Advanced Regression Methods Symposium on Updates on Clinical Research Methodology March 18, 2013

Lloyd Mancl, PhD Charles Spiekerman, PhD Oral Health Sciences Oral Health Sciences University of Washington University of Washington [email protected] [email protected] Outline • Introduce common regression methods for different types of outcomes – Logistic regression – Multiple linear regression – Cox proportional hazards regression – Poisson or log-linear regression • Uses of regression: – Adjust for confounding – Assess for effect modification or interaction – Account for non-independent outcomes

Uses for Multiple regression analysis • Used to adjust for confounding – In observational studies, groups of interest can differ on other variables that may be related to the outcome. – In an RCT, randomization may not result in balanced groups. • Used to assess simultaneously the associations for several explanatory variables, as well as, interactions between variables – In observational studies, you may be interested in how several explanatory variables are related to an outcome. – In an RCT, we are often interested in testing if treatment is modified by another variable (i.e., interaction or moderation). – In designed experiments, we typically test for interactions between the different study factors. • Used to develop a prediction equation – Not commonly used for this purpose; not covered in this workshop.

Multiple regression analysis Models the association between one outcome variable,

Y, and multiple variables of interest, X1, X2,…,Xk.

Multiple Linear regression model

Y = α + β1X1 + β2X2 + … + βkXk + random error

Generalized Linear Model

G(Y) ~ α + β1X1 + β2X2 + … + βkXk (basically, a more complicated version of the outcome, Y, is related to a linear combination of the variables of interest) Common Multiple Regression Methods

Outcome variable Regression method Regression results Slopes & difference Quantitative/continuous Linear regression between means Binary (2 categories) Logistic regression Odds ratio Poisson or Count or count rate Relative risk or rate ratio log-linear regression

Cox proportional hazards Time to an event Hazard ratio regression

Ordinal (>2 categories) Ordinal logistic regression Odds ratio

Multinomial logistic Nominal (>2 categories) Odds ratio regression Regression method depends on the outcome • Continuous or quantitative outcome – linear regression – Amount of attachment loss (mm) – Change in dmfs • Binary outcome – logistic regression – Any new decay – Incident TMD • Time to event outcome - Cox proportional hazards regression – Time to tooth loss – Time to pulp cap failure • Count outcome – Poisson or log-linear regression – Number of new caries – Rate of new caries

Example: heart disease and periodontitis • NHANES II –observational study that examined a large number of participants at a baseline visit and followed them for over 10 years to ascertain morbid events. • We are interested in assessing the association between periodontal disease evaluated at baseline and the occurrence of heart disease (CHD) within 10 years of study entry. Logistic regression

In this analysis the outcome variable, CHD incidence, is a binary variable, so a regression method we could employ is logistic regression

CHD risk by exposure group Group Healthy Periodontal Odds 95% Conf. Int. gums disease Ratio CHD incidence 4.9% 13.5% 3.0 (2.5, 3.7)

Logistic regression uses the Odds Ratio as an estimate of association between your independent variable and the outcome Confounding

• Confounding occurs when there is a third variable that is strongly related to both the dependent and the independent variable. • This can bias an estimate of association. ? Y X1 X2 • With respect to Periodontitis and CHD, an obvious potential confounder is Age. • We can adjust for the potential confounding effects by entering Age into the logistic regression model as an additional independent variable CHD incidence by periodontal disease

Logistic Regression with Periodontal disease as the only independent variable Independent Variable Odds Ratio 95% Conf. Int. Healthy Gums 1 -

Periodontal Disease 3.0 (2.5, 3.7)

Logistic Regression model with Age added Independent Variable Odds Ratio 95% Conf. Int. Healthy Gums 1 - Periodontal Disease 1.6 (1.3, 2.0) Age (10 year increment) 2.1 (1.9, 2.2) CHD incidence by periodontal disease

Logistic Regression model with Age added Independent Variable Odds Ratio 95% Conf. Int. Healthy Gums 1 - Periodontal Disease 1.6 (1.3, 2.0) Age (10 year increment) 2.1 (1.9, 2.2)

