Handling interaction in stata

Handling interactions in Stata Stata, especially with continuous predictors di t Patrick Royston & Willi Sauerbrei German Stata Users’ meeting, g, Berlin,, June 2012 Interactions – general concepts • General idea of a (two-way) (two way) interaction in multiple regression is effect modification: • η(x1,x x2) = f1(x1) + f2(x2) + f3(x1,x x2) • Often, η(x1,x2) = E(Y | x1,x2), with obvious extension to GLM, GLM Cox regression, regression etc etc • Simplest case: η(x1,x2) is linear in the x’s and f3((x1,,x2) is s tthe ep product oduct o of tthe e x’s: s • η(x1,x2) = β1x1 + β2x2 + β3x1x2 • Can extend to more general, non non-linear linear functions The simplest type of interaction • Binary x binary • E.g in the MRC RE01 trial in kidney cancer • 12 month % survival since randomisation • Substantial treatment effect in patients with low WCC • Little or no treatment effect in those with high g WCC • But really, WCC is a continuous variable … Treatment group White cell count low (10) MPA 34% (se 4) 24% (se 4) Interferon 49% (se 4) 21% (se 7) Overview • Interactions and factor variables (Stata 11/12) • Note: I am not an expert on factor variables! I sometimes use them them • General interactions between continuous covariates in observational studies • Focus on continuous covariates … • … because people don don’tt appear to know how to handle them! • Special case: interactions between treatment and continuous covariates in randomized controlled trials Interactions and factor variables Scope • We introduce the topic with a brief introduction to factor variables • In this part part, we consider only linear interactions: • Binary x binary (2 x table) • Binary x continuous • Continuous x continuous Factor variables: brief notes • Implemented via prefixes (unary operators) and binary interaction operators • see help fvvarlist • There are four factor-variable operators: Operator Description i unary operator to specify indicators (dummies) c unary operator to treat as continuous # binary operator to specify interactions ## binary operator to specify factorial interactions • Dummy variables are ‘virtual’ – not created per se • Names of regression parameters easily found by inspecting the post-estimation post estimation result matrix e(b) (b) Factor variables: i i prefix • Example from Stata manual [U]11.4.3: [U]11 3: li list t group i.group i i in 1/5 group 1b.group 2.group 3.group 1 0 0 3 0 Example dataset • MRC RE01 trial in advanced kidney cancer • Of 347 patients, only censored, the rest died • For F simplicity, i li it as a continuous ti response variable, Y, we use months to death, _t • Ignore the small amount of censoring • There are several prognostic factors that may influence time to death • Some are binary, some categorical, some continuous Example: factor variable parameters regress _t t i.who Source | SS df MS -+ -Model | 7780.81413 3890.40707 R id l | 85126 Residual 85126.5686 5686 344 247 247.460955 460955 -+ -Total | 92907.3828 346 268.518447 Number of obs F( 2, 344) Prob > F R-squared R d Adj R-squared Root MSE = = = = = = 347 15.72 0.0000 0837 0.0837 0.0784 15.731 -_t | Coef Std Err t P>|t| [95% Conf Interval] -+ -who | | -4.2783 2.026215 -2.11 0.035 -8.263629 -.2929707 | -12.94365 12.94365 2.354542 -5.50 5.50 0.000 -17.57476 17.57476 -8.312534 8.312534 | _cons | 19.17358 1.622518 11.82 0.000 15.98227 22.36488 matrix i li list e(b) (b) e(b)[1,4] y1 0b 0b 1 2 who who who -4.2782996 -12.943646 _cons 19.173577 values for interactions Results: P P-values mfpigen, select(0.05): mfpigen select(0 05): logit all10 cigs sysbp /// age ht wt chol (gradd1 gradd2 gradd3) *FP transformations were selected; otherwise, linear Graphical presentation of age x chol interaction fracgen cigs 5, center(mean) fracgen sysbp -2 -2, center(mean) fracgen wt -2 3, center(mean) mfpigen, linadj(cigs_1 sysbp_1 sysbp_2 > wt_1 wt_2 ht i.jobgrade) df(1) > fplot(%10 35 65 90): logit all10 age chol Alternatively: logit all10 c.age##c.chol cigs_1 sysbp_1 sysbp_2 > wt_1 wt_2 ht i.jobgrade sliceplot age chol, sliceat(10 35 65 90) percent • sliceplot is a new user-written command -5 -3 40 45 25 10 50 age 55 60 