2008.63 6.04 1917.00 2006.00 2010.00 2013.00 2015.00 ▁▁▁▁▇
fecal_coliform_value 4760 0.00 NaN NA NA NA NA NA NA
distance_to_primary_road 0 1.00 5021.53 5648.34 0.01 719.36 2972.78 7314.73 26909.86 ▇▂▁▁▁
distance_to_secondary_road 0 1.00 3750.47 3938.63 0.15 460.90 2554.25 5791.94 19559.48 ▇▃▁▁▁
distance_to_tertiary_road 0 1.00 1259.28 1680.04 0.02 121.25 521.77 1834.42 10966.27 ▇▂▁▁▁
distance_to_city 0 1.00 16663.99 10960.82 53.05 7930.75 15030.41 24255.75 47934.34 ▇▇▆▃▁
distance_to_town 0 1.00 16726.59 12452.65 30.00 6876.92 12204.53 27739.46 44020.64 ▇▅▃▃▂
rehab_priority 2654 0.44 489.33 1658.81 0.00 7.00 91.50 376.25 29697.00 ▇▁▁▁▁
water_point_population 4 1.00 513.58 1458.92 0.00 14.00 119.00 433.25 29697.00 ▇▁▁▁▁
local_population_1km 4 1.00 2727.16 4189.46 0.00 176.00 1032.00 3717.00 36118.00 ▇▁▁▁▁
crucialness_score 798 0.83 0.26 0.28 0.00 0.07 0.15 0.35 1.00 ▇▃▁▁▁
pressure_score 798 0.83 1.46 4.16 0.00 0.12 0.41 1.24 93.69 ▇▁▁▁▁
usage_capacity 0 1.00 560.74 338.46 300.00 300.00 300.00 1000.00 1000.00 ▇▁▁▁▅
days_since_report 0 1.00 2692.69 41.92 1483.00 2688.00 2693.00 2700.00 4645.00 ▁▇▁▁▁
staleness_score 0 1.00 42.80 0.58 23.13 42.70 42.79 42.86 62.66 ▁▁▇▁▁
location_id 0 1.00 235865.49 6657.60 23741.00 230638.75 236199.50 240061.25 267454.00 ▁▁▁▁▇
cluster_size 0 1.00 1.05 0.25 1.00 1.00 1.00 1.00 4.00 ▇▁▁▁▁
lat_deg_original 4760 0.00 NaN NA NA NA NA NA NA
lon_deg_original 4760 0.00 NaN NA NA NA NA NA NA
count 0 1.00 1.00 0.00 1.00 1.00 1.00 1.00 1.00 ▁▁▇▁▁

Things to note in the generated results

  • A frequency count of the data type of columns - character, logical, numeric - is provided.

  • Variables with excessive missing values should not be used for linear and logistic regression modeling. For instance, install_year will give us an idea of the age of the water point and presumably older water points tend to be non_functional as compared to the newer ones. However, we don’t use install_year for our model due to the high proportion of missing values (~24% missing) in the column.

  • Variables with a few missing values and assessed to be useful for the model can be included. We can remove the records with missing values from the data base. For our case, since water_point_population and local_population_1km only have 4 missing records, we will remove the 4 records and include the 2 variables for subsequent analysis. Using the results above, the number of missing values for each selected variable is as follow:

    • Status - 0 missing

    • distance_to_primary_road - 0 missing

    • distance_to_secondary_road - 0 missing

    • distance_to_tertiary_road - 0 missing

    • distance_to_city - 0 missing

    • distance_to_town - 0 missing

    • water_point_population - 4 missing

    • local_population_1km - 4 missing

    • usage_capacity - 0 missing

    • is_urban - 0 missing

    • water_source_clean - 0 missing

      We use the following code chunk to filter out records with missing values for water_point_population and local_population columns. After running this code, we should observe that the number of records has by 4 from 4,760 to 4,756.

Osun_wp_sf_clean <- Osun_wp_sf %>%
  filter_at(vars(water_point_population,
                 local_population_1km,
                 ),
            all_vars(!is.na(.)))
  • We note that usage_capacity is recognised as a numeric variable in R whereas it is more of a categorical variable denoting the type of water point. We change its data type to factor using the following code. After running this code, we should observe that usage capacity has been changed to “factor” type with 2 levels.
Osun_wp_sf_clean <- Osun_wp_sf_clean %>%
  mutate(usage_capacity = as.factor(usage_capacity))

7.7 Correlation Analysis

We first extract the selected variables from the Osun_wp_sf_clean and remove the geometry information from the data in order to construct a correlation matrix.

Osun_wp <- Osun_wp_sf_clean %>%
  select(c(7,35:39,42:43,46:47,57)) %>%
  st_set_geometry(NULL)

Then, we plot the matrix for all the numeric variables (excluding the dependent variable).

cluster_vars.cor = cor(
  Osun_wp[,2:7])

corrplot.mixed(cluster_vars.cor,
               lower = 'ellipse',
               upper = "number",
               tl.pos = "lt",
               diag = "l",
               tl.col= "black"
               )

Based on the results above, there is no sign of multicollinearity among the 6 continuous variables since none of the absolute correlation value of the variable pairs is above 0.85. We will go ahead and use all the 6 variables for modelling.

