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Project Example: Classifiying Flower STAT 3010 3/21/2022 Contents Introduction 2 Data Structure and Visualisation 2 Summary by Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Boxplots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Scatter plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Overall Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 If we only had one ANOVA 7 Test for Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Test for Equal Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 ANOVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Multiple Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Conclusion 10 library(tidyverse) library(psych) library(knitr) library(cowplot) library(GGally) library(rstatix) 1 Introduction Figure 1: Setosa In this project we are going to try and determine the physical differences between three sub-species of Iris using measurement from their flowers. The both length and width of the sepal and the petal of 50 flowers of each species was measured and record in this study. Data Structure and Visualisation General summary kable(summary(iris)) #overall summary Sepal.Length Sepal.Width Petal.Length Petal.Width Species Min. :4.300 Min. :2.000 Min. :1.000 Min. :0.100 setosa :50 1st Qu.:5.100 1st Qu.:2.800 1st Qu.:1.600 1st Qu.:0.300 versicolor:50 Median :5.800 Median :3.000 Median :4.350 Median :1.300 virginica :50 Mean :5.843 Mean :3.057 Mean :3.758 Mean :1.199 NA 3rd Qu.:6.400 3rd Qu.:3.300 3rd Qu.:5.100 3rd Qu.:1.800 NA Max. :7.900 Max. :4.400 Max. :6.900 Max. :2.500 NA db <- describeBy(iris,group=iris$Species) #Summaries by Group Summary by Species Setosa kable(db$setosa) vars n mean sd mediantrimmed mad min max range skew kurtosis se Sepal.Length 1 50 5.006 0.3524897 5.0 5.0025 0.29652 4.3 5.8 1.5 0.1129778 - 0.4508724 0.0498496 Sepal.Width 2 50 3.428 0.3790644 3.4 3.4150 0.37065 2.3 4.4 2.1 0.03872950.5959507 0.0536078 Petal.Length 3 50 1.462 0.1736640 1.5 1.4600 0.14826 1.0 1.9 0.9 0.10009540.6539303 0.0245598 2 vars n mean sd mediantrimmed mad min max range skew kurtosis se Petal.Width 4 50 0.246 0.1053856 0.2 0.2375 0.00000 0.1 0.6 0.5 1.17963281.2587179 0.0149038 Species* 5 50 1.000 0.0000000 1.0 1.0000 0.00000 1.0 1.0 0.0 NaN NaN 0.0000000 Versicolor kable(db$versicolor) vars n mean sd mediantrimmed mad min max range skew kurtosis se Sepal.Length 1 50 5.936 0.51617115.90 5.9375 0.51891 4.9 7.0 2.1 0.0991393 - 0.6939138 0.0729976 Sepal.Width 2 50 2.770 0.31379832.80 2.7800 0.29652 2.0 3.4 1.4 - 0.3413644 - 0.5493203 0.0443778 Petal.Length 3 50 4.260 0.46991104.35 4.2925 0.51891 3.0 5.1 2.1 - 0.5706024 - 0.1902555 0.0664554 Petal.Width 4 50 1.326 0.19775271.30 1.3250 0.22239 1.0 1.8 0.8 - 0.0293338 - 0.5873144 0.0279665 Species* 5 50 2.000 0.00000002.00 2.0000 0.00000 2.0 2.0 0.0 NaN NaN 0.0000000 Virginica kable(db$virginica) vars n mean sd mediantrimmed mad min max range skew kurtosis se Sepal.Length 1 50 6.588 0.63587966.50 6.5725 0.59304 4.9 7.9 3.0 0.1110286 - 0.2032597 0.0899270 Sepal.Width 2 50 2.974 0.32249663.00 2.9625 0.29652 2.2 3.8 1.6 0.3442849 0.3803832 0.0456079 Petal.Length 3 50 5.552 0.55189475.55 5.5100 0.66717 4.5 6.9 2.4 0.5169175 - 0.3651161 0.0780497 Petal.Width 4 50 2.026 0.27465012.00 2.0325 0.29652 1.4 2.5 1.1 - 0.1218119 - 0.7539586 0.0388414 Species* 5 50 3.000 0.00000003.00 3.0000 0.00000 3.0 3.0 0.0 NaN NaN 0.0000000 Boxplots p <- ggplot(iris,aes(x=Species,fill=Species)) p1 <- p+geom_boxplot(aes(y=Sepal.Length))+ theme(legend.position = "none") p2 <- p+geom_boxplot(aes(y=Sepal.Width))+ theme(legend.position = "none") p3 <- p+geom_boxplot(aes(y=Petal.Length))+ theme(legend.position = "none") p4 <- p+geom_boxplot(aes(y=Petal.Width))+ theme(legend.position = "none") plot_grid(p1,p2,p3,p4) 3 56 7 8 setosa versicolor virginica Species Se pa l.L en gt h 