Estimation of Gasoline Yields from Crude Oil

Description

Operational data of the proportion of crude oil converted to gasoline after distillation and fractionation.

Usage

data("GasolineYield", package = "betareg")

Format

A data frame containing 32 observations on 6 variables.

yield

proportion of crude oil converted to gasoline after distillation and fractionation.

gravity

crude oil gravity (degrees API).

pressure

vapor pressure of crude oil (lbf/in2).

temp10

temperature (degrees F) at which 10 percent of crude oil has vaporized.

temp

temperature (degrees F) at which all gasoline has vaporized.

batch

factor indicating unique batch of conditions gravity, pressure, and temp10.

Details

This dataset was collected by Prater (1956), its dependent variable is the proportion of crude oil after distillation and fractionation. This dataset was analyzed by Atkinson (1985), who used the linear regression model and noted that there is “indication that the error distribution is not quite symmetrical, giving rise to some unduly large and small residuals” (p. 60).

The dataset contains 32 observations on the response and on the independent variables. It has been noted (Daniel and Wood, 1971, Chapter 8) that there are only ten sets of values of the first three explanatory variables which correspond to ten different crudes and were subjected to experimentally controlled distillation conditions. These conditions are captured in variable batch and the data were ordered according to the ascending order of temp10.

Source

Taken from Prater (1956).

References

Atkinson, A.C. (1985). Plots, Transformations and Regression: An Introduction to Graphical Methods of Diagnostic Regression Analysis. New York: Oxford University Press.

Cribari-Neto, F., and Zeileis, A. (2010). Beta Regression in R. Journal of Statistical Software, 34(2), 1–24. doi:10.18637/jss.v034.i02

Daniel, C., and Wood, F.S. (1971). Fitting Equations to Data. New York: John Wiley and Sons.

Ferrari, S.L.P., and Cribari-Neto, F. (2004). Beta Regression for Modeling Rates and Proportions. Journal of Applied Statistics, 31(7), 799–815.

Prater, N.H. (1956). Estimate Gasoline Yields from Crudes. Petroleum Refiner, 35(5), 236–238.

See Also

betareg

Examples

library("betareg")

## IGNORE_RDIFF_BEGIN
data("GasolineYield", package = "betareg")

gy1 <- betareg(yield ~ gravity + pressure + temp10 + temp, data = GasolineYield)
summary(gy1)

Call:
betareg(formula = yield ~ gravity + pressure + temp10 + temp, data = GasolineYield)

Quantile residuals:
    Min      1Q  Median      3Q     Max 
-1.9010 -0.6829 -0.0385  0.5531  2.1314 

Coefficients (mean model with logit link):
              Estimate Std. Error z value Pr(>|z|)    
(Intercept) -2.6949422  0.7625693  -3.534 0.000409 ***
gravity      0.0045412  0.0071419   0.636 0.524871    
pressure     0.0304135  0.0281007   1.082 0.279117    
temp10      -0.0110449  0.0022640  -4.879 1.07e-06 ***
temp         0.0105650  0.0005154  20.499  < 2e-16 ***

Phi coefficients (precision model with identity link):
      Estimate Std. Error z value Pr(>|z|)    
(phi)   248.24      62.02   4.003 6.26e-05 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

Type of estimator: ML (maximum likelihood)
Log-likelihood: 75.68 on 6 Df
Pseudo R-squared: 0.9398
Number of iterations: 147 (BFGS) + 4 (Fisher scoring) 

## Ferrari and Cribari-Neto (2004)
gy2 <- betareg(yield ~ batch + temp, data = GasolineYield)
## Table 1
summary(gy2)

Call:
betareg(formula = yield ~ batch + temp, data = GasolineYield)

Quantile residuals:
    Min      1Q  Median      3Q     Max 
-2.1396 -0.5698  0.1202  0.7040  1.7506 

Coefficients (mean model with logit link):
              Estimate Std. Error z value Pr(>|z|)    
(Intercept) -6.1595710  0.1823247 -33.784  < 2e-16 ***
batch1       1.7277289  0.1012294  17.067  < 2e-16 ***
batch2       1.3225969  0.1179020  11.218  < 2e-16 ***
batch3       1.5723099  0.1161045  13.542  < 2e-16 ***
batch4       1.0597141  0.1023598  10.353  < 2e-16 ***
batch5       1.1337518  0.1035232  10.952  < 2e-16 ***
batch6       1.0401618  0.1060365   9.809  < 2e-16 ***
batch7       0.5436922  0.1091275   4.982 6.29e-07 ***
batch8       0.4959007  0.1089257   4.553 5.30e-06 ***
batch9       0.3857930  0.1185933   3.253  0.00114 ** 
temp         0.0109669  0.0004126  26.577  < 2e-16 ***

Phi coefficients (precision model with identity link):
      Estimate Std. Error z value Pr(>|z|)    
(phi)    440.3      110.0   4.002 6.29e-05 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

Type of estimator: ML (maximum likelihood)
Log-likelihood:  84.8 on 12 Df
Pseudo R-squared: 0.9617
Number of iterations: 51 (BFGS) + 3 (Fisher scoring) 

## Figure 2
par(mfrow = c(3, 2))
plot(gy2, which = 1, type = "pearson", sub.caption = "")
plot(gy2, which = 1, type = "deviance", sub.caption = "")
plot(gy2, which = 5, type = "deviance", sub.caption = "")
plot(gy2, which = 4, type = "pearson", sub.caption = "")
plot(gy2, which = 2:3)

par(mfrow = c(1, 1))

## exclude 4th observation
gy2a <- update(gy2, subset = -4)
gy2a

Call:
betareg(formula = yield ~ batch + temp, data = GasolineYield, subset = -4)

Coefficients (mean model with logit link):
(Intercept)       batch1       batch2       batch3       batch4       batch5  
   -6.35647      1.88688      1.37039      1.62512      1.08066      1.15158  
     batch6       batch7       batch8       batch9         temp  
    1.05766      0.56522      0.50066      0.38523      0.01146  

Phi coefficients (precision model with identity link):
(phi)  
577.8  

summary(gy2a)

Call:
betareg(formula = yield ~ batch + temp, data = GasolineYield, subset = -4)

Quantile residuals:
    Min      1Q  Median      3Q     Max 
-2.0153 -0.8176  0.0897  0.6948  2.0746 

Coefficients (mean model with logit link):
              Estimate Std. Error z value Pr(>|z|)    
(Intercept) -6.3564713  0.1716020 -37.042  < 2e-16 ***
batch1       1.8868782  0.1001837  18.834  < 2e-16 ***
batch2       1.3703911  0.1042352  13.147  < 2e-16 ***
batch3       1.6251199  0.1028326  15.804  < 2e-16 ***
batch4       1.0806596  0.0897855  12.036  < 2e-16 ***
batch5       1.1515826  0.0906857  12.699  < 2e-16 ***
batch6       1.0576556  0.0929172  11.383  < 2e-16 ***
batch7       0.5652219  0.0956100   5.912 3.39e-09 ***
batch8       0.5006625  0.0953210   5.252 1.50e-07 ***
batch9       0.3852258  0.1037500   3.713 0.000205 ***
temp         0.0114588  0.0003945  29.050  < 2e-16 ***

Phi coefficients (precision model with identity link):
      Estimate Std. Error z value Pr(>|z|)    
(phi)    577.8      146.7   3.938 8.22e-05 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

Type of estimator: ML (maximum likelihood)
Log-likelihood: 86.62 on 12 Df
Pseudo R-squared: 0.9662
Number of iterations: 51 (BFGS) + 4 (Fisher scoring) 

## IGNORE_RDIFF_END