7.8 Build a Global (and non-spatial) Logistic Regression Model

Logistic Regression is a type of Generalised Linear Model (GLM) and we use the glm() of R stats to fit the model.

model <- glm(status ~ distance_to_primary_road +
               distance_to_secondary_road +
               distance_to_tertiary_road +
               distance_to_city + 
               distance_to_town +
               water_point_population +
               local_population_1km +
               usage_capacity +
               is_urban +
               water_source_clean,
             data = Osun_wp_sf_clean,
             family = binomial(link='logit'))

Instead of typing Model to view the results, we use the blr_regress() of blorr to produce a more informative report to help us examine the results of the model

blr_regress(model)
                             Model Overview                              
------------------------------------------------------------------------
Data Set    Resp Var    Obs.    Df. Model    Df. Residual    Convergence 
------------------------------------------------------------------------
  data       status     4756      4755           4744           TRUE     
------------------------------------------------------------------------

                    Response Summary                     
--------------------------------------------------------
Outcome        Frequency        Outcome        Frequency 
--------------------------------------------------------
   0             2114              1             2642    
--------------------------------------------------------

                                 Maximum Likelihood Estimates                                   
-----------------------------------------------------------------------------------------------
               Parameter                    DF    Estimate    Std. Error    z value     Pr(>|z|) 
-----------------------------------------------------------------------------------------------
              (Intercept)                   1      0.3887        0.1124      3.4588       5e-04 
        distance_to_primary_road            1      0.0000        0.0000     -0.7153      0.4744 
       distance_to_secondary_road           1      0.0000        0.0000     -0.5530      0.5802 
       distance_to_tertiary_road            1      1e-04         0.0000      4.6708      0.0000 
            distance_to_city                1      0.0000        0.0000     -4.7574      0.0000 
            distance_to_town                1      0.0000        0.0000     -4.9170      0.0000 
         water_point_population             1      -5e-04        0.0000    -11.3686      0.0000 
          local_population_1km              1      3e-04         0.0000     19.2953      0.0000 
           usage_capacity1000               1     -0.6230        0.0697     -8.9366      0.0000 
              is_urbanTRUE                  1     -0.2971        0.0819     -3.6294       3e-04 
water_source_cleanProtected Shallow Well    1      0.5040        0.0857      5.8783      0.0000 
   water_source_cleanProtected Spring       1      1.2882        0.4388      2.9359      0.0033 
-----------------------------------------------------------------------------------------------

 Association of Predicted Probabilities and Observed Responses  
---------------------------------------------------------------
% Concordant          0.7347          Somers' D        0.4693   
% Discordant          0.2653          Gamma            0.4693   
% Tied                0.0000          Tau-a            0.2318   
Pairs                5585188          c                0.7347   
---------------------------------------------------------------

Things to note from the report:

  • distance_to_primary_road and distance_to_secondary_road have p-values that > 0.05, we can remove these 2 variables from our model since they are not statistically significant

  • For categorical variables, positive Estimate (or coefficient) value implies an above average correlation and a negative value implies below average correlation. The magnitude of the coefficient does not matter for categorical variables;

  • For continuous variables, positive Estimate value implies direct correlation and a negative Estimate value implies an inverse correlation. The magnitude of the Estimate value provides the strength of the correlation.

To appreciate the performance of the model, we generate the confusion matrix using blr_confusion_matrix() of blorr.

# Probability cut-off threshold for Class 1 is set at 0.5
blr_confusion_matrix(model,cutoff = 0.5)
Confusion Matrix and Statistics 

          Reference
Prediction FALSE TRUE
         0  1301  738
         1   813 1904

                Accuracy : 0.6739 
     No Information Rate : 0.4445 

                   Kappa : 0.3373 

McNemars's Test P-Value  : 0.0602 

             Sensitivity : 0.7207 
             Specificity : 0.6154 
          Pos Pred Value : 0.7008 
          Neg Pred Value : 0.6381 
              Prevalence : 0.5555 
          Detection Rate : 0.4003 
    Detection Prevalence : 0.5713 
       Balanced Accuracy : 0.6680 
               Precision : 0.7008 
                  Recall : 0.7207 

        'Positive' Class : 1

The accuracy of the model is 0.6716 is a good start and it is better than a random guess with 0.5 accuracy.

7.9 Build a Geographically Weighted Logistic Regression Model

Now, we take into account the geographic information of the water points in our model.

7.9.1 Convert the water point sf data frame to sp data frame

First, we convert the Osun_wp_sf_clean data frame from sf to sp for GW modelling. This is because GWmodel is a relatively older package which can only work with sp data frames.

We have to use the Osun_wp_sf_clean instead of the Osun_sp_sf data frame as the latter contains missing values in the variables and this would result in error when running the GW model. 
Osun_wp_sp <- Osun_wp_sf_clean %>%
  select(c(status,
            distance_to_primary_road,
            distance_to_secondary_road,
            distance_to_tertiary_road,
            distance_to_city,
            distance_to_town,
            water_point_population,
            local_population_1km,
            usage_capacity,
            is_urban,
            water_source_clean)) %>%
  as_Spatial()

