Recipe 4: Completely Randomized Design
Corn Silk: A Completely Randomized Design
Max Winkelman
Rensselaer Polytechnic Institute
October 16 2014
Version 1
1. Setting
Corn silk
The data analyzed in this recipe is a csv file that contains various measurements of the phytochemical parameters of corn silk taken from Sarepoua et al[1].
Install the ‘Cornsilk.csv’ file
corn <- read.csv("~/RPI/Classes/Design of Experiments/R/Cornsilk.csv")
#reads in the csv file "Cornsilk.csv" form My Documents and stores it in the variable "corn"
corn$Type = as.factor(corn$Type)
#makes the column "Type" a factor
corn$Varieties = as.factor(corn$Varieties)
#makes the column "Varieties" a factorFactors and Levels
Factor: Corn Type and Corn Variety
Levels: Type: Purple Waxy Corn, White Waxy Corn, and Super Sweet Corn and Variety P1-P5, W1-W3, and S1 and S2
#Summary of Data
head(corn)## Type Varieties Silkingstage Milkystage Maturitystage Mean
## 1 Purple waxy corn P1 123.8 179.6 149.1 150.8
## 2 Purple waxy corn P1 127.6 182.8 146.5 152.3
## 3 Purple waxy corn P1 120.0 176.4 151.7 149.4
## 4 Purple waxy corn P2 112.8 169.2 114.7 132.2
## 5 Purple waxy corn P2 114.9 166.8 117.8 133.2
## 6 Purple waxy corn P2 110.7 171.6 111.6 131.3#displays the first 6 sets of variables
tail(corn)## Type Varieties Silkingstage Milkystage Maturitystage Mean
## 25 Super sweet corn S1 85.5 75.2 65.5 75.4
## 26 Super sweet corn S1 83.1 78.4 63.5 75.0
## 27 Super sweet corn S1 87.9 72.0 67.5 75.8
## 28 Super sweet corn S2 93.4 78.3 69.4 80.4
## 29 Super sweet corn S2 90.2 81.4 71.6 81.1
## 30 Super sweet corn S2 96.6 75.2 67.2 79.7#displays the last 6 sets of variables
summary(corn)## Type Varieties Silkingstage Milkystage
## Purple waxy corn:15 P1 : 3 Min. : 83.1 Min. : 54.4
## Super sweet corn: 6 P2 : 3 1st Qu.: 89.8 1st Qu.: 59.3
## White waxy corn : 9 P3 : 3 Median :105.9 Median :112.9
## P4 : 3 Mean :110.7 Mean :121.3
## P5 : 3 3rd Qu.:122.8 3rd Qu.:178.8
## S1 : 3 Max. :176.0 Max. :212.3
## (Other):12
## Maturitystage Mean
## Min. : 24.4 Min. : 58.8
## 1st Qu.: 48.0 1st Qu.: 64.8
## Median : 91.6 Median :106.2
## Mean : 97.8 Mean :109.9
## 3rd Qu.:146.0 3rd Qu.:150.4
## Max. :190.3 Max. :192.9
## #displays a summary of the variablesContinuous variables:
The continuous variables in this file are the total phenolic content (ug GAE/g dried sample) from corn in the Silking Stage (R1), the Milky Stage (R4), and the Maturity Stage (R6). The mean total phenolic content of all three of these variables is also a continuous variable. The categorical variables in this file are “Type of Corn”" and “Varieties.”
Response variables:
The total phenolic content from corn in the Silking Stage (R1), the Milky Stage (R4), and the Maturity Stage (R6) can all be considered response variables. For this specific recipe, the response variable will be the mean the total phenolic content.
The Data: How is it organized and what does it look like?
The csv file “Cornsilk.csv” contains modified data from Table 2 from Sarepoua et al and is oganized into columns that contain the five variables described above with slightly altered variable names that were changed for ease of coding[1]. The original table that was taken from Sarepoua et al contains the variables Type of Corn, Varieties, and Total phenolic content, which is organized into the three stages, R1, R4, and R6, and the mean value of the three stages. R1, R4, and R6 contain the mean values taken from three measurements made during the course of the experiment, as well as the standard deviation. The csv file used in this recipe contains the upper and lower standard deviation bounds in addition to the means.
