Interaction between two continuous variables

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Statistical programs, like SPSS, do not always have "point-and-click" commands for every possible statistical test. This page is a description of how to test the interaction between two continuous variables. Below, an [[#What is an interaction? | explanation of interactions]] is presented,  then the '''[[#Three Steps | three steps to conduct the interaction]]''' is described, and examples are given to help in understanding the steps involved.
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Statistical programs, like SPSS, do not always have "point-and-click" commands for every possible statistical test. This page is a description of how to test the interaction between two continuous variables. Three approaches are described below:<br>
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(1) '''[[#Three Steps Using SPSS | three steps to conduct the interaction using commands within SPSS]]''', and<br>
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(2) '''[[#Interaction! software | Interaction! software]]''' by Daniel S. Soper that performs statistical analysis and graphics for interactions between dichotomous, categorical, and continuous variables.<br>
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(3)  '''[[#R commands | R commands]]''' for executing the analysis.
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==What is an interaction?==
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<nowiki>*</nowiki>For a description of what is an interaction and main effects, please see the accompanying page about [[What is an Interaction?]].
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*Interactions are when the effect of two, or more, variables is not simply additive. This page describes the interaction between two variables. It is possible to examine the interactions of three or more variables but this is beyond the scope of this page.
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*Example of interaction - One possible interaction is the effect of energy bars and energy drinks on time to run the 1500 meters. The quantity of energy bars and energy drinks represent two variables. The dependent variable is the time taken to run 1500 meters.
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*# ''Example 1'' - An interaction occurs if running speed improves by more than just the additive effect of having either an energy bar or an energy drink. For example, imagine eating a certain amount of energy bars increases running speed by 5 seconds, and drinking energy drinks increases running speed by 3 seconds. An interaction occurs if the joint effect of energy bars and energy drinks increases running speed by more than 8 seconds, such as liquid in the drink amplifying the ability to digest the energy in the bar leading to faster times.
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*# ''Example 2'' - Another example of an interaction effect would be if running time worsened by the joint effect of energy bars and energy drinks -- perhaps the person feels bloated from eating ''and'' drinking and so are unable to run quickly.
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*# ''Example 3'' - A third and final example of an interaction is that alone neither variable may have an effect on running speed, such as imagining that an energy bar by itself, or an energy drink by itself, is unable to increase running speed. But, there might be an interaction effect that influences running speed when you eat the bar ''and'' drink the drink, such as the energy bar having a chemical that unleashes the power of the energy drink to increase running speed.
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*For those more technically minded, here is the algebra. An interaction effect reflects the effect of the interaction controlling for the two predictors themselves.
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*#In the following examples:
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*#:energy bar = X1,
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*#:energy drink = X2
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*#:the interaction = X1*X2,
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*#:Y = running speed
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*#Here is the formula for: Running speed = intercept + b1energu drink + b2energy bar + b3(bar * drink) + ei
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*#:Y<sub>''i''</sub> = ''b''<sub>0</sub> + ''b''<sub>1</sub>X1<sub>''i''</sub> + ''b''<sub>2</sub>X2<sub>''i''</sub> + ''b''<sub>3</sub>(X1<sub>''i''</sub> X2<sub>''i''</sub>) + ''e<sub>i</sub>''
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*#This formula can be rewritten as
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*#:Y<sub>''i''</sub> = (''b''<sub>0</sub> + ''b''<sub>2</sub>X<sub>2''i''</sub>) + (''b''<sub>1</sub>+ ''b''<sub>3</sub>X<sub>2''i''</sub>) X<sub>1''i''</sub> + ''e<sub>i</sub>''
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*#:where (''b''<sub>1</sub>+ ''b''<sub>3</sub>X<sub>2''i''</sub>) represents the effect of X<sub>1</sub> on Y at specific levels of X<sub>2</sub>
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*#:and ''b<sub>3</sub>'' indicates how much the slope of X<sub>1</sub> changes as X<sub>2</sub> goes up or down one unit.
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*#It is then possible to factor out X<sub>2</sub>
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*#:Y<sub>''i''</sub> = (''b''<sub>0</sub> + ''b''<sub>1</sub>X<sub>1''i''</sub>) + (''b''<sub>2</sub>+ ''b''<sub>3</sub>X<sub>1''i''</sub>) X<sub>2''i''</sub> + ''e<sub>i</sub>''
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*#:where (''b''<sub>2</sub>+ ''b''<sub>3</sub>X<sub>1''i''</sub>) represents the effect of X<sub>2</sub> on Y at specific levels of X<sub>1</sub>
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*#:and ''b<sub>3</sub>'' indicates how much the slope of X<sub>2</sub> changes as X<sub>2</sub> goes up or down one unit.
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==Three Steps==
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__TOC__
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==Three Steps using SPSS==
