ࡱ> 5@ 0дbjbj22 "XXU[2222222F@@@@AlF*L(AAAAAmBmBmBJJJJJJJ$RMROJ)2mBmBmBmBmBJ22AAK7I7I7ImBB2A2AJ7ImBJ7I*7IaI22aIAA Zn]@FBaIJK0*LaIXPG<XPaIFF2222XP2aI4mBmB7ImBmBmBmBmBJJFF$%-I FF%Psychology 522/622 Lab Lecture #3 Interactions in Multiple Linear Regression The goal of this lab is to illustrate interactions of a continuous variable with a categorical variable in multiple regression. DATAFILE: CHURCHES.SAV The variables of interest are as follows: height: Height of the nave of the cathedral (in feet) length: Total length of cathedral (in feet) type: Architectural type of the cathedral (0 = Romanesque, 1 = Gothic) Height will be our dependent variable for all of the models. We are interested in how cathedral length and type relate to cathedral height. More specifically, we are interested in whether the relationship between cathedral length and height depends on cathedral type. (Alternatively, one might be interested in how the difference in cathedral height for Romanesque and Gothic cathedrals depends on cathedral length.) Because were interested in whether the effect of one variable (e.g., length) on the DV (e.g., height) depends on its relationship with another variable (e.g., type) we know well need to run a moderated regression model. Model A: Simple regression of height on length In this regression, height is the dependent variable and length is the independent variable. Analyze(Regression(Linear Move Height to the DV box, Move Length to the IV box Well keep the output from this model simple and not ask for the descriptives and plots like we might in other cases. REGRESSION /MISSING LISTWISE /STATISTICS COEFF OUTS R ANOVA /CRITERIA=PIN(.05) POUT(.10) /NOORIGIN /DEPENDENT height /METHOD=ENTER length . Regression  R2 = .41; this indicates that length accounts for 41% of the variance in height  F(1, 23) = 15.88, p < .01; so we know that, in addition to the relatively large R2, the effect of length on height is also statistically significant.  Lets write out the regression equation for practice: v = 37.54 + .087(length) How do we interpret this regression coefficient (AKA, slope)? For every one foot increase in cathedral length, cathedral height increases by about .09 feet. Model B: Regression of height on length and type In this regression, height again is the dependent variable and now both length and type are predictors. Analyze(Regression(Linear Move Height to the DV box, Move Length and Type to the IV box REGRESSION /MISSING LISTWISE /STATISTICS COEFF OUTS R ANOVA /CRITERIA=PIN(.05) POUT(.10) /NOORIGIN /DEPENDENT height /METHOD=ENTER length type . Regression  If we compare this to model A, we see that R2 has increased to .47 (R2ModelA = .41). This is expected because as we add predictors, R2 will ideally become larger. In reality, R2 may not change, but it will certainly not decrease.  Here, F(2, 22) = 9.91, p < .01. So we know that together, length and type are significant predictors of height.  v = 26.62 + .10(length) + 8.35(type) Let s focus on the partial regression coefficients (aka, slopes). Interpret the (unstandardized) effect of length on height: Holding cathedral type constant, for each additional foot in length there is an expected increase in height of .101 feet. This partial regression slope is statistically significant. Interpret the (unstandardized) effect of type on height: **Important: As is the case with any categorical variable, we need to recall how the type variable is coded when we interpret it. In this case, Romanesque = 0, Gothic = 1.** Holding length constant, Gothic cathedrals are 8.345 feet higher than Romanesque cathedrals; however, this difference is not statistically significant. Question: If type is a categorical variable, which it is, why didnt we create dummy codes?? Answer: there are only 2 levels. We dont need to dummy code when the categorical variable has just 2 levels. This analysis does not include an interaction term. Therefore, it assumes that the relationship between length and height is the same for Gothic and Romanesque cathedrals. This may not be true. In other words, this analysis assumes that the difference in height between Gothic and Romanesque cathedrals is the same irrespective of the length of the cathedral. Lets think about what this means in terms of our regression equations: