dummy variables and multiplicative regressionpsfaculty.ucdavis.edu/bsjjones/dummy.pdfbradford s....

Bradford S. Jones, UC-Davis, Dept. of Political Science

Dummy Variables and Multiplicative Regression

Brad Jones1

1Department of Political ScienceUniversity of California, Davis

February 24, 2010

Jones POL 212: Research Methods

Bradford S. Jones, UC-Davis, Dept. of Political Science Today: Dummy variables and multiplicative regression

Today: Dummy variables and multiplicative regression



Dummy Variables

I Begin with garden variety regression:

Yi = β0 + β1X1i + β2X2i ,

I If X1 and X2 are continuous, then the natural interpretationfor a slope coefficient is forthcoming: for an incremental (orunit) change in X , the expected value of Y changes (increasesor decreases) by about β amount.

I What about “qualitative” or nominal outcomes? Such avariable may represent gender, racial classifications, partyclassifications, or so on. Fortunately, such covariates pose fewproblems for the regression model. The main issues involvecoding categorical variables and interpreting them.



Dummy Variables

I Consider only dichotomous categorical variables:

Yi = β0 + β1D1i ,

the predicted value of Y when D = 1 is given by

β0 + β11.

When D = 0, the predicted value of Y is given by

β0 + β10

or β0.I In a bivariate model with one dummy variable, there will only

be two predicted values.I Note also that since our dummy variable has a natural 0 point

(which we arbitrarily defined), the constant term as a naturalinterpretation: it gives us the predicted Y for the conditionwhen D = 0.



Dummy Variables

I Dummy variables, constructed in this way, give nothing morethan differences in intercepts.

I In the previous model, the coefficient β1 tells us the expectedincrease (if the coefficient is positive) in Y for observationswhere D = 1.

I Note also that since the model above has no continuous (orquantitative) covariates, there are no slope coefficients. Thus,if you plotted Y with respect to D, you would simply get “twodots.”

I Sometimes, we include categorical covariates to account forvariation in Y that is a function of some group-specific orattribute-specific factor.



Dummy Variables

I Consider a model with one quantitative variable and onedummy variable:

Yi = β0 + β1X1i + β2D1i ,

where β1 is the coefficient for our quantitative variable X andβ2 is the coefficient for our dummy variable D.

I Unlike the previous model which yielded only differences inintercepts, this model will give us different intercepts butparallel slopes.

I Suppose that D = 1, then the regression response function forthis condition is:

β0 + β1X1i + β21.



Dummy Variables

I Rearranging, we find the following:

(β0 + β2) + β1X1i .

In contrast, when D = 0, the regression response function isgiven by

β0 + β1X1i + β20,

which is equivalent to

(β0) + β1X1i .

I It is clear, the “effect” of the dummy variable in the model isto act as a “contrast” or “offset” for the two groups.

I When D = 1, the coefficient β2 represents the increase in they -intercept attributable to D; when D = 0, the constant term(i.e. β0) gives us the intercept for this condition.

I The important thing to note, however, is the following: theeffect of X1 on Y is the same for both groups.



Dummy Variables

I The dummy variable only reflects a difference in interceptswhile the slope coefficient will be equivalent for the twogroups.

I Example using some simulated data.

I Fitted model:y = 78.9 = .95x

I Fitted model and fitted values plot (next two slides)



Dummy Variables

Estimate Std. Error t value Pr(>|t|)(Intercept) 78.8394 1.6539 47.67 0.0000

x1 0.9520 0.0518 18.39 0.0000



Dummy Variables



Dummy Variables

I Model has standard interpretation as with any OLS model.

I No attempt is made to account for group differences.

I One way to approach this might be for separate models bygroup (model for D = 1 and for D = 0)

I Model 1 then Model 2:



Dummy Variables


x1 0.9500 0.1335 7.11 0.0057


x1 0.9340 0.1178 7.93 0.0042



Dummy Variables



Dummy Variables

I Apparently no group-based differences.