Osun_wp_sp
class       : SpatialPointsDataFrame 
features    : 4756 
extent      : 182502.4, 290751, 340054.1, 450905.3  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=4 +lon_0=8.5 +k=0.99975 +x_0=670553.98 +y_0=0 +a=6378249.145 +rf=293.465 +towgs84=-92,-93,122,0,0,0,0 +units=m +no_defs 
variables   : 11
names       : status, distance_to_primary_road, distance_to_secondary_road, distance_to_tertiary_road, distance_to_city, distance_to_town, water_point_population, local_population_1km, usage_capacity, is_urban, water_source_clean 
min values  :      0,        0.014461356813335,          0.152195902540837,         0.017815121653488, 53.0461399623541, 30.0019777713073,                      0,                    0,           1000,        0,           Borehole 
max values  :      1,         26909.8616132094,           19559.4793799085,          10966.2705628969,  47934.343603562, 44020.6393368124,                  29697,                36118,            300,        1,   Protected Spring

7.9.2 Derive a Fixed Bandwidth for the GWLR Model

bw.fixed <- bw.ggwr(status ~ distance_to_primary_road +
                      distance_to_secondary_road +
                      distance_to_tertiary_road +
                      distance_to_city + 
                      distance_to_town +
                      water_point_population +
                      local_population_1km +
                      usage_capacity +
                      is_urban +
                      water_source_clean,
                data = Osun_wp_sp,
                family = "binomial",
                approach = "AIC",
                kernel = "gaussian",
                adaptive = FALSE,
                longlat = FALSE)
Take a cup of tea and have a break, it will take a few minutes.
          -----A kind suggestion from GWmodel development group
 Iteration    Log-Likelihood:(With bandwidth:  95768.67 )
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Fixed bandwidth: 95768.67 AICc value: 5684.357 
 Iteration    Log-Likelihood:(With bandwidth:  59200.13 )
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Fixed bandwidth: 59200.13 AICc value: 5646.785 
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Fixed bandwidth: 36599.53 AICc value: 5575.148 
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Fixed bandwidth: 22631.59 AICc value: 5466.883 
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Fixed bandwidth: 5366.266 AICc value: 4990.587 
 Iteration    Log-Likelihood:(With bandwidth:  3328.371 )
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Fixed bandwidth: 3328.371 AICc value: 4798.288 
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Fixed bandwidth: 2068.882 AICc value: 4837.017 
 Iteration    Log-Likelihood:(With bandwidth:  4106.777 )
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Fixed bandwidth: 4106.777 AICc value: 4873.161 
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Fixed bandwidth: 2847.289 AICc value: 4768.192 
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Fixed bandwidth: 2549.964 AICc value: 4762.212 
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Fixed bandwidth: 2366.207 AICc value: 4773.081 
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Fixed bandwidth: 2663.532 AICc value: 4762.568 
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Fixed bandwidth: 2479.775 AICc value: 4764.294 
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Fixed bandwidth: 2597.255 AICc value: 4761.809 
# Adaptaive is set to "FALSE" as we are computing fixed width
# longlat is set to "FALSE" as we are using projected CRS (instead of coordinate points)
bw.fixed
[1] 2599.672

The derived bandwidth is 2599.672 metres.

7.9.3 Fit the Fixed Bandwidth and data into the GWLR model

We fit the model using the bandwidth obtained above.

gwlr.fixed <- ggwr.basic(status ~ distance_to_primary_road +
                           distance_to_secondary_road +
                           distance_to_tertiary_road +
                           distance_to_city + 
                           distance_to_town +
                           water_point_population +
                           local_population_1km +
                           usage_capacity +
                           is_urban +
                           water_source_clean,
                       data = Osun_wp_sp,
                       bw=2599.672,
                       family = "binomial",
                       kernel = "gaussian",
                       adaptive = FALSE,
                       longlat = FALSE)
Warning in proj4string(data): CRS object has comment, which is lost in output; in tests, see
https://cran.r-project.org/web/packages/sp/vignettes/CRS_warnings.html
Warning in proj4string(regression.points): CRS object has comment, which is lost in output; in tests, see
https://cran.r-project.org/web/packages/sp/vignettes/CRS_warnings.html
 Iteration    Log-Likelihood
=========================
       0        -1958 
       1        -1676 
       2        -1526 
       3        -1443 
       4        -1405 
       5        -1405

We call the model to view the results

gwlr.fixed
   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2022-12-18 00:02:04 
   Call:
   ggwr.basic(formula = status ~ distance_to_primary_road + distance_to_secondary_road + 
    distance_to_tertiary_road + distance_to_city + distance_to_town + 
    water_point_population + local_population_1km + usage_capacity + 
    is_urban + water_source_clean, data = Osun_wp_sp, bw = 2599.672, 
    family = "binomial", kernel = "gaussian", adaptive = FALSE, 
    longlat = FALSE)

   Dependent (y) variable:  status
   Independent variables:  distance_to_primary_road distance_to_secondary_road distance_to_tertiary_road distance_to_city distance_to_town water_point_population local_population_1km usage_capacity is_urban water_source_clean
   Number of data points: 4756
   Used family: binomial
   ***********************************************************************
   *              Results of Generalized linear Regression               *
   ***********************************************************************

Call:
NULL

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-124.555    -1.755     1.072     1.742    34.333  