vGothic = 26.62 + .10(length) + 8.35(type) vGothic = 26.62 + .10(length) + 8.35(1) vGothic = 34.97 + .10(length) vRomanesque = 26.62 + .10(length) + 8.35(type) vRomanesque = 26.62 + .10(length) + 8.35(0) vRomanesque = 26.62 + .10(length) Although the intercepts are different in these equations, the partial regression coefficient for length is the same for both Romanesque and Gothic. Lets see whether it stays the same when we include an interaction term. Model C: Regression of height on length and type with a lengthXtype interaction Here we create and include an interaction term in the model. The interaction term interact is the simply the product of height and type. This variable has conveniently already been created for you in the dataset, but here are the instructions for creating it yourself: Transform(Compute Variable In the Target Variable box enter interact (**Note: this variable name is fine here, but in practice I prefer to use the actual product term, i.e., length*type. This makes things much less confusing, especially if youre dealing with more than one interaction term). In the Numeric expression box enter length*type Remember that you can also drag and drop variables from the column on the left. This might help you avoid typing errors ( Click OK. Now lets run a moderated regression using the term we just created. Analyze(Regression(Linear Height should already be in the DV box, and length and type should already be in the IV box. Lets move length*type into the IV box as well. Statistics(Select collinearity diagnostics. Click Continue, Click OK. REGRESSION /MISSING LISTWISE /STATISTICS COEFF OUTS R ANOVA tol /CRITERIA=PIN(.05) POUT(.10) /NOORIGIN /DEPENDENT height /METHOD=ENTER length type interact . Regression   Again, as in Model B, R2 improved by adding terms.  Since our F ratio was significant in Model B, we would also expect it to be significant here. We havent taken any predictors out, weve added. So if F was significant before, it will be significant again here.  None of the partial regression slopes are statistically significant. This is a bit strange because length has been a significant predictor in prior analyses. Why? Multicollinearity! Note that the tolerance statistics are pretty bad. This is because we did not center our variables. Well compare our results later when we conduct the analysis with centered variables. For practice, well go ahead and describe our findings. Blength: The partial regression slope for length is positive. This number reflects the relationship between length and height when type=0 (i.e., for Romanesque cathedrals); therefore it also describes the relationship between length and height when the interaction=0 (i.e., if type=0, typeXlength must = 0). Since type and typeXlength are set to zero, they have become constants, so that takes care of the holding all else constant part of our interpretation. So now we can say: Descriptively, there is a small (.023 feet) increase in height for each additional foot of length in Romanesque cathedrals. Btype: The partial regression slope for type is negative. This number reflects the difference in height between Romanesque and Gothic cathedrals when length=0 (!); therefore it also describes the relationship between length and height when the interaction=0. Lets break down this interpretation. Generally speaking, we would say something like this: holding length constant, for every unit increase in type we expect a decrease of 34.60 feet in height. Colloquially: holding length constant, as you move up in type you move down in height. So what does it mean to move up in type? It means moving from Romanesque to Gothic. Dont forget that our model assumes that length=0. Thus, descriptively, Gothic cathedrals with no length have less height (i.e., are shorter) than Romanesque churches with no length. (Strange interpretation but that is what the model is saying. This is why we need to center!) Binteract: The partial slope for the interaction is positive. There are two ways to interpret this interaction. First, cathedral type moderates the relationship between length and height. Second, length moderates the differences in cathedral height between Gothic and Romanesque cathedrals. One interpretation or both may be useful depending on the focus of the