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Using Stata Graphics as a Method of Understanding and Presenting Interaction Effects

Using Stata Graphics as a Method of Understanding and Presenting Interaction Effects. Joanne M. Garrett, PhD University of North Carolina at Chapel Hill July 11, 2005. Problems with Understanding Interaction. Interaction difficult concept to explain not a single answer (point estimate)

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Using Stata Graphics as a Method of Understanding and Presenting Interaction Effects

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  1. Using Stata Graphics as a Method of Understanding and Presenting Interaction Effects Joanne M. Garrett, PhD University of North Carolina at Chapel Hill July 11, 2005

  2. Problems with Understanding Interaction • Interaction difficult concept to explain • not a single answer (point estimate) • linear combination of betas • Often ignored because of difficulty • Graph of interaction may be more intuitive for: • students learning the concept • presentations at professional meetings

  3. Low Birth Weight Study* * “Loosely” adapted from data from Applied Logistic Regression, David W. Hosmer, Jr. and Stanley Lemeshow

  4. Example 1: Low birth weight and interaction between smoking and race (1=black, 0=white) • Logistic regression model: Add smk by race interaction • Create interaction term and run logistic regression: • . gen smkxrace = smk * race • . logistic low smk race smkxrace

  5. Example 1: Low birth weight and interaction between smoking and race (1=black, 0=white) • -------------------------------------------------- • low | OR z P>|z| [95% CI] • ---------+---------------------------------------- • smk | 5.76 4.14 0.000 2.51 13.19 • race | 5.43 4.13 0.000 2.43 12.15 • smkxrace | .320 -2.08 0.038 .109 .936 • -------------------------------------------------- • Significant interaction: Relationship differs between smoking and low birth wt depending on mother’s race • Question: How to interpret the interaction OR?

  6. Example 1: Low birth weight and interaction between smoking and race (1=black, 0=white) • Convert OR’s to beta coefficients and solve by categories of race: . logit • -------------------------------------------------- • low | Coef. z P>|z| [95% CI] • ---------+---------------------------------------- • smk | 1.751 4.14 0.000 0.921 2.580 • race | 1.693 4.13 0.000 0.889 2.497 • smkxrace | -1.141 -2.08 0.038 -2.216 -0.066 • _cons| -2.303 • -------------------------------------------------- • White: OR = e[β1 + β3(race)] = e[1.751 – 1.141(0)] = 5.76 • Black: OR = e[β1 + β3(race)] = e[1.751 – 1.141(1)] = 1.84

  7. Example 1: Low birth weight and interaction between smoking and race (1=black, 0=white) White: . lincom smk + 0*smkxrace ---------------------------------------------- low | OR z P>|z| [95% CI] -----+---------------------------------------- (1) | 5.76 4.14 0.000 2.51 13.2 ---------------------------------------------- Black: . lincom smk + 1*smkxrace ---------------------------------------------- low | OR z P>|z| [95% CI] -----+---------------------------------------- (1) | 1.84 1.75 0.081 .928 3.65 ----------------------------------------------

  8. Example 1: Low birth weight and interaction between smoking and race (1=black, 0=white) • Interpretation: • Among whites, women who smoke have 5.8 times the odds of having a low birth wt baby • Among blacks, there is no relationship between smoking and having a low birth wt baby (OR=1.8, but not statistically significant)

  9. Example 1: Low birth weight and interaction between smoking and race (1=black, 0=white) • Misinterpretations: • Black mothers have less “risk” for low birth wt babies compared to white mothers • It’s okay for black mothers to smoke • Alternative: • Solve the equation for values of smk and race • Graph the individual probabilities (“predxcat”) • . predxcat low, xvar(race smk) graph bar

  10. Example 1: Low birth weight and interaction between smoking and race (1=black, 0=white) +-------------------------------------------------+ | race smk numobs prob lower upper | |-------------------------------------------------| | 0:White 0:No 88 0.091 0.046 0.171 | | 0:White 1:Yes 104 0.365 0.279 0.462 | | 1:NonWh 0:No 142 0.352 0.278 0.434 | | 1:NonWh 1:Yes 44 0.500 0.356 0.644 | +-------------------------------------------------+ Likelihood ratio test of interaction for race * smk: LR Chi2(1) = 4.56 Prob > Chi2 = 0.0328

  11. Example 1: Low birth weight and interaction between smoking and race (1=black, 0=white)

  12. Example 2: Low birth weight and interaction between smoking and age (years) • Logistic regression model: Add smk by age interaction • Create interaction term and run logistic regression: • . gen smkxage = smk * age • . logistic low smk age smkxage

  13. Example 2: Low birth weight and interaction between smoking and age (years) • ------------------------------------------------- • low | OR z P>|z| [95% CI] • ---------+--------------------------------------- • smk | 0.382 -0.90 0.367 .047 3.110 • age | 0.920 -2.61 0.009 .865 0.980 • smkxage | 1.076 1.97 0.049 1.001 1.157 • ------------------------------------------------- • Significant interaction: Relationship differs between smoking and low birth wt depending on mother’s age • Question: How to interpret the interaction OR for a continuous interaction variable?

