Assignment in regression1
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1) The document describes regression analysis performed on a dataset containing measurements of girth, height, and volume of 31 black cherry trees. 2) Scatter plots and correlation coefficients show strong positive correlations between volume and girth, and weaker correlations between other variables. 3) Linear regression models were fitted with log-transformed variables, achieving high R-squared values between 0.65 and 0.98, indicating the models explain a large percentage of the variability in the data.
Assignment in regression1
- 1. Regression using R Dipika Patra 7 August 2016 Regression analysis Data Description The dataset prvides measurments of the grith, height and volume of timber in 31 felled black cherry trees. Note that grith is the diameter of the tree measured at 4 ft 6 inches above the ground. The dataset is given below: trees ## Girth Height Volume ## 1 8.3 70 10.3 ## 2 8.6 65 10.3 ## 3 8.8 63 10.2 ## 4 10.5 72 16.4 ## 5 10.7 81 18.8 ## 6 10.8 83 19.7 ## 7 11.0 66 15.6 ## 8 11.0 75 18.2 ## 9 11.1 80 22.6 ## 10 11.2 75 19.9 ## 11 11.3 79 24.2 ## 12 11.4 76 21.0 ## 13 11.4 76 21.4 ## 14 11.7 69 21.3 ## 15 12.0 75 19.1 ## 16 12.9 74 22.2 ## 17 12.9 85 33.8 ## 18 13.3 86 27.4 ## 19 13.7 71 25.7 ## 20 13.8 64 24.9 ## 21 14.0 78 34.5 ## 22 14.2 80 31.7 ## 23 14.5 74 36.3 ## 24 16.0 72 38.3 ## 25 16.3 77 42.6 ## 26 17.3 81 55.4 ## 27 17.5 82 55.7 ## 28 17.9 80 58.3 ## 29 18.0 80 51.5 ## 30 18.0 80 51.0 ## 31 20.6 87 77.0 1
- 2. Correlation cor(trees) ## Girth Height Volume ## Girth 1.0000000 0.5192801 0.9671194 ## Height 0.5192801 1.0000000 0.5982497 ## Volume 0.9671194 0.5982497 1.0000000 Graphical Display: To illustrate linked two dimentional scatter plots we refer to use “pairs” comand with “panel.smooth” argument.It shows two dimentional scatter plot of the each pairs of observations. pairs(trees, panel = panel.smooth, main = "trees data") Girth 65 70 75 80 85 8121620 657585 Height 8 10 12 14 16 18 20 10 20 30 40 50 60 70 10305070 Volume trees data As we know, Volume of cylinder= (22/7)* (Grith)ˆ2 * Height i.e. log(Volume)= constant+log(Height)+2log(Grith) To plot the data in logarithmic graph we refer to use “plot” command with argument “log”. plot(Volume ~ Girth, data = trees, log = "xy") 2
- 3. 10 12 14 16 18 20 102030406080 Girth Volume Regression Steps: Dependent variable “volume”: To regress the “Volume” variable with respect to the independent variable “Height” & “Grith” it is reasonable to fit linear regression on the logarithmic value of the given data. summary(fm1 <- lm(log(Volume) ~ log(Girth)+ log(Height), data = trees)) ## ## Call: ## lm(formula = log(Volume) ~ log(Girth) + log(Height), data = trees) ## ## Residuals: ## Min 1Q Median 3Q Max ## -0.168561 -0.048488 0.002431 0.063637 0.129223 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -6.63162 0.79979 -8.292 5.06e-09 *** ## log(Girth) 1.98265 0.07501 26.432 < 2e-16 *** ## log(Height) 1.11712 0.20444 5.464 7.81e-06 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 3
