Homework Week 11
Ben Herndon
11/13/2021
6.8)
Hypotheses:
H0_A: Time is not significant
H1_A: Time is significant
H0_B: Culture Medium is not significant
H1_B: Culture Medium is significant
H0_AB: Interaction is not significant
H1_AB: Interaction is significant
I) Read In Data
Time <- c(rep(-1,6),rep(1,6),rep(-1,6),rep(1,6))
Medium <- c(rep(-1,6),rep(-1,6),rep(1,6),rep(1,6))
obs8 <- c(21,22,23,28,20,26,25,26,24,25,29,27,37,39,38,38,35,36,31,34,29,33,30,35)
dat8 <- data.frame(Time,Medium,obs8)II) ANOVA and Models
model8 <- lm(obs8~Time*Medium, data = dat8)
coef(model8)## (Intercept) Time Medium Time:Medium
## 29.625000 -0.625000 4.958333 -1.958333halfnormal(model8)## Warning in halfnormal.lm(model8): halfnormal not recommended for models with
## more residual df than model df##
## Significant effects (alpha=0.05, Lenth method):## [1] Medium Time:Medium e8
interaction.plot(Medium,Time,obs8)
anova(model8)plot(model8)
## hat values (leverages) are all = 0.1666667
## and there are no factor predictors; no plot no. 5
Conclusions:
We can conclude based on the half-normal plot, the interaction plot, and the ANOVA table that Culture Medium and Interactions are significant. Therefore we reject the null hypothesis for B and AB and fail to reject the null hypothesis for A.
The residuals and NPP show that the data has relatively the same variance between A, B, and AB, and that the data is normal, respectively.
6.12)
Hypotheses:
H0_A: Arsenic Flow Rate is not significant
H1_A: Arsenic Flow Rate is significant
H0_B: Deposition Time is not significant
H1_B: Deposition Time is significant
H0_AB: Interaction is not significant
H1_AB: Interaction is significant
Read In Data
Arsenic <- c(rep(-1,4),rep(1,4),rep(-1,4),rep(1,4))
Deposition <- c(rep(-1,4),rep(-1,4),rep(1,4),rep(1,4))
Thickness <- c(14.037,16.165,13.972,13.907,13.880,13.860,14.032,13.914,14.821,14.757,14.843,14.878,14.888,14.921,14.415,14.932)
dat12 <- data.frame(Arsenic,Deposition,Thickness)a) Factor Effects
model12 <- lm(Thickness~Arsenic*Deposition, data = dat12)
coef(model12)## (Intercept) Arsenic Deposition Arsenic:Deposition
## 14.513875 -0.158625 0.293000 0.140750The effect estimates are as follows:
(1) = 14.513875
A (Arsenic) = -0.158625
B (Deposition) = 0.293000
AB (Arsenic:Deposition) = 0.140750
b) ANOVA
anova(model12)Based on the ANOVA table, we can see that the p-value for both factors and their interactions are greater than our alpha of 0.05. Therefore we fail to reject all three null hypotheses and conclude that Arsenic Flow Rate, Deposition Time, and their interaction are not significant
c) Regression Equation
\[ y (Thickness)=14.513875-0.158625(A)+0.293(B)+0.14075(AB)+Error \]
d) Analyze Residuals
plot(model12)
## hat values (leverages) are all = 0.25
## and there are no factor predictors; no plot no. 5
The Residuals vs. Fitted Plot and the Normality Plot both show an outlier. Furthermore, the Residuals vs. Fitted Plot shows non-consistant variance.
e)
I would run a log or natural log transformation on the data to normalize the variance and hopefully reduce the size of the outlier.
