1 / 32

Jane E. Miller, PhD

Calculating interaction effects from OLS coefficients: Interaction between 1 categorical and 1 continuous independent variable. Jane E. Miller, PhD. Overview. General equation for a model with main effects and interactions Coding of main effects and interaction terms

maia-watts
Download Presentation

Jane E. Miller, PhD

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Calculating interaction effects from OLS coefficients: Interaction between1 categorical and 1 continuous independent variable Jane E. Miller, PhD

  2. Overview • General equation for a model with main effects and interactions • Coding of main effects and interaction terms • Solving for the interaction pattern based on estimated coefficients • Intercept • Slope • Graphical depiction of the sum of coefficients for particular combinations of the independent variables

  3. Review: Contingency of coefficients in an interaction model Y = β0 + β1X1 + β2X2 + β3X1 _X2, • Inclusion of the interaction term X1_ X2means that the βis on the main effects terms X1 andX2no longer apply to all values of X1 and X2. • The main effects and interactions βis for X1 and X2 are contingent upon one another and cannot be considered separately.

  4. Review: Implications for interpreting main effects and interaction βs Y = β0 + β1X1 + β2X2 + β3X1 _X2, • In the interaction model: • β1 estimates the effect of X1 onY when X2 = 0, • β2 estimates the effect of X2 on Y when X1 = 0, • β3 must also be considered in order to calculate the shape of the overall pattern among X1, X2, and Y. • E.g., when X1 and X2 take on other values.

  5. Review: Some possible patterns of association between IPR, race, and birth weight Blacks & whites have sameslopebut differentinterceptsof IPR/BW curves No racial difference in IPR/BW relation: interceptand slopesame for blacks & whites. BW BW White Black IPR IPR Blacks & whites have differentslopeandinterceptsof IPR/BW curves Blacks & whites have sameinterceptbut different slope of IPR/BW curves BW BW IPR IPR

  6. General equation for predicted value of DV based on an interaction model • The general equation to calculate the predicted value of the dependent variable includes • main effects coefficients • interaction term coefficients • values of the independent variables = β0+ (βNHB × NHB) + (βIPR× IPR)+(βNHB_IPR× NHB_IPR)

  7. Calculating overall effect of interaction for specific case characteristics = β0+ (βNHB × NHB) + (βIPR× IPR)+(βNHB_IPR× NHB_IPR) • Each coefficient is multiplied by the value of the associated variable for cases with the characteristics of interest. • To see which coefficients pertain to which cases, fill in values of variables for different combinations of race and the income-to-poverty ratio (IPR).

  8. Example: Estimated coefficients IPR = family income ($) / Federal Poverty Level for a family of that size and age composition. Reference category: Non-Hispanic whites.

  9. Interpreting the intercept • The intercept β0 from an OLS model is an estimate of the level of the dependent variable when continuous variables take the value 0, for infants in the reference category for all categorical variables. • In a model where • The dependent variable is birth weight in grams. • The reference category is specified to be non-Hispanic white infants. • β0 is an estimate of birth weight when IPR = 0, for non-Hispanic white infants.

  10. Review: Coding of main effect and interaction term variables: race and income Reference category E.g., IPR = 0.5 means family income is half the Federal Poverty Level (FPL); IPR = 2.0 means family income is twice the FPL. For a two-category race variable (non-Hispanic white = reference category).

  11. Calculating the value of the intercept for one group = β0+ (βNHB × NHB) + (βIPR× IPR)+(βNHB_IPR× NHB_IPR) The intercept for non-Hispanic whites is calculated: = β0 + (βNHB × 0)+ (βIPR× 0.0)+ (βNHB_IPR× 0.0) =β0 Thus, the intercept for non-Hispanic white infants (when IPR = 0) collapses to include only β0 because all of the other coefficients in the formula are multiplied by a value of 0.

  12. Interpreting the IPR/birth weight pattern • IPR is a continuous variable • The coefficient is an estimate of the effect on the dependent for a 1-unit increase in the continuous IV, with categorical variables set to their reference category values. • So βIPR estimates the increment in birth weight for every one-unit increase in IPR (e.g., from family income at the poverty line to twice the poverty line) • It is the slope of the IPR/birth weight curve for infants in the reference category, in this case, non-Hispanic white infants.

  13. Calculating values for the IPR/birth weight curve for whiteinfants = β0+ (βNHB × NHB) + (βIPR× IPR)+(βNHB_IPR× NHB_IPR) = β0+ (βNHB × 0)+ (βIPR×1.5)+(βNHB_IPR× 0) = β0+ (βIPR ×1.5) Because non-Hispanic whites are the reference category for race, the equation collapses to include only the IPR main effect (βIPR) because the other coefficients are multiplied by 0. = β0+ (βIPR × IPR)

  14. Calculating values for the IPR/birth weight curve for whiteinfants = β0+ (βNHB × NHB) + (βIPR× IPR)+(βNHB_IPR× NHB_IPR) = β0+ (βNHB × 0)+ (βIPR×3.0)+(βNHB_IPR× 0) = β0+βIPR ×3.0

  15. Interpreting the race main effect • The main effect βNHB estimates the difference in birth weight between non-Hispanic black infants and those in the reference category (non-Hispanic whites), when continuous variables are set at the value 0. • It is an estimate of the difference in intercept between black and white infants when IPR is 0.