I Separate models yield relatively similar slopes . . .

I though we still have two models.

I Incorporate D into the model returns:



Dummy Variables


x1 0.9444 0.0840 11.24 0.0000d 0.3794 3.1682 0.12 0.9080



Dummy Variables



Dummy Variables

I The single model returns fitted values essentially identical tothe separate models approach.

I There are no apparent differences due to the “groups.”

I “Controlling” for the group is superfluous: x1 gives similarslopes for the two groups.

I Suppose we went the other way and ignored x1?

I That is: y = β0 + β1D



Dummy Variables


d 27.2000 8.5112 3.20 0.0127



Dummy Variables

I d is significantly different from 0 which suggests substantialgroup differences.

I d is picking up on the fact that the average value of Y | D ishigher for one group versus the other.

I However, the linear association between Y and X1 (which isnot included) is virtually identical across the two groups.

I Once we account for the variable X , the factor D has noimpact.∗

I Another example.

I Consider this plot:



Dummy Variables



Dummy Variables

I Two sets of observations both having a positive associationbetween Y and X2.

I Correlation for “group 1”: .97; for “group 2”: .99

I Individually, knowing X2 for each group almost perfectlypredicts the outcome Y .

I Try the separate regressions approach again(y = f (X2 | D = 1) and y = f (X2 | D = 0)



Dummy Variables


x2 0.9500 0.1335 7.11 0.0057


x2 0.7000 0.0306 22.91 0.0002



Dummy Variables



Dummy Variables

I Separate regressions obviously confirm the correlationalanalysis.

I Since we know more data are better than less data, supposewe pool the groups (like we did in the previous example).

I That is: regress y on x2

I The model:



Dummy Variables


x2 −0.2182 0.1516 −1.44 0.1879



Dummy Variables

I A reversal result is obtained.

I Individually, the groups exhibit a strong, positive association.

I When pooled, we observed a weak, negative association.

I This is an example of Simpson’s Paradox . . . a reversal result.

I Inclusion of relevant information is omitted in this example.

I And that information would be information relative to thegroup.

I First consider fitted values from the previous models:



Dummy Variables



Dummy Variables

I Simpson’s Paradox: When data from two or more groups arecombined, if the direction of the relationship changes in thecombined data set, then this is Simpson’s paradox. Moresuccinctly, as Fox puts it: “that marginal and partialrelationships can differ in sign is called “Simpson’s Paradox.”The marginal relationship between x2 and y is negative butthe partial relationship (controlling for d) is positive.

I Suppose we account for d in the regression?

I y = β0 + β1X + β2d



Dummy Variables

Estimate Std. Error t value Pr(>|t|)(Intercept) −0.9500 8.8414 −0.11 0.9174

x2 0.8250 0.0791 10.43 0.0000d 84.9500 5.9704 14.23 0.0000



Dummy Variables

I After accounting for d (or “controlling” for it as this issometimes called), the positive relationship between x2 and yis retrieved.

I This model will produce parallel slopes, which is equivalent tosaying, there will be two regression functions, one for d=1,one for d=0.

I Fitted function illustrates the parallel slopes result.



Dummy Variables



Dummy Variables

I After accounting for d (or “controlling” for it as this issometimes called), the positive relationship between x2 and yis retrieved.

I This model will produce parallel slopes, which is equivalent tosaying, there will be two regression functions, one for d=1,one for d=0.

I Fitted function illustrates the parallel slopes result.



Multiplicative Terms

I The parallel slopes property is a function of the model beinglinear in the parameters.

I Sometimes this may not hold.

I For example, across groups, the intercept and the slope mayvary.

I This may give rise to “multiplicative” models.

I Important: in the general case, multiplicative models willyield much different coefficients than non-mulitplicativemodels.

I Continue with illustration using simulated data.





x1 0.9520 0.0518 18.39 0.0000




I To this point, the linear models we have considered have allbeen interpreted in terms of “additive” relationships.

