factors influencing the human development index (hdi) using multiple linear regression
DESCRIPTION
Identified the most crucial factors that influence Human Development Index through regression analysis using Minitab softwareTRANSCRIPT
Factors influencing the Human Development Index (HDI) using
Multiple linear regression
ADITYA PANUGANTI
1202062944
Industrial Engineering
Year of data: 2008Source: UN Development Programme Database
Objective and Dataset description• To find which of the following variables have an effect on the Human
Development Index (HDI)
DESCRIPTION OF THE REGRESSOR VARIABLE TYPE LEVELS
Adult Literacy rate LIT NUMERICAL Combined gross enrollment ratio in
education (both sexes) GRO NUMERICAL
Gender Inequality index GEN NUMERICAL Mean years of schooling (people
whose age is more than 25) SCH NUMERICAL
Expected years of schooling (for a child at entrance age) EXP NUMERICAL
Life Expectancy at birth LIF NUMERICAL Gross Domestic product per capita
(GDP) GDP NUMERICAL
Gross National income per capita (GNI) GNI NUMERICAL
Maternal Mortality ratio MAT NUMERICAL
Intensity of Deprivation DEP CATEGORICAL 2Homicide rate HOM NUMERICAL
Under Five age mortality rate MOR NUMERICAL
Continent indicator variable CON1-Africa; CON2-Asia; CON3-Europe; CON4-N.America; CON5-Oceania; CON6-S.America CATEGORICAL 6
Fitting the full model without interaction terms
• The regression equation for full model is• y = 0.0596 + 0.00440 LIF + 0.000007 GDP - 0.000748 GRO + 0.0158 SCH +
0.0080 GEN+ 0.0159 EXP - 0.000004 GNI + 0.000003 MAT - 0.000051 HOM - 0.000540 MOR+ 0.000176 LIT - 0.0185 DEP + 0.0023 CON1 - 0.0117 CON2 - 0.0100 CON3+ 0.00431 CON4 - 0.0268 CON5
• Difficult to interpret the coefficients of the above regression equation.• Hence standardized the regression coefficients using Unit Normal
scaling
Fitting the full model after Standardization
• The regression equation is
y = 0.684 + 0.0404 LIF + 0.100 GDP - 0.0117 GRO + 0.0408 SCH + 0.00136 GEN + 0.0443 EXP - 0.0627 GNI + 0.00089 MAT - 0.00068 HOM - 0.0196 MOR+ 0.00259 LIT - 0.0185 DEP + 0.0023 CON1 - 0.0117 CON2 - 0.0100 CON3+ 0.00431 CON4 - 0.0268 CON5
• Model Statistics:
R-Sq = 98.5% R-Sq(adj) = 98.2%
Analysis of Variance (ANOVA)
Source DF SS MS F P
Regression 17 2.21784 0.13046 325.49 0.000
Residual Error 84 0.03367 0.00040
Total 101 2.25150
Signs of Multicollinearity• Inference from Variance Inflation Factor (VIFs):
VIF of GDP = 560.116 and VIF of GNI = 533.109 (Indicating Severe Multicollinearity)
VIF of EXP = 18.368 and VIF of GRO = 16.456 (just over 10; Indicating Multicollinearity)
• Inference from Correlation matrix: LIF GDP GRO SCH GEN EXP GNI MAT
GDP 0.595
GRO 0.719 0.630
SCH 0.603 0.553 0.776
GEN -0.677 -0.705 -0.758 -0.743
EXP 0.692 0.636 0.956 0.774 -0.798
GNI 0.584 0.999 0.618 0.539 -0.688 0.620
Dropped GNI from the model.No change in R-sq and R-sq(adj) statistics before and after dropping the model
R-Sq = 98.5% R-Sq(adj) = 98.2%
• To confirm Multicollinearity between EXP and GRO, did a further analysis using Principal Component Analysis.
• Found the condition number to be (Condition number = λmax/ λmin =7.8001/0.0327 = 238.53
>100, indicating moderate multicollinearity
Dropped EXP also from the model and check the Model summary statistics- a slight reduction in R-sq and R-sq(adj) .
