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.
*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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<nowiki>*</nowiki>(FYI - Chart 1b shows an ordinal interaction. A disordinal interaction occurs when the two lines cross).
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&lt;nowiki>*&lt;/nowiki>(FYI - Chart 1b shows an ordinal interaction. A disordinal interaction occurs when the two lines cross).
==Graphical representations of Interactions==
==Graphical representations of Interactions==
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<table width=100% cellpadding=20>
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&lt;table width=100% cellpadding=20>
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<td width=50% >
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&lt;td width=50% >
This column shows the four possibilities WITHOUT an interaction:
This column shows the four possibilities WITHOUT an interaction:
#Two Main Effects
#Two Main Effects
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#Main Effect for Variable A
#Main Effect for Variable A
#Main Effect for Variable B
#Main Effect for Variable B
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</td>
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&lt;/td>
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<td width=50%  valign=top>
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&lt;td width=50%  valign=top>
This column shows the four possibilities WITH an interaction:
This column shows the four possibilities WITH an interaction:
#Two Main Effects
#Two Main Effects
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#Main Effect for Variable A
#Main Effect for Variable A
#Main Effect for Variable B
#Main Effect for Variable B
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</td>
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&lt;/td>
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</table>
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&lt;/table>
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<table>
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&lt;table>
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<td>
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&lt;td>
There are other graphical representations of Interactions and Main Effects such as effects being reduced (e.g., imagine all the graphs below being flipped down) or effects going in the opposite direction (e.g., imagine all the graphs being flipped to the left), but for simplicity sake they are not displayed because you can discern those graphs after identifying the major types described below.
There are other graphical representations of Interactions and Main Effects such as effects being reduced (e.g., imagine all the graphs below being flipped down) or effects going in the opposite direction (e.g., imagine all the graphs being flipped to the left), but for simplicity sake they are not displayed because you can discern those graphs after identifying the major types described below.
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</td>
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&lt;/td>
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</table>
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&lt;/table>
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<table>
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&lt;table>
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<td>
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&lt;td>
[[Image:NoI2MEc.JPG]]
[[Image:NoI2MEc.JPG]]
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</td>
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&lt;/td>
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<td>
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&lt;td>
[[Image:I2ME.JPG]]  
[[Image:I2ME.JPG]]  
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</td>
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&lt;/td>
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</table>
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&lt;/table>
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<table>
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&lt;table>
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<td>
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&lt;td>
[[Image:NoINoME.JPG]]
[[Image:NoINoME.JPG]]
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</td>
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&lt;/td>
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<td>
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&lt;td>
[[Image:INoME.JPG]]  
[[Image:INoME.JPG]]  
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</td>
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&lt;/td>
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</table>
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&lt;/table>
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<table>
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&lt;table>
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<td>
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&lt;td>
[[Image:NoIMEb.JPG]]
[[Image:NoIMEb.JPG]]
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</td>
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&lt;/td>
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<td>
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&lt;td>
[[Image:IMEb.JPG]]  
[[Image:IMEb.JPG]]  
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</td>
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&lt;/td>
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</table>
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&lt;/table>
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<table>
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&lt;table>
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<td>
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&lt;td>
[[Image:NoIMEd.JPG]]
[[Image:NoIMEd.JPG]]
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</td>
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&lt;/td>
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<td>
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&lt;td>
[[Image:IMEd.JPG]]  
[[Image:IMEd.JPG]]  
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</td>
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&lt;/td>
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</table>
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&lt;/table>
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*#:the interaction = X1*X2,  
*#:the interaction = X1*X2,  
*#:Y = running speed
*#:Y = running speed
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*#Here is the formula for: Running speed = intercept + b1energy drink + b2energy bar + b3(bar * drink) + e<sub>i</sub>
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*#Here is the formula for: Running speed = intercept + b1energy drink + b2energy bar + b3(bar * drink) + e&lt;sub>i&lt;/sub>
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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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*#:Y&lt;sub>''i''&lt;/sub> = ''b''&lt;sub>0&lt;/sub> + ''b''&lt;sub>1&lt;/sub>X1&lt;sub>''i''&lt;/sub> + ''b''&lt;sub>2&lt;/sub>X2&lt;sub>''i''&lt;/sub> + ''b''&lt;sub>3&lt;/sub>(X1&lt;sub>''i''&lt;/sub> X2&lt;sub>''i''&lt;/sub>) + ''e&lt;sub>i&lt;/sub>''
*#This formula can be rewritten as
*#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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*#:Y&lt;sub>''i''&lt;/sub> = (''b''&lt;sub>0&lt;/sub> + ''b''&lt;sub>2&lt;/sub>X&lt;sub>2''i''&lt;/sub>) + (''b''&lt;sub>1&lt;/sub>+ ''b''&lt;sub>3&lt;/sub>X&lt;sub>2''i''&lt;/sub>) X&lt;sub>1''i''&lt;/sub> + ''e&lt;sub>i&lt;/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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*#:where (''b''&lt;sub>1&lt;/sub>+ ''b''&lt;sub>3&lt;/sub>X&lt;sub>2''i''&lt;/sub>) represents the effect of X&lt;sub>1&lt;/sub> on Y at specific levels of X&lt;sub>2&lt;/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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*#:and ''b&lt;sub>3&lt;/sub>'' indicates how much the slope of X&lt;sub>1&lt;/sub> changes as X&lt;sub>2&lt;/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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*#It is then possible to factor out X&lt;sub>2&lt;/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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*#:Y&lt;sub>''i''&lt;/sub> = (''b''&lt;sub>0&lt;/sub> + ''b''&lt;sub>1&lt;/sub>X&lt;sub>1''i''&lt;/sub>) + (''b''&lt;sub>2&lt;/sub>+ ''b''&lt;sub>3&lt;/sub>X&lt;sub>1''i''&lt;/sub>) X&lt;sub>2''i''&lt;/sub> + ''e&lt;sub>i&lt;/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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*#:where (''b''&lt;sub>2&lt;/sub>+ ''b''&lt;sub>3&lt;/sub>X&lt;sub>1''i''&lt;/sub>) represents the effect of X&lt;sub>2&lt;/sub> on Y at specific levels of X&lt;sub>1&lt;/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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*#:and ''b&lt;sub>3&lt;/sub>'' indicates how much the slope of X&lt;sub>2&lt;/sub> changes as X&lt;sub>2&lt;/sub> goes up or down one unit.

