setwd("C:/Users/lhomm/OneDrive/Documents/R")
Dat <- read.table ("http://users.stat.ufl.edu/~rrandles/sta4210/Rclassnotes/data/textdatasets/KutnerData/Chapter%2024%20Data%20Sets/CH24PR09.txt")
colnames (Dat) <- c("Work", "Fee", "Scope", "Supervision")
attach(Dat)
Fee <- factor(Dat$Fee)
levels(Fee) <- c("High", "Med", "Low")
Scope <- factor (Dat$Scope)
levels(Scope) <- c("In", "Out")
Supervision <- factor(Dat$Supervision)
levels(Supervision) <- c("Local", "Travel")
library(DescTools)
interaction.plot(Fee[Supervision == "Local"], Scope[Supervision ==
"Local"], Work[Supervision == "Local"], ylim = c(55, 130),
xlab = "Feel", ylab = "Work Index", main = "Supervision: Local")
interaction.plot(Fee[Supervision == "Travel"], Scope[Supervision ==
"Travel"], Work[Supervision == "Travel"], ylim = c(55,
130), xlab = "Fee", ylab = "Work Index", main = "Supervision: Travel")
### For supervision: local it sppears that the lines parallel. While they don't overlap they are very close. So while we can definitivly say that for supervision: local we can safely and definitivly say that Fee has a main effect we can only tennetivly say the same for scope; we need more information.###
### For supervision: travel the two lines are parallel but far from overlapping so we can say with more assurance that both factors fee and scope have main effects. ###
### There may be evidence of an interaction between scope and supervision. ###
interaction.plot(Fee[Scope == "In"], Supervision[Scope ==
"In"], Work[Scope == "In"], ylim = c(55, 130),
xlab = "Fee", ylab = "Work Index", main = "Scope: In")
interaction.plot(Fee[Scope == "Out"], Supervision[Scope ==
"Out"], Work[Scope == "Out"], ylim = c(55,
130), xlab = "Fee", ylab = "Work Index", main = "Scope: Out")
### For scope: in its clear that fee has a main effect it is a little less clear that scope also does but I am leaning towards saying that it does have a main effect as the two lines are close but still not overlappint. ###
### For scope: out it seems as though both factors have main effects. ###
### There me be an interaction between scope and interaction. ###
Anova1 <- aov (Work ~ Fee*Scope*Supervision)
Anova1
## Call:
## aov(formula = Work ~ Fee * Scope * Supervision)
##
## Terms:
## Fee Scope Supervision Fee:Scope Fee:Supervision
## Sum of Squares 10044.271 1833.977 3832.400 1.601 0.788
## Deg. of Freedom 2 1 1 2 2
## Scope:Supervision Fee:Scope:Supervision Residuals
## Sum of Squares 574.775 3.943 266.138
## Deg. of Freedom 1 2 36
##
## Residual standard error: 2.718954
## Estimated effects may be unbalanced
Anova2 <- anova(lm(Work ~ Fee + Scope + Supervision + Fee*Scope + Fee*Supervision + Scope*Supervision + Fee*Scope*Supervision))
Anova2
## Analysis of Variance Table
##
## Response: Work
## Df Sum Sq Mean Sq F value Pr(>F)
## Fee 2 10044.3 5022.1 679.3364 < 2.2e-16 ***
## Scope 1 1834.0 1834.0 248.0792 < 2.2e-16 ***
## Supervision 1 3832.4 3832.4 518.4027 < 2.2e-16 ***
## Fee:Scope 2 1.6 0.8 0.1083 0.8977
## Fee:Supervision 2 0.8 0.4 0.0533 0.9482
## Scope:Supervision 1 574.8 574.8 77.7489 1.602e-10 ***
## Fee:Scope:Supervision 2 3.9 2.0 0.2667 0.7674
## Residuals 36 266.1 7.4
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(Anova1)
## Df Sum Sq Mean Sq F value Pr(>F)
## Fee 2 10044 5022 679.336 < 2e-16 ***
## Scope 1 1834 1834 248.079 < 2e-16 ***
## Supervision 1 3832 3832 518.403 < 2e-16 ***
## Fee:Scope 2 2 1 0.108 0.898
## Fee:Supervision 2 1 0 0.053 0.948
## Scope:Supervision 1 575 575 77.749 1.6e-10 ***
## Fee:Scope:Supervision 2 4 2 0.267 0.767
## Residuals 36 266 7
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
