LMR Exercise 7.3

FemLab appears to have the lowest P value

## 
## Call:
## lm(formula = divorce ~ unemployed + femlab + marriage + birth + 
##     military, data = divusa)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.8611 -0.8916 -0.0496  0.8650  3.8300 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  2.48784    3.39378   0.733   0.4659    
## unemployed  -0.11125    0.05592  -1.989   0.0505 .  
## femlab       0.38365    0.03059  12.543  < 2e-16 ***
## marriage     0.11867    0.02441   4.861 6.77e-06 ***
## birth       -0.12996    0.01560  -8.333 4.03e-12 ***
## military    -0.02673    0.01425  -1.876   0.0647 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.65 on 71 degrees of freedom
## Multiple R-squared:  0.9208, Adjusted R-squared:  0.9152 
## F-statistic: 165.1 on 5 and 71 DF,  p-value: < 2.2e-16

The Plot shows dash slope gets shallower in areas where the points are closer to the line.

## 
## Call:
## lm(formula = divorce ~ femlab, data = divusa)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.7264 -1.6385  0.1595  1.2211  8.0442 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -3.65527    0.92798  -3.939 0.000182 ***
## femlab       0.43867    0.02302  19.056  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.361 on 75 degrees of freedom
## Multiple R-squared:  0.8288, Adjusted R-squared:  0.8265 
## F-statistic: 363.1 on 1 and 75 DF,  p-value: < 2.2e-16
##           (Intercept) I(femlab + rnorm(50)) 
##            -3.5182228             0.4352488
##               (Intercept) I(femlab + 2 * rnorm(50)) 
##                -3.1736670                 0.4284811

## 
## Naive model:
## lm(formula = divorce ~ femlab, data = divusa, x = TRUE)
## 
## SIMEX-Variables: divorce
## Number of Simulations: 1000
## 
## Coefficients:
## (Intercept)       femlab  
##      -3.641        0.438

\[\verb!The predicted value of ! \hat{\beta} \verb! is: 0.4!\]

The eigen values are large in range and condition numbers are large. There is low variance inflation is low.

##             year divorce unemployed femlab marriage birth military
## year        1.00    0.88      -0.23   0.99    -0.62 -0.58     0.01
## divorce     0.88    1.00      -0.21   0.91    -0.53 -0.72     0.02
## unemployed -0.23   -0.21       1.00  -0.26    -0.27 -0.31    -0.40
## femlab      0.99    0.91      -0.26   1.00    -0.65 -0.60     0.05
## marriage   -0.62   -0.53      -0.27  -0.65     1.00  0.67     0.26
## birth      -0.58   -0.72      -0.31  -0.60     0.67  1.00     0.14
## military    0.01    0.02      -0.40   0.05     0.26  0.14     1.00
## [1] 1.252019e+05 6.470014e+00
## [1]   1.0000 139.1082
## (Intercept)      femlab 
##    1.995224    1.000000

Creating random perbutation with sqrt of max vif value gives us a good p value for marriage , birth and femlab. The correlation matrix confirms that these value have the highest correlation. This linear model has a RSquared value of 0.813.

## 
## Call:
## lm(formula = divorce + sqrt(max(vif(x))) * rnorm(38) ~ unemployed + 
##     femlab + marriage + birth + military, data = divusa)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.4289 -1.5154  0.1035  1.3819  5.0485 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -1.51105    4.73761  -0.319   0.7507    
## unemployed  -0.07458    0.07807  -0.955   0.3427    
## femlab       0.42210    0.04270   9.886 5.54e-15 ***
## marriage     0.14893    0.03408   4.370 4.17e-05 ***
## birth       -0.13016    0.02177  -5.979 8.21e-08 ***
## military    -0.03703    0.01989  -1.862   0.0668 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.304 on 71 degrees of freedom
## Multiple R-squared:  0.864,  Adjusted R-squared:  0.8544 
## F-statistic: 90.21 on 5 and 71 DF,  p-value: < 2.2e-16
##             (Intercept) femlab
## (Intercept)           1     NA
## femlab               NA      1

The new linear model with just the high correlated values result in an higher RSquared value of 0.9106.

## 
## Call:
## lm(formula = divorce ~ marriage + birth + femlab, data = divusa)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.6923 -1.1934 -0.0534  1.2265  3.6701 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -1.54545    2.21247  -0.699    0.487    
## marriage     0.12609    0.02199   5.735 2.07e-07 ***
## birth       -0.11627    0.01412  -8.235 5.10e-12 ***
## femlab       0.41337    0.02275  18.174  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.695 on 73 degrees of freedom
## Multiple R-squared:  0.9141, Adjusted R-squared:  0.9106 
## F-statistic: 258.9 on 3 and 73 DF,  p-value: < 2.2e-16