Revision as of 23:04, 23 November 2010



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Contents


Interactions and Main Effects


<nowiki>*</nowiki>(FYI - Chart 1b shows an ordinal interaction. A disordinal interaction occurs when the two lines cross).

Graphical representations of Interactions

<table width=100% cellpadding=20> <td width=50% > This column shows the four possibilities WITHOUT an interaction:

  1. Two Main Effects
  2. No Main Effects
  3. Main Effect for Variable A
  4. Main Effect for Variable B

</td> <td width=50% valign=top> This column shows the four possibilities WITH an interaction:

  1. Two Main Effects
  2. No Main Effects
  3. Main Effect for Variable A
  4. Main Effect for Variable B

</td> </table> <table> <td> There are other graphical representations of Interactions and Main Effects such as effects being reduced (e.g., imagine all the graphs below being flipped down) or effects going in the opposite direction (e.g., imagine all the graphs being flipped to the left), but for simplicity sake they are not displayed because you can discern those graphs after identifying the major types described below. </td> </table> <table> <td> NoI2MEc.JPG </td> <td> I2ME.JPG </td> </table> <table> <td> NoINoME.JPG </td> <td> INoME.JPG </td> </table> <table> <td> NoIMEb.JPG </td> <td> IMEb.JPG </td> </table> <table> <td> NoIMEd.JPG </td> <td> IMEd.JPG </td> </table>



Statistical formula behind Interactions




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