  14. Example 2: Low birth weight and interaction between smoking and age (years) • Convert OR’s to beta coefficients and solve for selected values of age: . logit • ------------------------------------------------- • low | Coef. z P>|z| [95% CI] • --------+---------------------------------------- • smk | -.963 -0.90 0.367 -3.058 1.131 • age | -.083 -2.61 0.009 -0.145 -0.021 • smkxage | .073 1.97 0.049 0.001 0.146 • _cons| .805 • ------------------------------------------------- • Age=15: OR = e[β1 + β3(age)] = e[–0.963 + 0.073(15)] = 1.14 • Age=35: OR = e[β1 + β3(age)] = e[–0.963 + 0.073(35)] = 4.93

  15. Example 2: Low birth weight and interaction between smoking and age (years) Age=15: . lincom smk + 15*smkxage ----------------------------------------------- low | OR z P>|z| [95% CI] -----+----------------------------------------- (1) | 1.14 0.32 0.751 .503 2.59 ----------------------------------------------- Age=35: . lincom smk + 35*smkxage ----------------------------------------------- low | OR z P>|z| [95% CI] -----+----------------------------------------- (1) | 4.93 2.59 0.010 1.47 16.5 -----------------------------------------------

  16. Example 2: Low birth weight and interaction between smoking and age (years) • Interpretation: • Among 15 year olds, women who smoke have 1.1 times the odds of having a low birth wt baby • Among 35 year olds, women who smoke have 4.9 times the odds of having a low birth wt baby

  17. Example 2: Low birth weight and interaction between smoking and age (years) • Misinterpretations: • 15 year olds are not at risk for low birth wt babies • It’s okay for 15 year olds to smoke • Alternative: • Solve the equation and graph the probabilities for different levels of smoke and age (“predxcon”) • . predxcon low, xvar(age) from(15) to(35) inc(2) • class(smk) graph

  18. Example 2: Low birth weight and interaction between smoking and age (years) • -> smk = 0 • +---------------------------------+ • | age pred_y lower upper | • |---------------------------------| • | 15 .392 .272 .527 | • | 17 .353 .259 .461 | • | 19 .317 .243 .400 | • | (etc) ... ... ... | • +---------------------------------+ • -> smk = 1 • +---------------------------------+ • | age pred_y lower upper | • |---------------------------------| • | 15 .424 .285 .577 | • | 17 .419 .303 .546 | • | (etc) ... ... ... | • +---------------------------------+ • Likelihood ratio test for interaction of age * smk: • LR Chi2(1) = 3.88 • Prob > Chi2 = 0.05

  19. Example 2: Low birth weight and interaction between smoking and age (years) p=0.05

  20. Example 3: Birth weight (grams) and interaction between smoking and race • Linear regression model: • Create interaction term and run linear regression: • . gen smkxrace = smk * race • . regress low smk race smkxrace • Or: . predxcat bwt, xvar(race smk) graph bar

  21. Example 3: Birth weight (grams) and interaction between smoking and race p=0.006

  22. Example 4: Birth weight (grams) and interaction between smoking and age • Linear regression model: • Create interaction term and run linear regression: • . gen smkxage = smk * age • . regress low smk age smkxage • Or: . predxcon bwt, xvar(age) from(15) to(35) inc(2) • class(smk) graph

  23. Example 4: Birth weight (grams) and interaction between smoking and age p=0.001

  24. Example 5: Birth weight (grams) and interaction between smoking and age, age2, age3 • Linear regression model: add quadratic & cubic terms • Create interaction terms and run linear regression: • . gen age2 = age^2 • . gen age3 = age^3 • . gen smkxage = smk * age • . gen smkxage2 = smk * age2 • . gen smkxage3 = smk * age3 • . regress low smk age age2 age3 smkxage • smkxage2 smkxage3 • Or: . predxcon bwt, xvar(age) from(15) to(35) inc(2) • class(smk) graph poly(3)

  25. Example 5: Birth weight (grams) and interaction between smoking and age, age2, age3 p=0.045

  26. Conclusions • Interaction can be a difficult concept for people unfamiliar with the methodology • Examining a graph of an interaction is an easier way to get an intuitive feel for the effect • A useful technique for explaining interaction to students hearing it for the first time, before introducing mathematical models • A simple way to present study results at meetings, even to a statistically savvy audience

  27. Calculating and Graphing Predicted Values: (when “X” is categorical) • . predxcat yvar, xvar(xvar1 xvar2) • yvar – dependent variable • continuous – defaults to linear regression • binary (0,1) – defaults to logistic regression • xvar(xvar) – nominal variable for categories of estimated • means or proportions • xvar(xvar1 xvar2)– categories of all combinations of xvar1 • and xvar2; tests interaction • adjust(cov_list)– adjusts for any covariates

  28. Calculating and Graphing Predicted Values: (when “X” is categorical) • graph – display graph (otherwise shows list of predicted • values only) • bar – bar graph (instead of symbols – default) • model – for display purposes only; displays regression model • Some other options:level(#) • cluster(cluster_id) • savepred(ds_name)

  29. Calculating and Graphing Predicted Values: (when “X” is continuous) • . predxcon yvar, xvar(xvar) from(#) to(#) inc(#) graph • yvar – dependent variable • continuous – defaults to linear regression • binary (0,1) – defaults to logistic regression • xvar(xvar) – continuous independent variable; probabilities • calculated for each value of X • from(#) – bottom value for xvar • to(#) – top value for xvar • inc(#) – increment desired between bottom and top values • adjust(cov_list) – adjusts for any covariates

  30. Calculating and Graphing Predicted Values: (when “X” is continuous) • graph – display graph (otherwise shows list of predicted • values only) • class(class_var) – adds an xvar by class_var interaction term • poly(2 or 3) – polynomial terms added: 2=squared 3=squared and cubic • model – for display purposes only; displays regression model • Some other options:level(#) • cluster(cluster_id) • nolist • savepred(ds_name)

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