  16. Calculating the intercept for different values of the categorical variable As we saw a moment ago, for the intercept for non-Hispanic whites is calculated: = β0 + (βNHB × 0)+ (βIPR× 0.0)+ (βNHB_IPR× 0.0) =β0 For non-Hispanic blacks, the intercept is calculated: = β0 + (βNHB ×1) + (βIPR× 0.0)+ (βNHB_IPR× 0.0) =β0 + βNHB

  17. More on the race main effect • It is an estimate of the difference in intercept between black and white infants when IPR is 0. = β0 + βNHB = 3,106 + (– 177) = 2,929 • In other words, black infants born to families with an IPR of zero have a predicted birth weight of 2,929 grams. • or 177 grams LOWER than that of their white counterparts.

  18. Calculating values for the IPR/birth weight curve for whiteinfants = β0+ (βNHB × NHB) + (βIPR× IPR)+(βNHB_IPR× NHB_IPR) = β0+ (βNHB × 0) + (βIPR× IPR)+(βNHB_IPR× 0) = β0+ (βIPR × IPR) Because non-Hispanic whites are the reference category for race, the equation collapses to include only the IPR main effect (βIPR) because the other coefficients are multiplied by 0.

  19. Calculating values for the IPR birth weight curve for black infants = β0+ (βNHB× 1)+ (βIPR× 1.5)+(βNHB_IPR× 1.5) For Non-Hispanic blacks, the equation includes all three terms (βNHB, βIPR, andβNHB_IPR) because each of those coefficients is multiplied by a non-zero value.

  20. Interpreting the coefficient on the interaction between race and IPR • The slope • for blacks =βIPR+ βNHB_IPR= 23 + (–5) = 18 • for whites = βIPR= 23 • The race_IPR coefficient tests whether the slope of the IPR/birth weight pattern is different for non-Hispanic black infants than for their non-Hispanic white counterparts. • βNHB_IPRis thus the estimated difference in slope for blacks compared to whites.

  21. More on the race/IPR interaction • The estimated coefficients mean that each 1-unit increase in IPR is associated with • 23 grams more birth weight among non-Hispanic white infants. • 18 grams more birth weight among non-Hispanic black infants. • Thos values are the slopes of the respective IPR/BW curves for the two racial/ethnic groups.

  22. Preparing to graph the slope of IPR/birthweight by race • For infants in the reference category (non-Hispanic white), • Multiply selected values of IPR by βIPRand add to β0 to obtain predicted birth weight at interesting values of IPR. • For non-Hispanic black infants, • Multiply selected values of IPR by βIPR+ βNHB_IPRthen add to β0+βNHB .

  23. Calculated birth weight by racefor selected values of IPR β0 = 3,106; βIPR = 23; βNHB = –177; βNHB_IPR = –5

  24. 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 voice-over explanation

  25. Predicted birth weight by race/ethnicity and IPR = βIPR= 23 = slope of IPR/ BW curve for ref cat * 3,300 = β0 = intercept = 3,106 = predicted BW for ref cat * Birth weight (grams) 3,200 = βIPR + βNHB_IPR = 23 – 5 = 18 = slope of IPR/ BW curve for non-Hispanic black infants 3,100 3,000 2,900 = β0 + βNHB= 3,106 + (– 177) = 2,929 = intercept for black infants 2,800 6 0 1 2 4 IPR * Ref cat = Reference category = non-Hispanic white infants.

  26. Overall shape of the race/IPR/ birth weight pattern • Based on this set of βs, black infants have • a lower birth weight than whites at all IPR levels. • Negative coefficient on the NHB main effect yields a lower intercept for blacks than for whites. • a slower rate of birth weight increase as IPR rises. • Negative coefficient on NHB_IPR, which yields a shallower slope of the IPR/birth weight curve for blacks than for whites. • Thus the deficit in birth weight for blacks widens with increasing IPR.

  27. Using the three-way chart to verify your multivariate results • Check the pattern calculated from the estimated coefficients against the simple three-way chart. • If the shapes are wildly inconsistent with one another, probably reflects an error in either • How you specified the model, or • How you calculated the overall pattern from the coefficients. • Small changes in the shape or size of the pattern may occur due to controlling for other variables in your multivariate model.

  28. Summary • An interaction between a continuous and a categorical independent variable will yield differences in the intercept and/or slope of the association between the continuous IV and the DV. • Calculating the overall shape of an interaction requires adding together the pertinent main effects and interaction term βs for combinations of the categorical IV and selected values of the continuous IV in the interaction. • A spreadsheet can be helpful for storing and organizing the βs, input values, and formulas.

  29. Suggested resources • Chapters 9 and 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.

  30. Supplemental online resources • Podcasts • Introduction to interactions • Creating variables to test for interactions • Specifying models to test for interactions • Interpreting multivariate regression coefficients • Spreadsheet template for calculating overall effect of an interaction between a categorical and a continuous independent variable.

  31. Suggested practice exercises • Study guide to The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. • Question #4 in the problem set for Chapter 16 • Suggested course extensions for Chapter 16 • “Applying statistics and writing” exercise #2.

  32. Contact information Jane E. Miller, PhD jmiller@ifh.rutgers.edu Online materials available at http://press.uchicago.edu/books/miller/multivariate/index.html

More Related