Description Format Details Source Examples
The Gasoline
data frame has 32 rows and 6 columns.
This data frame contains the following columns:
a numeric vector giving the percentage of crude oil converted to gasoline after distillation and fractionation
a numeric vector giving the temperature (degrees F) at which all the gasoline is vaporized
the inferred crude oil sample number  a factor with levels
A
to J
a numeric vector giving the crude oil gravity (degrees API)
a numeric vector giving the vapor pressure of the crude oil (lbf/in^2)
a numeric vector giving the crude oil 10% point ASTM—the temperature at which 10% of the crude oil has become vapor.
Prater (1955) provides data on crude oil properties and
gasoline yields. Atkinson (1985)
uses these data to illustrate the use of diagnostics in multiple
regression analysis. Three of the covariates—API
,
vapor
, and ASTM
—measure characteristics of the
crude oil used to produce the gasoline. The other covariate —
endpoint
—is a characteristic of the refining process.
Daniel and Wood (1980) notice that the covariates characterizing
the crude oil occur in only ten distinct groups and conclude that the
data represent responses measured on ten different crude oil samples.
Prater, N. H. (1955), Estimate gasoline yields from crudes, Petroleum Refiner, 35 (5).
Atkinson, A. C. (1985), Plots, Transformations, and Regression, Oxford Press, New York.
Daniel, C. and Wood, F. S. (1980), Fitting Equations to Data, Wiley, New York
Venables, W. N. and Ripley, B. D. (1999) Modern Applied Statistics with SPLUS (3rd ed), Springer, New York.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16  require(lattice)
str(Gasoline)
xyplot(yield ~ endpoint  Sample, Gasoline, aspect = 'xy',
main = "Gasoline data", xlab = "Endpoint (degrees F)",
ylab = "Percentage yield",
type = c("g", "p", "r"),
index.cond = function(x,y) coef(lm(y~x))[2],
layout = c(5,2))
print(m1 < lmer(yield ~ endpoint + (1Sample), Gasoline), corr = FALSE)
m2 < lmer(yield ~ endpoint + (endpointSample), Gasoline, verbose = 1)
print(m2)
Gasoline$endptC < with(Gasoline, endpoint  mean(endpoint))
m3 < lmer(yield ~ endpoint + (endptCSample), Gasoline, verbose = 1)
print(m3)
xyplot(endptC ~ `(Intercept)`, ranef(m3)[[1]], type = c("g", "p", "r"),
aspect = 1)

Loading required package: lme4
Loading required package: Matrix
Attaching package: 'MEMSS'
The following objects are masked from 'package:datasets':
CO2, Orange, Theoph
Loading required package: lattice
'data.frame': 32 obs. of 6 variables:
$ yield : num 6.9 14.4 7.4 8.5 8 2.8 5 12.2 10 15.2 ...
$ endpoint: num 235 307 212 365 218 235 285 205 267 300 ...
$ Sample : Factor w/ 10 levels "A","B","C","D",..: 5 6 8 1 7 9 4 10 3 5 ...
$ API : num 38.4 40.3 40 31.8 40.8 41.3 38.1 50.8 32.2 38.4 ...
$ vapor : num 6.1 4.8 6.1 0.2 3.5 1.8 1.2 8.6 5.2 6.1 ...
$ ASTM : num 220 231 217 316 210 267 274 190 236 220 ...
Linear mixed model fit by REML ['lmerMod']
Formula: yield ~ endpoint + (1  Sample)
Data: Gasoline
REML criterion at convergence: 175.4306
Random effects:
Groups Name Std.Dev.