### H0: i, j, k: Alpha*Beta*Gamma_i_j_k = 0 H1: At least one Alpha*Beta*Gamma does not equal 0 ###
Decsion_Rule <- qf(0.01, 2, 36, lower.tail=FALSE) ### Decision Rule ###
Decsion_Rule
## [1] 5.247894
summary(Anova1)
## Df Sum Sq Mean Sq F value Pr(>F)
## Fee 2 10044 5022 679.336 < 2e-16 ***
## Scope 1 1834 1834 248.079 < 2e-16 ***
## Supervision 1 3832 3832 518.403 < 2e-16 ***
## Fee:Scope 2 2 1 0.108 0.898
## Fee:Supervision 2 1 0 0.053 0.948
## Scope:Supervision 1 575 575 77.749 1.6e-10 ***
## Fee:Scope:Supervision 2 4 2 0.267 0.767
## Residuals 36 266 7
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
### With an F-value of .7674 ~ .767 being much smaller than our decision rule we don't have enough evidence for three way interaction so we reject HO. The P-value above .1/.05/.01 gives extra weight to our conclusions. ###
### Hypthesis ###
### H0: i, j: Alpha*Beta_i_j = 0 H1: At least one Alpha*beta_i_j does not equal 0 ###
### H0: i, k,: Alpha*Gamma_i_k = 0 H1: At least one Alpha*Gamma_i_k does not equal 0 ###
### H0: j, k,: Beta*Gamma_j_k = 0 H1: At least one Beta*Gamma_j_k does not equal 0 ###
### Decision Rules ###
Decsion_Rule <- qf(0.01, 2, 36, lower.tail=FALSE) ### Decision Rule ###
Decsion_Rule
## [1] 5.247894
Decsion_Rule <- qf(0.01, 2, 36, lower.tail=FALSE) ### Decision Rule ###
Decsion_Rule
## [1] 5.247894
Decsion_Rule <- qf(0.01, 1, 36, lower.tail=FALSE) ### Decision Rule ###
Decsion_Rule
## [1] 7.395597
### Modles ###
Anova5 <- aov (Work ~ Fee*Scope)
Anova5
## Call:
## aov(formula = Work ~ Fee * Scope)
##
## Terms:
## Fee Scope Fee:Scope Residuals
## Sum of Squares 10044.271 1833.977 1.601 4678.044
## Deg. of Freedom 2 1 2 42
##
## Residual standard error: 10.55377
## Estimated effects may be unbalanced
Anova6 <- anova(lm(Work ~ Fee + Scope + Fee*Scope))
Anova6
## Analysis of Variance Table
##
## Response: Work
## Df Sum Sq Mean Sq F value Pr(>F)
## Fee 2 10044.3 5022.1 45.0893 3.498e-11 ***
## Scope 1 1834.0 1834.0 16.4656 0.0002108 ***
## Fee:Scope 2 1.6 0.8 0.0072 0.9928389
## Residuals 42 4678.0 111.4
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(Anova5)
## Df Sum Sq Mean Sq F value Pr(>F)
## Fee 2 10044 5022 45.089 3.5e-11 ***
## Scope 1 1834 1834 16.466 0.000211 ***
## Fee:Scope 2 2 1 0.007 0.992839
## Residuals 42 4678 111
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova7 <- aov (Work ~ Fee*Supervision)
Anova7
## Call:
## aov(formula = Work ~ Fee * Supervision)
##
## Terms:
## Fee Supervision Fee:Supervision Residuals
## Sum of Squares 10044.271 3832.400 0.788 2680.434
## Deg. of Freedom 2 1 2 42
##
## Residual standard error: 7.988733
## Estimated effects may be unbalanced
Anova8 <- anova(lm(Work ~ Fee + Scope + Fee*Supervision))
Anova8
## Analysis of Variance Table
##
## Response: Work
## Df Sum Sq Mean Sq F value Pr(>F)
## Fee 2 10044.3 5022.1 243.2582 < 2.2e-16 ***
## Scope 1 1834.0 1834.0 88.8327 8.099e-12 ***
## Supervision 1 3832.4 3832.4 185.6307 < 2.2e-16 ***
## Fee:Supervision 2 0.8 0.4 0.0191 0.9811
## Residuals 41 846.5 20.6
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(Anova7)
## Df Sum Sq Mean Sq F value Pr(>F)
## Fee 2 10044 5022 78.692 6.23e-15 ***
## Supervision 1 3832 3832 60.050 1.26e-09 ***
## Fee:Supervision 2 1 0 0.006 0.994
## Residuals 42 2680 64
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova9 <- aov (Work ~ Scope*Supervision)
Anova9