research. We ll look at both interpretations. Interpretation 1: Cathedral type moderates the relationship between length and height It is useful to write out the regression equation to aid interpretation. Here it is. vHeight = 63.44 + .023(Length)  34.60(Type) + .093(Length*Type) For the first interpretation of the interaction, we write two regression equations, one for Gothic cathedrals and one for Romanesque cathedrals. Romanesque (type = 0) vHeight = 63.44 + .023(Length)  34.60(0) + .093(Length*0) = 63.44 + .023(Length) Gothic (type = 1) vHeight = 63.44 + .023(Length)  34.60(1) + .093(Length*1) = 63.44 + .023(Length)  34.60 + .093(Length) = 28.84 + .116(Length) Let s compare the y-intercepts and slopes for these two equations. The relationship between length and height is more positive for Gothic cathedrals than for Romanesque cathedrals. This difference (.116-.023) is .093, which equals Binteract. Remember that when we did not have the interaction term in the model, Blength for both the Romanesque and Gothic equations was .10. Now they are different from one another (albeit not statistically). The mean heights (aka, y-intercepts, aka, constants) differ by cathedral type. For Romanesque, y-intercept = 63.44, for Gothic, y-intercept = 28.84. Again, going back to our strange interpretation, this is because Romanesque cathedrals are taller when length equals zero. The unstandardized partial regression coefficient (aka, slope) for type is the difference between these two intercepts, 63.44 28.84 = 34.60. However, remember that in the larger regression model, these differences are not statistically significant. Interpretation 2: Cathedral length moderates the relationship between type and height For the second interpretation of the interaction, we write two regression equations, one for small length cathedrals (i.e., length is 1 SD below the mean length) and for large length cathedrals (i.e., length is 1 SD above the mean on length). This is standard practice, and if you ever need someone to cite when doing it (e.g., for your thesis), use Aiken and West (1991). Well need to run descriptive statistics for cathedral length.  Cathedrals 1 SD below the mean on length are 425.48-109.26 = 316.22 feet long. Cathedrals 1 SD above the mean on length are 425.48+109.26 = 534.74 feet long. Let s plug these values into the regression equations: Small Length Cathedrals vHeight = 63.44 + .023(316.22)  34.60(Type) + .093(316.22*Type) = 63.44 + 7.27  34.60(Type) + 29.41(Type) = 70.71  5.19(Type) Large Length Cathedrals vHeight = 63.44 + .023(534.74)  34.60(Type) + .093(534.74*Type) = 63.44 + 12.30  34.60(Type) + 49.73(Type) = 75.74 + 15.13(Type) Again, let s compare the intercepts and slopes for these two equations. The relationship between cathedral type and height is different for cathedrals of different lengths. In other words, the difference between Gothic and Romanesque cathedral heights is different for cathedrals of different lengths. Exactly how does this relationship differ? Lets focus on mean height (i.e., intercepts). If we substitute 0 (Romanesque) in for type, we can calculate mean height for small and large length cathedrals. For small length Romanesque, intercept = 70.71; for large length Romanesque, intercept = 75.74. So, Romanesque cathedrals (when Type = 0) are taller when the cathedral length is large. We can also interpret these equations for Gothic cathedrals by substituting 1 for type in the equations. For small length Gothic, intercept (i.e., mean height) = 65.52; for large length Gothic, intercept (mean height) = 90.87. Gothic cathedrals, then, are shorter in height when the cathedral length is large. However, remember that in the larger regression model, these differences are not statistically significant. Model D: Regression of height on length and type with a lengthXtype interaction (with length centered) We now consider moderated multiple regression where we center the continuous predictor variable, length. Centering We need to do two things before proceeding. First, we are going to standardize cathedral length. This will make length (which we will call zlength for standardized length) have a mean of zero and a variance of one (these are features of any standardized variable). The mean of zero is important for our purposes, but