I That is, the relationship between two covariates, X1 and X2 inthe context of the model

Y = β0 + β1X1 + β2X2,

is additive.

I This implies that the partial effect of each independentvariable is the same, regardless of the specific value at whichthe other covariate is held constant.

I Model we estimated returns:

Y = 88.81− .766(X1)




I X1 is a quantitative covariate.

I The interpretation of the model is standard: for a 1 unitincrease in X1, the expected value of Y decreases by about.766.

I Fitted function:




I Now, suppose there were two distinct subpopulations relatedto Y .

I To account for these two groups, we create a dummy variablecalled D1 such that D1 = 1 denotes the first group andD1 = 0 denotes the second group.

I Estimate model with X1 and D1.





x1 −0.6652 0.1458 −4.56 0.0003d1 −5.9046 7.9237 −0.75 0.4663




I The coefficient for D1 is the partial regression coefficient forthe dummy variable.

I This is interpreted as giving us the expected change in Ywhen D1 = 1; that is, for group 1, the expected value of Ydecreases by about -5.91.

I Here, D1 is going to give us an offset between the two groups.

I Must produce parallel slopes (see fitted function):




I The idea of “additivity” is important here: our interpretationof X1 does not change as the value of D1 changes.

I Similarly, our interpretation of D1 does not change as afunction of X1.

I The relationship between D1 and Y is accounted for by theoffset in the two parallel slopes (which is equal to -5.91).

I The “effects” are additive because to understand the expectedvalue of Y for both variables, we simply “add” one coefficientto the other.

I The expected value of Y given D1 = 1 and X1 = 53.1 (itsmean) is Y = 86.43 +−.665(53.1)− 5.91(1) which is 45.21.




I This property ensures parallel slopes when one of theregressors is dummy variable (the slope will be offset by thevalue of the dummy variable).

I It’s all good if the property holds.

I This would imply that the effect of some covariate on thedependent variable would be influenced by changes in thevalue of another covariate; that is, the interpretation of, say,X1 on Y would be conditional on some other covariate, sayD1.

I Conditional relationships, if they exist (and they often dogiven the kinds of theories we posit) are not immediatelymodeled in the additive model.




I Suppose we reestimated our regression model, but this time,estimated a separate model for observations in group 1 andobservations in group 2.

I For group 1, our model gives us

Y = 62.11− .43(X1),

and for group 2, our model gives us

Y = 118.35− 1.81(X1).

I For group 1, the slope coefficient tells us that for a unitchange in X1, the expected change in Y is -.43; for group 2,that expected change is -1.81 (a little over 4 times that ofgroup 1).

I Note that if the additivity property held, the relationshipbetween X1 and Y would be nearly the same for the twogroups. (Why?)




I To see this, return to our definition of additivity: the partialeffect of each independent variable is the same, regardless ofthe specific value at which the other covariate is heldconstant.

I In the “two models” approach, we are essentially holding D1

constant and we can see our interpretation of X1 changes,depending on which value D1 assumes.

I This is suggestive that the additivity property may not hold.

I Consider the fitted functions:




I The point to note is how the slopes change depending onwhich group we are looking at.

I The slope for group 1 is considerably flatter than that forgroup 2 (which is consistent with our separate regressionmodels).

I In combining the two graphs, this point is made even clearer:the slopes are not parallel and the interpretation of X1 is notthe same for each group.

I This suggests that the relationship between X1 and Y may beconditional on D1; that is, D1 may moderate or condition therelationship.




I To see this, suppose we estimated a model treating X1 as afunction of D1. In doing this, we obtain

X1 = 28 + 50.2(D1),

where the the expected value of X for group 1 is 78.2; theexpected value of X for group 2 is 28.

I The average level of X1 is much higher for group 1 than group2.

I It seems that X1 is related to D1.

I In terms of our additivity assumption, it will not be the casethat we can correctly interpret X1 by holding D1 constant: thepartial effect changes, depending on which group we’relooking at.