2.0 2.5 3.0 3.5 4.0 4.5 setosa versicolor virginica Species Se pa l.W id th 2 4 6 setosa versicolor virginica Species Pe ta l.L en gt h 0.0 0.5 1.0 1.5 2.0 2.5 setosa versicolor virginica Species Pe ta l.W id th From the boxplots we can see there are clear difference between the species based on several variables. This is strong evidence that we will be able to find significant difference the groups Scatter plots q <- ggplot(iris,aes(colour=Species)) q1 <-q+geom_point(aes(x=Sepal.Length,y=Sepal.Width))+labs(title="Sepal Length vs Sepal Width")+theme(legend.position = 'None') q2 <-q+geom_point(aes(x=Petal.Length,y=Petal.Width))+labs(title="Petal Length vs Petal Width")+theme(legend.position = 'None') plot_grid(q1,q2) 4 2.0 2.5 3.0 3.5 4.0 4.5 5 6 7 8 Sepal.Length Se pa l.W id th Sepal Length vs Sepal Width 0.0 0.5 1.0 1.5 2.0 2.5 2 4 6 Petal.Length Pe ta l.W id th Petal Length vs Petal Width Overall Summary ggpairs(iris,aes(colour=Species)) 5 Corr: 0.118 setosa: 0.743*** versicolor: 0.526*** virginica: 0.457*** Corr: 0.872*** setosa: 0.267. versicolor: 0.754*** virginica: 0.864*** Corr: 0.428*** setosa: 0.178 versicolor: 0.561*** virginica: 0.401** Corr: 0.818*** setosa: 0.278. versicolor: 0.546*** virginica: 0.281* Corr: 0.366*** setosa: 0.233 versicolor: 0.664*** virginica: 0.538*** Corr: 0.963*** setosa: 0.332* versicolor: 0.787*** virginica: 0.322* Sepal.Length Sepal.Width Petal.Length Petal.Width Species Sepal.Length Sepal.W idth P etal.Length P etal.W idth Species 5 6 7 8 2.0 2.5 3.0 3.5 4.0 4.5 2 4 6 0.0 0.5 1.0 1.5 2.0 2.5 setosaversicolorvirginica 0.0 0.4 0.8 1.2 2.0 2.5 3.0 3.5 4.0 4.5 2 4 6 0.0 0.5 1.0 1.5 2.0 2.5 0.0 2.5 5.0 7.5 0.0 2.5 5.0 7.5 0.0 2.5 5.0 7.5 6 If we only had one ANOVA We wish to find the measurement that is most different among our given Species. So will use Anova to determine if there are differences in the measures for each Species. But first we check some the assumptions of the anova. Test for Normality iris %>% group_by(Species) %>% shapiro_test(Sepal.Width) # test for normality in each group ## # A tibble: 3 x 4 ## Species variable statistic p ## ## 1 setosa Sepal.Width 0.972 0.272 ## 2 versicolor Sepal.Width 0.974 0.338 ## 3 virginica Sepal.Width 0.967 0.181 Since our p-values are greater the α = 0.05 we fail to reject the null hypothesis and proceed with the assumption of normality. g_iris<- iris %>% group_by(Species) g_iris %>% shapiro_test(Sepal.Length)# test for normality in each group sepal length ## # A tibble: 3 x 4 ## Species variable statistic p ## ## 1 setosa Sepal.Length 0.978 0.460 ## 2 versicolor Sepal.Length 0.978 0.465 ## 3 virginica Sepal.Length 0.971 0.258 g_iris %>% shapiro_test(Petal.Length)# test for normality in each group sepal length ## # A tibble: 3 x 4 ## Species variable statistic p ## ## 1 setosa Petal.Length 0.955 0.0548 ## 2 versicolor Petal.Length 0.966 0.158 ## 3 virginica Petal.Length 0.962 0.110 g_iris %>% shapiro_test(Petal.Width)# test for normality in each group sepal length ## # A tibble: 3 x 4 ## Species variable statistic p ## ## 1 setosa Petal.Width 0.800 0.000000866 ## 2 versicolor Petal.Width 0.948 0.0273 ## 3 virginica Petal.Width 0.960 0.0870 Here we find that the normality assumption is only violated for one of our groups. Since the violation is only in one group we will proceed. Test for Equal Variance bartlett.test(Sepal.Width~Species,iris) ## ## Bartlett test of homogeneity of variances 7 ## ## data: Sepal.Width by Species ## Bartlett’s K-squared = 2.0911, df = 2, p-value = 0.3515 Since our p-value<α we fail to reject the null and conclude that our variance are not significantly different from each other. bartlett.test(Sepal.Length~Species,iris) ## ## Bartlett test of homogeneity of variances ## ## data: Sepal.Length by Species ## Bartlett's K-squared = 16.006, df = 2, p-value = 0.0003345 