6.21)
Hypotheses:
H0_A: Putt Length is not significant
H1_A: Putt Length is significant
H0_B: Putter Type is not significant
H1_B: Putter Type is significant
H0_C: Putt Break is not significant
H1_C: Putt Break is significant
H0_D: Putt Slope is not significant
H1_D: Putt Slope is significant
H0_AB: Interaction AB is not significant
H1_AB: Interaction AB is significant
H0_AC: Interaction AC is not significant
H1_AC: Interaction AC is significant
H0_AD: Interaction AD is not significant
H1_AD: Interaction AD is significant
H0_BC: Interaction BC is not significant
H1_BC: Interaction BC is significant
H0_BD: Interaction BD is not significant
H1_BD: Interaction BD is significant
H0_CD: Interaction CD is not significant
H1_CD: Interaction CD is significant
H0_ABC: Interaction ABC is not significant
H1_ABC: Interaction ABC is significant
H0_ABD: Interaction ABD is not significant
H1_ABD: Interaction ABD is significant
H0_ACD: Interaction ACD is not significant
H1_ACD: Interaction ACD is significant
H0_BCD: Interaction BCD is not significant
H1_BCD: Interaction BCD is significant
H0_ABCD: Interaction ABCD is not significant
H1_ABCD: Interaction ABCD is significant
Read In Data
Length <- c(rep(-1,7),rep(1,7),rep(-1,7),rep(1,7),rep(-1,7),rep(1,7),rep(-1,7),rep(1,7),rep(-1,7),rep(1,7),rep(-1,7),rep(1,7),rep(-1,7),rep(1,7),rep(-1,7),rep(1,7))
Type <- c(rep(-1,7),rep(-1,7),rep(1,7),rep(1,7),rep(-1,7),rep(-1,7),rep(1,7),rep(1,7),rep(-1,7),rep(-1,7),rep(1,7),rep(1,7),rep(-1,7),rep(-1,7),rep(1,7),rep(1,7))
Break <- c(rep(-1,7),rep(-1,7),rep(-1,7),rep(-1,7),rep(1,7),rep(1,7),rep(1,7),rep(1,7),rep(-1,7),rep(-1,7),rep(-1,7),rep(-1,7),rep(1,7),rep(1,7),rep(1,7),rep(1,7))
Slope <- c(rep(-1,7),rep(-1,7),rep(-1,7),rep(-1,7),rep(-1,7),rep(-1,7),rep(-1,7),rep(-1,7),rep(1,7),rep(1,7),rep(1,7),rep(1,7),rep(1,7),rep(1,7),rep(1,7),rep(1,7))
Distance <- c(10,18,14,12.5,19,16,18.5,
0,16.5,4.5,17.5,20.5,17.5,33,
4,6,1,14.5,12,14,5,
0,10,34,11,25.5,21.5,0,
0,0,18.5,19.5,16,15,11,
5,20.5,18,20,29.5,19,10,
6.5,18.5,7.5,6,0,10,0,
16.5,4.5,0,23.5,8,8,8,
4.5,18,14.5,10,0,17.5,6,
19.5,18,16,5.5,10,7,36,
15,16,8.5,0,0.5,9,3,
41.5,39,6.5,3.5,7,8.5,36,
8,4.5,6.5,10,13,41,14,
21.5,10.5,6.5,0,15.5,24,16,
0,0,0,4.5,1,4,6.5,
18,5,7,10,32.5,18.5,8)
dat21 <- data.frame(Length,Type,Break,Slope,Distance)a) Halfnormal Data and ANOVA
model21 <- lm(Distance~Length*Type*Break*Slope, data = dat21)
halfnormal(model21)## Warning in halfnormal.lm(model21): halfnormal not recommended for models with
## more residual df than model df##
## Significant effects (alpha=0.05, Lenth method):## [1] Length e95 e28 e44 e49 Type e84 e32 e78
anova(model21)Based on the Halfnormal plot and the ANOVA table, we can see that the only significant factors are the length of the putt and the type of putter. Therefore, we can reject the null hypothesis for A and B, and we fail to reject the rest of the null hypotheses.
b) Residuals
I) Complete Data Plots
plot(model21)
## hat values (leverages) are all = 0.1428571
## and there are no factor predictors; no plot no. 5
II) Significant Data Plots
model21s <- lm(Distance~Length*Type, data = dat21)
plot(model21s)
## hat values (leverages) are all = 0.03571429
## and there are no factor predictors; no plot no. 5
The Normality Plot has outliers, and the Residual vs Fitted Plot shows inconsistant variance. As both of these violate the assumptions of ANOVA, this model is inadequate.
