setwd("C:/Users/lhomm/OneDrive/Documents/R")
library(asbio)
## Loading required package: tcltk
library(car)
## Loading required package: carData
Dat <- read.table("http://users.stat.ufl.edu/~rrandles/sta4210/Rclassnotes/data/textdatasets/KutnerData/Chapter%2020%20Data%20Sets/CH20PR05.txt")
colnames(Dat) <- c("Ideas", "Group", "Size")
Dat$Group <- factor(Dat$Group)
Dat$Size <- factor(Dat$Size)
### The Model Y_ij = Mu + ALphai + Beta_2 +DAlpha_iBeta_j + Epsilon_ij. ###
### Ho: D equals 0 H1: D does not equal 0. ###
Decsion_Rule <- qf(0.01, 1, 2, lower.tail=FALSE) ### Decision Rule ###
Decsion_Rule
## [1] 98.50251
Add1 <- tukey.add.test(Dat$Ideas, Dat$Group, Dat$Size)
Add1
##
## Tukey's one df test for additivity
## F = 0.5987716 Denom df = 2 p-value = 0.5199941
### With an F-Value well below our decision rule and a P-Val larger than .1/.05/.01 we fail to reject Ho that D equals 0. ###
### To remediate we can try transofrmations. ###
IdeasLog <- log(Dat$Ideas)
Add2 <- tukey.add.test(IdeasLog, Dat$Group, Dat$Size)
Add2
##
## Tukey's one df test for additivity
## F = 1.5349516 Denom df = 2 p-value = 0.3410452
IdeasSqrt <- sqrt(Dat$Ideas)
Add3 <- tukey.add.test(IdeasSqrt, Dat$Group, Dat$Size)
Add3
##
## Tukey's one df test for additivity
## F = 0.9833965 Denom df = 2 p-value = 0.4258719
Ideas1Over <- 1/Dat$Ideas
Add4 <- tukey.add.test(Ideas1Over, Dat$Group, Dat$Size)
Add4
##
## Tukey's one df test for additivity
## F = 3.3169367 Denom df = 2 p-value = 0.2101623
### None of the transformations managed to correct the error and produce a large enough F-Statistic or a small enough P-value although it does seem that the 1 over transformation was the most effective. ###
Dat2 <- read.table("http://users.stat.ufl.edu/~rrandles/sta4210/Rclassnotes/data/textdatasets/KutnerData/Chapter%2019%20Data%20Sets/CH19PR14.txt")
colnames(Dat2) <- c("Relief", "Compound1", "Compound2", "Count")
Dat2$Compound1 <- factor(Dat2$Compound1)
Dat2$Compound2 <- factor(Dat2$Compound2)
Relieflm <- lm(Relief ~ Compound1 * Compound2, data = Dat2)
Relieflm
##
## Call:
## lm(formula = Relief ~ Compound1 * Compound2, data = Dat2)
##
## Coefficients:
## (Intercept) Compound12 Compound13
## 2.475 2.975 3.500
## Compound22 Compound23 Compound12:Compound22
## 2.125 2.100 1.350
## Compound13:Compound22 Compound12:Compound23 Compound13:Compound23
## 2.175 1.575 5.175
summary(Relieflm)
##
## Call:
## lm(formula = Relief ~ Compound1 * Compound2, data = Dat2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.4250 -0.1750 0.0125 0.1875 0.3500
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.4750 0.1227 20.177 < 2e-16 ***
