EDPS-942 Assignment 2

Questions / Answers

1. First regress Unemp on Income, then add Educ to the model as a second predictor.

1. Using the results write out the prediction equation for the Model 2 regression (be specific). Predicted Unemployment = 12.75+2.706e-05(Income)-1.057e-01(Education)
2. What is the proportion of variability in Unemp that is accounted for by (only) Educ? R=0.3704 R²=.608605. In other words, roughly 60 percent of the variability in Unemp is accounted for by only Educ.
3. Does the addition of the Educ predictor lead to a significant change in R²? How do you know? The SLR between Unemp and Income produces an R² of 0.03244 whereas the MR including Educ increases the R² to 0.1696. Pvalue increased from 0.2205 to 0.01525 as well
4. What is the correlation between Unemp and Income controlling for Educ? 0.08240113
5. Which of the two predictores in the two-predictor model should be considered relatively more important? How do you know? In the two-predictor model, Education has a larger t-value and is significant compared to income.
6. Of the two models, which model should we select as the “best” model? Why is this the “best” model? Model 2 has a higher R² value meaning that it explains more variability in Unemployment than model one. Therefore, using both Income and Education as predictors (model 2) is a better model than just Income. Model 2 also has a greater F statistic (4.597) and significant p-value (0.01525) compared to model 2 (F=1.542, p-value: 0.2205).

2. Regress Unemp on Educ, Income and Poor.

1. Does the model fit the data? How do you know? Yes. F-statistic is significant (pvalue = 0.0003174)
2. Plot Cook’s D against studentized deleted residuals. DONE
3. Identify any outliers that exist. Yes. To find influential outliers I looked for an states with Cook’s distance greater than 3 times the mean Cook’s Distance value. Also used a threshold vale of 4 / the number of observations (48) with all states having a Cook’s Distance value above this threshold as an influential outlier. Both methods indicated the same 4 states as outliers: California, Main, Rhode Island and West Virginia.
4. If any outliers exist determine if they are influential using the F statistic approach (i.e., Dc1 > 50% point from the F distribution) for determining if the case is influential (the proportion to be cut off is 0.5). There are no influential outliers using the F statisti approach.

1. a-c & e

#library(ggm,ppcor,gvmla)
data<- read.csv("F:/Grad School/FALL 2019/EDPS 942_Correlation Methods/Assignmnet 2/data.csv")
fit1 <- lm(Unemp ~ Income, data=data)
summary(fit1)

Call:
lm(formula = Unemp ~ Income, data = data)

Residuals:
    Min      1Q  Median      3Q     Max 
-2.6896 -0.7229 -0.0954  0.7641  3.1944 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  7.244e+00  1.353e+00   5.356 2.64e-06 ***
Income      -5.203e-05  4.189e-05  -1.242    0.221    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.258 on 46 degrees of freedom
Multiple R-squared:  0.03244,   Adjusted R-squared:  0.01141 
F-statistic: 1.542 on 1 and 46 DF,  p-value: 0.2205

fit2 <- lm(Unemp ~ Income + Educ, data=data)
summary(fit2)

Call:
lm(formula = Unemp ~ Income + Educ, data = data)

Residuals:
     Min       1Q   Median       3Q      Max 
-2.10082 -0.86434  0.01264  0.65182  2.95255 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  1.275e+01  2.382e+00   5.350 2.85e-06 ***
Income       2.706e-05  4.879e-05   0.555  0.58188    
Educ        -1.057e-01  3.876e-02  -2.727  0.00908 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.178 on 45 degrees of freedom
Multiple R-squared:  0.1696,    Adjusted R-squared:  0.1327 
F-statistic: 4.597 on 2 and 45 DF,  p-value: 0.01525

ppcor::spcor.test(data$Unemp,data$Educ,data$Income)#semi-partial only educ

ppcor::spcor.test(data$Unemp,data$Income,data$Educ)#semi-partial only income

1.d

#Partial controling for Education
ppcor::pcor.test(data$Unemp,data$Income,data$Educ)

#alternative version
reg1 <- lm(Unemp~Educ,data=data)   # run linear regression
resid1 <- resid(reg1)     # find the residuals - Unemp free of Educ
reg2 <- lm(Income ~ Educ,data=data)   # second regression
resid2 <- resid(reg2)     # second set of residuals - Income free of Educ
cor(resid1, resid2)       # correlation of residuals - partial correlation

[1] 0.08240113

2.a

fit2.1<- lm(Unemp ~ Educ+Income+Poor, data=data)
summary(fit2.1)

Call:
lm(formula = Unemp ~ Educ + Income + Poor, data = data)

Residuals:
     Min       1Q   Median       3Q      Max 
-2.26480 -0.69444 -0.07819  0.50527  2.88918 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)   
(Intercept)  2.763e+00  3.630e+00   0.761  0.45056   
Educ        -4.125e-02  3.967e-02  -1.040  0.30412   
Income       1.014e-04  4.902e-05   2.070  0.04439 * 
Poor         2.045e-01  6.001e-02   3.408  0.00141 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.06 on 44 degrees of freedom
Multiple R-squared:  0.343, Adjusted R-squared:  0.2982 
F-statistic: 7.658 on 3 and 44 DF,  p-value: 0.0003174

2.b-c

residfit2.1<-resid(fit2.1)
plot(cooks.distance(fit2.1),rstudent(fit2.1))+
abline(v= 4/nrow(data), col="red")+
abline(v=0.072,col="red",lty=2)

integer(0)

  abline(v=.801)

#finding outliers
data %>% mutate(cooks=cooks.distance(fit2.1))%>% group_by() %>% summarise(mean=mean(cooks))

data %>% mutate(cooks=cooks.distance(fit2.1))%>% filter(cooks>=3*0.024)#Dashed line indicating states with Cook's value 3 times greater than mean Cook's value. 

data %>% mutate(cooks=cooks.distance(fit2.1))%>% filter(cooks>=4/nrow(data))#Solid line indicating states over a threshold value of 4 / number of observations. 

#both methods indicate the same outlier states. California, Main, Rhode Island and West Virginia 

2.d

# Excel function  =FINV(0.5,3,44) = 0.801 providing the DCi >50% OR
qf(.5,3,44)

[1] 0.8010265

data %>% mutate(cooks=cooks.distance(fit2.1))%>% filter(cooks>=.801)#no influential outliers

gvmodel<-gvlma(fit2.1)
summary(gvmodel)

Call:
lm(formula = Unemp ~ Educ + Income + Poor, data = data)

Residuals:
     Min       1Q   Median       3Q      Max 
-2.26480 -0.69444 -0.07819  0.50527  2.88918 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)   
(Intercept)  2.763e+00  3.630e+00   0.761  0.45056   
Educ        -4.125e-02  3.967e-02  -1.040  0.30412   
Income       1.014e-04  4.902e-05   2.070  0.04439 * 
Poor         2.045e-01  6.001e-02   3.408  0.00141 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.06 on 44 degrees of freedom
Multiple R-squared:  0.343, Adjusted R-squared:  0.2982 
F-statistic: 7.658 on 3 and 44 DF,  p-value: 0.0003174


ASSESSMENT OF THE LINEAR MODEL ASSUMPTIONS
USING THE GLOBAL TEST ON 4 DEGREES-OF-FREEDOM:
Level of Significance =  0.05 

Call:
 gvlma(x = fit2.1) 

influencePlot(fit2.1)