Randomization
In the experiment conducted by Sarepoua et al, 10 corn varieties were evaluated using a completely randomized design that contained three replicates at the Vegetable Farm of Khon Kaen University in Khon Kaen, Thailand from May to July of 2012. The total phenolic, total flavonoids, total anthocyanin, and antioxidant activity were recorded by DPPH free-radical-scavenging assays [1].The total phenolic data was selected to be analyzed in this specific recipe.
2. Experimental Design
In this sample recipe, the effect of corn type and variety on the mean phenolic content will be investigated. An anova will be performed to determine if the variation of phenolic content in silk can be attributed to the type and the variety of corn. The two factors in this experiment will be the Type of Corn (Purple waxy corn, white waxy corn, and super sweet corn) and the variety of corn (P1-P5, W1-W3, and S1 and S2). The mean phenolic content will be the response variable that is used in the anova. It is hypothesized that the mean phenolic content of silk from corns from all types and varieties will be equal.
What is the rationale for this design?
An anova was selected to analyze the data from Cornsilk.csv because this recipe describes a two-factor, multilevel experiment. This test will be conducted with the assumption that the population from which the samples originated from are normally distributed and equality of variance amongst sample means.
Randomize: What is the Randomization Scheme?
The corn silk total phenolic content varieties were evaluated using a completely randomized design [1].
Replicate: Are there replicates and/or repeated measures?
There are both replicates and repeated measures in this experiment. Each measurement of total phenolic content was made three times and at each of the three stages of corn development.
Block: Did you use blocking in the design?
Yes, the 10 varieties of corn were grouped into 3 blocks based on corn type.
3. Statistical Analysis
Exploratory Data Analysis: Graphics and Descriptive Summary
#Boxplot
boxplot(Mean~Type,data=corn, xlab="Type of Corn", ylab="Total phenolic content (ug GAE/g dried sample)")
title("Effect of Corn Type on Phenolic Content")
#boxplot of the horse power data from each model year
boxplot(Mean~Varieties,data=corn, xlab="Variety of Corn", ylab="Total phenolic content (ug GAE/g dried sample)", las=2)
title("Effect of Corn Variety on Phenolic Content")
#boxplot of the horse power data from each model yearThe two boxplots above display the distribution of total phenolic content that can be attributed to the corn type and the corn variety. No statistical inference can be made from these boxplots without performing a statistical test. By visual inspection, the median phenolic content of purple waxy corn is much higher that the medians of the other two types. Due to the lack of a large sample size when comparing the variety of corn, the medians of each corn variety appear to be relatively distinct from each other, except for W1, W2, and W3, which appear to have similar medians.
ANOVA Testing
An analysis of variance (ANOVA) will be used to determine the statistical significance between the total phenolic content means. The null hypothesis for all three ANOVA tests is that the mean phenolic content vectors of all samples are equal to each other. The first anova test will analyze the phenolic content variance as a result of the variation of corn type. The second anova test will analyze the phenolic content variance as a result of the variation of the corn variety. The third anova test will analyze the phenolic content variance as a result of the interaction between type and variety. If the null hypothesis is rejected, the alternative hypothesis, which states that the mean vectors are not equal to each other, is accepted. Afterwards, a Tukey’s Honestly Significant Difference test will be performed to due which means are significantly different.
