Prestige Data Analysis
Wemi, Renan, Luis, Ignacio
December 12, 2017
Materials and Methods
- Obtained dataset from “car”: Prestige;
- Pair plot with all data from Prestige:
## The following object is masked from package:datasets:
##
## women
- Fitting using all variables (prestige vs. education, income, women and census);
##
## Call:
## lm(formula = prestige ~ education + income + women + census +
## type)
##
## Residuals:
## Min 1Q Median 3Q Max
## -12.9863 -4.9813 0.6983 4.8690 19.2402
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.213e+01 8.018e+00 -1.513 0.13380
## education 3.933e+00 6.535e-01 6.019 3.64e-08 ***
## income 9.946e-04 2.601e-04 3.824 0.00024 ***
## women 1.310e-02 3.018e-02 0.434 0.66524
## census 1.156e-03 6.183e-04 1.870 0.06471 .
## typeprof 1.077e+01 4.676e+00 2.303 0.02354 *
## typewc 2.877e-01 3.139e+00 0.092 0.92718
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.037 on 91 degrees of freedom
## (4 observations deleted due to missingness)
## Multiple R-squared: 0.841, Adjusted R-squared: 0.8306
## F-statistic: 80.25 on 6 and 91 DF, p-value: < 2.2e-16- Simplified models: removed non-significant variables (“type” was a marginally significant) step by step in order to find the optimal model.
4.1: Removing women variable
##
## Call:
## lm(formula = prestige ~ education + income + census + type)
##
## Residuals:
## Min 1Q Median 3Q Max
## -13.0873 -4.9935 0.7435 4.9617 19.4891
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.144e+01 7.823e+00 -1.462 0.1472
## education 3.947e+00 6.498e-01 6.075 2.76e-08 ***
## income 9.365e-04 2.221e-04 4.217 5.79e-05 ***
## census 1.125e-03 6.113e-04 1.840 0.0691 .
## typeprof 1.091e+01 4.645e+00 2.348 0.0210 *
## typewc 5.605e-01 3.062e+00 0.183 0.8551
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.006 on 92 degrees of freedom
## (4 observations deleted due to missingness)
## Multiple R-squared: 0.8407, Adjusted R-squared: 0.8321
## F-statistic: 97.12 on 5 and 92 DF, p-value: < 2.2e-164.2: Removing census variable
##
## Call:
## lm(formula = prestige ~ education + income + type)
##
## Residuals:
## Min 1Q Median 3Q Max
## -14.9529 -4.4486 0.1678 5.0566 18.6320
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.6229292 5.2275255 -0.119 0.905
## education 3.6731661 0.6405016 5.735 1.21e-07 ***
## income 0.0010132 0.0002209 4.586 1.40e-05 ***
## typeprof 6.0389707 3.8668551 1.562 0.122
## typewc -2.7372307 2.5139324 -1.089 0.279
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.095 on 93 degrees of freedom
## (4 observations deleted due to missingness)
## Multiple R-squared: 0.8349, Adjusted R-squared: 0.8278
## F-statistic: 117.5 on 4 and 93 DF, p-value: < 2.2e-16When we removed census variable, type was not longer significant.
4.3: Removing type variable
##
## Call:
## lm(formula = prestige ~ education + income)
##
## Residuals:
## Min 1Q Median 3Q Max
## -19.4040 -5.3308 0.0154 4.9803 17.6889
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -6.8477787 3.2189771 -2.127 0.0359 *
## education 4.1374444 0.3489120 11.858 < 2e-16 ***
## income 0.0013612 0.0002242 6.071 2.36e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.81 on 99 degrees of freedom
## Multiple R-squared: 0.798, Adjusted R-squared: 0.7939
## F-statistic: 195.6 on 2 and 99 DF, p-value: < 2.2e-16These selection models was made manually, however we tried automatic stepwise procedure and we came out with a different result.
## Start: AIC=389.16
## prestige ~ education + income + women + census + type
##
## Df Sum of Sq RSS AIC
## - women 1 9.33 4515.2 387.36
## <none> 4505.9 389.16
## - census 1 173.13 4679.0 390.86
## - type 2 669.16 5175.0 398.73
## - income 1 724.12 5230.0 401.77
## - education 1 1793.65 6299.5 420.00
##
## Step: AIC=387.36
## prestige ~ education + income + census + type
##
## Df Sum of Sq RSS AIC
## <none> 4515.2 387.36
## - census 1 166.09 4681.3 388.90
## + women 1 9.33 4505.9 389.16
## - type 2 660.45 5175.6 396.74
## - income 1 872.78 5388.0 402.68
## - education 1 1811.26 6326.4 418.42##
## Call:
## lm(formula = prestige ~ education + income + census + type)
##
## Residuals:
## Min 1Q Median 3Q Max
## -13.0873 -4.9935 0.7435 4.9617 19.4891
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.144e+01 7.823e+00 -1.462 0.1472
## education 3.947e+00 6.498e-01 6.075 2.76e-08 ***
## income 9.365e-04 2.221e-04 4.217 5.79e-05 ***
## census 1.125e-03 6.113e-04 1.840 0.0691 .
## typeprof 1.091e+01 4.645e+00 2.348 0.0210 *
## typewc 5.605e-01 3.062e+00 0.183 0.8551
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.006 on 92 degrees of freedom
## (4 observations deleted due to missingness)
## Multiple R-squared: 0.8407, Adjusted R-squared: 0.8321
## F-statistic: 97.12 on 5 and 92 DF, p-value: < 2.2e-16Model validation
Questions
- Is at least one of the predictors \(X_1\), \(X_2\) , \(. . .\) , \(X_p\) useful in predicting the response?
- Yes, education and income are useful for predicting the prestige of an individual.
- Do all the predictors help to explain \(Y\), or is only a subset of the predictors useful?
- No, for example women percentage, census and type were not a good predictors.
- How well does the model fit the data?
- Adjusted R-squared: 0.7939 Through model validation we checked that the structure of the choosen model was appropriate for the dataset considered.
- Given a set of predictor values, what response value should we predict, and how accurate is our prediction?
- More income and education more prestige. the equation: \[\mathrm{Prestige} = -6.84777 + 4.13744\star\mathrm{Education} + 0.00136\star\mathrm{Income} \pm 7.81.\]
This plot represents prestige vs. education with circles area depending on the income. We choose education for the \(x-\mathrm{axis}\) because of it is more representative than income.