Question

Question: Diet of ducks bred for broiling. Corn is high in starch content; consequently, it is considered excellent feed for domestic chickens. Does corn possess the same potential in feeding ducks bred for broiling? This was the subject of research published in Animal Feed Science and Technology (April 2010). The objective of the study was to establish a prediction model for the true metabolizable energy (TME) of corn regurgitated from ducks. The researchers considered 11 potential predictors of TME: dry matter (DM), crude protein (CP), ether extract (EE), ash (ASH), crude fiber (CF), neutral detergent fiber (NDF), acid detergent fiber (ADF), gross energy (GE), amylose (AM), amylopectin (AP), and amylopectin/amylose (AMAP). Stepwise regression was used to find the best subset of predictors. The final stepwise model yielded the following results:

$\widehat{TME}=7.70+2.14\left(AMAP\right)+0.16\left(NDF\right)$, R² = 0.988, s = .07, Global F p-value = .001

a. Determine the number of t-tests performed in step 1 of the stepwise regression.

b. Determine the number of t-tests performed in step 2 of the stepwise regression.

c. Give a full interpretation of the final stepwise model regression results.

d. Explain why it is dangerous to use the final stepwise model as the “best” model for predicting TME.

e. Using the independent variables selected by the stepwise routine, write a complete second-order model for TME.

f. Refer to part e. How would you determine if the terms in the model that allow for curvature are statistically useful for predicting TME?

Short answer

Answer

a. There are 11 independent variables to be considered for the model. For step 1 of the stepwise regression, 11 1-variable models will be fitted to the data.

b. 10 2-variable models are fitted.

c. The final stepwise model here is which means that only two variables neutral detergent fiber (NDF) and amylopectin/amylose (AMAP) are finalized through the stepwise model. Both the β parameters are positive indicating that the variables have a positive relationship with y.

d. Precautions while using stepwise model - First, an extremely large number of t-tests have been conducted, leading to a high probability of making one or more Type I or Type II errors. Second, the stepwise model does not include any higher-order or interaction terms.

e. A complete second-order model for TME can be written as

$\widehat{TME}={\beta }_0+{\beta }_1\left(AMAP\right)+{\beta }_2\left(NDF\right)+{\beta }_3{\left(AMAP\right)}^2+{\beta }_4{\left(NDF\right)}^2+{\beta }_5\left(AMAP\right)\left(NDF\right)$

f. To check if the terms in the model allow for curvature or not can be done using hypothesis testing where the null and alternate hypothesis would be

H₀:${\beta }_3={\beta }_4=0$and H_a: At least one of the β parameter is nonzero.

Step-by-step solution

Step 1 of stepwise regression

There are 11 independent variables to be considered for the model. For step 1 of the stepwise regression, 11 1-variable models will be fitted to the data.

Step 2 of stepwise regression

Since there are 11 independent variables, (k-1) no of models are 2-variable models are fitted in step 2 of stepwise regression.

So, 10 2-variable models are fitted.

Final stepwise model

The final stepwise model here is $\widehat{TME}=7.70+2.14\left(AMAP\right)+0.16\left(NDF\right)$which means that only two variables neutral detergent fiber (NDF) and amylopectin/amylose (AMAP) are finalized through the stepwise model. Both the β parameters are positive indicating that the variables have a positive relationship with y.

Precautions while using stepwise model

Precautions while using stepwise model -

First, an extremely large number of t-tests have been conducted, leading to a high probability of making one or more Type I or Type II errors. Second, the stepwise model does not include any higher-order or interaction terms.

Complete second-order model

A complete second-order model for TME can be written as$\widehat{TME}={\beta }_0+{\beta }_1\left(AMAP\right)+{\beta }_2\left(NDF\right)+{\beta }_3{\left(AMAP\right)}^2+{\beta }_4{\left(NDF\right)}^2+{\beta }_5\left(AMAP\right)\left(NDF\right)$

Hypothesis testing

To check if the terms in the model allow for curvature or not can be done using hypothesis testing where the null and alternate hypothesis would be

H₀:and H_a: ${\beta }_3={\beta }_4=0$At least one of the β parameter is nonzero.