Introduction

This contains more details on the multiple quantile regression conducted on the health economics data set.

Any questions can be directed to Brendan Lam at

Variables Included:

- Sex
- Age
- Days Seen from Injury
- Median Income
- Distance from Postal Code
- Number of Symptoms
- Number of Previous Concussions
- Psychiatric History (Y/N)
- Number of Physical Conditions
- Symptom Duration
- Number of Medications
- Number of Therapies Used

Exploring Data

Looking at the distribution of each variable with histograms

Investigating patterns of missing data

Missing data was handled with single imputation

##       health_econ_data.Psychiatric_hist health_econ_data.DistanceFromPostalCode 
##                                       3                                       2 
##       health_econ_data.NumberofSymptoms    health_econ_data.NumbPreviousConcuss 
##                                       2                                       2 
##     health_econ_data.DaysSeenFromInjury               health_econ_data.Duration 
##                                       1                                       1

Quantile Multiple Regression

Quantile regression for the 0.2th quantile/20th percentile

## 
## Call: rq(formula = `Healthcare Cost` ~ ., tau = 0.2, data = qmr_df)
## 
## tau: [1] 0.2
## 
## Coefficients:
##                                 Value     Std. Error t value   Pr(>|t|) 
## (Intercept)                     258.73584  93.26602    2.77417   0.00576
## SexM                             16.77248  31.39393    0.53426   0.59343
## `Psychiatric History`Yes        -60.62878  43.38568   -1.39744   0.16297
## Age                               6.05633   4.54911    1.33132   0.18376
## Income                           -0.00008   0.00071   -0.11791   0.90619
## Distance                          0.03220   0.23377    0.13773   0.89051
## `Number of Symptoms`             36.11734  15.30434    2.35994   0.01870
## `Number of Physical Conditions`   8.27630  20.50314    0.40366   0.68665
## `Number of Meds`                  9.53920  11.68967    0.81604   0.41491
## Duration                          0.10618   0.10146    1.04656   0.29587
## `Number Other Therapy`            1.88976  14.35727    0.13162   0.89534

Quantile regression for the 0.5th quantile/50th percentile

## 
## Call: rq(formula = `Healthcare Cost` ~ ., tau = 0.5, data = qmr_df)
## 
## tau: [1] 0.5
## 
## Coefficients:
##                                 Value     Std. Error t value   Pr(>|t|) 
## (Intercept)                     252.92062 185.97806    1.35995   0.17453
## SexM                            -77.53343  70.02007   -1.10730   0.26875
## `Psychiatric History`Yes        -71.64854  72.40118   -0.98960   0.32290
## Age                              13.59783   8.28776    1.64071   0.10156
## Income                            0.00045   0.00134    0.33311   0.73921
## Distance                          0.26271   0.46821    0.56109   0.57501
## `Number of Symptoms`             81.91633  26.64760    3.07406   0.00224
## `Number of Physical Conditions`  16.57820  54.14332    0.30619   0.75960
## `Number of Meds`                  3.38419  15.30344    0.22114   0.82508
## Duration                          0.41307   0.10269    4.02249   0.00007
## `Number Other Therapy`           28.81546  18.95451    1.52024   0.12915

Quantile regression for the 0.8th quantile/80th percentile

## 
## Call: rq(formula = `Healthcare Cost` ~ ., tau = 0.8, data = qmr_df)
## 
## tau: [1] 0.8
## 
## Coefficients:
##                                 Value      Std. Error t value    Pr(>|t|)  
## (Intercept)                      514.62155  395.67708    1.30061    0.19406
## SexM                            -145.31391  111.34699   -1.30505    0.19254
## `Psychiatric History`Yes          42.50050  171.46228    0.24787    0.80435
## Age                               29.59869   18.61816    1.58978    0.11259
## Income                            -0.00024    0.00208   -0.11343    0.90974
## Distance                           0.14424    0.64486    0.22368    0.82311
## `Number of Symptoms`             147.46042   44.58039    3.30774    0.00102
## `Number of Physical Conditions`   26.41524   64.29961    0.41081    0.68140
## `Number of Meds`                 -57.62770   21.01749   -2.74189    0.00635
## Duration                           0.90221    0.27885    3.23546    0.00130
## `Number Other Therapy`            64.64587   32.85124    1.96784    0.04970

Coefficient contrasts to see if the strength of predictors differs over different quantiles.

## Quantile Regression Analysis of Deviance Table
## 
## Model: `Healthcare Cost` ~ Sex + `Psychiatric History` + Age + Income + Distance + `Number of Symptoms` + `Number of Physical Conditions` + `Number of Meds` + Duration + `Number Other Therapy`
## Tests of Equality of Distinct Slopes: tau in {  0.2 0.5 0.8  }
## 
##                                 Df Resid Df F value   Pr(>F)   
## SexM                             2     1381  1.5900 0.204293   
## `Psychiatric History`Yes         2     1381  0.3113 0.732566   
## Age                              2     1381  0.9967 0.369363   
## Income                           2     1381  0.1519 0.859074   
## Distance                         2     1381  0.1634 0.849237   
## `Number of Symptoms`             2     1381  3.6875 0.025281 * 
## `Number of Physical Conditions`  2     1381  0.0434 0.957488   
## `Number of Meds`                 2     1381  5.3811 0.004700 **
## Duration                         2     1381  6.0265 0.002478 **
## `Number Other Therapy`           2     1381  2.1379 0.118290   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Visualizing each coefficient across quantiles for all predictors