Question

The tuna fish data from Exercise 11.16 were analyzed as a completely randomized design with four treatments. However, we could also view the experimental design as a \(2 \times 2\) factorial experiment with unequal replications. The data are shown below. \begin{array}{l|cc|rr} & {\text { 0il }} && {\text { Water }} \\ \hline \text { Light Tuna } & 2.56 & .62 & .99 & 1.12 \\ & 1.92 & .66 & 1.92 & .63 \\ & 1.30 & .62 & 1.23 & .67 \\ & 1.79 & .65 & .85 & .69 \\ & 1.23 & .60 & .65 & .60 \\ & & .67 & .53 & .60 \\ & & & 1.41 & .66 \\ & & & 1.49 & 1.29 \\ & & .27 & & 1.29 & 1.00 \\ & 1.22 & & 1.27 & 1.27 \\ & 1.19 & & 1.35 & 1.28 \\ \hline \text { White Tuna } & 1.22 & & 1.35 & \\ & & & \end{array} The data can be analyzed using the model $$ y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{1} x_{2}+\epsilon $$ where \(x_{1}=0\) if oil, 1 if water \(x_{2}=0\) if light tuna, 1 if white tuna a. Show how you would enter the data into a computer spreadsheet, entering the data into columns for \(y, x_{1}, x_{2},\) and \(x_{1} x_{2}\) b. The printout generated by MINITAB is shown below. What is the least-squares prediction equation? c. Is there an interaction between type of tuna and type of packing liquid? d. Which, if any, of the main effects (type of tuna and type of packing liquid) contribute significant information for the prediction of \(y ?\) e. How well does the model fit the data? Explain.

Short answer

#Answer# In the study of tuna fish using a factorial ANOVA 2x2 design, the x_1 variable represents the type of packing liquid (0 for oil and 1 for water) and the x_2 variable represents the type of tuna (0 for light tuna and 1 for white tuna).

Step-by-step solution

Creating the data columns

Using the data provided, we will create the following columns: \(y, x_1, x_2,\) and \(x_{1}x_{2}\). The values for \(y\) represent the given data, while the values for \(x_1\), \(x_2\), and \(x_{1}x_{2}\) are derived based on the definitions provided in the exercise: x_{1} = 0 if oil, 1 if water x_{2} = 0 if light tuna, 1 if white tuna Here's the created spreadsheet: \begin{array}{c|c|c|c} y & x_{1} & x_{2} & x_{1}.x_{2} \\ \hline 2.56 & 0 & 0 & 0 \\ 1.92 & 0 & 0 & 0 \\ 1.30 & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots \\ 1.35 & 1 & 1 & 1 \\ \end{array} b. The printout generated by MINITAB is missing from the provided information. However, to answer this question, find the coefficients for each term in the equation from the MINITAB output. c. To check for interactions between the type of tuna and type of packing liquid, look at the p-value for the interaction term (x_{1}x_{2}) in the MINITAB output. If the p-value is less than a predetermined significance level (e.g., 0.05), then there is a significant interaction present. d. To determine the main effects of type of tuna and type of packing liquid, look at the p-values for x_{1} and x_{2} in the MINITAB output. If the individual p-values are less than a predetermined significance level (e.g., 0.05) for each factor, then these main effects contribute significant information for the prediction of y. e. To check how well the model fits the data, we generally look at the R-squared value provided in the MINITAB output. If the R-squared value is close to 1, it indicates that the model fits the data well, whereas a value closer to 0 indicates a poor fit.

Key concepts

ANOVA

ANOVA, or Analysis of Variance, is a statistical method used to determine if there are significant differences between the means of different groups. In the context of a factorial experiment, such as in the given tuna fish data example, ANOVA helps to identify whether the different treatments (type of tuna and type of packing liquid) have significant impacts on the response variable, which in this case is the measured outcomes (y-values).

When conducting ANOVA for a factorial experiment, we break down the total variation observed in the data into components attributed to different factors and their interactions. This analysis allows us to assess whether the observed variations are due to the treatment applied or merely random noise. By comparing the group means using ANOVA, we can infer whether changing the type of tuna or the packing liquid (main effects), or their combination (interaction effects), creates significant differences in results, leading to better understanding and optimization of these factors.

Interaction Effects

Interaction effects occur when the effect of one factor differs depending on the level of another factor. In a factorial experiment, these interactions reveal insights about whether the variables studied together produce effects that are different from those expected based on their individual contributions alone.

In our example with tuna fish, interaction effects between the type of tuna (light or white) and the packing liquid (oil or water) explore whether the combination of these variables impacts the y-values more than they would independently. For instance, if the interaction term in your statistical model \( \beta_{3} x_{1} x_{2} \) is significant, it indicates that the combination of tuna type and packing liquid results in a unique effect not predictable from the individual effects alone.

This concept is crucial, as it highlights situations where one variable modifies how another variable influences the outcome, offering a nuanced understanding of the dynamics at play. Identifying significant interaction effects allows researchers and practitioners to pinpoint scenarios where specific combinations should be further investigated or optimized.

Main Effects

Main effects are the individual impacts that each factor in a study has on the response variable. In factorial experiments, analyzing main effects allows us to understand how each factor, independently of the others, influences the outcome.

For the tuna fish data, assessing main effects involves looking at how the type of tuna affects the y-values on average and separately analyzing how the type of packing liquid influences these values as well. To evaluate the main effects, we examine the statistical significance of factors like \( x_{1} \) (packing liquid type) and \( x_{2} \) (tuna type) individually in the regression model.

A significant main effect indicates that the factor has a meaningful impact on the response variable. Discovering these effects is essential for researchers and practitioners to make informed decisions about which factors should be prioritized or adjusted to enhance the outcome measures in their experiments.

Model Fit

Model fit refers to how well a statistical model represents the observed data. It is an assessment of the ability of a model to accurately capture the underlying data dynamics.

To evaluate the fit of the model in our factorial experiment, we primarily use metrics like the R-squared value. This value indicates the proportion of variance in the response variable that is explained by the model. A higher R-squared value, close to 1, suggests that the model accounts for a large portion of the variability in the data, indicating a good fit.

When R-squared values are low, it suggests that the model may not be capturing the necessary patterns in the data well, and further refinements or additional factors might need to be considered. Assessing model fit helps ensure that the conclusions drawn from the analysis are reliable and can inform decision-making processes effectively.