65 15 -1 05 -2 Pr(death) -4 Logit(pr(de eath)) -5 -3 05 15 -2 Pr(death h) -4 Logit(pr(dea L ath)) 25 -1 Graphical presentation of age x chol intn intn chol 15 40 chol 45 10 50 age 55 15 60 65 37 Check of chol x age interaction -2 -4 -6 -8 -8 2.5 7.5 10 12.5 2.5 7.5 10 12.5 -4 -6 -8 -6 -4 2 Q4: slope -0.01 (SE 0.04) Q3: slope 0.14 (SE 0.04) -8 Logit(pr(death)) -6 -4 -2 Q2: slope 0.22 (SE 0.05) Q1: slope 0.17 (SE 0.06) 2.5 7.5 10 12.5 chol 2.5 7.5 10 12.5 38 Interactions with continuous covariates in randomized trials 39 MFPI method (Royston & Sauerbrei 2004) • Continuous covariate x of interest interest, binary treatment variable t and other covariates z • Independent of x and t, t use MFP to select an ‘adjustment’ (confounder) model z* from z • Find best FP2 function of x (in all patients) adjusting for z* and t ã Test est FP2(x) ( ) ì t interaction te act o (2 ( d d.f.)) • Estimate β’s in each treatment group • Standard test for equality of β β’s s • May also consider simpler FP1 or linear g by y AIC functions – choose e.g 40 MFPI in Stata • MFPI is implemented as a user command, command mfpi • mfpi is available on SSC • Details are given by Royston & Sauerbrei, Stata Journal 9(2): 230-251 230 251 (2009) • Program was updated in 2012 to support acto variables a ab es factor 41 Treatment effect function • Have estimated two FP2 functions – one per treatment group • Plot the difference between functions against x to show the interaction • i.e i e the treatment effect at different x • Pointwise 95% CI shows how strongly the te act o is s supported suppo ted at different d e e t values a ues of o x interaction • i.e variation in the treatment effect with x 42 Example: MRC RE01 trial in kidney cancer • Main analysis: Interferon improves survival • HR: 0.76 (0.62 - 0.95), P = 0.015 • Is I the h treatment effect ff similar i il iin all ll patients? i ? • Nine possible covariates available for the investigation of treatment-covariate interactions – only one is significant (WCC) 43 1.0 00 Kaplan Meier showing treatment effect Kaplan-Meier 0.5 50 0.2 25 0.0 00 Prop portion a alive 0.7 75 (1) MPA (2) Interferon At risk 1: 175 55 22 11 At risk 2: 172 73 36 20 12 24 36 48 60 72 Follow-up (months) 44 The mfpi command mfpi, select(0.05) fp2(wcc) with(trt) gendiff(d): stcox (whod1 whod2) t t_dt dt t t_mt mt rem mets haem Interactions with trt (347 observations) Flex-1 model (least flexible) Var Main Interact idf Chi2 P Deviance tdf AIC wcc FP2(-1 -.5) FP2(-1 -.5) 6.91 0.0316 3180.194 3194.194 idf = interaction degrees of freedom; tdf = total model degrees of freedom mfpi_plot wcc [ i [using variables i bl created d b by gendiff(d)] diff(d)] 45 -3 -2 Log relativve hazard -1 Treatment effect plot for wcc 10 15 White cell count 20 25 About 25% of patients, those with WCC > 10 seem not to benefit from interferon 46 Concluding remarks • MFPIgen and MFPI should help researchers detect, model and visualize interactions with continuous covariates • Usually, we are searching for interactions, so small P-values are required q • Other methods not considered •S STEPP – mainly a y graphical g ap ca •… 47 Thank you you 48 Cox (1984) paper: Interaction • Cox identifies types of variable that might appear in interactions: • Treatment variables • Can be modified or imposed • Treatments, Treatments e.g e g chemotherapy, chemotherapy surgery • Behaviours, e.g smoking, drinking • Intrinsic variables • Cannot be modified • Often demog demographic, aphic e.g e g sex, se age • Unspecific variables • e.g structural t t l bl blocks, k `random’ d ’ factors f t 49 ... Continuous x continuous interaction • Results are best explored graphically • Consider in more detail next Continuous x continuous interactions 21 Motivation: continuous x continuous intn intn... • We introduce the topic with a brief introduction to factor variables • In this part part, we consider only linear interactions: • Binary x binary (2 x table) • Binary x continuous • Continuous... consider linear by linear interactions • Not sensible if main effect of either variable is non-linear • Mi Mismodelling d lli th the main i effect ff t may iintroduce t d spurious interactions