Coefficients:
                                           Estimate Std. Error z value Pr(>|z|)
Intercept                                 3.887e-01  1.124e-01   3.459 0.000543
distance_to_primary_road                 -4.642e-06  6.490e-06  -0.715 0.474422
distance_to_secondary_road               -5.143e-06  9.299e-06  -0.553 0.580230
distance_to_tertiary_road                 9.683e-05  2.073e-05   4.671 3.00e-06
distance_to_city                         -1.686e-05  3.544e-06  -4.757 1.96e-06
distance_to_town                         -1.480e-05  3.009e-06  -4.917 8.79e-07
water_point_population                   -5.097e-04  4.484e-05 -11.369  < 2e-16
local_population_1km                      3.451e-04  1.788e-05  19.295  < 2e-16
usage_capacity1000                       -6.230e-01  6.972e-02  -8.937  < 2e-16
is_urbanTRUE                             -2.971e-01  8.185e-02  -3.629 0.000284
water_source_cleanProtected Shallow Well  5.040e-01  8.574e-02   5.878 4.14e-09
water_source_cleanProtected Spring        1.288e+00  4.388e-01   2.936 0.003325
                                            
Intercept                                ***
distance_to_primary_road                    
distance_to_secondary_road                  
distance_to_tertiary_road                ***
distance_to_city                         ***
distance_to_town                         ***
water_point_population                   ***
local_population_1km                     ***
usage_capacity1000                       ***
is_urbanTRUE                             ***
water_source_cleanProtected Shallow Well ***
water_source_cleanProtected Spring       ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 6534.5  on 4755  degrees of freedom
Residual deviance: 5688.0  on 4744  degrees of freedom
AIC: 5712

Number of Fisher Scoring iterations: 5


 AICc:  5712.099
 Pseudo R-square value:  0.1295351
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Fixed bandwidth: 2599.672 
   Regression points: the same locations as observations are used.
   Distance metric: A distance matrix is specified for this model calibration.

   ************Summary of Generalized GWR coefficient estimates:**********
                                                   Min.     1st Qu.      Median
   Intercept                                -8.7229e+02 -4.9955e+00  1.7600e+00
   distance_to_primary_road                 -1.9389e-02 -4.8031e-04  2.9618e-05
   distance_to_secondary_road               -1.5921e-02 -3.7551e-04  1.2317e-04
   distance_to_tertiary_road                -1.5618e-02 -4.2368e-04  7.6179e-05
   distance_to_city                         -1.8416e-02 -5.6217e-04 -1.2726e-04
   distance_to_town                         -2.2411e-02 -5.7283e-04 -1.5155e-04
   water_point_population                   -5.2208e-02 -2.2767e-03 -9.8875e-04
   local_population_1km                     -1.2698e-01  4.9952e-04  1.0638e-03
   usage_capacity1000                       -2.0772e+01 -9.7231e-01 -4.1592e-01
   is_urbanTRUE                             -1.9790e+02 -4.2908e+00 -1.6864e+00
   water_source_cleanProtected.Shallow.Well -2.0789e+01 -4.5190e-01  5.3340e-01
   water_source_cleanProtected.Spring       -5.2235e+02 -5.5977e+00  2.5441e+00
                                                3rd Qu.      Max.
   Intercept                                 1.2763e+01 1073.2156
   distance_to_primary_road                  4.8443e-04    0.0142
   distance_to_secondary_road                6.0692e-04    0.0258
   distance_to_tertiary_road                 6.6815e-04    0.0128
   distance_to_city                          2.3718e-04    0.0150
   distance_to_town                          1.9271e-04    0.0224
   water_point_population                    5.0102e-04    0.1309
   local_population_1km                      1.8157e-03    0.0392
   usage_capacity1000                        3.0322e-01    5.9281
   is_urbanTRUE                              1.2841e+00  744.3099
   water_source_cleanProtected.Shallow.Well  1.7849e+00   67.6343
   water_source_cleanProtected.Spring        6.7663e+00  317.4133
   ************************Diagnostic information*************************
   Number of data points: 4756 
   GW Deviance: 2795.084 
   AIC : 4414.606 
   AICc : 4747.423 
   Pseudo R-square value:  0.5722559 

   ***********************************************************************
   Program stops at: 2022-12-18 00:02:38

Things to note:

  • The report above has 2 sections -Global Logistic Regression (Global LR) model and Geographically Weighted Logistic Regression (GWLR) model results.

  • The Global LR model’s AICc is 5712.099 while the GWLR model’s AICc is 4747.423. This shows that the GWLR model better fit the data than the Global LR.

7.9.4 Model assessment and comparison

To assess the performance of the gwlr, we will convert the SDF object to a data frame by using the code chunk below

gwr.fixed <- as.data.frame(gwlr.fixed$SDF)

Next, we label yhat values (probability of water point being functional) greater or equal to 0.5 into 1 or else 0. The result of the logic comparison operation will be saved in a column called most.

gwr.fixed <- gwr.fixed %>%
  mutate(most = ifelse(
    gwr.fixed$yhat >= 0.5,T,F))

Then we construct a confusion matrix using confusionMatrix() of caret.

gwr.fixed$y <- as.factor(gwr.fixed$y)
gwr.fixed$most <- as.factor(gwr.fixed$most)
CM <- confusionMatrix(data=gwr.fixed$most,
                      reference=gwr.fixed$y,
                      positive="TRUE")

CM
Confusion Matrix and Statistics

          Reference
Prediction FALSE TRUE
     FALSE  1824  263
     TRUE    290 2379
                                          
               Accuracy : 0.8837          
                 95% CI : (0.8743, 0.8927)
    No Information Rate : 0.5555          
    P-Value [Acc > NIR] : <2e-16          
                                          
                  Kappa : 0.7642          
                                          
 Mcnemar's Test P-Value : 0.2689          
                                          
            Sensitivity : 0.9005          
            Specificity : 0.8628          
         Pos Pred Value : 0.8913          
         Neg Pred Value : 0.8740          
             Prevalence : 0.5555          
         Detection Rate : 0.5002          
   Detection Prevalence : 0.5612          
      Balanced Accuracy : 0.8816          
                                          
       'Positive' Class : TRUE

Things to note:

  • Accuracy, Sensitivity and Specificity scores of the GWLR model has improved as compared to the Global LR model

    Model Accuracy Sensitivity Specificity
    Global LR 0.6739 0.7207 0.6154
    GLWR Model 0.8837 0.9005 0.8628

  • Based on the above comparison, including spatial attributes will improve the explanatory power of the model. The results also show that the strategies to manage and maintain water points should be localised by taking into consideration the neighboring LGAs.

