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An experiment was conducted in order to study the size of squid eaten by sharks and tuna. The regressor variables are characteristics of the beak or mouth of the squid. The regressor variables and response considered for the study are: x_1 Rostral (beak) length in inches x_2 Wing length in inches x_3 Rostral to notch length x_4 Notch to wing length x_5 Width in inches y Weight in pounds The data are (a) Plot weight against any four of the other variables that you think might be strongly related to the weight of squid. Comment on the relationships you observe. (b) Select a model using forward regression. Give the final fitted model. Explain the forward regression process. (c) Select a model using backward elimination. Give the final fitted model. Explain the backward regression process. (d) Exclude x_2 from the possible predictors and compute adjusted R^2, PRESS, AIC and C_m for all possible models (except the model with only the constant term). On the basis of this information, reduce the pool of models from 15 to a small subset for further consideration. Select what you think is the best model, justifying your choice. Give the final fitted model.

Explanation / Answer

a. We can done by R-software

x=read.csv("Book1.csv")
par(mfrow=c(2,2))
plot(x[,1],x[,6],main="Rostral vs weight")
plot(x[,2],x[,6],main="wing vs weight")
plot(x[,3],x[,6],main="rostral to notch vs weight")
plot(x[,4],x[,6],main="notch vs weight")

Seeing the figure, see that all variable is independently show a linear relationship between variables.

(b)

min.model = lm(x[,6]~ 1, data=x)
mod<-formula(lm(x[,6]~.,x))
fwd.model = step(min.model, direction='forward', scope=mod)

Start: AIC=52.25
x[, 6] ~ 1

       Df Sum of Sq     RSS      AIC
+ Y     1    215.93   0.000 -1552.64
+ X5    1    204.16 11.767    -9.77
+ X1    1    199.15 16.779    -1.96
+ X3    1    197.35 18.573     0.27
+ X4    1    191.25 24.674     6.52
+ X2    1    190.28 25.645     7.37
<none>              215.925    52.25

Step: AIC=-1552.64
x[, 6] ~ Y

       Df Sum of Sq        RSS     AIC
+ X2    1 2.5302e-30 1.5743e-30 -1571.7
+ X1    1 1.3944e-30 2.7102e-30 -1559.8
+ X4    1 1.2596e-30 2.8449e-30 -1558.7
+ X3    1 9.9061e-31 3.1139e-30 -1556.7
+ X5    1 6.0724e-31 3.4973e-30 -1554.2
<none>               4.1045e-30 -1552.6

Step: AIC=-1571.72
x[, 6] ~ Y + X2

       Df Sum of Sq        RSS     AIC
<none>               1.5743e-30 -1571.7
+ X3    1 2.8557e-32 1.5458e-30 -1570.1
+ X4    1 1.9692e-32 1.5546e-30 -1570.0
+ X5    1 1.3235e-32 1.5611e-30 -1569.9
+ X1    1 1.0408e-33 1.5733e-30 -1569.7

When the AIC get minimum, we make a better suitable model

Forward regression inc;lude one by one variable and see the impact of the y variable

(c)

fullmodel = lm(x[,6]~x[,1]+x[,2]+x[,3]+x[,4]+x[,5] , data=x)
bac.model = step(fullmodel, direction='backward', trace=FALSE)
bac.model

In backward, reduce one by one variable from the full model and get the results.

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