Jane E. Miller, PhD

The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.

Calculating interaction effects from OLS coefficients:

Interaction between two categorical independent variables

Jane E. Miller, PhD


Overview• General equation for a model with main effects

and interactions• Review: Coding of main effects and interaction

terms• Solving for the interaction pattern based on

estimated coefficients• Graphical depiction of the interaction pattern


Estimated coefficientsOLS model of birth weight (grams) βMain effect terms

Non-Hispanic black (NHB) –168Less than high school (<HS) –54High school diploma (=HS) –62

Interaction termsNHB_<HS –39NHB_=HS +18

Reference category: Non-Hispanic whites with >HS education.All variables are dummy-coded: 1 = named value, 0 = other values.


Interpreting the main effects• The main effect terms estimate the difference in birth

weight relative to those in the reference category (non-Hispanic whites with more than complete high school education).– βNHB is an estimate of the difference in intercept between

non-Hispanic black infants and those in the reference category.

– β<HS and β=HS estimate the difference in intercept between infants in the reference category and those born to mothers with less than complete high school and complete high school, respectively.

– Units are those of the dependent variable, grams.


Interpreting the interaction between race and education

• The race_education interaction tests whether the difference in birth weight for <HS versus =HS is different for non-Hispanic black infants than for their non-Hispanic white counterparts.– We calculate the overall effect for NHB and <HS as =

βNHB+ β<HS + βNHB_<HS

– If the difference in birth weight across mothers’ education categories were the same for blacks as for whites, then the interaction term βNHB_<HS = 0.


Calculating overall effect of interaction for specific case characteristics

• The general equation to calculate how a case differs from the reference category:– main effects coefficients– interaction term coefficients– values of the independent variables

= (βNHB × NHB) + (β<HS × <HS) + (β=HS × =HS) +

(βNHB_<HS × NHB_<HS) + (βNHB_=HS × NHB_=HS)

• To see which βs pertain to which cases, fill in values of variables for different combinations of race and education.


Review: Coding of main effects and interaction term variables

Case characteristics

Main effects terms Interaction termsRace Education Race & educationNHB <HS =HS NHB_<HS NHB_=HS

Non-H white & <HS 0 1 0 0 0Non-H white & =HS 0 0 1 0 0Non-H white & >HS 0 0 0 0 0Non-H black & <HS 1 1 0 1 0Non-H black & =HS 1 0 1 0 1Non-H black & >HS 1 0 0 0 0

Reference category


Cases in the reference category for both independent variables

NHB <HS =HS NHB_<HS NHB_=HSNon-H white & >HS 0 0 0 0 0

General equation to calculate how a case differs from the reference category:

= (βNHB × NHB) + (β<HS × <HS) + (β=HS × =HS) + (βNHB_<HS × NHB_<HS) + (βNHB_=HS × NHB_=HS)

Fill in values of variables for non-Hispanic whites with >HS:

= (βNHB × 0) + (β<HS × 0) + (β=HS × 0) + (βNHB_<HS × 0) + (βNHB_=HS × 0) = 0


Cases in the reference category for both independent variables

• All of the coefficients fall out of the equation for non-Hispanic whites born to mothers with >HS because each β is multiplied by a value of 0.

• Thus, cases in the reference category for both race and education have a calculated overall effect of 0. – As it should be, because there is no difference between them

and themselves!

= (βNHB × 0) + (β<HS × 0) + (β=HS × 0) + (βNHB_<HS × 0) + (βNHB_=HS × 0) = 0


Cases in the reference categoryfor 1 but not both independent variables

Fill values of variables for non-Hispanic whites with =HS into the general equation:

The equation for non-Hispanic white infants born to mothers with a high school diploma collapses to include only β=HS because all of the other coefficients are multiplied by a value of 0.

NHB <HS =HS NHB_<HS NHB_=HSNon-H white & =HS 0 0 1 0 0

= (βNHB × 0) + (β<HS × 0) + (β=HS × 1) + (βNHB_<HS × 0) + (βNHB_=HS × 0) = β=HS


Cases not in the reference category for either independent variable

Fill in values of variables for non-Hispanic blacks with =HS:

NHB <HS =HS NHB_<HS NHB_=HSNon-H black & =HS 1 0 1 0 1

Thus, the equation for non-Hispanic black infants born to mothers with a high school diploma collapses to include

the main effects terms for both βNHB and β=HS andthe interaction term βNHB_=HS.

All the other βs fall out because they are multiplied by 0.