I This suggests that the relationship between D1, X1 and Y ismultiplicative.

I That is, the effect of X1 on Y may be multiplicatively higherin one group than when compared to the second group.

I This leads to consideration of an interactive model, which hasthe form

Y = β0 + β1D1 + β2X1 + β3(D1X1),

where the last term denotes the interaction term ormultiplicative term.

I It is easy to see where the model gets its name: we areinteracting, multiplicatively, X1 with D1.




I Note what the interaction term is giving you. This term is 0for group 2 (why?) and nonzero for group 1; hence, you’regetting an estimate of the slope of X1 for D1. For group 2,the relationship between X1 is simply given by the coefficientfor X1. Thus

YG1 = β0 + β1D1 + (β2 + β3)X1,

gives us the regression function for Group 1 and

YG2 = β0 + β2X1,

gives us the regression function for Group 2.I From this model, we will not only get differences in intercepts

(due to the offset between the two groups), but we will alsoget differences in the slope.

I For Group 1, the slope is β2 + β3 and for Group 2, the slope isβ2.




I Multiplicative term created by: D1 × X1.

I Estimate model with term gives:

Y = 118.35− 56.24D1 − 1.81X1 + 1.38D1X1.

I This model is not interpretable as an additive model (thoughit is a linear model).

I The reason is that the impact of X1 on Y is conditional onwhich group one is referencing.

I NOTE: sometimes analysts will refer to the coefficient for X1

as a “main effect.”

I That is, sometimes the interpretation is given that thecoefficient for X1 represents the constant effect of X1. It doesnot!!





x1 −1.8054 0.1843 −9.79 0.0000d1 −56.2423 8.4892 −6.63 0.0000

x1:d1 1.3756 0.2025 6.79 0.0000




I In the case of multiple regression, the “main effect”terminology is not justified although it is commonly assumedthat the coefficients for D1 and X1 are “main effects.”

I In the presence of an interaction, these coefficients in noinstance represent a constant effect of the independentvariable on the dependent variable.

I The use of the term main effect implies that the coefficientsare somehow interpretable alone, when they actually representa portion of the effect of the corresponding variable on thedependent variable.

I A better terminology is “constituent” terms (i.e. theconstituent terms for the multiplicative term are D1 and X1.




I Model decomposition.

I For Group 1, the regression function is given by

Y = 118.35− 56.24D1 + (−1.81 + 1.38)X1,

and for Group 2, the regression function is given by

Y = 118.35− 1.81X1.

I Consider the fitted functions:




I Here we see nonparallel slopes.

I It should be clear that the results we obtain using theinteraction term are identical to the results obtained using theseparate regressions approach (refer to previous plot).

I Not coincidental!

I In the interactive model, we are conditioning the relationshipof X1 on D1.

I This, in effect, gives us separate models within a single model.

I This is the intuition of an interactive model.




I Some observations:

I Recall in the model where we treated X1 as a function of D1

(i.e. regressed X1 on D1).

I There, we got the expected value of Y for each group.

I Consider the previous plot but this time inserting a verticalreference line at the mean of X1 for each group.




I The vertical reference line goes through the points 28, whichcorresponds to the mean of X1 for Group 2 and 78.2, whichcorresponds to the mean of X1 for Group 1.

I This figure illustrates the point made in the regression of X1

on D1: the separate slopes pass through the group means,which are equivalent to the regression estimates from thesubmodels considered.

I Extensions (but first start with the dummy variableinteraction).




I Posit the following model

Y = β0 + β1D1 + β2X1 + β3D1X1,

where the last term is an interaction term between thecovariates X1 and D1.

I The relationship between X1 and Y is conditional on thepresence or absence of D1.

I This gives rise to two submodels: one for group 1 and one forgroup 2. For group 1, the regression function is given by

Y = β0 + β11 + β2X1 + β31X1

= β0 + β1 + (β2 + β3)X1.

and for group 2, the regression function is given by

Y = β0 + β10 + β2X1 + β30X1

= β0 + β2X1.