There are three steps involved to calculate the interaction between two continuous variables.
There are three steps involved to calculate the interaction between two continuous variables.
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#'''Center''' the two continuous variables
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===► '''Center''' the two continuous variables===
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#*To increase interpretability of interactions numerous researchers (e.g. [[Aiken and West, 1991]]; [[Judd and McClelland, 1989]]) have recommended centering the predictor variables (X1 and X2). If the variables are not centered there are possible problems with multicolinearity, which means that if the IVs are not centered their product (used in computing the interaction) is highly correlated with the original IV.  
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*Why center the variables?
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#*You center the continuous variables by subtracting the mean score from each data-point. In other words, use SPSS, or another statistical program, to find the mean value of the variable. Then, use the "Compute" command in SPSS to create a new variable that is the original values minus the mean. Then, repeat the procedure for the second variable.
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*#To increase interpretability of interactions numerous researchers (e.g. [[Aiken and West, 1991]]; [[Judd and McClelland, 1989]]) have recommended centering the predictor variables (X1 and X2).  
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#*As a concrete example, imagine you have 200 subjects (N=200) for which you have their IQ score and the length of time they studied for an exam. Thus, there are two continuous variables (X1=IQ, X2=time spent studying), and your dependent variable is the test score (Y=test score).
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*#If the variables are not centered there are possible problems with multicolinearity, which means that if the IVs are not centered their product (used in computing the interaction) is highly correlated with the original IV.  
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#*Imagine that the average IQ score is 100. To center the IQ variable, 100 needs to be subtracted from every every subject's IQ score. So if a subject has an IQ of 115, their centered IQ score is 15. If a subject has an IQ of 90, their centered IQ score is -10. For easy reference, lets called the newly centered IQ score as "IQ_c".
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*How to center the variables?
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#*To check your transformation has been performed correctly you should compute the mean of your IQ_c  variable. If the centering process has worked the mean score for IQ_c should be 0. It is important that the mean score you subtract is as accurate as possible. Typically this means your mean score should be entered to say at least 4 decimal places (though the number of decimal places needed will depend on your data). If you have rounded your mean score your centered variable may not have a mean of zero.  
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*#You center the continuous variables by subtracting the mean score from each data-point. In other words, use SPSS, or another statistical program, to find the mean value of the variable. Then, use the "Compute" command in SPSS to create a new variable that is the original values minus the mean.  
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*#Then, repeat the procedure for the second variable.
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*As a concrete example,  
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*#Suppose you have 200 subjects (N=200) for which you have their IQ score and the length of time they studied for an exam. Thus, there are two continuous variables (X1=IQ, X2=time spent studying), and your dependent variable is the test score (Y=test score).
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*#Imagine that the average IQ score is 100. To center the IQ variable, 100 needs to be subtracted from every every subject's IQ score. So if a subject has an IQ of 115, their centered IQ score is 15. If a subject has an IQ of 90, their centered IQ score is -10. For easy reference, lets called the newly centered IQ score as "IQ_c".
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*#To check your transformation has been performed correctly you should compute the mean of your IQ_c  variable. If the centering process has worked the mean score for IQ_c should be 0. It is important that the mean score you subtract is as accurate as possible. Typically this means your mean score should be entered to say at least 4 decimal places (though the number of decimal places needed will depend on your data). If you have rounded your mean score your centered variable may not have a mean of zero.
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*There is a macro available that will center the variables
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*#Macros are useful when you need to perform the same statistical procedure for lots of variables or imagine in the future you will be performing the same analysis over and over again. In other words macros may take some initial time to learn but in the long run will save you time.
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*#See [http://www2.jura.uni-hamburg.de/instkrim/kriminologie/Mitarbeiter/Enzmann/Software/Enzmann_Software.html this website] and download the file.
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*#Open and select run all from the pull-down menu.
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*#At the bottom of the downloaded file is the following text
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*#:/* --------------------------------------------------------- */.<br>
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*#:/* The macro is called by:                                  <br>
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*#:/* Center IDVar  = variable containing casenumbers          <br>
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*#:/*      /VARS  = variables                                <br>