having a variance of one makes it easier to interpret the interaction as we did in the steps above. One way to do this is via the compute menu, where you would subtract the mean of the length variable from the length variable. Another (faster) way of doing this is with the descriptives menu. Analyze(Descriptive Statistics(Descriptives Move length over to the variables box. Click on the Save standardized values as variables option. Click OK. DESCRIPTIVES VARIABLES=length /SAVE /STATISTICS=MEAN STDDEV MIN MAX . This creates a new variable zlength and adds it to your data set. Next, we need to compute the interaction term which is the product of Type and zlength. Any time you have an interaction term that contains a continuous variable, you need to create the interaction term using the centered variable. You do not standardize the interaction term. You standardize the individual variables and then create an interaction term. To differentiate this interaction from the previous interaction, well call this Zinteract. You can create this interaction term using the same steps that we used earlier in the assignment to create the first interaction term, length*type. COMPUTE Zinteract = type*Zlength . EXECUTE . Running Model D Now we conduct our regression analysis with height as the outcome and Type, Zlength, and Zinteract as predictors. Analyze(Regression(Linear Youll need to take length and interact out of the IV box and replace them with zlength and zinteract. Everything else should be set up like we want it. Click OK. REGRESSION /MISSING LISTWISE /STATISTICS COEFF OUTS R ANOVA tol /CRITERIA=PIN(.05) POUT(.10) /NOORIGIN /DEPENDENT height /METHOD=ENTER Zlength type Zinteract . Regression   Note that R2 here in Model D is the same as it was in Model C. Of course it is! Its the same model, with the same data. Weve just centered one of the IVs and the interaction term in order to help with multicollinearity problems and to make our interpretation more meaningful.  Again, same info here as in Model C.  Aha! Now things are changing The tolerance statistics are a little better, but not much. However, every little bit counts! Importantly, tolerance was much improved for type. This is good because we would not expect that type should share so much variance with length, as it did before we centered. Both Romanesque and Gothic cathedrals can be long and/or short, right? The two variables should be relatively independent. Concerning collinearity between zlength and zinteract, this is somewhat expected. They both contain the variable zlength, right? So wed expect that there might be some collinearity problems here. Centering typically helps immensely with this issue, but sometimes not as much as we would like it to. None of the predictors are significantly related to height. Collinearity may still be wreaking havoc; however, our interpretation of the slopes (irrespective of their statistical significance) does improve. BzLength: The partial regression slope for zlength is positive. This number reflects the relationship between standardized length and height when type=0 (i.e., for Romanesque cathedrals) and therefore interact=0. Thus descriptively, there is a 2.5 foot increase in height for a standard deviation increase in length in Romanesque cathedrals. Btype: The partial regression slope for type is positive (recall it was negative in the prior analysis). This number reflects the difference in height between Romanesque and Gothic cathedrals when zlength=0 (the mean length of the cathedrals) and therefore interact=0. Thus, descriptively, Gothic cathedrals of average length are taller than Romanesque churches of average length. (This is a much cleaner and less strange interpretation than when length was not centered about its mean.) Btype*zlength: The partial slope for the interaction is positive. As before, there are two ways to interpret this interaction. First, cathedral type moderates the relationship between standardized length and height. Second, standardized length moderates the differences in cathedral height between Gothic and Romanesque cathedrals. One interpretation or both may be useful depending on the focus of the research. We pursue both interpretations. It is useful to write out the regression equation to aid interpretation. Here