7.9.5 Visualise the results of the GWLR model

We will first extract the administrative boundary details into a new data frame.

Osun_wp_sf_selected <- Osun_wp_sf_clean %>%
  select(c(ADM2_EN, ADM2_PCODE,ADM1_EN,ADM1_PCODE,status))

Next, we will combine the new data frame with the model results

gwr_sf.fixed <- cbind(Osun_wp_sf_selected, gwr.fixed)

Now we will plot the actual status of functional and non-functional water points (left map) and place the status generated by the gwlr model next to it (right map) for ease of comparison.

tmap_mode("view")
tmap mode set to interactive viewing
prob_T <- tm_shape(Osun) + 
  tm_polygons(alpha = 0.1) +
  tm_shape(gwr_sf.fixed) + 
  tm_dots(col = "most",
          border.col = "gray60",
          border.lwd = 1) +
  tm_layout(main.title = "Predicted Status of Water Points",
            main.title.position = "center",
            main.title.size = 1.0) +
  tm_view(set.zoom.limits = c(9,12))
  

tmap_arrange(actual_status, prob_T, 
             asp=1, ncol=2, nrow = 1,
             sync = TRUE)

We can observe that the location of functional (TRUE) and non-functional (FALSE) water points on both plots are almost identical (justifying the 88% accuracy 😜) .

7.10 Revised Global LR and GWLR models by removing the statistically non-significant dependent variables.

In Section 7.8 above, we discovered that distance_to_primary_road and distance_to_secondary_road are not statistically significant variables and can be excluded from the model. We will now update the Global LR and GWLR models by excluding the 2 variables. We will be largely repeating the steps covered in Sections 7.8 and 7.9.

7.10.1 Build a Global Logistic Regression Model without non-significant dependent variables

We fit the model in step 1 and then generate the model results in step 2

# Step 1:
revised_model <- glm(status ~ distance_to_tertiary_road +
               distance_to_city +
               distance_to_town +
               is_urban +
               usage_capacity +
               water_source_clean +
               water_point_population +
               local_population_1km,
             data = Osun_wp_sf_clean,
             family = binomial(link = "logit"))

# Step 2"
blr_regress(revised_model)
                             Model Overview                              
------------------------------------------------------------------------
Data Set    Resp Var    Obs.    Df. Model    Df. Residual    Convergence 
------------------------------------------------------------------------
  data       status     4756      4755           4746           TRUE     
------------------------------------------------------------------------

                    Response Summary                     
--------------------------------------------------------
Outcome        Frequency        Outcome        Frequency 
--------------------------------------------------------
   0             2114              1             2642    
--------------------------------------------------------

                                 Maximum Likelihood Estimates                                   
-----------------------------------------------------------------------------------------------
               Parameter                    DF    Estimate    Std. Error    z value     Pr(>|z|) 
-----------------------------------------------------------------------------------------------
              (Intercept)                   1      0.3540        0.1055      3.3541       8e-04 
       distance_to_tertiary_road            1      1e-04         0.0000      4.9096      0.0000 
            distance_to_city                1      0.0000        0.0000     -5.2022      0.0000 
            distance_to_town                1      0.0000        0.0000     -5.4660      0.0000 
              is_urbanTRUE                  1     -0.2667        0.0747     -3.5690       4e-04 
           usage_capacity1000               1     -0.6206        0.0697     -8.9081      0.0000 
water_source_cleanProtected Shallow Well    1      0.4947        0.0850      5.8228      0.0000 
   water_source_cleanProtected Spring       1      1.2790        0.4384      2.9174      0.0035 
         water_point_population             1      -5e-04        0.0000    -11.3902      0.0000 
          local_population_1km              1      3e-04         0.0000     19.4069      0.0000 
-----------------------------------------------------------------------------------------------

 Association of Predicted Probabilities and Observed Responses  
---------------------------------------------------------------
% Concordant          0.7349          Somers' D        0.4697   
% Discordant          0.2651          Gamma            0.4697   
% Tied                0.0000          Tau-a            0.2320   
Pairs                5585188          c                0.7349   
---------------------------------------------------------------
  • We observe that the p-value of the remaining dependent variables are all < 0.05, indicating that they are statisitically significant.