## Compound12 2.9750 0.1735 17.150 4.82e-16 ***
## Compound13 3.5000 0.1735 20.176 < 2e-16 ***
## Compound22 2.1250 0.1735 12.250 1.55e-12 ***
## Compound23 2.1000 0.1735 12.106 2.03e-12 ***
## Compound12:Compound22 1.3500 0.2453 5.503 7.91e-06 ***
## Compound13:Compound22 2.1750 0.2453 8.866 1.76e-09 ***
## Compound12:Compound23 1.5750 0.2453 6.420 7.05e-07 ***
## Compound13:Compound23 5.1750 0.2453 21.094 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2453 on 27 degrees of freedom
## Multiple R-squared: 0.9957, Adjusted R-squared: 0.9944
## F-statistic: 774.9 on 8 and 27 DF, p-value: < 2.2e-16
### Alphaihat estimates the main effect of compound1. ###
Anova1 <- aov(Relief ~ Compound1 * Compound2, data = Dat2)
Anova1
## Call:
## aov(formula = Relief ~ Compound1 * Compound2, data = Dat2)
##
## Terms:
## Compound1 Compound2 Compound1:Compound2 Residuals
## Sum of Squares 220.020 123.660 29.425 1.625
## Deg. of Freedom 2 2 4 27
##
## Residual standard error: 0.2453267
## Estimated effects may be unbalanced
summary(Anova1)
## Df Sum Sq Mean Sq F value Pr(>F)
## Compound1 2 220.02 110.01 1827.9 <2e-16 ***
## Compound2 2 123.66 61.83 1027.3 <2e-16 ***
## Compound1:Compound2 4 29.43 7.36 122.2 <2e-16 ***
## Residuals 27 1.63 0.06
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova2 <- Anova(Anova1, type = "III")
Anova2
## Anova Table (Type III tests)
##
## Response: Relief
## Sum Sq Df F value Pr(>F)
## (Intercept) 24.503 1 407.118 < 2.2e-16 ***
## Compound1 28.502 2 236.783 < 2.2e-16 ***
## Compound2 11.902 2 98.875 3.762e-13 ***
## Compound1:Compound2 29.425 4 122.227 < 2.2e-16 ***
## Residuals 1.625 27
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(Anova2)
## Sum Sq Df F value Pr(>F)
## Min. : 1.625 Min. : 1.0 Min. : 98.88 Min. :0
## 1st Qu.:11.902 1st Qu.: 2.0 1st Qu.:116.39 1st Qu.:0
## Median :24.503 Median : 2.0 Median :179.50 Median :0
## Mean :19.191 Mean : 7.2 Mean :216.25 Mean :0
## 3rd Qu.:28.502 3rd Qu.: 4.0 3rd Qu.:279.37 3rd Qu.:0
## Max. :29.425 Max. :27.0 Max. :407.12 Max. :0
## NA's :1 NA's :1
### While the F-values and the sums of squares for compount 1 and 2 differ between tests the conclusions are fundamentally the same. Both tests have large F-values and P-values all below .01 so in both cases we conclude that there is signifigant interaction between compound 1 and 2.
Decsion_Rule <- qf(0.05, 1, 2, lower.tail=FALSE) ### Decision Rule ###
Decsion_Rule
## [1] 18.51282
### Ho: There is no interaction between Compound 1 and Compound 2 in the model. ####
### H1: There is interaction between Compound 1 and Compound 2 in the model. ###
### Ho: Compound1 equals 0. ###
### H1: Compound1 does not equal 0. ###
### Ho: Compound2 equals 0. ###
### H1: Compound2 does not equal 0. ###
Anova1
## Call:
## aov(formula = Relief ~ Compound1 * Compound2, data = Dat2)
##
## Terms:
## Compound1 Compound2 Compound1:Compound2 Residuals
## Sum of Squares 220.020 123.660 29.425 1.625
## Deg. of Freedom 2 2 4 27
##
## Residual standard error: 0.2453267
## Estimated effects may be unbalanced
summary(Anova1)
## Df Sum Sq Mean Sq F value Pr(>F)
## Compound1 2 220.02 110.01 1827.9 <2e-16 ***
## Compound2 2 123.66 61.83 1027.3 <2e-16 ***
## Compound1:Compound2 4 29.43 7.36 122.2 <2e-16 ***
## Residuals 27 1.63 0.06
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Interaction1_C1XC2_Comparison <- TukeyHSD(Anova1, which = ("Compound1:Compound2"))