it is. vHeight = 73.29 + 2.53(zLength) + 4.91(Type) + 10.15(zLength*Type) Interpretation 1: Cathedral type moderates the relationship between length and height For the first interpretation of the interaction, we write two regression equations, one for Gothic cathedrals and one for Romanesque cathedrals. Romanesque vHeight = 73.29 + 2.53(zLength) + 4.91(0) + 10.15(zLength*0) = 73.29 + 2.53(zLength) Gothic vHeight = 73.29 + 2.53(zLength) + 4.91(1) + 10.15(zLength*1) = 73.29 + 2.53(zLength) + 4.91 + 10.15(zLength) = 78.20 + 12.68(zLength) Lets compare the y-intercepts and slopes for these two equations. -The relationship between zlength and height is more positive for Gothic cathedrals, 12.68(zlength), than for Romanesque cathedrals, 2.53(zlength). The difference here, 12.68-2.53= 10.15, is the same as the unstandardized slope for the interaction. -The mean heights (i.e., y-intercepts) differ by cathedral type. Gothic cathedrals (78.20ft) are taller than Romanesque cathedrals (73.29ft) for cathedrals of average length (recall that by standardizing length, the mean of zlength = 0. The coefficient associated with zlength drops out because it is multiplied by zero). The slope for type is the difference between these two intercepts, 78.20-73.29 = 4.91. However, remember that in the larger regression model, these differences are not statistically significant. Interpretation 2: Cathedral length moderates the relationship between type and height For the second interpretation of the interaction, we write two regression equations, one for small length cathedrals (i.e., length is 1 SD below the mean length) and for large length cathedrals (i.e., length is 1 SD above the mean on length). Note that because we have standardized our variables (mean = 0, variance = 1), one SD below the mean is a value of -1, and one SD above the mean is a value of 1. This is especially nice when we write out our regression equations because it means less work compared to using non-centered variables ( Small Length Cathedrals vHeight = 73.29 + 2.53(-1) + 4.91(Type) + 10.15(-1*Type) = 73.29 - 2.53 + 4.91(Type) - 10.15(Type) = 70.76  5.24(Type) Large Length Cathedrals vHeight = 73.29 + 2.53(1) + 4.91(Type) + 10.15(1*Type) = 73.29 + 2.53 + 4.91(Type) + 10.15(Type) = 75.82 + 15.06(Type) Lets compare the y-intercepts and slopes for these two equations. Descriptively, the relationship between cathedral type and height is different for cathedrals of different lengths (remember that the interaction term was not statistically significant). In other words, the difference between Gothic and Romanesque cathedral heights is different for cathedrals of different lengths. Exactly how does this relationship differ? Lets focus on mean height (i.e., intercepts) #3u   L M y z ~    j  H I J P Q S [ y z ü~zvvrh'zh"HhU hMhU5>*CJ\]aJhM5>*CJ\]aJhc5>*CJ\]aJ hMhM5>*CJ\]aJ h4,8h:hMh4,8h4,856\]h4,8hSh\h:5\h\h\5\h\h h:hayg,#NO  M z   I J y z ( gdMgdcgd:$a$gd:gdaygд 789DEFGH-ƩơoWIIhMhM6B*]ph/hLIhM5B*CJOJQJ\^JaJph8j'hLIh*5B*CJOJQJU\^JaJphhMhMB*phhMB*H*phhMB*ph8jhLIh*5B*CJOJQJU\^JaJph)h*5B*CJOJQJ\^JaJphhUh*CJOJQJ^JaJh* jhMhMh: 89DF.01g24prTUogdc 7$8$H$gdMgd:gd* 7$8$H$gd*-./01fpr39>BSTU­sbbbQMIMIMIEMAh*hch"Hh: hchU5>*CJ\]aJ hch*5>*CJ\]aJ hchc5>*CJ\]aJhc5>*CJ\]aJhchcB*phh*B*phhcB*ph)hJ5B*CJOJQJ\^JaJph)h*5B*CJOJQJ\^JaJph8j hLIh*5B*CJOJQJU\^JaJphhMh*B*phU\]ghLMXYZ~bdαΩmΩ__hchc6B*]ph8j+hLIh*5B*CJOJQJU\^JaJphhchcB*phhcB*H*phhcB*H*phhcB*ph8jhLIh*5B*CJOJQJU\^JaJph)h*5B*CJOJQJ\^JaJphh*h*CJOJQJ^JaJ jhchco -LMXZdh[\ gdw^gdJ 7$8$H$gdc 7$8$H$gd*gd:gdcdfhY7RZ[\jo-  XdDX)ƾƾƾƶƯƫƫƫʫʦʦʜʦʦʜʀhDhD6] hD5\hDhD5H*\hDhD5\ hDH*h hJhw^hh*hJhb?hw^hD)h*5B*CJOJQJ\^JaJph8jhLIh*5B*CJOJQJU\^JaJph0  VB. I U!" " "R"l""%#?#@#K# 7$8$H$gd-gd1Mgd:gdDgdw^=\- . 7 8 k s t T!!""Y"Z"d"e"##?#@######}`}8jhh-5B*CJOJQJU\^JaJph)h-5B*CJOJQJ\^JaJphh-CJOJQJ^JaJ jhghg jJhDh1M jhDh-h*hD5>*CJ\]aJ hDh5>*CJ\]aJ hDhD5>*CJ\]aJhD hDhD$K#_###########/$1$%%x&y&&&)),,b.d.gd1MgdH"gd-gdggd: 7$8$H$gd-###$$.$/$0$1$%%%%%%%&"&y&&&&&&&&&&]'_'i'''''ƼƱƱwsososososodXsososososohghgCJH*aJhghgCJaJh-hg8j,hh-5B*CJOJQJU\^JaJph8j'hh-5B*CJOJQJU\^JaJphhghgB*phhgB*H*phhgB*ph)h-5B*CJOJQJ\^JaJph8j"hh-5B*CJOJQJU\^JaJph"'''((((())))C)J))*K***,+W+X++++++++,,:,Y,x,},,,,,,,,,- -6-U---...4.d./////////00(0>0@0B0ؼܳܬ h-h\ h1MH*hphphp>*h1MCJH*aJhgh1MCJaJh\h1MhH"CJH*aJhghH"CJaJhH"h-hgDd.