# ANOVA
#Corn Type
model_type = aov(Mean~Type,data=corn)
anova(model_type)## Analysis of Variance Table
##
## Response: Mean
## Df Sum Sq Mean Sq F value Pr(>F)
## Type 2 53032 26516 109 1.2e-13 ***
## Residuals 27 6561 243
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#performs an anova test
#Corn Variety
model_variety = aov(Mean~Varieties,data=corn)
anova(model_variety)## Analysis of Variance Table
##
## Response: Mean
## Df Sum Sq Mean Sq F value Pr(>F)
## Varieties 9 59546 6616 2850 <2e-16 ***
## Residuals 20 46 2
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#performs an anova test
#Corn Type and Variety
model_type_variety = aov(Mean~Type*Varieties,data=corn)
anova(model_type_variety)## Analysis of Variance Table
##
## Response: Mean
## Df Sum Sq Mean Sq F value Pr(>F)
## Type 2 53032 26516 11423 <2e-16 ***
## Varieties 7 6515 931 401 <2e-16 ***
## Residuals 20 46 2
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#performs an anova testANOVA Results: The anova test that analyzed the variation in phenolic content as a result of the variation in the corn type produced a p-value of 1.2e-13. This indicates that there is a very small probability that the variation of phenolic content can be solely attributed to randomization. It is highly likely that the corn type has an effect on the phenolic content mean. The anova test that analyzed the variation in phenolic content as a result of the variation in the variety of corn produced a p-value of less than 2e-16. This indicates that there is a very small probability that the variation of phenolic content can be solely attributed to randomization. It is highly likely that the variety of corn has an effect on the phenolic content mean. The anova test that analyzed the variation in phenolic content as a result of interaction of corn type and variety produced a p-value of less than 2e-16. This indicates that there is a small probability that the variation of phenolic content can be solely attributed to randomization. It is likely that the interaction of the type and variety of corn has an effect on the phenolic content mean. However, without a post-hoc analysis, it is impossible to determine precisely which means are significantly distinct from the others.
Post-Hoc Analysis
Tukey’s Honestly Significantly Difference is a multiple comparison procedure that is used after an ANOVA to determine which specific sample means are significantly different from the others. In this recipe, Tukey’s HSD is used to determine which corn type and variety produced significantly different phenolic content means.
#Tukey's HSD
#Corn Type
TukeyHSD(model_type, ordered = FALSE, conf.level = 0.95)## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = Mean ~ Type, data = corn)
##
## $Type
## diff lwr upr p adj
## Super sweet corn-Purple waxy corn -73.69 -92.36 -55.023 0.0000
## White waxy corn-Purple waxy corn -89.78 -106.08 -73.486 0.0000
## White waxy corn-Super sweet corn -16.09 -36.46 4.282 0.1421#performs a THSD test for the type of corn
#Corn Variety
TukeyHSD(model_variety, ordered = FALSE, conf.level = 0.95)## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = Mean ~ Varieties, data = corn)