## Call:
## aov(formula = Work ~ Scope * Supervision)
##
## Terms:
## Scope Supervision Scope:Supervision Residuals
## Sum of Squares 1833.977 3832.400 574.775 10316.741
## Deg. of Freedom 1 1 1 44
##
## Residual standard error: 15.31246
## Estimated effects may be unbalanced
Anova10 <- anova(lm(Work ~ Scope + Supervision + Scope*Supervision))
Anova10
## Analysis of Variance Table
##
## Response: Work
## Df Sum Sq Mean Sq F value Pr(>F)
## Scope 1 1834.0 1834.0 7.8218 0.0076268 **
## Supervision 1 3832.4 3832.4 16.3449 0.0002093 ***
## Scope:Supervision 1 574.8 574.8 2.4514 0.1245872
## Residuals 44 10316.7 234.5
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(Anova9)
## Df Sum Sq Mean Sq F value Pr(>F)
## Scope 1 1834 1834 7.822 0.007627 **
## Supervision 1 3832 3832 16.345 0.000209 ***
## Scope:Supervision 1 575 575 2.451 0.124587
## Residuals 44 10317 234
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
### Conclusions###
### For the model with the only two way interaction being Fee*scope the P-value was larger than .1/.05/.01 and the F-value was smaller than the decision rule. ###
### For the model with Fee*Supervision as the only two way ineraction the P-value was smaller than .1/.05/.01 and the F-value was larger than the decision rule. ###
### For the model with the only two way ineraction being Scope*Supervision the p-value is smaller than .1/.05/.01 and the F-value is larger than the decision rule. ###
### It appears that there is no signifigant interaction between fee and scope but there is signifigant interaction between fee and scope and scope and supervision. ####
### We will now check combinations of two way interactions. ###
### Models ###
Anova3 <- aov (Work ~ Fee*Scope*Supervision)
Anova3
## Call:
## aov(formula = Work ~ Fee * Scope * Supervision)
##
## Terms:
## Fee Scope Supervision Fee:Scope Fee:Supervision
## Sum of Squares 10044.271 1833.977 3832.400 1.601 0.788
## Deg. of Freedom 2 1 1 2 2
## Scope:Supervision Fee:Scope:Supervision Residuals
## Sum of Squares 574.775 3.943 266.138
## Deg. of Freedom 1 2 36
##
## Residual standard error: 2.718954
## Estimated effects may be unbalanced
Anova4 <- anova(lm(Work ~ Fee + Scope + Supervision + Fee*Scope + Fee*Supervision + Scope*Supervision))
Anova4
## Analysis of Variance Table
##
## Response: Work
## Df Sum Sq Mean Sq F value Pr(>F)
## Fee 2 10044.3 5022.1 706.6086 < 2.2e-16 ***
## Scope 1 1834.0 1834.0 258.0384 < 2.2e-16 ***
## Supervision 1 3832.4 3832.4 539.2142 < 2.2e-16 ***
## Fee:Scope 2 1.6 0.8 0.1126 0.8938
## Fee:Supervision 2 0.8 0.4 0.0554 0.9462
## Scope:Supervision 1 574.8 574.8 80.8702 5.989e-11 ***
## Residuals 38 270.1 7.1
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(Anova3)
## Df Sum Sq Mean Sq F value Pr(>F)
## Fee 2 10044 5022 679.336 < 2e-16 ***
## Scope 1 1834 1834 248.079 < 2e-16 ***
## Supervision 1 3832 3832 518.403 < 2e-16 ***
## Fee:Scope 2 2 1 0.108 0.898
## Fee:Supervision 2 1 0 0.053 0.948
## Scope:Supervision 1 575 575 77.749 1.6e-10 ***
## Fee:Scope:Supervision 2 4 2 0.267 0.767
## Residuals 36 266 7
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova11 <- aov (Work ~ Fee*Scope + Fee*Supervision)
Anova11
## Call:
## aov(formula = Work ~ Fee * Scope + Fee * Supervision)
##
## Terms:
## Fee Scope Supervision Fee:Scope Fee:Supervision
## Sum of Squares 10044.271 1833.977 3832.400 1.601 0.788
## Deg. of Freedom 2 1 1 2 2
## Residuals
## Sum of Squares 844.856
## Deg. of Freedom 39
##
## Residual standard error: 4.654349
## Estimated effects may be unbalanced
Anova12 <- anova(lm(Work ~ Fee + Scope + Supervision + Fee*Scope + Fee*Supervision))
Anova12