Residual plots and Model Adequacy
Both normality and Residual vs fitted plots look good, satisfying the normality and constant variance conditions
Indicator Interactions
• Considered interaction terms of DEP and other numerical variables.• 24 variables in all including all the interaction terms• S = 0.0220704 R-Sq = 98.3% R-Sq(adj) = 97.8%; R-Sq(pred) = 96.80%• Residual plots:
0.0500.0250.000-0.025-0.050
99.9
99
90
50
10
1
0.1
Residual
Perc
ent
1.00.80.60.40.2
0.050
0.025
0.000
-0.025
-0.050
Fitted Value
Residual
0.060.040.020.00-0.02-0.04-0.06
30
20
10
0
Residual
Fre
quency
1009080706050403020101
0.050
0.025
0.000
-0.025
-0.050
Observation Order
Residual
Normal Probability Plot Versus Fits
Histogram Versus Order
Residual Plots for y
Outliers and Influential points
ObsHii
(0.49484)Outliers or
not
Student residuals
(>2)
Outliers or not
DFFIT (0.97014)
Influential or not
94 0.322922 3.642607124 Outlier 2.747085588 Influential
95 0.174703 2.499454616 Outlier 1.191860945 Influential
70 0.514913 Outlier 2.612075845 Outlier 2.800604995
62 0.825517 Outlier 0.957486148 2.08152168 Influential
38 0.108995 3.082041891 Outlier 1.14891402 Influential
Other outliers in graph
• Fitting each of the datapoints 45, 50, 80 and checking if there is any changes in summary stats
• These points are not contributing to any leverage, nor being influential; except for the fact that they are outliers; also R-sq not changing much, therefore we are leaving them in the model.
Residual plots after taking off the outliers and influential points
0.0500.0250.000-0.025-0.050
99.9
99
90
50
10
1
0.1
Residual
Perc
ent
1.00.80.60.4
0.04
0.02
0.00
-0.02
-0.04
Fitted Value
Resi
dual
0.0300.0150.000-0.015-0.030-0.045
20
15
10
5
0
Residual
Fre
quency
9080706050403020101
0.04
0.02
0.00
-0.02
-0.04
Observation Order
Resi
dual
Normal Probability Plot Versus Fits
Histogram Versus Order
Residual Plots for y
• Normal probability plot looks good but the Residuals vs fit looks like a double bow shaped.
• To confirm this, we have used box cox transformation which showed us that there is a need in the transformation on ‘y’
Box-Cox Transformation
lambda=> -1 -0.5 0 0.15 0.25 0.375 0.5 0.75 1 2
SS res = 0.07918 0.05952 0.04749 0.04484 0.04326 0.04148 0.03989 0.03721 0.03511 0.03147
Suggests lambda = 2, implies transform y y2
5.02.50.0-2.5-5.0
0.6
0.5
0.4
0.3
0.2
0.1
Lambda
StD
ev
Lower CL Upper CL
Limit
Estimate 2.09
Lower CL 1.15Upper CL 3.06
Rounded Value 2.00
(using 95.0% confidence)
Lambda
Box-Cox Plot of y
Residual plots after transformation
0.0500.0250.000-0.025-0.050
99.9
99
90
50
10
1
0.1
Residual
Perc
ent
0.80.60.40.20.0
0.050
0.025
0.000
-0.025
-0.050
Fitted Value
Resi
dual
0.040.020.00-0.02-0.04-0.06
40
30
20
10
0
Residual
Fre
quency
9080706050403020101
0.050
0.025
0.000
-0.025
-0.050
Observation Order
Resi
dual
Normal Probability Plot Versus Fits
Histogram Versus Order
Residual Plots for y2
Can find some outliers in the Normal probability plot
Outliers and Influential points
ObsHii
(0.49485)Outlier or
notStudent
residuals (>2)Outlier or not
DFFIT (0.99483)
Influential
82 0.319989 1.567901961 1.07554448 Influential
76 0.329424 1.663755784 1.166118631 Influential
55 0.137185 3.046901559 Outlier 1.214935585
45 0.10912 2.940944269 Outlier 1.029270034
44 0.844545 Outlier 0.063698129 0.148468899
39 0.243545 2.155012184 Outlier 1.222778345 Influential
Residual plots after taking off the outliers and influential points
0.0500.0250.000-0.025-0.050
99.9
99
90
50
10
1
0.1
Residual
Perc
ent
0.80.60.40.20.0
0.050
0.025
0.000
-0.025
-0.050
Fitted Value
Resi
dual
0.040.020.00-0.02-0.04
40
30
20
10
0
Residual
Fre
quency
9080706050403020101
0.050