##
## $Varieties
## diff lwr upr p adj
## P2-P1 -18.6000 -23.0052 -14.1948 0.0000
## P3-P1 2.3667 -2.0385 6.7718 0.6678
## P4-P1 -18.2333 -22.6385 -13.8282 0.0000
## P5-P1 38.2667 33.8615 42.6718 0.0000
## S1-P1 -75.4333 -79.8385 -71.0282 0.0000
## S2-P1 -70.4333 -74.8385 -66.0282 0.0000
## W1-P1 -89.6333 -94.0385 -85.2282 0.0000
## W2-P1 -86.1667 -90.5718 -81.7615 0.0000
## W3-P1 -91.2667 -95.6718 -86.8615 0.0000
## P3-P2 20.9667 16.5615 25.3718 0.0000
## P4-P2 0.3667 -4.0385 4.7718 1.0000
## P5-P2 56.8667 52.4615 61.2718 0.0000
## S1-P2 -56.8333 -61.2385 -52.4282 0.0000
## S2-P2 -51.8333 -56.2385 -47.4282 0.0000
## W1-P2 -71.0333 -75.4385 -66.6282 0.0000
## W2-P2 -67.5667 -71.9718 -63.1615 0.0000
## W3-P2 -72.6667 -77.0718 -68.2615 0.0000
## P4-P3 -20.6000 -25.0052 -16.1948 0.0000
## P5-P3 35.9000 31.4948 40.3052 0.0000
## S1-P3 -77.8000 -82.2052 -73.3948 0.0000
## S2-P3 -72.8000 -77.2052 -68.3948 0.0000
## W1-P3 -92.0000 -96.4052 -87.5948 0.0000
## W2-P3 -88.5333 -92.9385 -84.1282 0.0000
## W3-P3 -93.6333 -98.0385 -89.2282 0.0000
## P5-P4 56.5000 52.0948 60.9052 0.0000
## S1-P4 -57.2000 -61.6052 -52.7948 0.0000
## S2-P4 -52.2000 -56.6052 -47.7948 0.0000
## W1-P4 -71.4000 -75.8052 -66.9948 0.0000
## W2-P4 -67.9333 -72.3385 -63.5282 0.0000
## W3-P4 -73.0333 -77.4385 -68.6282 0.0000
## S1-P5 -113.7000 -118.1052 -109.2948 0.0000
## S2-P5 -108.7000 -113.1052 -104.2948 0.0000
## W1-P5 -127.9000 -132.3052 -123.4948 0.0000
## W2-P5 -124.4333 -128.8385 -120.0282 0.0000
## W3-P5 -129.5333 -133.9385 -125.1282 0.0000
## S2-S1 5.0000 0.5948 9.4052 0.0184
## W1-S1 -14.2000 -18.6052 -9.7948 0.0000
## W2-S1 -10.7333 -15.1385 -6.3282 0.0000
## W3-S1 -15.8333 -20.2385 -11.4282 0.0000
## W1-S2 -19.2000 -23.6052 -14.7948 0.0000
## W2-S2 -15.7333 -20.1385 -11.3282 0.0000
## W3-S2 -20.8333 -25.2385 -16.4282 0.0000
## W2-W1 3.4667 -0.9385 7.8718 0.2061
## W3-W1 -1.6333 -6.0385 2.7718 0.9390
## W3-W2 -5.1000 -9.5052 -0.6948 0.0155#performs a THSD test for the variety of corn
#Corn Type and Variety
TukeyHSD(model_type_variety, ordered = FALSE, conf.level = 0.95)## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = Mean ~ Type * Varieties, data = corn)
##
## $Type
## diff lwr upr p adj
## Super sweet corn-Purple waxy corn -73.69 -75.56 -71.83 0
## White waxy corn-Purple waxy corn -89.78 -91.41 -88.16 0
## White waxy corn-Super sweet corn -16.09 -18.12 -14.06 0
##
## $Varieties
## diff lwr upr p adj
## P2-P1 -18.6000 -23.0052 -14.1948 0.0000
## P3-P1 2.3667 -2.0385 6.7718 0.6678
## P4-P1 -18.2333 -22.6385 -13.8282 0.0000
## P5-P1 38.2667 33.8615 42.6718 0.0000
## S1-P1 -1.7400 -6.1452 2.6652 0.9139
## S2-P1 3.2600 -1.1452 7.6652 0.2702
## W1-P1 0.1489 -4.2563 4.5541 1.0000
## W2-P1 3.6156 -0.7896 8.0207 0.1677
## W3-P1 -1.4844 -5.8896 2.9207 0.9651
## P3-P2 20.9667 16.5615 25.3718 0.0000
## P4-P2 0.3667 -4.0385 4.7718 1.0000
## P5-P2 56.8667 52.4615 61.2718 0.0000
## S1-P2 16.8600 12.4548 21.2652 0.0000
## S2-P2 21.8600 17.4548 26.2652 0.0000
## W1-P2 18.7489 14.3437 23.1541 0.0000
## W2-P2 22.2156 17.8104 26.6207 0.0000
## W3-P2 17.1156 12.7104 21.5207 0.0000
## P4-P3 -20.6000 -25.0052 -16.1948 0.0000
## P5-P3 35.9000 31.4948 40.3052 0.0000
## S1-P3 -4.1067 -8.5118 0.2985 0.0806
## S2-P3 0.8933 -3.5118 5.2985 0.9990
## W1-P3 -2.2178 -6.6229 2.1874 0.7380
## W2-P3 1.2489 -3.1563 5.6541 0.9885
## W3-P3 -3.8511 -8.2563 0.5541 0.1191
## P5-P4 56.5000 52.0948 60.9052 0.0000
## S1-P4 16.4933 12.0882 20.8985 0.0000
## S2-P4 21.4933 17.0882 25.8985 0.0000
## W1-P4 18.3822 13.9771 22.7874 0.0000
## W2-P4 21.8489 17.4437 26.2541 0.0000
## W3-P4 16.7489 12.3437 21.1541 0.0000
## S1-P5 -40.0067 -44.4118 -35.6015 0.0000
## S2-P5 -35.0067 -39.4118 -30.6015 0.0000
## W1-P5 -38.1178 -42.5229 -33.7126 0.0000
## W2-P5 -34.6511 -39.0563 -30.2459 0.0000
## W3-P5 -39.7511 -44.1563 -35.3459 0.0000
## S2-S1 5.0000 0.5948 9.4052 0.0184
## W1-S1 1.8889 -2.5163 6.2941 0.8696
## W2-S1 5.3556 0.9504 9.7607 0.0100
## W3-S1 0.2556 -4.1496 4.6607 1.0000
## W1-S2 -3.1111 -7.5163 1.2941 0.3244
## W2-S2 0.3556 -4.0496 4.7607 1.0000
## W3-S2 -4.7444 -9.1496 -0.3393 0.0285
## W2-W1 3.4667 -0.9385 7.8718 0.2061
## W3-W1 -1.6333 -6.0385 2.7718 0.9390
## W3-W2 -5.1000 -9.5052 -0.6948 0.0155#performs a THSD test for the interaction of the type and variety of cornTukey’s HSD returns a matrix that contains statistical parameters for the interaction of an individual sample mean with every other sample mean being statistically analyzed. The comparisons between super sweet corn and purple waxy corn and white waxy corn and purple waxy corn both return confidence intervals that do not contain the value zero and have p adj values of 0.00. This indicates that there is difference between the sample means. However, the comparison between white waxy corn and super sweet corn produced a confidence interval that did contact zero and a p adj value of 0.1421. This indicates that there is no difference between the sample means and that the variation can be attributed to randomization. The same method of interperetation can be done for the Tukey’s HSD matrices returned from the corn variety and the type-variety interaction. These matrices are too large to individually describe each comparison. It should be noted that since the p adj values for the comparisons between W1, W2, and W3 were all above 0.05, indicating that the prediction made from the boxplot that the means were similar was correct.
Estimation of Parameters
Summary of all factors and levels
#Summary
pwc<-corn$Type=="Purple waxy corn"
summary(corn[pwc,])## Type Varieties Silkingstage Milkystage
## Purple waxy corn:15 P1 :3 Min. :111 Min. :144
## Super sweet corn: 0 P2 :3 1st Qu.:115 1st Qu.:168
## White waxy corn : 0 P3 :3 Median :124 Median :180
## P4 :3 Mean :131 Mean :178
## P5 :3 3rd Qu.:129 3rd Qu.:186
## S1 :0 Max. :176 Max. :212
## (Other):0
## Maturitystage Mean
## Min. :112 Min. :131
## 1st Qu.:133 1st Qu.:133
## Median :146 Median :151
## Mean :147 Mean :152
## 3rd Qu.:151 3rd Qu.:154
## Max. :190 Max. :193
## #displays a summary of the data of the type "Purple Waxy Corn"
wwc<-corn$Type=="White waxy corn"
summary(corn[wwc,])## Type Varieties Silkingstage Milkystage
## Purple waxy corn:0 W1 :3 Min. : 84.1 Min. :54.4
## Super sweet corn:0 W2 :3 1st Qu.: 87.1 1st Qu.:55.0
## White waxy corn :9 W3 :3 Median : 89.6 Median :56.8
## P1 :0 Mean : 91.8 Mean :56.9
## P2 :0 3rd Qu.: 98.5 3rd Qu.:58.8
## P3 :0 Max. :101.1 Max. :59.6
## (Other):0
## Maturitystage Mean
## Min. :24.4 Min. :58.8
## 1st Qu.:28.6 1st Qu.:59.6
## Median :36.1 Median :61.2
## Mean :36.7 Mean :61.8
## 3rd Qu.:45.8 3rd Qu.:64.0
## Max. :49.4 Max. :65.3
## #displays a summary of the data of the type "White Waxy Corn"
ssc<-corn$Type=="Super sweet corn"
