Table Of Content
In the table, a yes means that there was statistically significant difference for one of the main effects or interaction, and a no means that there was not a statisically significant difference. As you can see, just by adding one more independent variable, the number of possible outcomes quickly become more complicated. When you conduct a 2x2 design, the task for analysis is to determine which of the 8 possibilites occured, and then explain the patterns for each of the effects that occurred. In the middle panel, independent variable “B” has a stronger effect at level 1 of independent variable “A” than at level 2. This is like the hypothetical driving example where there was a stronger effect of using a cell phone at night than during the day. In the bottom panel, independent variable “B” again has an effect at both levels of independent variable “A”, but the effects are in opposite directions.
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Advantages and disadvantages of factorial experiments
Because experiments from the POD are time consuming, a half fraction design of 8 trial was used. The default factors are named "A", "B", "C", and "D" and have respective high and low levels of 1 and -1. The name of the factors can be changed by simply clicking in the box and typing a new name.
2. Multiple Independent Variables¶
While simple psychology experiments look at how one independent variable affects one dependent variable, researchers often want to know more about the effects of multiple independent variables. Factorial designs require the experimenter to manipulate at least two independent variables. Imagine you are trying to figure out which of two light switches turns on a light. The dependent variable is the light (we measure whether it is on or off). The first independent variable is light switch #1, and it has two levels, up or down. The second independent variable is light switch #2, and it also has two levels, up or down.
Non-Manipulated Independent Variables
Additional modifications to the design include randomizing and renumbering the design. These are very straightforward modifications which affect the ordering of the trials. For information about the "Fold design" and "Add axial points", consult the "Help" menu. The only option in this menu is the number of replicates to add.
Examples of Factorial Designs
If a quadratic effect is expected for a factor, a more complicated experiment should be used, such as a central composite design. Optimization of factors that could have quadratic effects is the primary goal of response surface methodology. This is a nice example to illustrate the purpose of a screening design. You want to test a number of factors to see which ones are important. But B appears not to be important either as a main effect or within any interaction. B was the rate of gas flow across the edging process and it does not seem to be an important factor in this process, at least for the levels of the factor used in the experiment.
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ANOVAs are an amazing statistical tool because they also allow is to compare means for a combination of IVs (not just IV levels). This happens in a factorial design, when each level of each IV is combined so that a set of participants experiences the combination of levels of each IV. In addition, SuperGym offers 4 different workout plans, A through D, none of which are directly catered to any of the different types. Create an experimental factorial design that could be used to test the effects of the different workout plans on the different types of people at the gym. After the complete DOE study has been performed, Minitab can be used to analyze the effect of experimental results (referred to as responses) on the factors specified in the design.
Contrasts, main effects and interactions
As another example, researcher Susan Knasko was interested in how different odors affect people’s behavior [Kna92]. She conducted an experiment in which the independent variable was whether participants were tested in a room with no odor or in one scented with lemon, lavender, or dimethyl sulfide (which has a cabbage-like smell). Although she was primarily interested in how the odors affected people’s creativity, she was also curious about how they affected people’s moods and perceived health—and it was a simple enough matter to measure these dependent variables too. Although she found that creativity was unaffected by the ambient odor, she found that people’s moods were lower in the dimethyl sulfide condition, and that their perceived health was greater in the lemon condition. It would seem almost wasteful to measure a single dependent variable. Even if you are primarily interested in the relationship between an independent variable and one primary dependent variable, there are usually several more questions that you can answer easily by including multiple dependent variables.
1. Multiple Dependent Variables¶
And you picked your low and high level as illustrated above, then you would have missed capturing the true relationship. Your conclusion would probably be that there is no effect of that factor. You need to have some understanding of what your factor is to make a good judgment about where the levels should be. In the end, you want to make sure that you choose levels in the region of that factor where you are actually interested and are somewhat aware of a functional relationship between the factor and the response.
1 - The Simplest Case
Just the form of these variances tells us something about the efficiency of the two-factor design. A benefit of a two factor design is that the marginal means have n × b number of replicates for factor A and n × a for factor B. The factorial structure, when you do not have interactions, gives us the efficiency benefit of having additional replication, the number of observations per cell times the number of levels of the other factor. This benefit arises from factorial experiments rather than single factor experiments with n observations per cell. An alternative design choice could have been to do two one-way experiments, one with a treatments and the other with b treatments, with n observations per cell.
These interaction plots show us the three sets of two-way cell means, each of the three are plotted in two different ways. Where na is the number of observations in each level of factor B. Overall now, we are thinking that our effect of distraction (IV) on spotting the difference (DV) by finding the difference in performance between the two conditions might be affected by more than just being distracted. This may lead us to want to know how to make people better at ignoring distracting things. Our first stab at science found that distraction affected paying attention, but we want to expand on that finding. Maybe now we want to know what makes people worse at ignoring things?
Of course, the researchers could also test, for example, 4 levels of concentration for the additive, and this would give 4 x 4 or 16 tanks, meaning 48 tanks in total. For each combination of time, temperature and operator, there are three observations. Now we have a case where there are three factors and three observations per cell. We can test the hypotheses that the marginal means are all equal, or in terms of the definition of our effects that the \(\alpha_i\)'s are all equal to zero, and the hypothesis that the \(\beta_j\)'s are all equal to zero.
As a further example, the effects of three input variables can be evaluated in eight experimental conditions shown as the corners of a cube. Having included all the terms back into a full model we have shown how to produce a normal plot. Remember that all of these effects are 1 degree of freedom contrasts of the original data, each one of these is a linear combination of the original observations, which are normally distributed with constant variance. Then these 15 linear combinations or contrasts are also normally distributed with some variance. If we assume that none of these effects are significant, the null hypothesis for all of the terms in the model, then we simply have 15 normal random variables, and we will do a normal random variable plot for these.
Factorial experiments can be used when there are more than two levels of each factor. However, the number of experimental runs required for three-level (or more) factorial designs will be considerably greater than for their two-level counterparts. Factorial designs are therefore less attractive if a researcher wishes to consider more than two levels. Let's look at the situation where we have one observation per cell. We need to think about where the variation occurs within this design.
However, the limits of the model should be tested before the model is used to predict responses at many different operating conditions. This main total effect value for each variable or variable combination will be some value that signifies the relationship between the output and the variable. For instance, if your value is positive, then there is a positive relationship between the variable and the output (i.e. as you increase the variable, the output increases as well). Notice, however, that the values are all relative to one another.
You can always spot an interaction in the graphs because when there are lines that are not parallel an interaction is present. If you observe the main effect graphs above, you will notice that all of the lines within a graph are parallel. In contrast, for interaction effect graphs, you will see that the lines are not parallel. If these values represent "low" and "high" settings of a treatment, then it is natural to have 1 represent "high", whether using 0 and 1 or −1 and 1. This is illustrated in the accompanying table for a 2×2 experiment.
It is possible to test more than two factors, but this becomes unwieldy very quickly. Researchers often use factorial designs to understand the causal influences behind the effects they are interested in improving. Effects are the change in a measure (DV) caused by a manipulation (IV levels). In the Graphs menu shown above, the three effects plots for "Normal", "Half Normal", and "Pareto" were selected. These plots are different ways to present the statistical results of the analysis.
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