Next, we evaluate the performance metrics of the model using blr_confusion_matrix() of blorr.

blr_confusion_matrix(revised_model,cutoff = 0.5)
Confusion Matrix and Statistics 

          Reference
Prediction FALSE TRUE
         0  1300  743
         1   814 1899

                Accuracy : 0.6726 
     No Information Rate : 0.4445 

                   Kappa : 0.3348 

McNemars's Test P-Value  : 0.0761 

             Sensitivity : 0.7188 
             Specificity : 0.6149 
          Pos Pred Value : 0.7000 
          Neg Pred Value : 0.6363 
              Prevalence : 0.5555 
          Detection Rate : 0.3993 
    Detection Prevalence : 0.5704 
       Balanced Accuracy : 0.6669 
               Precision : 0.7000 
                  Recall : 0.7188 

        'Positive' Class : 1
  • We note there’s no substantial change in the Accuracy, Sensitivity and Specificity scores from the previous Global Logistic Regression model

7.10.2 Derive the revised Fixed Bandwidth for the GWLR Model

revised_bw.fixed <- bw.ggwr(status ~ distance_to_tertiary_road +
                      distance_to_city + 
                      distance_to_town +
                      water_point_population +
                      local_population_1km +
                      usage_capacity +
                      is_urban +
                      water_source_clean,
                data = Osun_wp_sp,
                family = "binomial",
                approach = "AIC",
                kernel = "gaussian",
                adaptive = FALSE,
                longlat = FALSE)
Take a cup of tea and have a break, it will take a few minutes.
          -----A kind suggestion from GWmodel development group
 Iteration    Log-Likelihood:(With bandwidth:  95768.67 )
=========================
       0        -2890 
       1        -2837 
       2        -2830 
       3        -2829 
       4        -2829 
       5        -2829 
Fixed bandwidth: 95768.67 AICc value: 5681.18 
 Iteration    Log-Likelihood:(With bandwidth:  59200.13 )
=========================
       0        -2878 
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       2        -2812 
       3        -2810 
       4        -2810 
       5        -2810 
Fixed bandwidth: 59200.13 AICc value: 5645.901 
 Iteration    Log-Likelihood:(With bandwidth:  36599.53 )
=========================
       0        -2854 
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       2        -2777 
       3        -2774 
       4        -2774 
       5        -2774 
       6        -2774 
Fixed bandwidth: 36599.53 AICc value: 5585.354 
 Iteration    Log-Likelihood:(With bandwidth:  22631.59 )
=========================
       0        -2810 
       1        -2732 
       2        -2711 
       3        -2707 
       4        -2707 
       5        -2707 
       6        -2707 
Fixed bandwidth: 22631.59 AICc value: 5481.877 
 Iteration    Log-Likelihood:(With bandwidth:  13998.93 )
=========================
       0        -2732 
       1        -2635 
       2        -2604 
       3        -2597 
       4        -2596 
       5        -2596 
       6        -2596 
Fixed bandwidth: 13998.93 AICc value: 5333.718 
 Iteration    Log-Likelihood:(With bandwidth:  8663.649 )
=========================
       0        -2624 
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       2        -2459 
       3        -2447 
       4        -2446 
       5        -2446 
       6        -2446 
       7        -2446 
Fixed bandwidth: 8663.649 AICc value: 5178.493 
 Iteration    Log-Likelihood:(With bandwidth:  5366.266 )
=========================
       0        -2478 
       1        -2319 
       2        -2250 
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       7        -2220 
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       9        -2220 
Fixed bandwidth: 5366.266 AICc value: 5022.016 
 Iteration    Log-Likelihood:(With bandwidth:  3328.371 )
=========================
       0        -2222 
       1        -2002 
       2        -1894 
       3        -1838 
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       6        -1814 
Fixed bandwidth: 3328.371 AICc value: 4827.587 
 Iteration    Log-Likelihood:(With bandwidth:  2068.882 )
=========================
       0        -1837 
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       2        -1357 
       3        -1261 
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       5        -1222 
Fixed bandwidth: 2068.882 AICc value: 4772.046 
 Iteration    Log-Likelihood:(With bandwidth:  1290.476 )
=========================
       0        -1403 
       1        -1016 
       2       -807.3 
       3       -680.2 
       4       -680.2 
Fixed bandwidth: 1290.476 AICc value: 5809.721 
 Iteration    Log-Likelihood:(With bandwidth:  2549.964 )
=========================
       0        -2019 
       1        -1753 
       2        -1614 
       3        -1538 
       4        -1506 
       5        -1506 
Fixed bandwidth: 2549.964 AICc value: 4764.056 
 Iteration    Log-Likelihood:(With bandwidth:  2847.289 )
=========================
       0        -2108 
       1        -1862 
       2        -1736 
       3        -1670 
       4        -1644 
       5        -1644 
Fixed bandwidth: 2847.289 AICc value: 4791.834 
 Iteration    Log-Likelihood:(With bandwidth:  2366.207 )
=========================
       0        -1955 
       1        -1675 
       2        -1525 
       3        -1441 
       4        -1407 
       5        -1407 
Fixed bandwidth: 2366.207 AICc value: 4755.524 
 Iteration    Log-Likelihood:(With bandwidth:  2252.639 )
=========================
       0        -1913 
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       2        -1465 
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Fixed bandwidth: 2252.639 AICc value: 4759.188 
 Iteration    Log-Likelihood:(With bandwidth:  2436.396 )
=========================
       0        -1980 
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Fixed bandwidth: 2436.396 AICc value: 4756.675 
 Iteration    Log-Likelihood:(With bandwidth:  2322.828 )
=========================
       0        -1940 
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Fixed bandwidth: 2322.828 AICc value: 4756.471 
 Iteration    Log-Likelihood:(With bandwidth:  2393.017 )
=========================
       0        -1965 
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Fixed bandwidth: 2393.017 AICc value: 4755.57 
 Iteration    Log-Likelihood:(With bandwidth:  2349.638 )
=========================
       0        -1949 
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Fixed bandwidth: 2349.638 AICc value: 4755.753 
 Iteration    Log-Likelihood:(With bandwidth:  2376.448 )
=========================
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       5        -1413 
Fixed bandwidth: 2376.448 AICc value: 4755.48 
 Iteration    Log-Likelihood:(With bandwidth:  2382.777 )
=========================
       0        -1961 
       1        -1683 
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       3        -1450 
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       5        -1416 
Fixed bandwidth: 2382.777 AICc value: 4755.491 
 Iteration    Log-Likelihood:(With bandwidth:  2372.536 )
=========================
       0        -1958 
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       5        -1411 
Fixed bandwidth: 2372.536 AICc value: 4755.488 
 Iteration    Log-Likelihood:(With bandwidth:  2378.865 )
=========================
       0        -1960 
       1        -1681 
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Fixed bandwidth: 2378.865 AICc value: 4755.481 
 Iteration    Log-Likelihood:(With bandwidth:  2374.954 )
=========================
       0        -1959 
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Fixed bandwidth: 2374.954 AICc value: 4755.482 
 Iteration    Log-Likelihood:(With bandwidth:  2377.371 )
=========================
       0        -1959 
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Fixed bandwidth: 2377.371 AICc value: 4755.48 
 Iteration    Log-Likelihood:(With bandwidth:  2377.942 )
=========================
       0        -1960 
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Fixed bandwidth: 2377.942 AICc value: 4755.48 
 Iteration    Log-Likelihood:(With bandwidth:  2377.018 )
=========================
       0        -1959 
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       5        -1413 
Fixed bandwidth: 2377.018 AICc value: 4755.48 
revised_bw.fixed
[1] 2377.371