////B0D0f1h1112<2>2b2d22<3j3l33~7gd $a$gdg ^`gd\ ^`gd1M$a$gd\gd: $`a$gd\gd-B0D0d1f1|111111111112,282:2<2>2J2`2d2f2r2t2222222222 3 3*3,383:3X3Z3f3h3j3l33333644444444p5q5t5555555F666Ž̸̸ huH*huhuH* huhuhuhkh hZ`h h-hgh\ h1MH*h1Mh:hgh-G6666777777777798@8888K9L9`9l999999Z:\:^::::;;;,;<;N;V;f;~;;;;;;;;;;;;< <òߑߝߝߝߝߝߝߝߝhYB h'SH* h``h``h`` h``5CJOJQJ\^JaJ h;d?5CJOJQJ\^JaJ/j4hh``5CJOJQJU\^JaJh;d?h'Shp hp>*hphp>*hZ`h hb?7~7777M9N99999\:^::::;;;<<L<N<<4= ^`gd'S^gd'S$a$gd'Sgd`` 7$8$H$gd``gdpgdZ` <<<<<(<6<N<P<\<z<<<<<<<<< = ==(=0=2=4=T=`=h=j=l=n==>x>>?*?=?a?b?e?????@9@:@=@J@*AJAA BBBB%B+B/B5BEBRBvBwB˽ˬˬ h-hg h>hC5>*CJ\]aJh>5>*CJ\]aJ h>h>5>*CJ\]aJ h-hZ`h5Rhb?hYB h'SH*hZ`h``h'SA4=l=n== BBvBwBBBBMEyEEEEF3F4FvFwFHH 7$8$H$gdCgdwLgd:$a$gdggd>gd'SgdZ`$a$gdZ` ^`gd'SwBBBBBBBBCCC-CLCWCwCzCCCCCCmDDDDD EEE:ELEMETEUEkElExEyEE3FBFQFFFFF^GG5HHHH I{IIIII9J:JJJJѽ)hC5B*CJOJQJ\^JaJph jhq<hq<hq<hq<5\hwLhwL6]hCCJOJQJ^JaJ jhwLhwLhwLhwL5\hgh>hC>HHHH I{II/J9J:JEJYJ~JJJJJJJJJ LL4L6L7Lgd& 7$8$H$gdq<gdC 7$8$H$gdCJJJJJKK L LLL3L4L5L6L7LVLsLLL O)OHOOOαΩwΩZVRNRNRNJN?hhCJaJhh&hq<hC8jaFhhC5B*CJOJQJU\^JaJph8j@hhC5B*CJOJQJU\^JaJphhq<hq<B*phhq<B*H*phhq<B*ph8jo<hhC5B*CJOJQJU\^JaJph)hC5B*CJOJQJ\^JaJph8j 8hhC5B*CJOJQJU\^JaJph7LULVL O OOO2Q3QSSTT1U2UVV4W6WXXZXpXrXXY ^`gd ^`gdgdgd&gdq<OOOP PPPP3Q4Q8Q9Q:QcQQQQQ$RhRjRRRRRRRRSSS*S+S,S@SkS}SSSSSS TIThTTTVVVVVVV$V,V0V* h-h& hH*h?hCJH*aJh&hhhCJaJhhCJH*aJKXXXXY YYYYY.Y0Y*hph1d`>*hk h-h? hH*h&hh?PYY,Y.YY Z.Z/ZsZm[t]u]]]__``z`` ^`gd1d`$a$gd?gd1d`gdk$a$gd&^gd$a$gd ^`gdgd& ^`gd____` ``L`T`l`v`x`z``````````a aa4a6aBava~aaaaaaaaaaaaaaabbbbZbcb_`ghˤ$;Bf,-L¾¾h_h}hjLhSc hjL6hgh">Uh=;hV[ h-h?h? h1d`H*h1d`h& h``h&I`aa2a4aaabbZb_`hi-.ܧݧxvgdkgd}gd=;$a$gd1d`$a$gd?gd& ^`gd1d`. If we substitute 0 (Romanesque) in for type, we can calculate mean height for small and large length cathedrals. For small length Romanesque, intercept = 70.76; for large length Romanesque, intercept = 75.82. So, Romanesque cathedrals (when Type = 0) are taller when the cathedral length is large. We can also interpret these equations for Gothic cathedrals by substituting 1 for type in the equations. For small length Gothic, intercept (i.e., mean height) = 65.52; for large length Gothic, intercept (mean height) = 90.88. Gothic cathedrals, then, are shorter in height when the cathedral length is large. Note that all of these predicted values (mean heights) are the same as the values we calculated when using our original, uncentered variables. However, remember that in the larger regression model, these differences are not statistically significant. Summary In summary, we did not find statistical evidence that the relationship between cathedral length and height varies across Romanesque and Gothic cathedrals. In the alternative form of the interpretation of the interaction, we did not find evidence that there is a difference in the difference in heights for Romanesque and Gothic cathedrals as a function of the length of the cathedral. However, we should be cautious in our interpretation since the small sample size (N=25) may have not given us sufficient power to detect this interaction. The collinearity issue made finding such an interaction even more difficult given its effect on the standard error of the slopes (i.e., the variance inflation factor). Thus the regression model that did not include the interaction term seems reasonable (i.e., we do not necessarily need the interaction term). Interpreting the regression equation without the interaction, we note that controlling for cathedral type, longer cathedrals tend to be taller. Duh. Controlling for cathedral length, Romanesque and Gothic cathedrals do not differ significantly in height. Appendix: Graphing Interactions A note on graphing: When you graph interactions, you use the unstandardized partial regression coefficients in the equations to calculate predicted values. So, in our example, you would calculate four values. We ve actually already calculated all of these, so I ve included them below: 1) vheight for Gothic cathedrals of small length (-1SD) = 65.52 2) vheight for Gothic cathedrals of large length (1SD) = 90.88 3) vheight Romanesque cathedrals of small length (-1SD) = 70.76 4) vheight Romanesque cathedrals of large length (1SD) = 75.82 I recommend using Excel to graph your interactions. Line graphs are typically