Interaction1_C1XC2_Comparison
## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = Relief ~ Compound1 * Compound2, data = Dat2)
##
## $`Compound1:Compound2`
## diff lwr upr p adj
## 2:1-1:1 2.975 2.3913187 3.5586813 0.0000000
## 3:1-1:1 3.500 2.9163187 4.0836813 0.0000000
## 1:2-1:1 2.125 1.5413187 2.7086813 0.0000000
## 2:2-1:1 6.450 5.8663187 7.0336813 0.0000000
## 3:2-1:1 7.800 7.2163187 8.3836813 0.0000000
## 1:3-1:1 2.100 1.5163187 2.6836813 0.0000000
## 2:3-1:1 6.650 6.0663187 7.2336813 0.0000000
## 3:3-1:1 10.775 10.1913187 11.3586813 0.0000000
## 3:1-2:1 0.525 -0.0586813 1.1086813 0.1033088
## 1:2-2:1 -0.850 -1.4336813 -0.2663187 0.0011424
## 2:2-2:1 3.475 2.8913187 4.0586813 0.0000000
## 3:2-2:1 4.825 4.2413187 5.4086813 0.0000000
## 1:3-2:1 -0.875 -1.4586813 -0.2913187 0.0007862
## 2:3-2:1 3.675 3.0913187 4.2586813 0.0000000
## 3:3-2:1 7.800 7.2163187 8.3836813 0.0000000
## 1:2-3:1 -1.375 -1.9586813 -0.7913187 0.0000005
## 2:2-3:1 2.950 2.3663187 3.5336813 0.0000000
## 3:2-3:1 4.300 3.7163187 4.8836813 0.0000000
## 1:3-3:1 -1.400 -1.9836813 -0.8163187 0.0000004
## 2:3-3:1 3.150 2.5663187 3.7336813 0.0000000
## 3:3-3:1 7.275 6.6913187 7.8586813 0.0000000
## 2:2-1:2 4.325 3.7413187 4.9086813 0.0000000
## 3:2-1:2 5.675 5.0913187 6.2586813 0.0000000
## 1:3-1:2 -0.025 -0.6086813 0.5586813 1.0000000
## 2:3-1:2 4.525 3.9413187 5.1086813 0.0000000
## 3:3-1:2 8.650 8.0663187 9.2336813 0.0000000
## 3:2-2:2 1.350 0.7663187 1.9336813 0.0000007
## 1:3-2:2 -4.350 -4.9336813 -3.7663187 0.0000000
## 2:3-2:2 0.200 -0.3836813 0.7836813 0.9596929
## 3:3-2:2 4.325 3.7413187 4.9086813 0.0000000
## 1:3-3:2 -5.700 -6.2836813 -5.1163187 0.0000000
## 2:3-3:2 -1.150 -1.7336813 -0.5663187 0.0000131
## 3:3-3:2 2.975 2.3913187 3.5586813 0.0000000
## 2:3-1:3 4.550 3.9663187 5.1336813 0.0000000
## 3:3-1:3 8.675 8.0913187 9.2586813 0.0000000
## 3:3-2:3 4.125 3.5413187 4.7086813 0.0000000
Interaction1_All_Comparison <- TukeyHSD(Anova1)
Interaction1_All_Comparison
## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = Relief ~ Compound1 * Compound2, data = Dat2)
##
## $Compound1
## diff lwr upr p adj
## 2-1 3.95 3.701676 4.198324 0
## 3-1 5.95 5.701676 6.198324 0
## 3-2 2.00 1.751676 2.248324 0
##
## $Compound2
## diff lwr upr p adj
## 2-1 3.30 3.0516759 3.548324 0
## 3-1 4.35 4.1016759 4.598324 0
## 3-2 1.05 0.8016759 1.298324 0
##
## $`Compound1:Compound2`
## diff lwr upr p adj
## 2:1-1:1 2.975 2.3913187 3.5586813 0.0000000
## 3:1-1:1 3.500 2.9163187 4.0836813 0.0000000
## 1:2-1:1 2.125 1.5413187 2.7086813 0.0000000
## 2:2-1:1 6.450 5.8663187 7.0336813 0.0000000
## 3:2-1:1 7.800 7.2163187 8.3836813 0.0000000
## 1:3-1:1 2.100 1.5163187 2.6836813 0.0000000
## 2:3-1:1 6.650 6.0663187 7.2336813 0.0000000
## 3:3-1:1 10.775 10.1913187 11.3586813 0.0000000
## 3:1-2:1 0.525 -0.0586813 