I For group 1, the slope is given by (β2 + β3) and the interceptis given by (β0 + β1)

I For group 2, the slope is given by β2 and the intercept is givenby β0.

I In no case is the “effect” between X1 and Y constant acrossgroups; hence there is no main effect.

I This is all easy to see in the case of a dummy variable (andeven easier to see if you interact two dummy variablestogether); however, what about the case with two quantitativevariables?




I Suppose we are interested in the following model,

Y = β0 + β1X1 + β2X2 + β3X1X2,

where the two covariates are quantitative variables and thelast term denotes the interaction between them.

I The intuition? The slope of X1 on Y is conditional on X2 andthe slope of X2 on Y is conditional on X1.

I Since there are no main effects, the relationship of oneindependent variable on the dependent variable is conditionalon the other independent variable.




I The question arises, naturally, as to how to interpret thismodel?

I In order to derive the conditional regression function for Yand X1 conditional on some specific value of X2 we obtain

Y = β0 + β2X2 + (β1 + β3X2)X1.

I The conditional regression function for Y and X2 conditionalon some specific value of X1, we obtain

Y = β0 + β1X1 + (β2 + β3X1)X2.




I In these models, the intercept gives us the expected value ofY when all the covariates are equal to 0.

I In the first model, the coefficient for β3 gives us the change inthe slope of Y on X1 associated with a unit change in X2.

I In the second model, β3 gives us the change in the slope of Yon X2 associated with a unit change in X1.

I Key point: the slope between one covariate is governed by theother covariate.

I To evaluate the impact one covariate has on the other, it isuseful to note that the coefficients β1 and β2 denote thebaselines around which the slopes vary.

I This is easy to see (on next slide).




I Suppose X2 = 0, then the first model above reduces to

Y = β0 + β20 + (β1 + β30)X1

= β0 + β1X1.

I Suppose X1 = 0, then the second model above reduces to

Y = β0 + β10 + (β2 + β30)X2

= β0 + β2X2.

I In both cases, the models represent the “baseline” case fromwhich the conditional slope coefficients can be evaluated.




I In the first model, the coefficients β0 and β1 give the interceptand slope for the regression of Y on X1 when X2 is equal to 0.

I In the second model, the coefficients β0 and β2 give theintercept and slope for the regression of Y on X2 when X1 isequal to 0.

I Note the conditional nature of the regression model wheninteractions are applied.

I In the standard least squares model without interactions, thecoefficients for the covariates tell us something much differentthan they do in the interactive case.

I In the interactive case, the coefficients are conditional on thevalue of some covariate; hence, the effect of one covariate, sayX1, on Y is not constant across the full range of values takenby the other covariate, X2.




I Note in passing: Suppose X1 = 0, then the first model abovereduces to

Y = β0 + β2X2 + (β1 + β3X2)0

= β0 + β2X2.

I Now suppose X2 = 0, then the second model above reduces to

Y = β0 + β1X1 + (β2 + β3X1)0

= β0 + β1X1.

I Implications??

I The reported regression coefficient β1 and β2 only has aunique interpretation when X2 or X1 is 0.




I Because of this conditional property, coefficient estimatesfrom an interactive model will in general be much differentthan estimates derived from a model possessing the additivityproperty.

I Note also, the importance of “0” in model decomposition.

I The standard error for the reported constituent term will onlymake sense at the point at which the conditioning variableassumes the value “0” (why?).

I If the Xk (constituent variables) do not have a natural 0point, then the reported standard errors on the constituentterms are utterly irrelevant.

I Implications?




I Having a 0 point was convenient because it gave thecoefficients a natural interpretation: the baseline againstwhich changes in the conditional slope could be evaluated.

I In the absence of a meaningful 0 term, the baseline “effects”are never observed in the data.