The derived bandwidth is 2377.371 metres.

7.10.3 Fit the revised Fixed Bandwidth and data to the GWLR model

We fit a revised model using the updated bandwidth obtained above.

revised_gwlr.fixed <- ggwr.basic(status ~ distance_to_tertiary_road +
                           distance_to_city + 
                           distance_to_town +
                           water_point_population +
                           local_population_1km +
                           usage_capacity +
                           is_urban +
                           water_source_clean,
                       data = Osun_wp_sp,
                       bw=2377.371,
                       family = "binomial",
                       kernel = "gaussian",
                       adaptive = FALSE,
                       longlat = FALSE)
Warning in proj4string(data): CRS object has comment, which is lost in output; in tests, see
https://cran.r-project.org/web/packages/sp/vignettes/CRS_warnings.html
Warning in proj4string(regression.points): CRS object has comment, which is lost in output; in tests, see
https://cran.r-project.org/web/packages/sp/vignettes/CRS_warnings.html
 Iteration    Log-Likelihood
=========================
       0        -1959 
       1        -1680 
       2        -1531 
       3        -1447 
       4        -1413 
       5        -1413

We review the results of the revised model.

revised_gwlr.fixed
   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2022-12-18 00:12:30 
   Call:
   ggwr.basic(formula = status ~ distance_to_tertiary_road + distance_to_city + 
    distance_to_town + water_point_population + local_population_1km + 
    usage_capacity + is_urban + water_source_clean, data = Osun_wp_sp, 
    bw = 2377.371, family = "binomial", kernel = "gaussian", 
    adaptive = FALSE, longlat = FALSE)

   Dependent (y) variable:  status
   Independent variables:  distance_to_tertiary_road distance_to_city distance_to_town water_point_population local_population_1km usage_capacity is_urban water_source_clean
   Number of data points: 4756
   Used family: binomial
   ***********************************************************************
   *              Results of Generalized linear Regression               *
   ***********************************************************************

Call:
NULL

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-129.368    -1.750     1.074     1.742    34.126  

Coefficients:
                                           Estimate Std. Error z value Pr(>|z|)
Intercept                                 3.540e-01  1.055e-01   3.354 0.000796
distance_to_tertiary_road                 1.001e-04  2.040e-05   4.910 9.13e-07
distance_to_city                         -1.764e-05  3.391e-06  -5.202 1.97e-07
distance_to_town                         -1.544e-05  2.825e-06  -5.466 4.60e-08
water_point_population                   -5.098e-04  4.476e-05 -11.390  < 2e-16
local_population_1km                      3.452e-04  1.779e-05  19.407  < 2e-16
usage_capacity1000                       -6.206e-01  6.966e-02  -8.908  < 2e-16
is_urbanTRUE                             -2.667e-01  7.474e-02  -3.569 0.000358
water_source_cleanProtected Shallow Well  4.947e-01  8.496e-02   5.823 5.79e-09
water_source_cleanProtected Spring        1.279e+00  4.384e-01   2.917 0.003530
                                            
Intercept                                ***
distance_to_tertiary_road                ***
distance_to_city                         ***
distance_to_town                         ***
water_point_population                   ***
local_population_1km                     ***
usage_capacity1000                       ***
is_urbanTRUE                             ***
water_source_cleanProtected Shallow Well ***
water_source_cleanProtected Spring       ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 6534.5  on 4755  degrees of freedom
Residual deviance: 5688.9  on 4746  degrees of freedom
AIC: 5708.9

Number of Fisher Scoring iterations: 5


 AICc:  5708.923
 Pseudo R-square value:  0.129406
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Fixed bandwidth: 2377.371 
   Regression points: the same locations as observations are used.
   Distance metric: A distance matrix is specified for this model calibration.