## Analysis of Variance Table
##
## Response: Work
## Df Sum Sq Mean Sq F value Pr(>F)
## Fee 2 10044.3 5022.1 231.8305 < 2.2e-16 ***
## Scope 1 1834.0 1834.0 84.6596 2.561e-11 ***
## Supervision 1 3832.4 3832.4 176.9102 4.488e-16 ***
## Fee:Scope 2 1.6 0.8 0.0370 0.9638
## Fee:Supervision 2 0.8 0.4 0.0182 0.9820
## Residuals 39 844.9 21.7
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(Anova11)
## Df Sum Sq Mean Sq F value Pr(>F)
## Fee 2 10044 5022 231.830 < 2e-16 ***
## Scope 1 1834 1834 84.660 2.56e-11 ***
## Supervision 1 3832 3832 176.910 4.49e-16 ***
## Fee:Scope 2 2 1 0.037 0.964
## Fee:Supervision 2 1 0 0.018 0.982
## Residuals 39 845 22
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova13 <- aov (Work ~ Fee*Scope + Scope*Supervision)
Anova13
## Call:
## aov(formula = Work ~ Fee * Scope + Scope * Supervision)
##
## Terms:
## Fee Scope Supervision Fee:Scope Scope:Supervision
## Sum of Squares 10044.271 1833.977 3832.400 1.601 574.775
## Deg. of Freedom 2 1 1 2 1
## Residuals
## Sum of Squares 270.868
## Deg. of Freedom 40
##
## Residual standard error: 2.602251
## Estimated effects may be unbalanced
Anova14 <- anova(lm(Work ~ Fee + Scope + Supervision + Fee*Scope + Scope*Supervision))
Anova14
## Analysis of Variance Table
##
## Response: Work
## Df Sum Sq Mean Sq F value Pr(>F)
## Fee 2 10044.3 5022.1 741.6350 < 2.2e-16 ***
## Scope 1 1834.0 1834.0 270.8293 < 2.2e-16 ***
## Supervision 1 3832.4 3832.4 565.9429 < 2.2e-16 ***
## Fee:Scope 2 1.6 0.8 0.1182 0.8888
## Scope:Supervision 1 574.8 574.8 84.8789 1.944e-11 ***
## Residuals 40 270.9 6.8
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(Anova13)
## Df Sum Sq Mean Sq F value Pr(>F)
## Fee 2 10044 5022 741.635 < 2e-16 ***
## Scope 1 1834 1834 270.829 < 2e-16 ***
## Supervision 1 3832 3832 565.943 < 2e-16 ***
## Fee:Scope 2 2 1 0.118 0.889
## Scope:Supervision 1 575 575 84.879 1.94e-11 ***
## Residuals 40 271 7
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova15 <- aov (Work ~ Fee*Supervision + Scope*Supervision)
Anova15
## Call:
## aov(formula = Work ~ Fee * Supervision + Scope * Supervision)
##
## Terms:
## Fee Supervision Scope Fee:Supervision
## Sum of Squares 10044.271 3832.400 1833.977 0.788
## Deg. of Freedom 2 1 1 2
## Supervision:Scope Residuals
## Sum of Squares 574.775 271.682
## Deg. of Freedom 1 40
##
## Residual standard error: 2.606155
## Estimated effects may be unbalanced
Anova16 <- anova(lm(Work ~ Fee + Scope + Supervision + Fee*Supervision + Scope*Supervision))
Anova16
## Analysis of Variance Table
##
## Response: Work
## Df Sum Sq Mean Sq F value Pr(>F)
## Fee 2 10044.3 5022.1 739.415 < 2.2e-16 ***
## Scope 1 1834.0 1834.0 270.019 < 2.2e-16 ***
## Supervision 1 3832.4 3832.4 564.249 < 2.2e-16 ***
## Fee:Supervision 2 0.8 0.4 0.058 0.9437
## Scope:Supervision 1 574.8 574.8 84.625 2.026e-11 ***
## Residuals 40 271.7 6.8
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(Anova15)
## Df Sum Sq Mean Sq F value Pr(>F)
## Fee 2 10044 5022 739.415 < 2e-16 ***
## Supervision 1 3832 3832 564.249 < 2e-16 ***
## Scope 1 1834 1834 270.018 < 2e-16 ***
## Fee:Supervision 2 1 0 0.058 0.944
## Supervision:Scope 1 575 575 84.625 2.03e-11 ***
## Residuals 40 272 7
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
