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But including multiple independent variables also allows the researcher to answer questions about whether the effect of one independent variable depends on the level of another. This is referred to as an interaction between the independent variables. As we will see, interactions are often among the most interesting results in psychological research. Notice that the number of possible conditions is the product of the numbers of levels. A 2 × 2 factorial design has four conditions, a 3 × 2 factorial design has six conditions, a 4 × 5 factorial design would have 20 conditions, and so on. Also notice that each number in the notation represents one factor, one independent variable.
2.1: Example with Main Effects and Interactions
Such an experiment allows the investigator to study the effect of each factor on the response variable, as well as the effects of interactions between factors on the response variable. We might also say an interaction occurs when the difference between the differences are different! There was a difference in spot-the-difference performance between the distraction and no-distraction condition, this is called the distraction effect (it is a difference measure). The reward manipulation changed the size of the distraction effect, that means there was difference in the size of the distraction effect. The distraction effect is itself a measure of differences.
2.3. Assigning Participants to Conditions¶
For information about these designs, please refer to the "Help" menu. The following Yates algorithm table using the data for the null outcome was constructed. As seen in the table, the values of the main total factorial effect are 0 for A, B, and AB. This proves that neither dosage or age have any effect on percentage of seizures. The quantities b1, b2, and so on are regression weights that indicate how large a contribution an independent variable makes, on average, to the dependent variable. Specifically, they indicate how much the dependent variable changes for each one-unit change in the independent variable.
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You are given the following table that relates the combination of these factors and the students scores over the course of a semester. Use the Yates method in order to determine the effect each variable on the students performance in the course. 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. In the main "Create Factorial Design" menu, click "OK" once all specifications are complete. The following table is obtained for a 2-level, 4 factor, full factorial design.
In our example above we showed you two bar graphs of the very same means for our 2x2 design. Even though the graphs plot identical means, they look different, so they are more or less easy to interpret by looking at them. Results from 2x2 designs are also often plotted with line graphs. There are four different graphs in Figure 9.7, using bars and lines to plot the very same means from before. We are showing you this so that you realize how you graph your data matters because it makes it more or less easy for people to understand the results. Also, how the data is plotted matters for what you need to look at to interpret the results.
To do this, we collapse, or average over the observations in the hat conditions. For example, looking only at the no shoes vs. shoes conditions we see the following averages for each subject. Regardless of whether the design is between subjects, within subjects, or mixed, the actual assignment of participants to conditions or orders of conditions is typically done randomly. The two contrast vectors for A depend only on the level of factor A.
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In any case, your mom has to consider both the fertilizer type and amount of water provided to the plants when determining the proper growing conditions. Make plots to determine the main or interaction effects of each factor. From this table, we can see that there is positive correlation for factors A and C, meaning that more sleep and more studying leads to a better test grade in the class. Factor B, however, has a negative effect, which means that spending time with your significant other leads to a worse test score. The lesson here, therefore, is to spend more time sleeping and studying, and less time with your boyfriend or girlfriend. As seen above, RPM is shown with a positive effect for number of theoretical stages, but a negative effect for wt% methanol in biodiesel.
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We present a slightly different way of thinking about the usefulness of factorial designs, and we think it is so important, it get’s its own section. Anytime all of the levels of each IV in a design are fully crossed, so that they all occur for each level of every other IV, we can say the design is a fully factorial design. Like Pareto plots, Half Normal plots show which factors have significant effects on the responses. The factors that have significant effects are shown in red and the ones without significant effects are shown in black. The further a factor is from the blue line, the more significant effect it has on the corresponding response. For wt% methanol in biodiesel, RPM is further from the blue line than pressure, which indicates that RPM has a more significant effect on wt% methanol in biodiesel than pressure does.
2. Multiple Independent Variables¶
The first run (as specified by the random run order) should be performed at the low levels of A and C and the high levels of B and D. The following Yates algorithm table using the data from second two graphs of the main effects section was constructed. Besides the first row in the table, the main total effect value was 10 for factor A and 20 for factor B. This means that both age and dosage affect percentage seizures. However, since the value for B is larger, dosage has a larger effect on percentage of seizures than age.
Replicates are repeats of each trial that help determine the reproducibility of the design, thus increasing the number of trials and accuracy of the DOE. To add replicates, click the "Replicate design" radio button in the "Modify Design" menu. In one study, potentiation of the startle reflex was measured during a low or high probability of receiving an electric shock.
Here are the results from the 2x2 repeated-measures ANOVA, using the aov function in R. The main effect of distraction compares the overall means for all scores in the no-distraction and distraction conditions, collapsing over the reward conditions. We find that the interaction concept is one of the most confusing concepts for factorial designs.
Although the researchers could have treated each of the seven ratings as a separate dependent variable, these researchers combined them into a single dependent variable by computing their mean. Often a researcher wants to know how an independent variable affects several distinct dependent variables. 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).
The only thing you are manipulating is the amount of coffee. To do another, second manipulation, you need to additionally manipulate something that is not coffee (like time of day in our previous example). These equations can be used as a predictive model to determine wt% methanol in biodiesel and number of theoretical stages achieved at different operating conditions without actually performing the experiments. However, the limits of the model should be tested before the model is used to predict responses at many different operating conditions.
Nevertheless, maybe the next time you lose your keys, you should stand up and look for them, rather than sitting down and not look for them. The design of the study was a 2x2 repeated-measures design. We will discuss some research designs, and the ANOVAs that are appropriate for their analysis. We will conduct the ANOVAs using R, and print out the ANOVA tables. This is what you do in the lab, and what most researchers do.
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Many researchers argue that the Stroop effect measures something about selective attention, the ability to ignore distracting information. If this theory is true, Dr. MO is very bad at ignoring distracting information! In the Stroop task, the target information that you need to pay attention to is the color of the word, not the word itself.
Instead, if we have done the calculations of the \(SS\)es correctly, they should be same as what we would get if we used R to calculate the \(SS\)es. Let’s make R do the work, and then compare to check our work. Even though this textbook meant to explain things in a step by step way, we guess you are tired from watching us work out the 2x2 ANOVA by hand. We have already shown you how to compute the SS for error before, so we will not do the full example here. Instead, we solve for SS Error using the numbers we have already obtained.
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