6.36)
Hypotheses:
H0_A: Factor A is not significant
H1_A: Factor A is significant
H0_B: Factor B is not significant
H1_B: Factor B is significant
H0_C: Factor C is not significant
H1_C: Factor C is significant
H0_D: Factor D is not significant
H1_D: Factor D is significant
H0_AB: Interaction AB is not significant
H1_AB: Interaction AB is significant
H0_AC: Interaction AC is not significant
H1_AC: Interaction AC is significant
H0_AD: Interaction AD is not significant
H1_AD: Interaction AD is significant
H0_BC: Interaction BC is not significant
H1_BC: Interaction BC is significant
H0_BD: Interaction BD is not significant
H1_BD: Interaction BD is significant
H0_CD: Interaction CD is not significant
H1_CD: Interaction CD is significant
H0_ABC: Interaction ABC is not significant
H1_ABC: Interaction ABC is significant
H0_ABD: Interaction ABD is not significant
H1_ABD: Interaction ABD is significant
H0_ACD: Interaction ACD is not significant
H1_ACD: Interaction ACD is significant
H0_BCD: Interaction BCD is not significant
H1_BCD: Interaction BCD is significant
H0_ABCD: Interaction ABCD is not significant
H1_ABCD: Interaction ABCD is significant
Read In Data
A36 <- c(-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1)
B36 <- c(-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1)
C36 <- c(-1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,1,1,1,1)
D36 <- c(-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1)
obs36 <- c(1.92,11.28,1.09,5.75,2.13,9.53,1.03,5.35,1.6,11.73,1.16,4.68,2.16,9.11,1.07,5.30)
dat36 <- data.frame(A36,B36,C36,D36,obs36)a) Factor Effects and Model
I) Factor Effects
model36 <- lm(obs36~A36*B36*C36*D36,data = dat36)
coef(model36)## (Intercept) A36 B36 C36 D36
## 4.680625 3.160625 -1.501875 -0.220625 -0.079375
## A36:B36 A36:C36 B36:C36 A36:D36 B36:D36
## -1.069375 -0.298125 0.229375 -0.056875 -0.046875
## C36:D36 A36:B36:C36 A36:B36:D36 A36:C36:D36 B36:C36:D36
## 0.029375 0.344375 -0.096875 -0.010625 0.094375
## A36:B36:C36:D36
## 0.141875The estimated Factor Effects are as follows:
\[ (1)=4.680625 \]
\[ A=3.160625 \]
\[ B=-1.501875 \]
\[ C=-0.220625 \]
\[ D= -0.079375 \]
\[ AB=-1.069375 \]
\[ AC=-0.298125 \]
\[ BC=0.229375 \]
\[ AD=-0.056875 \]
\[ BD=-0.046875 \]
\[ CD= 0.029375 \]
\[ ABC=0.344375 \]
\[ ABD=-0.096875 \]
\[ ACD=-0.010625 \]
\[ BCD=0.094375 \]
\[ABCD= 0.141875\]
II) Significant Factors
halfnormal(model36)##
## Significant effects (alpha=0.05, Lenth method):## [1] A36 B36 A36:B36 A36:B36:C36
Our halfnormal plot indicates that the significant factors are A,B, and AB.
III) Significant Factors Model
Our model is:
\[ y(Resistivity)=4.680625+3.160625(A)-1.501875(B)-1.069375(AB) \]
b)
New Hypotheses:
H0_A: Factor A is not significant
H1_A: Factor A is significant
H0_B: Facto B is not significant
H1_B: Factor B is significant
H0_AB: Interaction AB is not significant
H1_AB: Interaction AB is significant
I) Linear Model and ANOVA
model36n <- lm(obs36~A36*B36,data = dat36)
anova(model36n)Our ANOVA table supports our rejection of the null hypotheses for A,B, and AB.
