How many factors are in a 2x2 design?

Description Graphical representation of a two-level design with 3 factors Consider the two-level, full factorial design for three factors, namely the 23 design. This implies eight runs (not counting replications or center point runs). Graphically, we can represent the 23 design by the cube shown in Figure 3.1. The arrows show the direction of increase of the factors. The numbers `1' through `8' at the corners of the design box reference the `Standard Order' of runs (see Figure 3.1).

FIGURE 3.1  A 23 two-level, full factorial design; factors X1, X2, X3

The design matrix In tabular form, this design is given by:

TABLE 3.3  A 23 two-level, full factorial design table showing runs in `Standard Order'runX1X2X31-1-1-121-1-13-11-1411-15-1-1161-117-1118111

The left-most column of Table 3.3, numbers 1 through 8, specifies a (non-randomized) run order called the `Standard Order.' These numbers are also shown in Figure 3.1. For example, run 1 is made at the `low' setting of all three factors. Standard Order for a 2k Level Factorial Design Rule for writing a 2k full factorial in "standard order" We can readily generalize the 23 standard order matrix to a 2-level full factorial with k factors. The first (X1) column starts with -1 and alternates in sign for all 2k runs. The second (X2) column starts with -1 repeated twice, then alternates with 2 in a row of the opposite sign until all 2k places are filled. The third (X3) column starts with -1 repeated 4 times, then 4 repeats of +1's and so on. In general, the i-th column (Xi) starts with 2i-1 repeats of -1 folowed by 2i-1 repeats of +1. Example of a 23 Experiment Analysis matrix for the 3-factor complete factorial An engineering experiment called for running three factors; namely, Pressure (factor X1), Table speed (factor X2) and Down force (factor X3), each at a `high' and `low' setting, on a production tool to determine which had the greatest effect on product uniformity. Two replications were run at each setting. A (full factorial) 23 design with 2 replications calls for 8*2=16 runs.TABLE 3.4 Model or Analysis Matrix for a 23 ExperimentModel MatrixResponse
VariablesIX1X2X1*X2X3X1*X3X2*X3X1*X2*X3Rep
1Rep
2+1-1-1+1-1+1+1-1-3-1+1+1-1-1-1-1+1+1 0-1+1-1+1-1-1+1-1+1-1 0+1+1+1+1-1-1-1-1+2+3+1-1-1+1+1-1-1+1-1 0+1+1-1-1+1+1-1-1+2+1+1-1+1-1+1-1+1-1+1+1+1+1+1+1+1+1+1+1+6+5

The block with the 1's and -1's is called the Model Matrix or the Analysis Matrix. The table formed by the columns X1, X2 and X3 is called the Design Table or Design Matrix.

Orthogonality Properties of Analysis Matrices for 2-Factor Experiments Eliminate correlation between estimates of main effects and interactions When all factors have been coded so that the high value is "1" and the low value is "-1", the design matrix for any full (or suitably chosen fractional) factorial experiment has columns that are all pairwise orthogonal and all the columns (except the "I" column) sum to 0.

The orthogonality property is important because it eliminates correlation between the estimates of the main effects and interactions.

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Highlights

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  In simulation research, we are often interested in comparing the effects of more than one independent variable.

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  Factorial designs allow investigators to efficiently compare multiple independent variables (also known as factors).

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  An example and resources are described for using a two by two factorial design in simulation research.

Keywords: Simulation, Methods, Factorial design, ANOVA

Key Points

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  Simulation researchers are often interested in the effects of multiple independent variables.

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  Factorial designs allow investigators to efficiently examine multiple independent variables (also known as factors).

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  Factorial designs allow investigators to examine both main and interaction effects.

In simulation research, we are often interested in comparing the effects of more than one independent variable. For example, during the COVID-19 pandemic, educators experimented with the use of different synchronous communication platforms (e.g., online video conferencing and text chat) for debriefing after virtual simulation experiences. To contribute to the literature comparing the effects (participant outcomes) of these different synchronous communication platforms and the ideal group size when using these platforms for debriefing after virtual simulation experiences, we may want to design a research study. In this study, we want to examine the effects of both the type of communication platform and the ideal group size, and to understand the interactions between platform and group size on the dependent variables (participant outcomes).

It would be helpful to compare the effect of the two different synchronous communication platforms used for debriefing virtual simulation experiences on selected outcomes such as learning and learner satisfaction. Similarly, it would be helpful to compare the effect of two different group sizes, perhaps two to four students versus 10 to 12 students per group, during these debriefing sessions. It might also be interesting to identify if there are any interaction effects between these two variables: the type of communication platform and group size. For example, we could ask the following questions:

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  Are learning and learner satisfaction higher in the small group for those participants using online text chat but equal across group size for those using video conferencing?

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  Are learning and learner satisfaction consistently higher in the large groups regardless of which synchronous communication platform is used?

The question is how can we do all these comparisons in one experiment? Factorial designs allow investigators to efficiently compare multiple independent variables (also known as factors). The scenario described previously represents a two-by-two factorial design where synchronous communication platform has two conditions or possible options: online video conferencing and text chat. Group size also has two conditions: small (two to four participants) and large (10-12 participants). This creates a total of four conditions: small group online video conferencing (A), small group text chat (B), large group video conferencing (C), and large group text chat (D). We illustrate all four conditions using a table like this one.

Online Video ConferencingText ChatSmall group (2-4 participants)ABLarge group (10-12 participants)CD

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Two-way analysis of variance (ANOVA) can be used to assess the main effects of group size and synchronous communication platform as well as possible interactions between these variables. Additional analyses can also be completed to explore whether there is a cut-point at which group size is optimized. It is also possible to include a comparison of an additional group size, perhaps 18 to 20 participants, or an additional synchronous communication platform such as voice over internet protocol. However, each additional variable decreases the sample size in each box and makes the statistical analysis potentially less reliable. If the total sample size for the example in the table was 100 learners, we would have approximately 25 participants in each condition for comparison (100 learners/4 conditions = 25 learners per condition). However, if we added another group size and another synchronous communication platform, we would dilute that sample size over nine conditions and only have 11 or 12 participants in each condition for comparison (100 learners/9 conditions = 11.11 learners per condition).

Factorial designs are a simple, yet elegant, way of comparing the main effects of multiple independent variables and exploring possible interaction effects. We hope this example of a two-by-two factorial design will inspire you to efficiently compare the effects of two variables, each with two conditions, on simulation outcomes. Happy researching!

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