Sample (Intercept) 8.388
Residual 1.880
Number of obs: 32, groups: Sample, 10
Fixed Effects:
(Intercept) endpoint
33.3063 0.1576
iteration: 1
f(x) = 245.910051
iteration: 2
f(x) = 244.477085
iteration: 3
f(x) = 252.598440
iteration: 4
f(x) = 255.981895
iteration: 5
f(x) = 246.813046
iteration: 6
f(x) = 252.606218
iteration: 7
f(x) = 220.984353
iteration: 8
f(x) = 177.134711
iteration: 9
f(x) = 173.635328
iteration: 10
f(x) = 173.406852
iteration: 11
f(x) = 189.313666
iteration: 12
f(x) = 213.562479
iteration: 13
f(x) = 221.764819
iteration: 14
f(x) = 193.775636
iteration: 15
f(x) = 188.483959
iteration: 16
f(x) = 196.693322
iteration: 17
f(x) = 195.153555
iteration: 18
f(x) = 219.138072
iteration: 19
f(x) = 175.986510
iteration: 20
f(x) = 172.792337
iteration: 21
f(x) = 173.170826
iteration: 22
f(x) = 173.199688
iteration: 23
f(x) = 172.863689
iteration: 24
f(x) = 173.173220
iteration: 25
f(x) = 173.433493
iteration: 26
f(x) = 173.065043
iteration: 27
f(x) = 172.803232
iteration: 28
f(x) = 172.839403
iteration: 29
f(x) = 172.791633
iteration: 30
f(x) = 172.791856
iteration: 31
f(x) = 172.790692
iteration: 32
f(x) = 172.792796
iteration: 33
f(x) = 172.790421
iteration: 34
f(x) = 172.790311
iteration: 35
f(x) = 172.790302
iteration: 36
f(x) = 172.790382
iteration: 37
f(x) = 172.790302
boundary (singular) fit: see ?isSingular
Linear mixed model fit by REML ['lmerMod']
Formula: yield ~ endpoint + (endpoint  Sample)
Data: Gasoline
REML criterion at convergence: 172.7903
Random effects:
Groups Name Std.Dev. Corr
Sample (Intercept) 4.53713
endpoint 0.01063 1.00
Residual 1.79038
Number of obs: 32, groups: Sample, 10
Fixed Effects:
(Intercept) endpoint
31.8891 0.1543
convergence code 0; 1 optimizer warnings; 0 lme4 warnings
iteration: 1
f(x) = 270.772588
iteration: 2
f(x) = 257.199808
iteration: 3
f(x) = 284.266784
iteration: 4
f(x) = 280.843034
iteration: 5
f(x) = 291.077528
iteration: 6
f(x) = 284.390191
iteration: 7
f(x) = 245.877857
iteration: 8
f(x) = 184.987769
iteration: 9
f(x) = 176.935525
iteration: 10
f(x) = 177.210253
iteration: 11
f(x) = 221.398772
iteration: 12
f(x) = 213.799171
iteration: 13
f(x) = 269.318953
iteration: 14
f(x) = 254.116989
iteration: 15
f(x) = 252.809983
iteration: 16
f(x) = 183.418959
iteration: 17
f(x) = 204.972310
iteration: 18
f(x) = 211.966629
iteration: 19
f(x) = 180.028929
iteration: 20
f(x) = 174.507121
iteration: 21
f(x) = 205.515479
iteration: 22
f(x) = 182.380692
iteration: 23
f(x) = 180.642299
iteration: 24
f(x) = 177.523272
iteration: 25
f(x) = 178.446217
iteration: 26
f(x) = 174.423815
iteration: 27
f(x) = 174.398137
iteration: 28
f(x) = 174.410428
iteration: 29
f(x) = 174.406391
iteration: 30
f(x) = 174.389145
iteration: 31
f(x) = 174.390294
iteration: 32
f(x) = 174.388875
iteration: 33
f(x) = 174.389343
iteration: 34
f(x) = 174.389101
iteration: 35
f(x) = 174.389927
iteration: 36
f(x) = 174.389358
iteration: 37
f(x) = 174.388748
boundary (singular) fit: see ?isSingular
Linear mixed model fit by REML ['lmerMod']
Formula: yield ~ endpoint + (endptC  Sample)
Data: Gasoline
REML criterion at convergence: 174.3887
Random effects:
Groups Name Std.Dev. Corr
Sample (Intercept) 6.542021
endptC 0.008715 1.00
Residual 2.027737
Number of obs: 32, groups: Sample, 10
Fixed Effects:
(Intercept) endpoint
31.5621 0.1533
convergence code 0; 1 optimizer warnings; 0 lme4 warnings
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