1. Using the data in the ‘Part 1’ sheet, select two variables and conduct a two-way ANOVA. Justify your selection and discuss how this model could be used to identify sources of low production rate. Then, identify a third variable that you believe should be treated as a nuisance variable and create a blocked ANOVA to control for the effect of this variable.

QUESTION

1. Using the data in the ‘Part 1’ sheet, select two variables and conduct a two-way ANOVA. Justify your selection and discuss how this model could be used to identify sources of low production rate. Then, identify a third variable that you believe should be treated as a nuisance variable and create a blocked ANOVA to control for the effect of this variable.

The data in the ‘Part 2’ sheet has been limited to four observations per treatment combination. Using this data, create a 2k factorial model by restructuring the Supplier factor as two, two-level factors and treating Line as a four- level block. Use Minitab to generate a 25-2 model with one block and extract the relevant data from the provided dataset to complete the analysis. Discuss the validity and fit of the model and interpret the results including addressing aliased relationships.

Identify the treatment combinations that would be needed to complete the full fold-over for this design and briefly discuss how the fold-over model would help to interpret the results from Part 2. Complete the full fold-over, discuss the validity and fit of the model, and interpret the results.

ANSWER

Two-Way ANOVA

To conduct a two-way ANOVA, you need to select two variables that you believe may affect the production rate. These variables should be categorical or factors. For example, you might choose “Machine Type” and “Shift Time” as two variables to investigate their impact on production rate.

By conducting a two-way ANOVA, you can analyze how these variables individually and interactively affect the production rate. The main effects of each variable will reveal their independent contributions, while the interaction effect will indicate if the combined effect of the variables is significant.

This model can help identify sources of low production rate by examining the significance of the main effects and the interaction effect (Fujikoshi, 1993). If the main effects of either variable are significant, it suggests that the specific factor has a significant influence on the production rate. Additionally, if the interaction effect is significant, it implies that the combination of these factors has a synergistic or detrimental effect on production rate.

To control for a third variable that may confound the analysis, you can use a blocked ANOVA design. A nuisance variable could be something like “Operator” or “Production Line,” which might have an unintended impact on the production rate. By blocking this variable, you ensure that its influence is accounted for and controlled.

2k Factorial Model

A 2k factorial design is used to investigate the main effects and interaction effects of k factors, where each factor has two levels (e.g., high and low). In this case, you mentioned restructuring the “Supplier” factor into two, two-level factors. Additionally, you treat the “Line” factor as a four-level block (Causal Inference From 2          K          Factorial Designs by Using Potential Outcomes on JSTOR, n.d.).

To generate a 25-2 model, you would have five factors (2k) with two levels each, resulting in a total of 32 treatment combinations. However, as the data in the ‘Part 2’ sheet is limited to four observations per treatment combination, you’ll need to use a fractional factorial design to select a subset of these treatment combinations for analysis.

Using Minitab, you can create a fractional factorial design with the appropriate resolution (e.g., a 2^5-2 design). The resulting model will allow you to estimate the main effects, interaction effects, and potential aliasing among the factors.

The validity and fit of the model can be assessed by analyzing the significance of the effects and examining the goodness-of-fit measures, such as the R-squared value. It’s crucial to address any aliased relationships, which occur when some effects cannot be uniquely estimated due to the design’s confounding structure.

Interpreting the results involves examining the significance of the main effects and interactions to determine which factors have a significant impact on the response variable. Additionally, assessing the aliasing patterns can help identify potential confounding effects that may require further investigation.

 Full Fold-Over and Interpretation

The full fold-over is a technique used to resolve aliasing in fractional factorial designs. It involves conducting additional experiments to complete the missing treatment combinations and obtain unique estimates for all effects.

To complete the full fold-over, you would need to run additional experiments to cover the remaining treatment combinations that were not included in the initial fractional design. By doing so, you obtain a balanced design that allows for unbiased estimation of all effects  (Goldfarb et al., 2002).

The fold-over model helps interpret the results by providing more accurate estimates of the effects and reducing the confounding among factors. With the complete data set, you can analyze the main effects and interactions without the limitations of confounding, leading to more precise conclusions about the factors’ impact on the response variable.

When interpreting the results from the full fold-over model, you can assess the statistical significance of the effects, evaluate their magnitude, and consider any interactions among factors. This information can guide decision-making, process optimization, and identifying factors that contribute the most to the response variable.

In summary, conducting a two-way ANOVA allows you to analyze the impact of two variables on the production rate and identify potential sources of low production. A blocked ANOVA helps control for nuisance variables. In the 2k factorial design, restructuring factors and considering blocks enable investigation of main effects and interactions. Completing the full fold-over resolves aliasing, leading to more accurate interpretations of the results.

References

Causal inference from 2          K          factorial designs by using potential outcomes on JSTOR. (n.d.). https://www.jstor.org/stable/24775307 

Fujikoshi, Y. (1993). Two-way ANOVA models with unbalanced data. Discrete Mathematics, 116(1–3), 315–334. https://doi.org/10.1016/0012-365x(93)90410-u 

Goldfarb, J. W., & Shinnar, M. (2002). Field-of-view restrictions for artifact-free SENSE imaging. In Proceedings of the 10th Annual Meeting of the ISMRM.https://cds.ismrm.org/ismrm-2002/PDF9/2412.PDF