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Question: Estimating change-point dosage. A standard method for studying toxic substances and their effects on humans is to observe the responses of rodents exposed to various doses of the substance over time. In the Journal of Agricultural, Biological, and Environmental Statistics (June 2005), researchers used least squares regression to estimate the change-point dosage—defined as the largest dose level that has no adverse effects. Data were obtained from a dose-response study of rats exposed to the toxic substance aconiazide. A sample of 50 rats was evenly divided into five dosage groups: 0, 100, 200, 500, and 750 milligrams per kilogram of body weight. The dependent variable y measured was the weight change (in grams) after a 2-week exposure. The researchers fit the quadratic model $E\left(y\right)={\mathrm{β}}_0+{\mathrm{β}}_{1}x+{\mathrm{β}}_2{x}^2$, where x = dosage level, with the following results:

$\hat{\mathbf{y}}\mathbf{=}\mathbf{=}\mathbf{10}\mathbf{.}\mathbf{25}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{0053}\mathit{x}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{0000266}{\mathit{x}}^{\mathbf{2}}$

1. Construct a rough sketch of the least square’s prediction equation. Describe the nature of the curvature in the estimated model.
2. Estimate the weight change (y) for a rat given a dosage of 500 mg/kg of aconiazide.
3. Estimate the weight change (y) for a rat given a dosage of 0 mg/kg of aconiazide. (This dosage is called the control dosage level.)
4. Of the five dosage groups in the study, find the largest dosage level x that yields an estimated weight change that is closest to but below the estimated weight change for the control group. This value is the change-point dosage.

Step by step solution

01

Given Information

A sample of 50 rats was drawn which was evenly divided into five dosage groups: 0, 100, 200, 500, and 750 milligrams per kilogram of body weight. The quadratic model is given as$E\left(y\right)={\mathrm{β}}_0+{\mathrm{β}}_{1}x+{\mathrm{β}}_2{x}^2$ where y is the dependent variable (here, weight change after two-week exposure) and x is theindependent variable (here, dosage level) and the estimated model is given as$\hat{y}=10.25+0.0053x-0.0000266x^2$

02

Graph for least square prediction equation 

a.

The given equation:
$\hat{y}=10.25+0.0053x-0.0000266x^2$

Putting the value of x (dosage level) in the above equation we will get 5 different values of y (the weight change).Using these values the graph below is drawn.

Here the graph for the x² prediction equation will be a curvature to account for the term and since the sign of coefficient of x² term is negative the curvature will be downward sloping curve.

03

Estimation for y

b.

The weight change for a rat given a dosage of x = 500 gm is:

$$\begin{gathered} \hat{y}=10.25+0.0053x-0.0000266x^2 \\ \hat{y}=10.25+0.0053\text{(}500\text{)}-0.0000266{\text{(}500\text{)}}^2\hat{y}=6.25 \end{gathered}$$

So, the change in weight is 6.25 gm.

04

Control dosage level

The weight change for a rat given a dosage of x = 0 gm is

$$\begin{gathered} \hat{y}=10.25+0.0053x-0.0000266x^2 \\ \hat{y}=10.25+0.0053\text{(}0\text{)}-0.0000266{\text{(}0\text{)}}^2\hat{y}=10.25 \end{gathered}$$

Hence, the change in weight is 10.25 gm.

05

Change-point dosage level

d.

For the dosage level, x = 99 , the weight change for a rat is:

$$\begin{gathered} \hat{y}=10.25+0.0053x-0.0000266x^2 \\ \hat{y}=10.25+0.0053\text{(}99\text{)}-0.0000266{\text{(}99\text{)}}^2\hat{y}=10.51399 \end{gathered}$$

For x = 100 (control group) the weight change is 10.514.

Therefore, x = 99 is the change point level of dosage since the weight change (10.513) is also closest but less than the estimated weight change for the control group.

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