Data Analytics Assignment The random sample of fish with age in days, room temperature and length of fish in centimeter is given. > a data data age temp length 1 14 25 620 2 28 25 1315 3 41 25 2120 4 55 25 2600 5 69 25 3110 6 83 25 3535 7 97 25 3935 8 111 25 4465 9 125 25 4530 10 139 25 4570 11 153 25 4600 12 14 27 625 13 28 27 1215 14 41 27 2110 15 55 27 2805 16 69 27 3255 17 83 27 4015 18 97 27 4315 19 111 27 4495 20 125 27 4535 21 139 27 4600 22 153 27 4600 23 14 29 590 24 28 29 105 25 41 29 2140 26 55 29 2890 27 69 29 3920 28 83 29 3920 29 97 29 4515

30 111 29 4520 31 125 29 4525 32 139 29 4565 33 13 29 4566 34 14 31 590 35 2 31 1205 36 41 31 1915 37 55 31 2140 38 69 31 2710 39 83 31 3020 40 97 31 3030 41 111 31 3040 42 125 31 3180 43 139 31 3257 44 153 31 3214 > res=lm(age~ temp + length,data=data) > data age temp length 1 14 25 620 2 28 25 1315 3 41 25 2120 4 55 25 2600 5 69 25 3110 6 83 25 3535 7 97 25 3935 8 111 25 4465 9 125 25 4530 10 139 25 4570 11 153 25 4600 12 14 27 625 13 28 27 1215 14 41 27 2110 15 55 27 2805 16 69 27 3255 17 83 27 4015 18 97 27 4315 19 111 27 4495

20 125 27 4535 21 139 27 4600 22 153 27 4600 23 14 29 590 24 28 29 105 25 41 29 2140 26 55 29 2890 27 69 29 3920 28 83 29 3920 29 97 29 4515 30 111 29 4520 31 125 29 4525 32 139 29 4565 33 13 29 4566 34 14 31 590 35 2 31 1205 36 41 31 1915 37 55 31 2140 38 69 31 2710 39 83 31 3020 40 97 31 3030 41 111 31 3040 42 125 31 3180 43 139 31 3257 44 153 31 3214

> res=lm(age~ temp + length,data=data) > res Call: lm(formula = age ~ temp + length, data = data) Coefficients: (Intercept) temp length -59.66395 2.01300 0.02685

> summary(res) Call: lm(formula = age ~ temp + length, data = data) Residuals: Min 1Q Median 3Q Max -108.320 -13.108 -1.708 12.749 63.958 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -59.663949 55.586364 -1.073 0.289 temp 2.012999 1.888509 1.066 0.293 length 0.026852 0.003099 8.664 8.29e-11 *** --Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 27.53 on 41 degrees of freedom Multiple R-squared: 0.6476, Adjusted R-squared: 0.6304 F-statistic: 37.67 on 2 and 41 DF, p-value: 5.18e-10 #Leverage lev=hat(model.matrix(res)) > plot(lev) #reId

#Identifying the points > data[lev>0.10,] age temp length 1 14 25 620 2 28 25 1315 12 14 27 625 24 28 29 105 34 14 31 590 > plot(data$age,data$temp)

> points(data[1,]$age,data[1,]$temp, col='red') > points(data[2,]$age,data[2,]$temp, col='red') > points(data[12,]$age,data[12,]$temp, col='red') > points(data[24,]$age,data[24,]$temp, col='red') > points(data[34,]$age,data[34,]$temp, col='red')

#Studentizing the Data > r=rstudent(res) >r r 1 2 3 4 5 6 0.26250971 0.07754834 -0.24722205 -0.20405413 -0.19191693 -0.09579536 7 8 9 10 11 12 0.02508872 0.01657851 0.47608978 0.96979906 1.49281803 0.09626689 13 14 15 16 17 18 0.02566921 -0.37985245 -0.54842354 -0.47750170 -0.71690026 -0.49862860 19 20 21 22 23 24 -0.16143286 0.31464746 0.77190357 1.30930524 -0.02100444 1.03205025 25 26 27 28 29 30 -0.55662321 -0.78120867 -1.30662923 -0.77309032 -0.85579331 -0.33621356 31 32 33 34 35 36 0.17676149 0.65852096 -5.17888911 -0.17551866 -1.27202402 -0.49165419 37 38 39 40 41 42 -0.19343852 -0.24145258 -0.03088439 0.47980658 1.00006022 1.40180497

43 44 1.89173754 2.56193086 #calculating the cook's distance > cook= cooks.distance(res) > plot(cook,ylab="Cooks distances") > points(1,cook[1],col='blue') > points(2,cook[2],col='blue') > points(12,cook[12],col='blue') > points(24,cook[24],col='blue') > points(34,cook[34],col='blue')