I Hence, the coefficient estimate (which is interpreted in termsof the covariate having a 0 point) and its standard error canbe misleading, sometimes very misleading, because it is givingyou the conditional relationship under a condition that doesn’tactually hold in the data.

I Common Mistake: Analysts conclude constituent termparameters are “insignificant” based solely on inspectingregression output.

I A small t-value at the point of 0 on the conditioning variabledoes not mean the t will be larger over other ranges of theconditioning variable.




I Some observations: proper interpretation of the multiplicativeregression model requires a lot more work.

I To see why, motivate it with the following model:

Y = β0 + β1X + β2Z + β3XZ + ε

(some of what follows comes from Brambor et al [2006]).

I Z may be discrete or continuous.

I Suppose Z is continuous for the moment.

I Useful to consider “two-sided conditioning” in themultiplicative model.




I In our model, β1 gives the one-unit change in X on theexpected value of Y only when Z = 0.

I That is: (∂(Y )/∂(X1) | Z = 0) = β1

I Simplifying:

Y = β0 + β2Z + (β1 + β3Z )X1

I If Z is a binary variable, the simplification is morestraightforward:

Y = (β0 + β2) + (β1 + β3)X1

noting that Z is implied to be 1.I In either model, β1 only gives the “average effect” of X1 on Y

when Z = 0.I Also in either model, the relationship between X and Y is

conditioned by Z .




I The “flip-side”: β2 gives the one-unit change in Z on theexpected value of Y only when X1 = 0.

I That is: (∂(Y )/∂(Z ) | X1 = 0) = β2

I Simplifying:

Y = β0 + β1X1 + (β2 + β3X1)Z

I Again: β1 only gives the “average effect” of X1 on Y whenZ = 0.

I Also: the relationship between Z and Y is conditioned by X1.




I A multiplicative hypothesis: (language from Brambor et al)“An increase in X is associated with an increase in Y when Zis present, but not when Z is absent.”

I More general: “An increase in X is associated with anincrease in Y as Z departs from 0.”

I In other words, the relationship between X and Y isconditional on the value Z assumes.

I If both the constituent terms are continuous, the model is“fully” conditional meaning we cannot say anything about X1

(or Z ) without knowing something about Z (or X1).

I Common Mistake: Analysts often proceed as giving anunconditional interpretation. This cannot be sustained by themodel that is posited.




I Estimating a model (Quick illustration)

I Let X1 and X2 be two quantitative variables.

I Estimate: Y = β0 + β1X1 + β2X2

I Using the simulated data, we return:

Y = 6.43 + .18X1 + .99X2 − .011X1X2.





x1 0.1770 0.0586 3.02 0.0081x2 0.9860 0.0527 18.70 0.0000

x1:x2 −0.0108 0.0014 −7.53 0.0000




I Question: what do the parameters for X1 and X2 mean?

I Question: suppose either of these parameters are no differentfrom 0?

I Question: suppose one is and one is not?

I We will return to these questions in a bit.

I Back to some quick interpretation.




I Because the relationship between X1 and Y will beconditional on X2 (and vise versa), we will want to consider“conditional slope” coefficients.

I That is, the slope varies over the range of the conditioningvariable.

I Easy to see if we rearrange our model: the conditional slope(the term in parentheses) for Y regressed on X1 conditionalon X2 is

Y = 6.43 + .99X2 + (.18− .011X2)X1.

and the conditional slope for Y regressed on X2 conditionalon X1 is

Y = 6.43 + .18X1 + (.99− .011X1)X2.




I It should be clear that the “average effect” for X1 on Y isnever unconditionally related to X2. This is a fundamentaldifference between this and the additive model.

I For X1, the range is 21 to 100; for X2, the range is 40 to 100.

I Suppose we compute the predicted regression function of theslope of Y on X1 for the maximum and minimum and meanvalues of X2?

I Plot the predicted functions (next slide).




I Note is the slope of the three lines, which representhypothetical regression functions.

I These lines are created by substituting 40, 65, and 100 intothe first model above and then allowing X1 to freely vary.

