Many research designs have only one independent variable that the researchers change over time to see what effect the independent variable has on the dependent variable. An independent variable is simply a factor that the researcher manipulates in order to generate results, the results being called the dependent variable. For example, say a researcher wanted to determine the effect of television viewing on grades. Television viewing would be the independent variable, and the researcher could change that over time with respect to hours viewed per day; the dependent variable would be the grades the students earn as a result of being distracted by television viewing.
However, the above design actually does not fit all research projects, and that's where factorial design comes in. Factorial design permits a scientist to conduct an experiment using more than one independent variable. A factorial design study is essential when it's impossible and impractical to separate corresponding independent variables. One example can be seen in sociology research. Say a sociologist wanted to determine how effective educational methods are. The sociologist would also know that an educational method is not the only contributing factor to learning or lack of learning; instead, a student's socioeconomic background and environment also play contributing factors. Hence, the sociologist would have to conduct the experiment while manipulating the multiple factors of method, environment, and socioeconomic background in order to achieve the most accurate results.
Single subject research designs are a different type of design study. In these designs only one individual test subject can be studied or a group of test subjects that are exactly the same. They also usually require that only one independent variable can be changed to yield results; however, some studies can use two independent variables that are changed to yield results. One example of a single subject design using two independent variables could be a sociologist wanting to compare two different teaching methods on one test subject with a specific disability. The single test subject would be the dependent variable, whereas the two teaching methods would each be one separate independent variable
Hence, if you need to create a study that is both a factorial and a single subject design study, then you would need to a study like the above, but you could not use more than two independent variables.
In a correlation study, we are seeing if there is a correlation between an increase or decrease in one variable and a corresponding increase or decrease in the other variable. Hence, we would either increase or decrease the output of the independent variable to see if there was a decrease or increase in the dependent variable. If you also need to construct a study that is a correlation study, as well as a single subject study and a factorial study, then the last example mentioned above would also fit.
One could construct a single subject study to see how two teaching methods affected one test subject with a specific disability. As one either increased or decreased the output of the two teaching methods, one would see if there is an increase or decrease in the test subject's improved learning behavior.
First of all, a factorial design deals with an experiment with two or more factors. These factors have possible values and the experimental units can usually be a mix of these values. An example of this would be if someone was doing a survey of 4 (a,b,c,d) items that determine a cause, each with a yes or no answer that can be manipulated, the experimenter can manipulate these in order to make the factors independent. Since it is independent and there are 4 factors and there are only two levels or values (yes or no) you would do 2x2x2x2 or 2^4=16. This means you will have 16 different experimental conditions. A single study or a single subject research is when the subject is their own control instead of another group. I am not sure if this would be a proper example, but I believe an example would be if a person takes part in a program that leads them to evaluate how well they respond based on their interactions with the program. Finally, a correlation design. This simply determines if two variables are correlated or not. A correlation study can be done by evaluating a students attendance to the grade they receive and if this correlates. By collecting data and viewing if student's with higher attendance receive better grades, a correlation can be found.
I hope this was helpful!
Source: Have taken psychology