> str(res) List of 12 $ coefficients : Named num [1:3] -59.6639 2.013 0.0269 ..- attr(*, "names")= chr [1:3] "(Intercept)" "temp" "length" $ residuals : Named num [1:44] 6.69 2.03 -6.59 -5.48 -5.17 ... ..- attr(*, "names")= chr [1:44] "1" "2" "3" "4" ... $ effects : Named num [1:44] -526.74 -14.7 238.55 -6.37 -6.48 ... ..- attr(*, "names")= chr [1:44] "(Intercept)" "temp" "length" "" ... $ rank : int 3

$ fitted.values: Named num [1:44] 7.31 25.97 47.59 60.48 74.17 ... ..- attr(*, "names")= chr [1:44] "1" "2" "3" "4" ... $ assign : int [1:3] 0 1 2 $ qr :List of 5 ..$ qr : num [1:44, 1:3] -6.633 0.151 0.151 0.151 0.151 ... .. ..- attr(*, "dimnames")=List of 2 .. .. ..$ : chr [1:44] "1" "2" "3" "4" ... .. .. ..$ : chr [1:3] "(Intercept)" "temp" "length" .. ..- attr(*, "assign")= int [1:3] 0 1 2 ..$ qraux: num [1:3] 1.15 1.18 1.08 ..$ pivot: int [1:3] 1 2 3 ..$ tol : num 1e-07 ..$ rank : int 3 ..- attr(*, "class")= chr "qr" $ df.residual : int 41 $ xlevels : Named list() $ call : language lm(formula = age ~ temp + length, data = data) $ terms :Classes 'terms', 'formula' length 3 age ~ temp + length .. ..- attr(*, "variables")= language list(age, temp, length) .. ..- attr(*, "factors")= int [1:3, 1:2] 0 1 0 0 0 1 .. .. ..- attr(*, "dimnames")=List of 2 .. .. .. ..$ : chr [1:3] "age" "temp" "length" .. .. .. ..$ : chr [1:2] "temp" "length" .. ..- attr(*, "term.labels")= chr [1:2] "temp" "length" .. ..- attr(*, "order")= int [1:2] 1 1 .. ..- attr(*, "intercept")= int 1 .. ..- attr(*, "response")= int 1 .. ..- attr(*, ".Environment")= .. ..- attr(*, "predvars")= language list(age, temp, length) .. ..- attr(*, "dataClasses")= Named chr [1:3] "numeric" "numeric" "numeric" .. .. ..- attr(*, "names")= chr [1:3] "age" "temp" "length" $ model :'data.frame': 44 obs. of 3 variables: ..$ age : int [1:44] 14 28 41 55 69 83 97 111 125 139 ... ..$ temp : int [1:44] 25 25 25 25 25 25 25 25 25 25 ... ..$ length: int [1:44] 620 1315 2120 2600 3110 3535 3935 4465 4530 4570 ... ..- attr(*, "terms")=Classes 'terms', 'formula' length 3 age ~ temp + length .. .. ..- attr(*, "variables")= language list(age, temp, length) .. .. ..- attr(*, "factors")= int [1:3, 1:2] 0 1 0 0 0 1

.. .. .. ..- attr(*, "dimnames")=List of 2 .. .. .. .. ..$ : chr [1:3] "age" "temp" "length" .. .. .. .. ..$ : chr [1:2] "temp" "length" .. .. ..- attr(*, "term.labels")= chr [1:2] "temp" "length" .. .. ..- attr(*, "order")= int [1:2] 1 1 .. .. ..- attr(*, "intercept")= int 1 .. .. ..- attr(*, "response")= int 1 .. .. ..- attr(*, ".Environment")= .. .. ..- attr(*, "predvars")= language list(age, temp, length) .. .. ..- attr(*, "dataClasses")= Named chr [1:3] "numeric" "numeric" "numeric" .. .. .. ..- attr(*, "names")= chr [1:3] "age" "temp" "length" - attr(*, "class")= chr "lm"

# diagnosing residuals > par(mfrow=c(1,6)) > plot(data$age, res$res) > plot(data$temp, res$res) > plot(data$length, res$res)

# plot studentized residuals vs. fitted values

> spreadLevelPlot(res) Suggested power transformation: 0.5361655

> par(mfrow=c(1,6)) > plot(data$age, r) > plot(data$temp, r) > plot(data$length, r) > par(mfrow=c(1,1)) > plot(res$fitted, r) > plot(res$fitted, r)

> par(mfrow=c(1,1)) > qqnorm(res$res)

qqline(res$res)

> hist(res$res)

> # test of multicollinearity > vif(res) temp length 1.034884 1.034884 > #non-constant error variance test > ncvTest(res) Non-constant Variance Score Test Variance formula: ~ fitted.values Chisquare = 8.288245 Df = 1 p = 0.003990254 > # Test for Autocorrelated Errors > durbinWatsonTest(res) lag Autocorrelation D-W Statistic p-value 1 0.3505196 1.165927 0.002 Alternative hypothesis: rho != 0 Pairs: > pairs(~age+temp+length,data=data)