1.1086813 0.1033088
## 1:2-2:1 -0.850 -1.4336813 -0.2663187 0.0011424
## 2:2-2:1 3.475 2.8913187 4.0586813 0.0000000
## 3:2-2:1 4.825 4.2413187 5.4086813 0.0000000
## 1:3-2:1 -0.875 -1.4586813 -0.2913187 0.0007862
## 2:3-2:1 3.675 3.0913187 4.2586813 0.0000000
## 3:3-2:1 7.800 7.2163187 8.3836813 0.0000000
## 1:2-3:1 -1.375 -1.9586813 -0.7913187 0.0000005
## 2:2-3:1 2.950 2.3663187 3.5336813 0.0000000
## 3:2-3:1 4.300 3.7163187 4.8836813 0.0000000
## 1:3-3:1 -1.400 -1.9836813 -0.8163187 0.0000004
## 2:3-3:1 3.150 2.5663187 3.7336813 0.0000000
## 3:3-3:1 7.275 6.6913187 7.8586813 0.0000000
## 2:2-1:2 4.325 3.7413187 4.9086813 0.0000000
## 3:2-1:2 5.675 5.0913187 6.2586813 0.0000000
## 1:3-1:2 -0.025 -0.6086813 0.5586813 1.0000000
## 2:3-1:2 4.525 3.9413187 5.1086813 0.0000000
## 3:3-1:2 8.650 8.0663187 9.2336813 0.0000000
## 3:2-2:2 1.350 0.7663187 1.9336813 0.0000007
## 1:3-2:2 -4.350 -4.9336813 -3.7663187 0.0000000
## 2:3-2:2 0.200 -0.3836813 0.7836813 0.9596929
## 3:3-2:2 4.325 3.7413187 4.9086813 0.0000000
## 1:3-3:2 -5.700 -6.2836813 -5.1163187 0.0000000
## 2:3-3:2 -1.150 -1.7336813 -0.5663187 0.0000131
## 3:3-3:2 2.975 2.3913187 3.5586813 0.0000000
## 2:3-1:3 4.550 3.9663187 5.1336813 0.0000000
## 3:3-1:3 8.675 8.0913187 9.2586813 0.0000000
## 3:3-2:3 4.125 3.5413187 4.7086813 0.0000000
### With an F-Value larger than our decision rule and a P-value smaller than .01 we can reject HO that there is no interaction and conclude that there is interaction in the model. We also see that all but 3 comparisons were signifigant at .05 and .01 and of those three one's P-value was nearly .1. These comparisons add extra weight to our conclusions and give us further insight into specific differences. ###
Additive1 <- aov(Relief ~ Compound1 + Compound2, data = Dat2)
Additive1
## Call:
## aov(formula = Relief ~ Compound1 + Compound2, data = Dat2)
##
## Terms:
## Compound1 Compound2 Residuals
## Sum of Squares 220.02 123.66 31.05
## Deg. of Freedom 2 2 31
##
## Residual standard error: 1.000806
## Estimated effects may be unbalanced
summary(Additive1)
## Df Sum Sq Mean Sq F value Pr(>F)
## Compound1 2 220.02 110.01 109.83 8.51e-15 ***
## Compound2 2 123.66 61.83 61.73 1.55e-11 ***
## Residuals 31 31.05 1.00
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Additive2 <- anova(lm(Relief ~ Compound1 + Compound2, data = Dat2))
Additive2
## Analysis of Variance Table
##
## Response: Relief
## Df Sum Sq Mean Sq F value Pr(>F)
## Compound1 2 220.02 110.010 109.83 8.514e-15 ***
## Compound2 2 123.66 61.830 61.73 1.547e-11 ***
## Residuals 31 31.05 1.002
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(Additive2)
## Df Sum Sq Mean Sq F value
## Min. : 2.00 Min. : 31.05 Min. : 1.002 Min. : 61.73
## 1st Qu.: 2.00 1st Qu.: 77.36 1st Qu.: 31.416 1st Qu.: 73.76
## Median : 2.00 Median :123.66 Median : 61.830 Median : 85.78
## Mean :11.67 Mean :124.91 Mean : 57.614 Mean : 85.78
## 3rd Qu.:16.50 3rd Qu.:171.84 3rd Qu.: 85.920 3rd Qu.: 97.81
## Max. :31.00 Max. :220.02 Max. :110.010 Max. :109.83
## NA's :1
## Pr(>F)
## Min. :0
## 1st Qu.:0
## Median :0
## Mean :0
## 3rd Qu.:0
## Max. :0
## NA's :1
Additive1_C1_Comparison <- TukeyHSD(Additive1, which = ("Compound1"))