   ************Summary of Generalized GWR coefficient estimates:**********
                                                   Min.     1st Qu.      Median
   Intercept                                -3.7021e+02 -4.3797e+00  3.5590e+00
   distance_to_tertiary_road                -3.1622e-02 -4.5462e-04  9.1291e-05
   distance_to_city                         -5.4555e-02 -6.5623e-04 -1.3507e-04
   distance_to_town                         -8.6549e-03 -5.2754e-04 -1.6785e-04
   water_point_population                   -2.9696e-02 -2.2705e-03 -1.2277e-03
   local_population_1km                     -7.7730e-02  4.4281e-04  1.0548e-03
   usage_capacity1000                       -5.5889e+01 -1.0347e+00 -4.1960e-01
   is_urbanTRUE                             -7.3554e+02 -3.4675e+00 -1.6596e+00
   water_source_cleanProtected.Shallow.Well -1.8842e+02 -4.7295e-01  6.2378e-01
   water_source_cleanProtected.Spring       -1.3630e+03 -5.3436e+00  2.7714e+00
                                                3rd Qu.      Max.
   Intercept                                 1.3755e+01 2171.6375
   distance_to_tertiary_road                 6.3011e-04    0.0237
   distance_to_city                          1.5921e-04    0.0162
   distance_to_town                          2.4490e-04    0.0179
   water_point_population                    4.5879e-04    0.0765
   local_population_1km                      1.8479e-03    0.0333
   usage_capacity1000                        3.9113e-01    9.2449
   is_urbanTRUE                              1.0554e+00  995.1841
   water_source_cleanProtected.Shallow.Well  1.9564e+00   66.8914
   water_source_cleanProtected.Spring        7.0805e+00  208.3749
   ************************Diagnostic information*************************
   Number of data points: 4756 
   GW Deviance: 2815.659 
   AIC : 4418.776 
   AICc : 4744.213 
   Pseudo R-square value:  0.5691072 

   ***********************************************************************
   Program stops at: 2022-12-18 00:12:58

A comparison of the AICc of the models with and without the non-significant dependent variables is as follows:

Model With non-significant dependent variables Without non-significant dependent variables
Global LR 5712.099 5708.923
GLWR Model 4747.423 4744.213

There is only a marginal change in the AICc results of the models after we remove the non-significant variables.

We go on to assess the model performance of the revised GWLR model by constructing the confusion matrix using the confusionMatrix() of caret.

# Step 1: Convert the SDF object of the gwlr model into a data frame
revised_gwr.fixed <- as.data.frame(revised_gwlr.fixed$SDF)

# Step 2: Include a new column most that indicate if the modelled results
revised_gwr.fixed <- revised_gwr.fixed %>%
  mutate(most = ifelse(
    revised_gwr.fixed$yhat >= 0.5,T,F))

# Step 3: Generate the performance metrics
revised_gwr.fixed$y <- as.factor(revised_gwr.fixed$y)
revised_gwr.fixed$most <- as.factor(revised_gwr.fixed$most)
CM <- confusionMatrix(data=revised_gwr.fixed$most,
                      reference=revised_gwr.fixed$y,
                      positive="TRUE")

CM
Confusion Matrix and Statistics

          Reference
Prediction FALSE TRUE
     FALSE  1833  268
     TRUE    281 2374
                                          
               Accuracy : 0.8846          
                 95% CI : (0.8751, 0.8935)
    No Information Rate : 0.5555          
    P-Value [Acc > NIR] : <2e-16          
                                          
                  Kappa : 0.7661          
                                          
 Mcnemar's Test P-Value : 0.6085          
                                          
            Sensitivity : 0.8986          
            Specificity : 0.8671          
         Pos Pred Value : 0.8942          
         Neg Pred Value : 0.8724          
             Prevalence : 0.5555          
         Detection Rate : 0.4992          
   Detection Prevalence : 0.5582          
      Balanced Accuracy : 0.8828          
                                          
       'Positive' Class : TRUE

We tabulate the performance metrics of the 4 models as follow

Model Accuracy Sensitivity Specificity

Global LR

(With non-significant dependent variables)

 0.6739 0.7207 0.6154

GLWR Model

(With non-significant dependent variables)

 0.8837 0.9005 0.8628

Global LR

(Without non-significant dependent variables)

 0.6726 0.7188 0.6149

GLWR Model

(Without non-significant dependent variables)

 0.8846 0.8986 0.8671

As we can see from the above, the inclusion of statistically non-significant variables do not adversely affect the performance of logistic regression models (differences of <0.01), be it non-spatial or geographically weighted. For computational efficiency, we should exclude the dependent variables (i.e. noise) from the modelling process once they are determined to be non-significant. Also, from an explanatory modelling perspective, the results above provide evidence that the distance of water points to primary or secondary roads are not relevant to the functional status of the water points.

7.11 Conclusion

From the data that is used for modelling, it is evident from the generated AICc that Geographically Weighted models provide better explanatory power about the status of the water points as compared to a non-spatial (or Global) Logistic Regression models. The administrators of Osun State Nigeria could make use of the coefficient estimates derived for the 8 dependent variables of each water point to understand the factors that contribute to its functional status and device measures to prevent the water point from malfunctioning.

References