bartlett.test(Petal.Width~Species,iris) ## ## Bartlett test of homogeneity of variances ## ## data: Petal.Width by Species ## Bartlett's K-squared = 39.213, df = 2, p-value = 3.055e-09 bartlett.test(Petal.Length~Species,iris) ## ## Bartlett test of homogeneity of variances ## ## data: Petal.Length by Species ## Bartlett's K-squared = 55.423, df = 2, p-value = 9.229e-13 He we see heavy violations of the equality of variance since our ni large we can proceed with our anova. ANOVA summary(aov(Sepal.Width~Species,data=iris)) ## Df Sum Sq Mean Sq F value Pr(>F) ## Species 2 11.35 5.672 49.16 <2e-16 *** ## Residuals 147 16.96 0.115 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Our anova show us that there are statistically significant difference between our groups summary(aov(Sepal.Length~Species,data=iris)) ## Df Sum Sq Mean Sq F value Pr(>F) ## Species 2 63.21 31.606 119.3 <2e-16 *** ## Residuals 147 38.96 0.265 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 summary(aov(Petal.Length~Species,data=iris)) ## Df Sum Sq Mean Sq F value Pr(>F) ## Species 2 437.1 218.55 1180 <2e-16 *** ## Residuals 147 27.2 0.19 ## --- 8 ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 summary(aov(Petal.Width~Species,data=iris)) ## Df Sum Sq Mean Sq F value Pr(>F) ## Species 2 80.41 40.21 960 <2e-16 *** ## Residuals 147 6.16 0.04 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Just as we suspected when we observed the boxplots all our variable have difference between groups. We therefore proceed to perform multiple comparison testing. Multiple Comparison TukeyHSD(aov(Sepal.Width~Species,data=iris),conf.level = .95) ## Tukey multiple comparisons of means ## 95% family-wise confidence level ## ## Fit: aov(formula = Sepal.Width ~ Species, data = iris) ## ## $Species ## diff lwr upr p adj ## versicolor-setosa -0.658 -0.81885528 -0.4971447 0.0000000 ## virginica-setosa -0.454 -0.61485528 -0.2931447 0.0000000 ## virginica-versicolor 0.204 0.04314472 0.3648553 0.0087802 TukeyHSD(aov(Sepal.Length~Species,data=iris),conf.level = .95) ## Tukey multiple comparisons of means ## 95% family-wise confidence level ## ## Fit: aov(formula = Sepal.Length ~ Species, data = iris) ## ## $Species ## diff lwr upr p adj ## versicolor-setosa 0.930 0.6862273 1.1737727 0 ## virginica-setosa 1.582 1.3382273 1.8257727 0 ## virginica-versicolor 0.652 0.4082273 0.8957727 0 TukeyHSD(aov(Petal.Length~Species,data=iris),conf.level = .95) ## Tukey multiple comparisons of means ## 95% family-wise confidence level ## ## Fit: aov(formula = Petal.Length ~ Species, data = iris) ## ## $Species ## diff lwr upr p adj ## versicolor-setosa 2.798 2.59422 3.00178 0 ## virginica-setosa 4.090 3.88622 4.29378 0 ## virginica-versicolor 1.292 1.08822 1.49578 0 TukeyHSD(aov(Petal.Width~Species,data=iris),conf.level = .95) ## Tukey multiple comparisons of means 9 ## 95% family-wise confidence level ## ## Fit: aov(formula = Petal.Width ~ Species, data = iris) ## ## $Species ## diff lwr upr p adj ## versicolor-setosa 1.08 0.9830903 1.1769097 0 ## virginica-setosa 1.78 1.6830903 1.8769097 0 ## virginica-versicolor 0.70 0.6030903 0.7969097 0 In fact we find that each variable can be distinguished by their species hence we should be able tell which species or flower is by looking at any of the variables, but since petal length has the largest mean difference it would be the easiest variable to use. Conclusion In our investigation we sort to determine difference between species of iris through a looking at measurements taken from their flower. We found difference with respect to very measurement that look at but we found the largest difference to be found within Petal Length. Future work could look at build a model that predicts the species based on the given measurements and as our investigation shows there are significant differences between species based on these measurements. 10