- 4. ## ## Residual standard error: 0.08139 on 28 degrees of freedom ## Multiple R-squared: 0.9777, Adjusted R-squared: 0.9761 ## F-statistic: 613.2 on 2 and 28 DF, p-value: < 2.2e-16 library(ggplot2) qplot(Volume,exp(predict(fm1)), data= trees,xlab = "Observed Value of Volume",ylab="Predicted value of V 20 40 60 80 20 40 60 80 Observed Value of Volume PredictedvalueofVolume R square and adjusted R square with interpretation: The R-squared is the ratio of the response variable variation that is explained by a linear model. Mathematically,R-squared = Explained variation / Total variation R-squared is always between 0 and 1: 0 indicates that the model explains none of the variability of the response data around its mean. 1 indicates that the model explains all the variability of the response data around its mean. In general, the higher the R-squared, the better the model fits your data. In the above model R square =0.9777 i.e. the model explain 97.77 % variablity of the data set. The adjusted R-squared is a modified version of R-squared for the number of predictors in a model that compares the explanatory power of regression models that contain different numbers of predictors. In the above model adjusted R square = 0.9761 Fitted Model: log(Volume)=-6.6312+1.98255log(Grith)-1.11712log(Height) 4
- 5. Dependent variable Height: To regress the “Height” variable with respect to the independent variable “Volume” & “Grith” it is reasonable to fit linear regression on the logarithmic value of the given data. summary(fm2 <- lm(log(Height) ~ log(Volume)+log(Girth), data = trees)) ## ## Call: ## lm(formula = log(Height) ~ log(Volume) + log(Girth), data = trees) ## ## Residuals: ## Min 1Q Median 3Q Max ## -0.08628 -0.03084 -0.00146 0.02622 0.13465 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 4.91689 0.22497 21.856 < 2e-16 *** ## log(Volume) 0.46196 0.08454 5.464 7.81e-06 *** ## log(Girth) -0.82177 0.19043 -4.315 0.000179 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 0.05234 on 28 degrees of freedom ## Multiple R-squared: 0.6521, Adjusted R-squared: 0.6273 ## F-statistic: 26.24 on 2 and 28 DF, p-value: 3.804e-07 qplot(Height,exp(predict(fm2)), data= trees,xlab = "Observed Value of Height",ylab="Predicted value of H 5
- 6. 70 75 80 85 70 80 Observed Value of Height PredictedvalueofHeight R -Square Interpretation: In that model R square =0.6521 and Adjusted R square=0.6247 i.e. the model explain 65.21 % variablity of the data set. Fitted Model: log(Height)=4.91689+0.46196 log(volume)-0.82177 log(Grith) Dependent variable Grith: Next regress the “Grith” variable with respect to the independent variable “Volume” & “Height” it is reasonable to fit linear regression on the logarithmic value of the given data. summary(fm3 <- lm(log(Girth) ~ log(Volume)+log(Height), data = trees)) ## ## Call: ## lm(formula = log(Girth) ~ log(Volume) + log(Height), data = trees) ## ## Residuals: ## Min 1Q Median 3Q Max ## -0.070123 -0.027064 0.000029 0.022232 0.079494 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) 6
- 7. ## (Intercept) 3.07353 0.45083 6.817 2.09e-07 *** ## log(Volume) 0.48494 0.01835 26.432 < 2e-16 *** ## log(Height) -0.48606 0.11263 -4.315 0.000179 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 0.04025 on 28 degrees of freedom ## Multiple R-squared: 0.9723, Adjusted R-squared: 0.9703 ## F-statistic: 491.4 on 2 and 28 DF, p-value: < 2.2e-16 qplot(Girth,exp(predict(fm3)), data= trees,xlab = "Observed Value of Girth",ylab="Predicted value of Gir 8 12 16 20 8 12 16 20 Observed Value of Girth PredictedvalueofGirth R -Square Interpretation: In that model R square =0.9723 and Adjusted R square=0.9703 i.e. the model explain 97.23 % variablity of the data set. Fitted Model: log(Grith)=3.0753+0.48494 log(volume)-0.48606 log(Height) 7