I This gives us a sense of the conditional relationship and ittells us that as X2 decreases, the slope between Y and X1

tends to decrease, that is, flatten out.

I To see this numerically, note that the estimated slope forX2 = 100 is about -.92; for X2 = 40, the estimated slope isabout -.26. The dots in the graph represent the actualpredicted values of Y given the actual values of X1 and X2.




I Repeat the same exercise for the slope of Y on X2.

I The conclusion here is that as X1 increases, the slope betweenY and X2 tends to decrease (it becomes negative for theupper bound on X1).

I The range of the estimated slopes is -.11 (for X1 = 100) to.76 (for X1 = 21).

I Plot the function. Again, the dots represent the actualpredicted values of Y given the actual values of X1 and X2.

I Important! I’ve used the phrase “two-sided” conditioning butthere is obviously only 1 predicted value for any coordinate(X1,X2).




I The point of this exercise is to note that inclusion of, andinterpretation of interaction terms is straightforward.

I Yet people mess this up all the time.

I There are caveats: you need to be careful in interpreting theresults.

I Plots like the ones shown above are problematic becausemany (most!) of the points on the predicted lines are notactual data points—they are hypothetical.

I Side-trip into problem-land.




I Return to a model with a dummy variable D and aquantitative variable X .

I What follows is generalizable to all settings.

I Suppose we estimate a model with X1D1 included as apredictor?

I Further, imagine the regression coefficient for D1 is nodifferent from 0.

I Thus instead of Y = β0 + β1X1 + β2D1 + β3X1D1 weestimate Y = β0 + β1X1 + β3X1D1.




I Rearranging, we fit: Y = β0 + β1X1 ⇐⇒ D1 = 0 andY = β0 + (β1X1 + β3)X1 ⇐⇒ D1 = 1.

I This approach is not infrequently taken in the literature.

I BUT, it is almost always a bad idea!

I Consider the fitted function (using the simulated data andvariables X1 and D1).





x1 -1.0629 0.2747 -3.87 0.0012x1:d1 0.2049 0.1855 1.10 0.2848




I Implications?

I Omitting β2 forces the regression lines to meet at a commony -intercept.

I This is an awkward and probably unnatural functional form:two groups start at the same place and then diverge.

I Also makes the strong assumption that you unequivocallyknow β2 = 0 in the population.

I Suppose now you keep β2 but omit β1?





d1 -5.6909 17.2942 -0.33 0.7461d1:x1 -0.4298 0.2149 -2.00 0.0618




I We now have an equally awkward functional form: whenD1 = 1, the slope is non-zero; when D1 = 0, the slope is 0.

I Again note: this is NOT a function of the “real world” but afunction of the model’s specification.

I You are assuming you know that in the population there is norelationship between X1 and Y when condition D1 is absent.

I This is a strong assumption and one not usually justified.

I Despite temptation to do otherwise, always include theconstituent terms.




I The marginal “effect” or conditional slope is (β1 + β3X2).

I It is clear then that the relationship between Y and X1 isgoing to be conditioned by the level of X2.

I If the slope varies, the standard error must vary as well.

I This means the following: simply reporting coefficients from amultiplicative model (especially one where the two constituentterms are quantitative variables) will be inadequate.

I Much more work is needed.




I Understand that the standard error reported to you is thestandard error for the case when X1 and X2, respectively, arezero.

I That is, the standard errors are only applicable to a particularrange of the interactive effect.

I It is not at all uncommon to estimate a statistical interactionand find that the standard errors, relative to the coefficient,are very large, and indeed, statistically insignificant. Thisinsignificance can be misleading (although it won’t always bemisleading).

I Just as the relationship between X1 and Y in an interactivemodel is conditional on X2, so to are the standard errors.




I Because multiple terms are involved in the multiplicativeeffect, the standard error must be modified to account for thevariances and covariances of the β1 and β3.