II) Check for Adequacy
plot(model36n)
## hat values (leverages) are all = 0.25
## and there are no factor predictors; no plot no. 5
The residuals vs. fitted plot indicates that the variance of the data is not constant, which breaks the assumption of ANOVA.
The Normality Plot shows large skews at both ends, indicating that the data is not normal. This breaks the assumption of normality for ANOVA
This model is shown by the plots to be inadequate.
c) Data Transformation and Analysis
I) ln Transformation
OBS36 <- log(obs36)II) Effects
MODEL36 <- lm(OBS36~A36*B36*C36*D36)
coef(MODEL36)## (Intercept) A36 B36 C36 D36
## 1.185417116 0.812870345 -0.314277554 -0.006408558 -0.018077390
## A36:B36 A36:C36 B36:C36 A36:D36 B36:D36
## -0.024684570 -0.039723700 -0.004225796 -0.009578245 0.003708723
## C36:D36 A36:B36:C36 A36:B36:D36 A36:C36:D36 B36:C36:D36
## 0.017780432 0.063434408 -0.029875960 -0.003740235 0.003765760
## A36:B36:C36:D36
## 0.031322043III) Significant Factors
halfnormal(MODEL36)##
## Significant effects (alpha=0.05, Lenth method):## [1] A36 B36 A36:B36:C36
The halfnormal plot of the transformed data indicates that only A and B are significant.
IV) Checking for Adequacy
ModeL36n <- lm(OBS36~A36*B36)
plot(ModeL36n)
## hat values (leverages) are all = 0.25
## and there are no factor predictors; no plot no. 5
The plots using the transformed data are much imporved. They show normality, which is adequate, and the present more constant variance, also adequate.
Overall, the transformation of the data has produced a viable model.
d)
anova(ModeL36n)The halfnormal plot and ANOVA of the transformed data indicates that only A and B are significant. Therefore our new model is:
\[ y(resistivity)=1.185417116+0.812870345(A)-0.314277554(B) \]
6.39)
Hypotheses:
H0_A: Factor A is not significant
H1_A: Factor A is significant
H0_B: Factor B is not significant
H1_B: Factor B is significant
H0_C: Factor C is not significant
H1_C: Factor C is significant
H0_D: Factor D is not significant
H1_D: Factor D is significant
H0_E: Factor E is not significant
H1_E: Factor E is significant
H0_AB: Interaction AB is not significant
H1_AB: Interaction AB is significant
H0_AC: Interaction AC is not significant
H1_AC: Interaction AC is significant
H0_AD: Interaction AD is not significant
H1_AD: Interaction AD is significant
H0_AE: Interaction AE is not significant
H1_AE: Interaction AE is significant
H0_BC: Interaction BC is not significant
H1_BC: Interaction BC is significant
H0_BD: Interaction BD is not significant
H1_BD: Interaction BD is significant
H0_BE: Interaction BE is not significant
H1_BE: Interaction BE is significant
H0_CD: Interaction CD is not significant
H1_CD: Interaction CD is significant
H0_CE: Interaction CE is not significant
H1_CE: Interaction CE is significant
H0_DE: Interaction DE is not significant
H1_DE: Interaction DE is significant
H0_ABC: Interaction ABC is not significant
H1_ABC: Interaction ABC is significant
H0_ABD: Interaction ABD is not significant
H1_ABD: Interaction ABD is significant
H0_ABE: Interaction ABE is not significant
H1_ABE: Interaction ABE is significant
H0_ACD: Interaction ACD is not significant
H1_ACD: Interaction ACD is significant
H0_ACE: Interaction ACE is not significant