used to represent interactions, though in some cases a bar graph may be appropriate. If you dont know how to use Excel to create graphs, or you dont have another preferred program for creating graphs, it would be a good idea to learn now. Online Ive posted an Excel template that should be helpful to you in calculating the values that youll need to graph an interaction, and creating the actual graph. Sadly, we donܧݧTX flv dt~ƭQʴ˴ʹδϴдƻ۹۲h? hChC hjLhjLhjL jJhyU hc hc*CJ\aJh}+vδϴдgd:gd$gdk gdkt have enough time to go through an example of how to graph an interaction, but Id be happy to walk you through the template and share a few tricks of the trade during office hours or some other time. ( 1h/ =!"#$%'Dd #0  # A2ȣv& ȲD`!wȣv& Ȳ ' 6ExڝVOOQB- 聀1m@NDKC`h[ \,$ph6x$ ҃^-5,yofv{ /wxTRo{C=1*k4lbj7(1#ydzҤ~1Kh0!#ML2$ /埸jJ^JmIX(-Ɨ8S|y^ub30H6 *C#7~B@ )d%5KڊX>1rXvXe9b13CS$2DE1<94l32`#/|]4 bnZ k?X'F"X>f AFƧ"X'=.=F'&&ݡ8(T:7t?xFUxEADL(9";-e3GJrʵ,|f7Tȿ{kI(h8KEP FosY=cl])ug:gY3UDXִf[ e&}!NgT\EIrNzM;RAūX<H8@cǵzo}fnvqn2f_j✞ѲOtVR2F_p|oUd.]U[=6ejtimҮ<+ZI`~RKڬ(V&j/f*}א+a@JANiD@ vۛO-IR(BP#sobxQGbl<7Dd I1 0  # A2Za${Yvlk`!Za${Yvl17M`JxڽXoEx@2*N㤽%Q.pqq;@JR!![mqsSJ>J3SQg+%f޾}3o}3kJxyX^k'_)֋{/P"4x6xu-cY>0š19Xxv5ʡ g$]?W(c}wͤ7etWB^^D{0Njqİӿv!#g!,Y164rQEv 7.MMbQHc ;Q·b!|Oݳ+.P%_Tp=)cfFk /I LC aŝޙ+;^pVWv sX=D;LQ~~_E#TE3A/:jq7PUH(_G*'gX"bjNy"o<7Pc5=[j2LM&T2b^@[[j2$.#`{T OՎ@=a +cr:фw|+8ʏϙUXo6V/78۲89"b6ʩ6 V!2,Jܚ.V>(ƓDZ&i‡bh)c vh"1CU z9!^;X;0v-TX>&C|Ͼ\Kds'h[mZ0$jj˪71j]?gqF|HPa"1*=ɽ!wrE<Ƴ$!1_/! je!-' Ul_ھ8u3$z+9O9֙l OYdLg?/VCoUK|x(g3+Xuf uYRFHqGy^UfGr(ǽ\Evm 36b ?mrЊͬ2 3< :0}wQ|f=\leu7Ƅ,65! }Mx]*|ߠC _~eOP_K5B~YLsXC^#[M^g5Dd  0  # A2R qx2yE\ `!R qx2yE\6h,MxbSxڥWMlExc6R։ҔTMPpBobcWJF I|\q,!q "/9PB[e6f{|gq"f?c:{Y(Z Ǚ3XW,):>onӔݱG>? UTPV()ڿᎏQ򰬸vHcM5Sc(`㋕B㙮C,{SxR}_ F҉iYi9<_ YRnGdپ$@$OtV0K\m2Ε D.m~~%,sPHo!ۣks c :8Q`h!Qư \ y<lzLjk[u(i-ӥ{V~[;v+Jap۰g%ņOaϲt-?y{˃ N^#!{f( ^F)ax#7 y3(Zm:櫱qN}Htʷ=]w]kywgHw ]vxޡ>D>b̝ӂVunmUWN#v\<Oｽ߹Goݛ.b|5>{d\ƹ̉\V)2┚SF2ȼ/,Dd #0  # A2|v*0_KP?C`!||v*0_KP? ' 6JxڝOAg4+Fxhꥡ`(V Z"x(m \I&h`|3mɮai27o7f !!#Eb )z]qS˦yf u%1J?%vmzKnL裿L*lBcOt ] ƺ^ؤul-``gfZF!>?MgV'C g`MX{RۘvH6)<ͦe9dRmRb٥c!X*胢r\Yp7#"]u|r-@jNPE~M%D(uO)^Dq $=rGq Rs@g nOD7 cR: ̢[]OePADm>8R̹+)"jX _NʧT-. *R2$/9MI7[g˸nküa5SÚ)CRKiUͪrvt Yny:-eR--OKIaq5&;Q:Z߿0t<-U=ר>#7{0JHC17;0*y-CO+9- KeD^VVkN:W5a!_,^!H mv@w1|Tǯkn Q ޢ.XZNodd6ĉٮ<\w`,2 o.R~o1VEkX<$%AcA (%g+z:Dd I> 0  # A2)3wi;2㔪,o`!3wi;2㔪,17MnxڽXoE;vn E"XV&v~5P"&.8\QgX*KV[ġRSHZǷ3SQǕbiwμ7߾5g]~-N3ׁ+$nzyOs%{7,93C)eӿᎏQɅa[]nS?(&h11~`ѩt2c"dx"?>^9+2J0pQaM3ii[&ƕ0xPl}٩j``w1XOgre[k!`{,¯=o#Xڅ6Pɶ)16:bNbAKQt;(%6 k?-Y(!~<17''(п7FIs2>:!&j } ;/{3WØVd5İWOIwfL As9E3+/-h_HW}H_C+gauTWh[?-h족auV*96>b_@[뮶VC1ĉ1m:]\N8[Sӎc['>!G~vZt.Vn"{C-)qTc8dFwu7vPw(:$2Pi&WHfuW~ +oagoaZX>[nCMsFP"s<zv9̝u*M/_N8e涯lFkհPN^#Vi~A7U;NyHuPF.=Q3<0.|DLԵ3r=)W:ΒC:eꧾx3_8+bձ:>v캒[Wrvܺ8uRZzWNK2g1VS>[_rKw.B2[83)ܸn D2V6UX;5IQC(IPҌhkW0۫^p'qς%GW4o]VMg.c,Zb]fR.3)03Ks̨LST̀OMmߛ#xPC惮 W[/T]ѱn]ïJ!Jo9uo`at<)[BDo 53`xkr9]%s] 9);2AO-WDd  0  # A20U;^t2 zfgߙ3̰Y7cW[\tl@4bФh ۅpiM'3Kȋn[QɅa纽z#1S}#iG666Xv 7Nz8^XwDHm9ۇ+I:;8[+B˜/9 Ӷ$I{}R4KsL[BvUiZm~=lL0@qv)6҉)1L*Fa G`w)77Ҫ `j {ڮCZy&d*`xexSh`8CTKFPk+m\{Nu%2ⱸ*\`ksxdkT%8NPhMp=R%8D^XK4eLiUQQwW97VAYZ#.!=|_@+OVAy5yV zwv;AU> m e߲}WAQ+o52]ZqAen:uwj ߇w)ΆA2Ė<6,u |?d Z3zcSܥg;P;ɦg38C};hGD{eQYn9q{q[v)#Wa-<_/+.Z\fL-y3TiM57pӰW/$n>iNDd  0  # A2[ϒH`![ϒHH wК+XlxڥVOAni`E4ŏM/Ƥœx&O*DbW @śp&GDEC4A{Pۙndw޼o,%!B|tSxP$̔vև_'Qj8-0HFA|0?D< `ۧ܁(7_ >203MP[z3vAxdH8ʠp:.!y|siH1ځ^|/99U҄8݄)L>g^=/E듪s#]|& ّ#Om#X#5'3na5*dQ2L|0F{aT!n}Va--t7ĿT%/g#m;S%qT0yƑ5h8xrϨSHǏ2o6 I.CPA,Izm ֡(PC e{` y?l0)wLj5rp y1(kelDY(M<=Lк;)Qui'4Owupcd([kXߜK!׹f@D8E "%+@Vgْ{LUskI oPH,qZTL {[8ӂg~DT;}9) ?pE/4}Iҧ'J2"30Œ( zMHm0j 5F9v=ujVjVtq*te)9GʖBU_6(xDKm GXyEjh+ۭ.9D61ϕDxALow-5,M͛yfR$ODHڄzB]>qPۦqf y- O(a?Mb+aŜ0Fq%X6T0, )(a@rT8'oiIeŀn"8^&,mKsO!Eh wc~Ҏ*= s-; e#-z6ɠt?b>1DHv!29v5WcQ$d(>܇9Gh-֯fWZR nJ};A''xEHЇ"Hdjbj#>Hե)?ݨ3* HUДL0^ =%d&C09@G2̚KsسC\YN# Pj@r|1hWZ\6%%^䲺36^q Lu m7k5SӚ34`ϫtVOݑ'əߴ"%CWtKǢ2t5b& zZcz.Pa370J"HC07HSY=e7gY]ICr22 j>?,/]ڽ]ava6(qqt 3F5αiq.O Zn}ByX$Yg'YU~v\j"Z:ۮm\W\wӽGCxZ=ɜ5QAǗS 0_ ^佟-:'FDd I1 0   # A 2111HD7aT V'`!11HD7aT17M`JxڽXOG u~PV[&6@.)jժC@ȱS=(`)KVz赇+))VDzvgYԋ2yͼoͬ9d,;N],_.qhu'gb^w`D=G/[s;p/c!tΉ""/1nGgL.A}‴x,Ip/eKpisH Icږqk˼ 5v@&|1ncn'] e_lYڢJ[iڒ?1,)څ6P<`M$DtL)y78Ea>P|FU!W7<[Keп7GE ѱĐ*>Q0bA`^MF4{g66wav9:ޙ;-Oeh/헡}Pοk~ߠX=̨&t nEY̢̏cf>쮍gܵmVs*`*c`^ըw0T@*^v=~ݩ5 M.^ss\~P㽯lF )հPݺZC]j5Wv3n/xfװfC%f2K*LU${[g'g^(@2}\~Jܯְ)Y?