I The general formula to compute the standard errors of theconditional slopes is given by

s.e.β1+β3=

√var(β1) + X 2

2 var(β3) + 2X2cov(β1β3),

for the slope of Y on X1 conditional on X2.

I This may also be notated as σ ∂Y∂X

.

I Note also that:

s.e.β2+β3=

√var(β2) + X 2

1 var(β3) + 2X1cov(β2β3),

is the slope of Y on X2 conditional on X1.




I Note that this formula is not a constant; it is determined bythe level of the covariate upon which the slope is conditioned.In the first case, this is X2; in the second case, this is X1.

I Illustration using the simulated data.I Recall that we estimated the following model:

Y = 6.43 + .18X1 + .99X2 − .011X1X2.

I The conditional slope for Y regressed on X1 conditional on X2

was given by

Y = 6.43 + .99X2 + (.18− .011X2)X1,

and the conditional slope for Y regressed on X2 conditionalon X1 was given by

Y = 6.43 + .18X1 + (.99− .011X1)X2.




I The quantity of primary interest is the term in the last twoequations in parentheses.

I This gives us the conditional slope or marginal effect.

I What we want to do is compute the standard error for theseconditional slopes, which will vary as X2 (or X1) varies.

I Looking at the first submodel, suppose that X2 is set at 100.

I The conditional slope is about -.90 (rounding).

I The question to ask is whether or not this conditional slopecoefficient is different from 0?




I To answer this question, we need to compute the standarderror.

I In order to compute the standard error, we need to back outof our coefficient estimates, the variances and covariances ofthe parameters.

I In R use the syntax: vcov(lm object):



> vcov(idmod4)

(Intercept) x1 x2 x1:x2

(Intercept) 29.24396789 -1.165203e-01 -2.608762e-01 -1.918980e-03

x1 -0.11652031 3.430211e-03 2.078236e-03 -6.691408e-05

x2 -0.26087616 2.078236e-03 2.779359e-03 -1.065934e-05

x1:x2 -0.00191898 -6.691408e-05 -1.065934e-05 2.042136e-06




I Using elements from this matrix, obtain the followingvariances and covariances that can be substituted into theformula above.

I For the slope of Y on X1 conditional on X2 = 100:

s.e.β1+β3=

√.0034 + 1002(.000002042) + 2(100)(−.000067),

I This returns a standard error estimate of about .102. Giventhe conditional slope estimate for this case was about -.92, Isee that the t ratio is about -9, which would be significant atany level.

I Visualize this: I plot the conditional slope estimates as well asthe upper and lower 95 percent confidence limits around theestimate.




I As can be seen, the slope of Y on X1 conditional on X2 is ingeneral more precisely estimated than the slope of Y on X2,conditional on X1.

I The major point to understand here is that if you are going tointerpret these interactions, you need to be aware that therange of effects are not all going to be statistically differentfrom 0.

I To illustrate these points further, we could compute the tratios for each of the predicted slope coefficients.

I This could be done easily be deriving the conditional slopeand dividing it by its standard error.

I As these are t ratios, any t exceeding the critical t would bestatistically significant.

I For these data, I set α = .05 and compute the two-tail criticalt, which is 2.12.




I In the first figure, I find that all of the estimated t ratiosexceed the critical t of 2.12.

I This implies that the conditional slopes are statisticallydifferent from 0, and implies that across the full range of X2,the interaction effect seems to be significant.

I In the second figure, we see that once X1 exceeds 75 (or so),the estimated t ratios drop below the critical threshold (whichis denoted by the horizontal reference line at 2.12).

I For this range of data, the statistical interaction is notdifferent from 0.

I This is illustrative of the larger point that the statisticalinteraction need not be significant across the full range of theconditioning variable, in this case, X1.




I Useful resource: Matt Golder (soon-to-be at PSU) butcurrently at Florida State has very useful Stata modules tocompute marginal effects and standard errors.


dummy variables and multiplicative regressionpsfaculty.ucdavis.edu/bsjjones/dummy.pdfbradford s....

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