summary(corn[ssc,])## Type Varieties Silkingstage Milkystage
## Purple waxy corn:0 S1 :3 Min. :83.1 Min. :72.0
## Super sweet corn:6 S2 :3 1st Qu.:86.1 1st Qu.:75.2
## White waxy corn :0 P1 :0 Median :89.0 Median :76.8
## P2 :0 Mean :89.5 Mean :76.8
## P3 :0 3rd Qu.:92.6 3rd Qu.:78.4
## P4 :0 Max. :96.6 Max. :81.4
## (Other):0
## Maturitystage Mean
## Min. :63.5 Min. :75.0
## 1st Qu.:65.9 1st Qu.:75.5
## Median :67.3 Median :77.8
## Mean :67.5 Mean :77.9
## 3rd Qu.:68.9 3rd Qu.:80.2
## Max. :71.6 Max. :81.1
## #displays a summary of the data of the type "Super sweet Corn"
summary(corn[corn$Varieties=="P1",])## Type Varieties Silkingstage Milkystage
## Purple waxy corn:3 P1 :3 Min. :120 Min. :176
## Super sweet corn:0 P2 :0 1st Qu.:122 1st Qu.:178
## White waxy corn :0 P3 :0 Median :124 Median :180
## P4 :0 Mean :124 Mean :180
## P5 :0 3rd Qu.:126 3rd Qu.:181
## S1 :0 Max. :128 Max. :183
## (Other):0
## Maturitystage Mean
## Min. :146 Min. :149
## 1st Qu.:148 1st Qu.:150
## Median :149 Median :151
## Mean :149 Mean :151
## 3rd Qu.:150 3rd Qu.:152
## Max. :152 Max. :152
## #displays a summary of the data of the variety P1
summary(corn[corn$Varieties=="P2",])## Type Varieties Silkingstage Milkystage
## Purple waxy corn:3 P2 :3 Min. :111 Min. :167
## Super sweet corn:0 P1 :0 1st Qu.:112 1st Qu.:168
## White waxy corn :0 P3 :0 Median :113 Median :169
## P4 :0 Mean :113 Mean :169
## P5 :0 3rd Qu.:114 3rd Qu.:170
## S1 :0 Max. :115 Max. :172
## (Other):0
## Maturitystage Mean
## Min. :112 Min. :131
## 1st Qu.:113 1st Qu.:132
## Median :115 Median :132
## Mean :115 Mean :132
## 3rd Qu.:116 3rd Qu.:133
## Max. :118 Max. :133
## #displays a summary of the data of the variety P2
summary(corn[corn$Varieties=="P3",])## Type Varieties Silkingstage Milkystage
## Purple waxy corn:3 P3 :3 Min. :124 Min. :183
## Super sweet corn:0 P1 :0 1st Qu.:126 1st Qu.:184
## White waxy corn :0 P2 :0 Median :127 Median :185
## P4 :0 Mean :127 Mean :185
## P5 :0 3rd Qu.:128 3rd Qu.:186
## S1 :0 Max. :130 Max. :187
## (Other):0
## Maturitystage Mean
## Min. :145 Min. :153
## 1st Qu.:146 1st Qu.:153
## Median :147 Median :153
## Mean :147 Mean :153
## 3rd Qu.:149 3rd Qu.:154
## Max. :150 Max. :154
## #displays a summary of the data of the variety P3
summary(corn[corn$Varieties=="P4",])## Type Varieties Silkingstage Milkystage
## Purple waxy corn:3 P4 :3 Min. :113 Min. :144
## Super sweet corn:0 P1 :0 1st Qu.:114 1st Qu.:146
## White waxy corn :0 P2 :0 Median :116 Median :148
## P3 :0 Mean :116 Mean :148
## P5 :0 3rd Qu.:117 3rd Qu.:149
## S1 :0 Max. :118 Max. :151
## (Other):0
## Maturitystage Mean
## Min. :132 Min. :132
## 1st Qu.:133 1st Qu.:132
## Median :134 Median :133
## Mean :134 Mean :133
## 3rd Qu.:136 3rd Qu.:133
## Max. :137 Max. :133
## #displays a summary of the data of the variety P4
summary(corn[corn$Varieties=="P5",])## Type Varieties Silkingstage Milkystage