### For the model with all two way For the model with all two way ineractions only scope*Supervision had a P-value smaller than .1/.05/.01 and a large enough F-value. ###
### For the model with Fee*Scope and Fee*Supervision both had P-values smaller than .1/.05/.01 and F-values that were too small. ###
### For the model with Fee*Scope and Scope*Supervision only scope*Supervision had a P-value smaller than .1/.05/.01 and a large enough F-value. ###
### For the model with Fee*Supervision and Scope*Supervision only only scope*Supervision had a P-value smaller than .1/.05/.01 and a large enough F-value. ###
### It seems the only signifigant interaction is Scope*Supervision, ###
### Yes the results are consitent with the graphical representations. ###
Anova17 <- aov (Work ~ Fee + Scope*Supervision)
Anova17
## Call:
## aov(formula = Work ~ Fee + Scope * Supervision)
##
## Terms:
## Fee Scope Supervision Scope:Supervision Residuals
## Sum of Squares 10044.271 1833.977 3832.400 574.775 272.470
## Deg. of Freedom 2 1 1 1 42
##
## Residual standard error: 2.547032
## Estimated effects may be unbalanced
Anova18 <- anova(lm(Work ~ Fee + Scope + Supervision + Scope*Supervision))
Anova18
## Analysis of Variance Table
##
## Response: Work
## Df Sum Sq Mean Sq F value Pr(>F)
## Fee 2 10044.3 5022.1 774.140 < 2.2e-16 ***
## Scope 1 1834.0 1834.0 282.700 < 2.2e-16 ***
## Supervision 1 3832.4 3832.4 590.748 < 2.2e-16 ***
## Scope:Supervision 1 574.8 574.8 88.599 6.621e-12 ***
## Residuals 42 272.5 6.5
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(Anova17)
## Df Sum Sq Mean Sq F value Pr(>F)
## Fee 2 10044 5022 774.1 < 2e-16 ***
## Scope 1 1834 1834 282.7 < 2e-16 ***
## Supervision 1 3832 3832 590.7 < 2e-16 ***
## Scope:Supervision 1 575 575 88.6 6.62e-12 ***
## Residuals 42 272 6
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
t1 <- qt(1- (.1/12), 36)
t1
## [1] 2.51104
means <- c(122.38125, 111.43125, 116.93958, 92.14792, 121.41875,
110.46875, 115.97708, 91.18542, 91.225, 80.275, 85.78333,
60.99167)
c1 <- c(1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0)
c2 <- c(0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0)
c3 <- c(1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0)
c4 <- c(1/3, -1/3, 0, 0, 1/3, -1/3, 0, 0, 1/3, -1/3, 0, 0)
c5 <- c(0, 0, 1/3, -1/3, 0, 0, 1/3, -1/3, 0, 0, 1/3, -1/3)
c6 <- c(1/3, -1/3, -1/3, 1/3, 1/3, -1/3, -1/3, 1/3, 1/3, -1/3,
-1/3, 1/3)
MSE <- sum(((Anova17$residuals) ^ 2) / Anova17$df.residual)
A <- 3
B <- 2
C <- 2
n <- 48
### Mean1.. - Mean2.. = (0.58, 1.345) ###
### Mean2.. - Mean3.. = (29.8, 30.6) ###
### Mean1.. - Mean3.. = (30.8, 31.5) ###
### Mean.11 - Mean.12 = (10.7, 11.2) ###
### Mean.21 _ Mean.22 = (24.5, 24) ###
### Mean.11 - Mean.12 - Mean.21 + Mean.22 = (-14.46 , -13.23) ###
c7 <- c(0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0)
t2 <- qt(.975, 36)
Interval <-sum(c7 * means) + c(-1, 1) * t2 * sqrt(MSE/n * sum(c7 * c7))
Interval
## [1] -0.09192775 2.01692775
Tukey1 <- TukeyHSD(Anova1, which = "Scope:Supervision", conf.level = .9)
Tukey1
## Tukey multiple comparisons of means
## 90% family-wise confidence level
##
## Fit: aov(formula = Work ~ Fee * Scope * Supervision)
##
## $`Scope:Supervision`
## diff lwr upr p adj
## Out:Local-In:Local -5.441667 -8.079514 -2.803819 0.0001151
## In:Travel-In:Local -10.950000 -13.587847 -8.312153 0.0000000
## Out:Travel-In:Local -30.233333 -32.871181 -27.595486 0.0000000
## In:Travel-Out:Local -5.508333 -8.146181 -2.870486 0.0000960
## Out:Travel-Out:Local -24.791667 -27.429514 -22.153819 0.0000000
## Out:Travel-In:Travel -19.283333 -21.921181 -16.645486 0.0000000
### All of the contrasts are significant. In are better than out. Local is also better. High fee level is the best of the three fee levels. So the best option is one with local supervision, in house, and has a high fee. ###