Additive1_C1_Comparison
## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = Relief ~ Compound1 + Compound2, data = Dat2)
##
## $Compound1
## diff lwr upr p adj
## 2-1 3.95 2.9444139 4.955586 0.00e+00
## 3-1 5.95 4.9444139 6.955586 0.00e+00
## 3-2 2.00 0.9944139 3.005586 8.42e-05
Additive1_C2_Comparison <- TukeyHSD(Additive1, which = ("Compound2"))
Additive1_C2_Comparison
## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = Relief ~ Compound1 + Compound2, data = Dat2)
##
## $Compound2
## diff lwr upr p adj
## 2-1 3.30 2.29441388 4.305586 0.0000000
## 3-1 4.35 3.34441388 5.355586 0.0000000
## 3-2 1.05 0.04441388 2.055586 0.0392554
Additive1_All_Comparison <- TukeyHSD(Additive1)
Additive1_All_Comparison
## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = Relief ~ Compound1 + Compound2, data = Dat2)
##
## $Compound1
## diff lwr upr p adj
## 2-1 3.95 2.9444139 4.955586 0.00e+00
## 3-1 5.95 4.9444139 6.955586 0.00e+00
## 3-2 2.00 0.9944139 3.005586 8.42e-05
##
## $Compound2
## diff lwr upr p adj
## 2-1 3.30 2.29441388 4.305586 0.0000000
## 3-1 4.35 3.34441388 5.355586 0.0000000
## 3-2 1.05 0.04441388 2.055586 0.0392554
### In the addative without interaction the F-values, P-values, and pariwise comparisons all demontrast that Compound 1 and 2 both have signifigant main effects on the number of hours of relief from hay fever the respondents experienced. ###
JustC11 <- aov(Relief ~ Compound1, data = Dat2)
JustC11
## Call:
## aov(formula = Relief ~ Compound1, data = Dat2)
##
## Terms:
## Compound1 Residuals
## Sum of Squares 220.02 154.71
## Deg. of Freedom 2 33
##
## Residual standard error: 2.165221
## Estimated effects may be unbalanced
summary(JustC11)
## Df Sum Sq Mean Sq F value Pr(>F)
## Compound1 2 220.0 110.01 23.46 4.58e-07 ***
## Residuals 33 154.7 4.69
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
JustC12 <- anova(lm(Relief ~ Compound1, data = Dat2))
JustC12
## Analysis of Variance Table
##
## Response: Relief
## Df Sum Sq Mean Sq F value Pr(>F)
## Compound1 2 220.02 110.010 23.465 4.578e-07 ***
## Residuals 33 154.71 4.688
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(JustC12)
## Df Sum Sq Mean Sq F value
## Min. : 2.00 Min. :154.7 Min. : 4.688 Min. :23.47
## 1st Qu.: 9.75 1st Qu.:171.0 1st Qu.: 31.019 1st Qu.:23.47
## Median :17.50 Median :187.4 Median : 57.349 Median :23.47
## Mean :17.50 Mean :187.4 Mean : 57.349 Mean :23.47
## 3rd Qu.:25.25 3rd Qu.:203.7 3rd Qu.: 83.680 3rd Qu.:23.47
## Max. :33.00 Max. :220.0 Max. :110.010 Max. :23.47
## NA's :1
## Pr(>F)
## Min. :5e-07
## 1st Qu.:5e-07
## Median :5e-07
## Mean :5e-07
## 3rd Qu.:5e-07
## Max. :5e-07
## NA's :1
JustC1_C1_Comparison <- TukeyHSD(JustC11, which = ("Compound1"))