## Purple waxy corn:3 P5 :3 Min. :171 Min. :201
## Super sweet corn:0 P1 :0 1st Qu.:172 1st Qu.:204
## White waxy corn :0 P2 :0 Median :173 Median :207
## P3 :0 Mean :173 Mean :207
## P4 :0 3rd Qu.:175 3rd Qu.:210
## S1 :0 Max. :176 Max. :212
## (Other):0
## Maturitystage Mean
## Min. :184 Min. :185
## 1st Qu.:186 1st Qu.:187
## Median :187 Median :189
## Mean :187 Mean :189
## 3rd Qu.:189 3rd Qu.:191
## Max. :190 Max. :193
## #displays a summary of the data of the variety P5
summary(corn[corn$Varieties=="W1",])## Type Varieties Silkingstage Milkystage
## Purple waxy corn:0 W1 :3 Min. : 98.5 Min. :55.0
## Super sweet corn:0 P1 :0 1st Qu.: 99.2 1st Qu.:56.1
## White waxy corn :3 P2 :0 Median : 99.8 Median :57.3
## P3 :0 Mean : 99.8 Mean :57.3
## P4 :0 3rd Qu.:100.5 3rd Qu.:58.5
## P5 :0 Max. :101.1 Max. :59.6
## (Other):0
## Maturitystage Mean
## Min. :24.4 Min. :59.3
## 1st Qu.:25.4 1st Qu.:60.2
## Median :26.5 Median :61.2
## Mean :26.5 Mean :61.2
## 3rd Qu.:27.6 3rd Qu.:62.1
## Max. :28.6 Max. :63.1
## #displays a summary of the data of the variety W1
summary(corn[corn$Varieties=="W2",])## Type Varieties Silkingstage Milkystage
## Purple waxy corn:0 W2 :3 Min. :87.1 Min. :54.4
## Super sweet corn:0 P1 :0 1st Qu.:88.3 1st Qu.:55.6
## White waxy corn :3 P2 :0 Median :89.6 Median :56.8
## P3 :0 Mean :89.6 Mean :56.8
## P4 :0 3rd Qu.:90.8 3rd Qu.:58.0
## P5 :0 Max. :92.1 Max. :59.2
## (Other):0
## Maturitystage Mean
## Min. :45.8 Min. :64.0
## 1st Qu.:46.7 1st Qu.:64.3
## Median :47.6 Median :64.7
## Mean :47.6 Mean :64.7
## 3rd Qu.:48.5 3rd Qu.:65.0
## Max. :49.4 Max. :65.3
## #displays a summary of the data of the variety W2
summary(corn[corn$Varieties=="W3",])## Type Varieties Silkingstage Milkystage
## Purple waxy corn:0 W3 :3 Min. :84.1 Min. :54.6
## Super sweet corn:0 P1 :0 1st Qu.:85.0 1st Qu.:55.6
## White waxy corn :3 P2 :0 Median :85.9 Median :56.7
## P3 :0 Mean :85.9 Mean :56.7
## P4 :0 3rd Qu.:86.8 3rd Qu.:57.8
## P5 :0 Max. :87.7 Max. :58.8
## (Other):0
## Maturitystage Mean
## Min. :34.2 Min. :58.8
## 1st Qu.:35.1 1st Qu.:59.2
## Median :36.1 Median :59.6
## Mean :36.1 Mean :59.6
## 3rd Qu.:37.0 3rd Qu.:60.0
## Max. :38.0 Max. :60.3
## #displays a summary of the data of the variety W3
summary(corn[corn$Varieties=="S1",])## Type Varieties Silkingstage Milkystage
## Purple waxy corn:0 S1 :3 Min. :83.1 Min. :72.0
## Super sweet corn:3 P1 :0 1st Qu.:84.3 1st Qu.:73.6
## White waxy corn :0 P2 :0 Median :85.5 Median :75.2
## P3 :0 Mean :85.5 Mean :75.2
## P4 :0 3rd Qu.:86.7 3rd Qu.:76.8
## P5 :0 Max. :87.9 Max. :78.4
## (Other):0
## Maturitystage Mean
## Min. :63.5 Min. :75.0
## 1st Qu.:64.5 1st Qu.:75.2
## Median :65.5 Median :75.4
## Mean :65.5 Mean :75.4
## 3rd Qu.:66.5 3rd Qu.:75.6
## Max. :67.5 Max. :75.8
## #displays a summary of the data of the variety S1
summary(corn[corn$Varieties=="S2",])## Type Varieties Silkingstage Milkystage
## Purple waxy corn:0 S2 :3 Min. :90.2 Min. :75.2
## Super sweet corn:3 P1 :0 1st Qu.:91.8 1st Qu.:76.8
## White waxy corn :0 P2 :0 Median :93.4 Median :78.3
## P3 :0 Mean :93.4 Mean :78.3
## P4 :0 3rd Qu.:95.0 3rd Qu.:79.8
## P5 :0 Max. :96.6 Max. :81.4
## (Other):0
## Maturitystage Mean
## Min. :67.2 Min. :79.7
## 1st Qu.:68.3 1st Qu.:80.0
## Median :69.4 Median :80.4
## Mean :69.4 Mean :80.4
## 3rd Qu.:70.5 3rd Qu.:80.8
## Max. :71.6 Max. :81.1
## #displays a summary of the data of the variety S2Diagnostics/Model Adequacy Checking