0.025
0.000
-0.025
-0.050
Observation Order
Resi
dual
Normal Probability Plot Versus Fits
Histogram Versus Order
Residual Plots for y2
No need for any transformation, Box-Cox suggests λ = 1
Variable selection and Model building
Method Suggested Predictors Summary Statistics
Best subsets methodLIF GDP GRO MOR SCH GEN CON2
CON5 GDP_D HOM_D MOR_D(No of variables = 11)
S = 0.01869R-Sq = 99.1
R-sq (adj) = 99Mallows Cp = 10.6
Forward Selection
(α–to-enter: 0.15 )
GEN GRO LIF SCH GDP CON2 GDP_D LIF_D MOR CON5 MOR_D HOM_D DEP
HOM(No of variables = 14)
S = 0.0186R-Sq = 99.18
R-sq (adj) = 99.03Mallows Cp = 12.8
Backward Elimination
(α–to-remove: 0.15)
LIF GDP GRO MOR SCH GEN CON2 CON5 GDP_D HOM_D MOR_D
(No of variables = 11)
S = 0.0187R-Sq = 99.14
R-sq (adj) = 99.02Mallows Cp = 10.6
Stepwise Regression
( α–to-enter: 0.15, α–to-remove: 0.15)
GEN GRO LIF GDP SCH CON2 MOR GDP_D CON5 MOR_D HOM_D
(No of variables = 11)
S = 0.0187R-Sq = 99.14
R-sq (adj) = 99.02Mallows Cp = 10.6
Fit the selected model• Regression equation:
y2 = 0.476 - 0.0164 GEN + 0.0403 GRO + 0.0422 LIF + 0.0557 GDP + 0.0449 SCH - 0.0181 CON2 - 0.0388 MOR + 0.0523 GDP_D + 0.0289 CON5 + 0.0412 MOR_D - 0.0476 HOM_D
• Detected Multicollinearity using Principal component analysis
condition number = 134.837 (>100, Moderate Multicollinearity)
• Linear dependency equation: 0.107GRO+0.337LIF+0.798MOR-0.467MOR_D (dependency between the variables in the equation)
• Using correlation matrix found that the variable MOR has large correlation with LIF and MOR_D.
• Dropping MOR removed multicollinearity from model (condition number = 39.04617 (<100, No multicollinearity)
Residual plots after dropping MOR
0.080.040.00-0.04-0.08
99.9
99
90
50
10
1
0.1
Residual
Perc
ent
0.80.60.40.20.0
0.05
0.00
-0.05
Fitted Value
Resi
dual
0.040.020.00-0.02-0.04-0.06-0.08
20
15
10
5
0
Residual
Fre
quency
9080706050403020101
0.05
0.00
-0.05
Observation OrderResi
dual
Normal Probability Plot Versus Fits
Histogram Versus Order
Residual Plots for y2
• Presence of an outlier datapoint 72 • No need for any transformation, Box-Cox suggests λ = 1
Fit the model after dropping off the outlier
• The regression equation is
y2 = 0.482 - 0.0221 GEN + 0.0436 GRO + 0.0576 LIF + 0.0528 GDP + 0.0483 SCH - 0.0115 CON2 + 0.0556 GDP_D + 0.0182 CON5 + 0.0169 MOR_D - 0.0538 HOM_D
• R-sq = 99.1% R-sq(adj) = 99% R-sq(pred) = 98.73%
0.0500.0250.000-0.025-0.050
99.9
99
95
90
80706050403020
10
5
1
0.1
Residual
Perc
ent
Normal Probability Plot(response is y2)
0.90.80.70.60.50.40.30.20.1
0.04
0.03
0.02
0.01
0.00
-0.01
-0.02
-0.03
-0.04
Fitted Value
Resi
dual
Versus Fits(response is y2)
Model validation• Considered 118 countries for modelling
102 Estimation data and 16 prediction data
y[pred] y[actual]
0.784 0.8670.583 0.5990.793 0.870.796 0.8960.781 0.8810.776 0.870.695 0.770.831 0.8510.575 0.6160.378 0.4170.720 0.8160.341 0.4080.753 0.8470.815 0.90.315 0.4210.770 0.849
R2 [pred] by model = 83.65%
R2[pred] (L.S fit) = 98.73%
MS res by model = 0.006122809
Ms res (L.S fit) = 0.00036
Conclusion
• The reduced model has a better R-sq than the actual model and most of the variables are significant (low p-value) in the model.
• The following variables were found to be significant – Gender inequality index– Combined gross enrolment– Life expectancy at birth– GDP– Mean schooling years– Countries in continent 2– GDP& intensity of deprivation– Under 5 mortality rate& intensity of deprivation– Homicide rate& intensity of deprivation
Possible improvements
• More datapoints
• Ridge regression to eliminate multicollinearity
• Robust regression – to add more weight to the datapoints and retain them in the model.