JustC1_C1_Comparison
## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = Relief ~ Compound1, data = Dat2)
##
## $Compound1
## diff lwr upr p adj
## 2-1 3.95 1.7809738 6.119026 0.0002523
## 3-1 5.95 3.7809738 8.119026 0.0000003
## 3-2 2.00 -0.1690262 4.169026 0.0755297
JustC1_All_Comparison <- TukeyHSD(JustC11)
JustC1_All_Comparison
## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = Relief ~ Compound1, data = Dat2)
##
## $Compound1
## diff lwr upr p adj
## 2-1 3.95 1.7809738 6.119026 0.0002523
## 3-1 5.95 3.7809738 8.119026 0.0000003
## 3-2 2.00 -0.1690262 4.169026 0.0755297
### In the model with just Compound 1 the F-Value, P-value, and pair wise comparisons all suggest that the main effect of Compound 1 on the number of hours of relief from hay fever the respondents experienced. ###
JustC21 <- aov(Relief ~ Compound2, data = Dat2)
JustC21
## Call:
## aov(formula = Relief ~ Compound2, data = Dat2)
##
## Terms:
## Compound2 Residuals
## Sum of Squares 123.66 251.07
## Deg. of Freedom 2 33
##
## Residual standard error: 2.758293
## Estimated effects may be unbalanced
summary(JustC21)
## Df Sum Sq Mean Sq F value Pr(>F)
## Compound2 2 123.7 61.83 8.127 0.00135 **
## Residuals 33 251.1 7.61
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
JustC22 <- anova(lm(Relief ~ Compound2, data = Dat2))
JustC22
## Analysis of Variance Table
##
## Response: Relief
## Df Sum Sq Mean Sq F value Pr(>F)
## Compound2 2 123.66 61.830 8.1268 0.00135 **
## Residuals 33 251.07 7.608
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(JustC22)
## Df Sum Sq Mean Sq F value
## Min. : 2.00 Min. :123.7 Min. : 7.608 Min. :8.127
## 1st Qu.: 9.75 1st Qu.:155.5 1st Qu.:21.164 1st Qu.:8.127
## Median :17.50 Median :187.4 Median :34.719 Median :8.127
## Mean :17.50 Mean :187.4 Mean :34.719 Mean :8.127
## 3rd Qu.:25.25 3rd Qu.:219.2 3rd Qu.:48.275 3rd Qu.:8.127
## Max. :33.00 Max. :251.1 Max. :61.830 Max. :8.127
## NA's :1
## Pr(>F)
## Min. :0.00135
## 1st Qu.:0.00135
## Median :0.00135
## Mean :0.00135
## 3rd Qu.:0.00135
## Max. :0.00135
## NA's :1
JustC2_C2_Comparison <- TukeyHSD(JustC21, which = ("Compound2"))
JustC2_C2_Comparison
## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = Relief ~ Compound2, data = Dat2)
##
## $Compound2
## diff lwr upr p adj
## 2-1 3.30 0.5368592 6.063141 0.0163636
## 3-1 4.35 1.5868592 7.113141 0.0014032
## 3-2 1.05 -1.7131408 3.813141 0.6239689
JustC2_All_Comparison <- TukeyHSD(JustC21)
JustC2_All_Comparison
## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = Relief ~ Compound2, data = Dat2)
##
## $Compound2
## diff lwr upr p adj
## 2-1 3.30 0.5368592 6.063141 0.0163636
## 3-1 4.35 1.5868592 7.113141 0.0014032
## 3-2 1.05 -1.7131408 3.813141 0.6239689
## In the model with just Compound 1 the F-Value, P-value, and pair wise comparisons all suggest that the main effect of Compound 2 on the number of hours of relief from hay fever the respondents experienced. ###