H1_ACE: Interaction ACE is significant
H0_ADE: Interaction ADE is not significant
H1_ADE: Interaction ADE is significant
H0_BCD: Interaction BCD is not significant
H1_BCD: Interaction BCD is significant
H0_BCE: Interaction BCE is not significant
H1_BCE: Interaction BCE is significant
H0_BDE: Interaction BDE is not significant
H1_BDE: Interaction BDE is significant
H0_CDE: Interaction CDE is not significant
H1_CDE: Interaction CDE is significant
H0_ABCD: Interaction ABCD is not significant
H1_ABCD: Interaction ABCD is significant
H0_ABCE: Interaction ABCE is not significant
H1_ABCE: Interaction ABCE is significant
H0_ABDE: Interaction ABDE is not significant
H1_ABDE: Interaction ABDE is significant
H0_ACDE: Interaction ACDE is not significant
H1_ACDE: Interaction ACDE is significant
H0_BCDE: Interaction BCDE is not significant
H1_BCDE: Interaction BCDE is significant
H0_ABCDE: Interaction ABCDE is not significant
H1_ABCDE: Interaction ABCDE is significant
Read In Data
A39 <- c(-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1)
B39 <- c(-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1)
C39 <- c(-1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,1,1,1,1)
D39 <- c(-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1)
E39 <- c(-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)
obs39 <- c(8.11,5.56,5.77,5.82,9.17,7.8,3.23,5.69,8.82,14.23,9.2,8.94,8.68,11.49,6.25,9.12,7.93,5,7.47,12,9.86,3.65,6.4,11.61,12.43,17.55,8.87,25.38,13.06,18.85,11.78,26.05)
dat39 <- data.frame(A39,B39,C39,D39,E39,obs39)a) Analyze Data
Linear Model and Effects
mod39 <- lm(obs39~A39*B39*C39*D39*E39,data=dat39)
coef(mod39)## (Intercept) A39 B39 C39
## 10.1803125 1.6159375 0.0434375 -0.0121875
## D39 E39 A39:B39 A39:C39
## 2.9884375 2.1878125 1.2365625 -0.0015625
## B39:C39 A39:D39 B39:D39 C39:D39
## -0.1953125 1.6665625 -0.0134375 0.0034375
## A39:E39 B39:E39 C39:E39 D39:E39
## 1.0271875 1.2834375 0.3015625 1.3896875
## A39:B39:C39 A39:B39:D39 A39:C39:D39 B39:C39:D39
## 0.2503125 -0.3453125 -0.0634375 0.3053125
## A39:B39:E39 A39:C39:E39 B39:C39:E39 A39:D39:E39
## 1.1853125 -0.2590625 0.1709375 0.9015625
## B39:D39:E39 C39:D39:E39 A39:B39:C39:D39 A39:B39:C39:E39
## -0.0396875 0.3959375 -0.0740625 -0.1846875
## A39:B39:D39:E39 A39:C39:D39:E39 B39:C39:D39:E39 A39:B39:C39:D39:E39
## 0.4071875 0.1278125 -0.0746875 -0.3553125halfnormal(mod39)##
## Significant effects (alpha=0.05, Lenth method):## [1] D39 E39 A39:D39 A39 D39:E39 B39:E39
##
## [7] A39:B39 A39:B39:E39 A39:E39 A39:D39:E39
The significant factors are A,D, and E
The significant interations are AB, AD, AE, BE, DE, ABE, and ADE
b) Analyze Residuals
model39 <- lm(obs39~A39*B39*D39*E39,data = dat39)
plot(model39)
## hat values (leverages) are all = 0.5
## and there are no factor predictors; no plot no. 5
Using the significant factors and interactions, we can see that the model has both normality (from the NPP) and constant variance (from the Residuals vs. Fitted Graph). The model is adequate.
c) Analysis of Significant Factors and interactions
halfnormal(model39)##
## Significant effects (alpha=0.05, Lenth method):## [1] D39 E39 A39:D39 A39 D39:E39 B39:E39
##
## [7] A39:B39 A39:B39:E39 A39:E39 A39:D39:E39 e10
anova(model39)