Question

Ascorbic acid reduces goat stress. Refer to the Animal Science Journal (May, 2014) study on the use of ascorbic acid (AA) to reduce stress in goats during transportation from farm to market, Exercise 9.12 (p. 529). Recall that 24 healthy goats were randomly divided into four groups (A, B, C, and D) of six animals each. Goats in group A were administered a dosage of AA 30 minutes prior to transportation; goats in group B were administered a dosage of AA 30 minutes following transportation; group C goats were not given any AA prior to or following transportation; and, goats in group D were not given any AA and were not transported. Weight was measured before and after transportation and the weight loss (in kilograms) determined for each goat.

1. Write a model for mean weight loss, E(y), as a function of the AA dosage group (A, B, C, or D). Use group D as the base level.
2. Interpret the’s in the model, part a.
3. Recall that the researchers discovered that mean weight loss is reduced in goats administered AA compared to goats not given any AA. On the basis of this result, determine the sign (positive or negative) of as many of the’s in the model, part a, as possible.

Short answer

1. A dummy variable model for mean weight loss as a function of the AA dosage group (A, B, C, and D) can be written as $\mathbf{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{3}}$.
2. The value of${\mathbf{\beta }}_0$represents the mean weight loss at the base level, here the base level is represented by group D of the AA dosage level.${\mathbf{\beta }}_1$represents the mean weight loss when the AA dosage group observed in group A.${\mathbf{\beta }}_2$represents the mean weight loss when the AA dosage group observed in group B.${\mathbf{\beta }}_{\mathbf{3}}$represents the mean weight loss when the AA dosage group observed in group C.
3. The sign of${\mathbf{\beta }}_0$ will be positive. However, the sign of ${\mathbf{\beta }}_1$and ${\mathbf{\beta }}_{\mathbf{2}}$will be negative since the researcher discovered that the mean weight loss is reduced in goats who were given AA. The sign of${\mathbf{\beta }}_3$ will also be positive since group C was not given AA dosage.

Step-by-step solution

Dummy variable model

A dummy variable model for mean weight loss as a function of the AA dosage group (A, B, C, and D) can be written as$\mathbf{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{3}}$

Where${\mathbf{x}}_{\mathbf{1}}$ represents the group A of AA dosage

${\mathbf{x}}_{\mathbf{2}}$represents group B of the AA dosage

${\mathbf{x}}_{\mathbf{3}}$ represents group C of the AA dosage

Interpretation of

The value of${\mathbf{\beta }}_0$represents the mean weight loss at base level, here the base level is represented by group D of the AA dosage level.

${\mathbf{\beta }}_{\mathbf{1}}$represents the mean weight loss when the AA dosage group observed in group A.

${\mathbf{\beta }}_{\mathbf{2}}$represents the mean weight loss when the AA dosage group was observed in group B.

${\mathbf{\beta }}_{\mathbf{3}}$represents the mean weight loss when the AA dosage group was observed in group C.

Sign of β

The sign ${\mathbf{\beta }}_0$of will be positive. However, the sign of role="math" localid="1649844743384" ${\mathbf{\beta }}_{\mathbf{1}}$and ${\mathbf{\beta }}_{\mathbf{2}}$ will be negative since the researcher discovered that the mean weight loss is reduced in goats who were given AA. The sign of${\mathbf{\beta }}_3$ will also be positive since group C was not given AA dosage.