˞mw"^ZxQcev<_8u='^N_5IiD2cf)M-Vn!Y\6o%օ~=̅U_Ϙٔ5߯[7ojʿY k-&24tn}M-٫z`k`k5oY3kI|dN[2IiaF>$EXWpCTƫ܋y"~qD!O隧~ +Vc\ xqr}zÊ*}_:ÎhSyڹ{O "2y3o{S[ԜqDvSVmmnnrc:Dd ' 0   # A  2֖ Б v~6 -`!֖ Б v~6E0@b8XxڵoEg'!iP&饴I8p@-)H$SKUAJHm~ܹ[B!!Al H/\6 C̮NjZ7o>}gƻ3qc"8j3m(\Vyͦ w`1χ8.:q>;M^\6c4>5.,D;{H%M -w $& ??{t__ Ij>EI:p7z_umߢTHggmvSo5'icݲm }%6m"~X-j G?+QmPX^6ijxTYy=0<0}`XC/>X 8Ÿ}K' Wi"Vyш*ICUCOjh*5`^n P|L`m-ܗyxHX%WY+P#êTUOGh\)u[ϹR>SScΓ sccz4x$LJwU)\ ڜI#zltLa wy0/p<"QUMηp_2efǔyeιA&2ʜs2MFֻ2iR$džשwL2宔9() iV +e])-RT @])h]j҂&')G!(lu>Ԃ&Cl4ϭ4r5iAPXy$,xdդM\T Jh])W49I083"IEEty؍/j?W>k^or; /w[l6Sݡ,bÎ28:$ 84WWUE^̛O2`R5C51y!H&$IJF&1UͽSȽܳǢH]k@oLfB5Qvi̅31be5H_oCrCZYyEquY.ZeLf&Ʊ8JGdC%:/gDf:1m+æf=- jx"w~cԌlB5^ }Nr 8 Z;-Z&|59|O D+k+{S5+*{zMpk(3&V(C{&we8 t፳jޖ0^"q$? #QJLD&DglC_ EYx?·}ni?Ԡ:%ax]>^i`WU٭u?;j.UG- }KxۭzX>K!v$%8M׌4y8j$\mE`"El~qoQnݽG_ýo}5 ߑ:&7nݚ3NNޮ[W+@_;u[رuTa;x:~r*w߽4ݘc_vUK?hDř;6sιm&SDd 0   # A  2hIb8QwE4`!ohIb8Q| 4! DJh>=xڽAOAgۊ.SҴ 'S& $xTz2^~M%-a`yfofQ})G'3?h(Ǐ}dv]0f0)ߢ#=DaH {`H 0Nl,gӗxo֍~Yugs?mwΦiҬgsȿ)-ƋaMcXzU#aEi /8#)^jx5@U V>AR|Ay1l&U3}@:ғٱL1ZD+2FĶ'9+ 8Qik69EϤ OWR$Iq3)&u8=SGd8a.ܳJU_B6iz1s9ΙW YdoreODd  0  # A 2D ,]# d8`!D ,]# J wК+XmxڥVOAniE@4^Ii'M4hB8XڒR!\d6OPz+ AMogfw [7ٝ7>~ff) [&'H̔avƇi_'1jtk&a%F)~`~Bx|GvD10z,*m#IaaH!f( I;:Zd bg2X4UKQxT#3_x_-}e)ueb48o7xE#jt A7vyJWL^(-izP|܁dV!;rYW=U҄`ݏ0T*0gHC\E2(LWc\.}uB\un߼^VsZX}瞁_r*㊜'m;Z+C\|8m q=H&-^+s->h Z^ HecZ01iRFσWRbr MPWV=xuzlاMlPrByѿ(el9lr^Q&isMtSq[XXOz5W8y{Qf2uHk[cvKsYk. H;j3P1!YAJ絊>_3ڂu颊n1w!>2DA!Q hqmZL˞^#秌Le3ȆfVE-es^)XuaF3q:,^.P[7Zwo3 ٞ:X5Z5Zc *jRk!1VlXF&b)J&O0$o!5(c[KXBk7㜥FR"Z&[le'ۀ.4<Dd #0   # A  2X joB\'<`!X joB\' ' 6ZxڝAOAg-m)`S@Tp$x0&V[ \,$rhn1#m \IA1yݙn[kXL͛7fR'CHC 5) :\(qкM:i$P>lخ}+Ƅ>+܆/4nK~hDKNc`{|fPx"alR4c"K!ڂ]_!HJ`:WX6ecy 3IJEOU\Xt͈erD@v!tsTm E= %AGW--Ո5}c;%%([Z Hp"N ֐`:'/ pU$ &'&GRCTl*ОBх:*pYM:DASYDj03v 8n/?0?ĕs?Q /8Q8Whgٌkͳe\CF5n0Cی`!CҔU)=G:9 4h4IYeIr=1,ڹ^ʦlXT:kmO:lU960=&SxbVKջTFS0UF$m%Ta_݁ ~I<9$YcQnBnanب_:ugGQ6z[P?QD[̓M*?].5UvM]ݮ\W\wӽCCx^>`־ cXIB>yŤo3D']wsȶDd I> 0   # A  22h-r3tQSI@`!h-r3tQSI17MnxڽXAoE;n % !cGVv4D (D9fr;zhz6!!Rm?WPJ |How6kx+~;3훷̚c_q (3=8b'g'Ѐ޼5"rp Y 8Nz!$cp<IL/gRYTh>|pkw|yA=IkL^C8_2k5)*UCӕCЯ},&/;%ʼ+*m#VϠ7iA'wCT2,win`;ۦd2'cSO1cY,B46-FSI=>6`Yb9Qv>9S0$l ;ۤbqw61OTYsxkV*Q"#P{8##p]d :8o~e-ԕqq5z#/;QH/B&S}+7!ߖm[<J*:y7,= Xh6_LeWTg~3u1!{mgrڠY>/nC]sF š%U5x3hp^+w6;k4r%_9j녢q[~NMNdZ,]zk=vk>-Y8*Dd ' 0  # A2 98DJvY' Z.F`!z 98DJvY' Z.E0@b8HxڵoEg'/'6 j"Di,&-GRemkkWJUpHmp!@!!BH/\҈6HCήNjZ7o>}g^2c^*>8x?;˙^sgnx[K\݃/hxpnxNӺA8 O[VW;yLeCަ͟;89MS(|œqܸs*.|N~P{mmW$z\m$;-I>Zt~- jYF@ J`(. 'ZyFu***@EGB cZ JE'"1ee0άUZԬ`[hA/-pH`PתTWA4P%0@F 8 /) 8U%(m/IxAQ$4oC9<4ª2 (3i64C)h2)3,&sRyh2ei2JKm4&A0  |IMIʣ`&sRyh1y) MdNj2M( F4iɤm s%hhCII4(3,&R49CDL` mCdФMb$R07PeRҀ*`hR{U&* "U h^IJPuZaU:BS*ZN]|c*^0id{\7/ 5pt 8i62 Vu7v-#(Hy󺮚{+܏G  &Ҧϩ:By]䟭 ^bEz73LXBVy]ĢA 0fTc*+D߈crcZEyUP6cZΙ~ߙl6UGUǏ3ŠCe:/fxv>>oWMTR%:5݌krC߉y_(F,5nvKJW/-SWWf7o W(7U6{~Fˤ0*G}$`Q mc&L"UXYJ%z<0ofbL-h~C6 pH&u`H,*Nj2ݦ۔Yuv6Cd=]W{^{c;2,{[nuHY 9f1S4_|0S 5=qZuSiY9=.ҧ,Bj7;}DwWz]9jcu88FYW+RSWomQ]#?#{######$)$*$n$''I(J())***S*******5+f++++++,*,+,s,0000T1U1_133[4\4i444444>7?7b7l7m7}77 88888889919[9\9g9i9k9:::::::==O>P>??AANCOCCCCC@DADDDDDE4E5E#?#{######$)$*$n$''I(J(***S*******5+f++++++,*,+,0000T1_13\4i444444>7?7b7l7}7888889919[9\9g9i9::::==O>P>??AANCOCCCCC@DDDDD5E?ABDFHJKMW o K#d.~74=H7LY`vд369;=@CEGILNXд48@0(  B S  ?iuu x t{ ALz&-CNit  T"["?#F###%%c%j%%% & &&'('+'**++00113344:5A566G7P7X7_77777Y8`8e8n888A9H9N9W9]<i<r<y<~<<<<<===P>X>{>>??l@s@AAAACCCCCCDDDDEE+E2E=EDETE[EoEvEEEEEEE3F:FtF{FFFFFGG H'HKK@LGLoQyQT)TVVWWWWXXRXYXW[CMMSz~.7a e Z \ mS^?#I#0011u4~444X7a7b7k789N9Z9::UUW[33333333333333333333333333# Dg @Abval!!?##,+,Q4[4\4k4m7}788\9g9::D3EEEL'L@LyLQQPVoVW[W[Elizabeth McCuneGF*wLM$p__gk%-4,8:=;">?;d?D"HM1M'Sc,+Z`\@,/R$ggg g ttt!t"t#t$t*t+t,CDEKLNVWXZU[p@pp$@pp,@pp<@p.p0p2ph@p:p<p|@pVpXp@p`p@p@@pppX@ph@UnknownG: Times New Roman5Symbol3& : Arial;Wingdings5"System"qhǡFԴF(` M. M.!24d'['[3QH(?\l*Lab Lecture for Multiple Linear Regression Todd BodnerElizabeth McCuneOh+'0 ( D P \hpx+Lab Lecture for Multiple Linear Regressionoab  Todd BodneroddNormaldElizabeth McCuneult40zMicrosoft Word 10.0@@T@B g@n] M՜.+,0 hp|  PSU.'[A +Lab Lecture for Multiple Linear Regression Title  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[]^_`abcdefghijklmnopqrstuvwxyz{|}~Root Entry FPn]Data \M1TableXPWordDocument"SummaryInformation(DocumentSummaryInformation8CompObjj  FMicrosoft Word Document MSWordDocWord.Document.89q