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*#:/*      /DVARS = new variables.                            <br>
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*#:/* --------------------------------------------------------- */.<br>
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*#You should re-write that text to reflect your current study. For example, remove the "/*" because that is telling SPSS to ignore the enclosed text. Then, insert your variable names into the text, such as
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*#:CenterIDVar = subjects
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*#:/VARS  = IQ, study
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*#:/DVARS = IQ_c, study_c
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*#In the above example, ‘IQ’ and ‘study’ are the variable names in SPSS given to the IQ and time spent studying by the subjects. ‘Subjects’ indicates the variable containing the case numbers, in this case 1-200 as there were 200 subjects in the study.
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*#Highlight the text, and click run selection. A new SPSS data editor window should be created at the end of which should be 2 new SPSS variables IQ_c and study_c. You should now save this spss file with a new name.
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===► '''Create the interaction term'''===
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*How to create the interaction term?
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*#Simply multiply together the two new centered variables.
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*#In our example, multiple IQ_c x study_c
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*#In SPSS this is accomplished using the "compute" command and typing "IQ_c * study_c" in the open box.
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===► '''Conduct Regression'''===
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*How to conduct the regression analysis?
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*#In SPSS, click on "linear regression" and enter the test score variable as the DV.
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*#Enter the newly centered variables as the IVs in the regression analysis.
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*#Click "next" and enter both centered variables AND the new interaction variable as the IVs.
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*#Run the analysis.
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*#In the output, look at the second model in the "Coefficients" box. An interaction is depicted as a significant value for the interaction variable. A significant value for the centered variables can be conceptualized as a "main effect".
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*#If your interaction term is then significant it is recommended you produce plots to assist the interpretation of your interaction.
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==Interaction! software==
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*Given the tedious nature of using the [[#Three Steps using SPSS | three steps described above]] every time you need to test interactions between continuous variables, I was happy to find Windows-based software which analyzes statistical interactions between dichotomous, categorical, or continuous variables, AND plots the interaction graphs.
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*The software is called [http://www.danielsoper.com/Interaction/default.aspx Interaction!] from a graduate student in the Information Systems department at Arizona State University. I found it very easy to use. There is also a good [http://www.danielsoper.com/Interaction/help.aspx Help section] on the website.
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*There is an SPSS macro for conducting cross-product regressions [http://www.ilstu.edu/~wjschne/tests.html here].
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==R commands==
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*Assuming you have your data in a comma delimited text file called 'myGreatData.csv' and the first line (header) labels the three columns 'y, x1, x2', the following command will generate your regression.  Note that these commands are the minimum and assume the same things are true as are true in the SPSS example above (centering, assumptions of the regression are met, etc.).
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*#setwd( 'dataDir' ) #Set the working director to the path to your data file.  You could skip this step and just enter the full path into the next step.
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*#dat <- read.csv( 'myGreatData.csv', header = TRUE ) #load your data file into the variable 'dat'
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*#m <- lm( y ~ x1 * x2, data = 'dat') #do the regression
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*#summary(m) #view the results
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==Plotting the interaction==
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----
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◄ Back to [[Analyzing Data]] page

Latest revision as of 16:49, 5 August 2011

Statistical programs, like SPSS, do not always have "point-and-click" commands for every possible statistical test. This page is a description of how to test the interaction between two continuous variables. Three approaches are described below:
(1) three steps to conduct the interaction using commands within SPSS, and
(2) Interaction! software by Daniel S. Soper that performs statistical analysis and graphics for interactions between dichotomous, categorical, and continuous variables.
(3) R commands for executing the analysis.

*For a description of what is an interaction and main effects, please see the accompanying page about What is an Interaction?.


Contents


Three Steps using SPSS

There are three steps involved to calculate the interaction between two continuous variables.

Center the two continuous variables


Create the interaction term


Conduct Regression



Interaction! software

R commands


◄ Back to Analyzing Data page

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