Quantile-Quantile (Q-Q) plots are graphs used to verify the distributional assumption for a set of data. Based on the theoretical distribution, the expected value for each datum is determined. If the data values in a set follow the theoretical distribution, then they will appear as a straight line on a Q-Q plot. When an anova is performed, it is done so with the assumption that the test statistic follows a normal distribution. Visualization of a Q-Q plot will further confirm if that assumption is correct for the anova tests that were performed.
#Q-Q Plots
#Corn Type
qqnorm(residuals(model_type), main="Normal Q-Q Plot for Corn Type", ylab="Total phenolic content Residuals")
qqline(residuals(model_type))
#produces a Q-Q normal plot for the corn type residuals with a normal fit line
#Corn Type
qqnorm(residuals(model_variety), main="Normal Q-Q Plot for Corn Variety", ylab="Total phenolic content Residuals")
qqline(residuals(model_variety))
#produces a Q-Q normal plot for the corn variety residuals with a normal fit line
#Corn Type
qqnorm(residuals(model_type_variety), main="Normal Q-Q Plot for Interaction of Corn Type and Variety", ylab="Total phenolic content Residuals")
qqline(residuals(model_type_variety))
#produces a Q-Q normal plot for the corn type-variety interaction residuals with a normal fit lineAll three Normal Q-Q plots for the type and variety of corn have a linear relationship between the 25th and 75th percentile and two non-linear tails. Despite that fact that the tails do not follow a non-linear relationship, the distribution assumption that the samples follow a normal distribution can still be assumed since it is not unusual that tails deviate from the linear portion of the graph. The relatively linear relationship for all three data sets justifies the use of ANOVA to test for the significant difference.
Two Way Interaction Plots display the mean of the response for two-way combinations of factors, and can indicate if there is any interactions between them through visual inspection. Data sets that do not have any interaction will appear as perfectly parallel lines. Changes in slope and intersections are good indications of interactions.
interaction.plot(corn$Varieties,corn$Type,corn$Mean, xlab="Corn Variety", ylab="Means of Total Phenolic Content", main="Corn Interaction Plot", trace.label="Corn Type")
#creates a plot that shows the interaction of corn type and variety on a corn's phenolic contentIn the interaction plot above, there are clear observations of non-parallel lines, but no intersections. This still indicates that there are interactions between the type and variety of corn.
A Residuals vs. Fits Plot is a common graph used in residual analysis. It is a scatter plot of residuals as a function of fitted values, or the estimated responses. These plots are used to identify linearity, outliers, and error variances.
#Residual vs Fit
#Corn Type
plot(fitted(model_type),residuals(model_type), main="Residual vs Fitted Plot for Corn Type") 
#Corn Variety
plot(fitted(model_variety),residuals(model_variety), main="Residual vs Fitted Plot for Country of Origin")
#Corn Interaction
plot(fitted(model_type_variety),residuals(model_type_variety), main="Residual vs Fitted Plot for Interaction") 