= (βNHB × 1) + (β<HS × 0) + (β=HS × 1) + (βNHB_<HS × 0) + (βNHB_=HS × 1)

= βNHB + β=HS + βNHB_=HS


Equations to calculate overall effect

Subgroup

Equation

ValueSymbols Estimated β

Non-Hisp. white, <HS = β<HS = –54 = –54

Non-Hisp. white, = HS = β=HS = –62 = –62

Non-Hisp. white, >HS NA (ref cat) NA 0

Non-Hisp. black, <HS = βNHB + β<Hs + βNHB_<HS = (–168) + (–54) + (–39)

= –261

Non-Hisp. black, =HS = βNHB + β=Hs + βNHB_=HS = (–168) + (–62) + (+18)

= –212

Non-Hisp. black, >HS = βNHB = –168 = –168

Difference in birth weight (grams) compared to infants born to non-Hispanic white women with more than a high school education = reference category.


Interpreting the sign of the interaction terms: NHB_<HS

• βNHB_<HS = –39, meaning that infants in that group have lower estimated birth weight than would be predicted from their race and mother’s education alone, based on the main effects (βNHB + β<HS)

– All three βs (both main effects and interaction) have negative signs, meaning that they cumulate to a large deficit in birth weight for NHB <HS.

– The β on the interaction term reinforces (adds to) the predicted deficit based on race and education alone.


Calculating overall effect for non-Hispanic blacks with <HS education

–54–39

–54

–39

–168

–168

= βNHB + β<HS + βNHB_<HS = (–168) + (–54) + (–39) = –261

βNHB =

β<HS = βNHB_<HS =

Compared to infants born to non-Hispanic white women with more than a high school education = reference category.


Interpreting the sign of the interaction term: NHB_=HS

• On the other hand, βNHB_=HS = +18, meaning that infants in that group have higher estimated birth weight than would be predicted from their race and mother’s education alone, based on the main effects (βNHB and

β=HS).– Both main effects terms have negative signs, but the

interaction term has a positive (opposite) sign, so it partially offsets the deficit in birth weight predicted based on race and education alone.


Calculating overall effect for non-Hispanic blacks with =HS education

–62

–62

+18

–168

–168

= βNHB + β=HS + βNHB_=HS = (–168) + (–62) + (+18) = –212

βNHB =

β=HS = βNHB_=HS =

Note that interaction term has the OPPOSITE SIGN of the two main effects, partially offsetting their two negative effects on birth weight with a positive effect.

Compared to infants born to non-Hispanic white women with more than a high school education = reference category.

+18


Overall effects of race and mother’s education on birth weight

Solid = main effect term.Striped = interaction of education level w/ NHB.Compared to non-Hispanic whites with >HS education.

= -261


Predicted value and the intercept term• The intercept (or “constant”) term estimates the

value of the dependent variable Y for cases in the reference category.

• To calculate the predicted value of Y for each combination of the Xi, – add the estimated coefficient for the intercept (β0) to

the βs for each variable that pertains to the category of interest.


Examples: Predicted value• For instance, β0 = 3,042.8.

• So infants who are in the reference category for all variables are estimated to weigh 3,042.8 grams.

• This includes non-Hispanic whites born to women with >HS.– Reference category for race and mother’s education

• Those born to Mexican American women with less than a high school education: • β0 + βMA + β<HS + βMA_<HS

= 3,042.8 + (–104.2) + (–54.2) + 99.4 = 3,039.8 – 59.0 = 2,983.8 grams.


Use a spreadsheet to calculate and graph the interaction

• Spreadsheets can – Store

• The estimated coefficients• The input values of the independent variables• The correct generalized formula to calculate the predicted

values for many combinations of the IVs involved in the interaction

– Graph the overall pattern• See spreadsheet template and podcast


Summary• Calculating the overall shape of an interaction

pattern requires adding together the pertinent main effects and interaction term coefficients for each possible combination of the two categorical IVs in the interaction.– A spreadsheet can be helpful for storing and organizing

the coefficients and formulas.• Depending on the respective signs of those βs, the

interaction can either amplify or dampen the main effects on the component variables.


Suggested resources• Chapter 16 of Miller, J.E. 2013. The Chicago

Guide to Writing about Multivariate Analysis, 2nd Edition.

• Chapters 8 and 9 of Cohen et al. 2003. Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences, 3rd Edition. Florence, KY: Routledge.


Supplemental online resources• Podcast on creating interaction term variables• Spreadsheet template for calculating overall

effect of an interaction between two categorical variables.


Suggested practice exercises• Study guide to The Chicago Guide to Writing

about Multivariate Analysis, 2nd Edition.– Questions #3 and 5 in the problem set for Chapter 16– Suggested course extensions for Chapter 16

• “Applying statistics and writing” exercise #1.


Contact information

Jane E. Miller, [email protected]

Online materials available athttp://press.uchicago.edu/books/miller/multivariate/index.html

mailto:[email protected]

https://hsb.rutgers.edu/owa/redir.aspx?C=7303dc1e3af340ffbabeb7d8d0e92de8&URL=http%3A%2F%2Fpress.uchicago.edu%2Fbooks%2Fmiller%2Fmultivariate%2Findex.html

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Jane E. Miller, PhD