• By controlling for Age the estimated association of Periodontal disease with CHD is less strong. • The association is still statistically significant (confidence interval does not contain 1). • The Age output indicates 2.1 times higher odds of CHD associated with 10 years greater Age Effect Modification / Interaction • An interaction is when the association or relationship between an explanatory variable and the outcome variable depends on the value of another explanatory variable. • Also called effect modification or moderation. • In extreme cases, an interaction may completely reverse the relationship between the explanatory variable and outcome. • More commonly, the effect is stronger (or weaker) depending on the value of another explanatory variable. • Stratification can be used to identify an interaction. • Can use regression to test for an interaction by adding an interaction term/variable in the regression model, which is the product of two explanatory variables. Chewing Gum Study • Subjects randomly assigned to 3 different chewing gums • Outcome was continuous, change in DMFS • Linear regression used to compare the 3 groups, adjusting for baseline DMFS DMFS Change Baseline DMFS Group n Mean (SD) Mean (SD) A 25 -0.72 (5.37) 4.68 (1.02) B 35 -0.83 (3.57) 3.77 (0.55) C 40 2.63 (3.80) 3.67 (0.57) Linear regression results

Coefficient Standard Estimate Error P-value Intercept 1.30 0.90 .15 Group B (vs A) -0.50 1.01 .62 Group C (vs A) 2.91 0.98 .004 Baseline DMFS -0.43 0.10 <.001 Group main effect, p-value <.001

• Model estimates a constant group difference • DFMS change 2.91 greater for Group C than Group A Linear regression results

Coefficient Standard Estimate Error P-value Intercept 1.30 0.90 .15 Group B (vs A) -0.50 1.01 .62 Group C (vs A) 2.91 0.98 .004 Baseline DMFS -0.43 0.10 <.001 Group main effect, p-value <.001

• Model estimates a constant group difference • DFMS change 2.91 greater for Group C than Group A • DFMS change -0.50 less for Group B than Group A Scatterplot of change in DMFS versus baseline DMFS for the treatment groups (A, B, C)

15 Group A Group B Group C 10

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ChangeDMFS in -10

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0 5 10 15 20 25

Baseline DMFS • Group x baseline DMFS interaction added to the linear regression model to test if group differences are affected by baseline DMFS

Coefficient Standard Estimate Error P-value Intercept 2.96 0.97 .003 Group B (vs A) -1.53 1.33 .25 Group C (vs A) -0.79 1.26 .53 Baseline DMFS -0.79 0.14 <.001 Group B x Baseline DMFS 0.19 0.23 .42 Group C x Baseline DMFS 0.91 0.21 <.001

• Group x Baseline DMFS interaction, p-value <.001 • Difference between Group C and A increases with baseline DMFS Scatterplot of change in DMFS versus baseline DMFS for the treatment groups (A, B, C) 15 Group A Group B Group C 10

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ChangeDMFS in -10

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0 5 10 15 20 25

Baseline DMFS • Difference between Group C and A increases with baseline DMFS Effect Modification / Interaction • Test for interactions after all main effects are included in the regression model. • Typically, only assess for two-way interactions. • Usually only test interactions, when at least one of the variables has a significant main effect. • (Exceptions for designed experiments involving a small number of factors, where all possible interactions may be assessed). • Lower significance level (e.g., p<0.01) may be used, if testing a large number of interactions, to control type I error due to multiple comparisons.

Survival Analysis: Time to event data

• In some studies the outcome of interest is the time until an event – Time to implant failure – Time to tooth loss – Time to death • Analyses of this type of data are commonly called “Survival Analysis”

Censored events

• In most time to event studies a non-trivial portion of the events will not be observed because they don’t occur during the period of observation. • These unobserved events are considered “censored” • For the censored events we don’t know the actual time until the event, but we do know that the time until event is at least as great as the time until the patient was last seen. • Survival analysis uses the information on the complete observations and the censored observations in a smart way. Censored events

Enrollment date Event date

Time Start of End of enrollment study • The 2nd, 4th and 5th patients have censored times to event. • The 1st, 3rd and 6th patients we know the exact times to event. Periodontal disease and tooth loss

• One hundred periodontal patients under maintenance care*. • Interest in assessing factors associated with tooth loss. • Outcome is time to loss of tooth. • We will look at oral hygiene, patient age, and smoking.

*M. McGuire & M. Nunn, J Periodontology, 1996; 67:666-674.

Kaplan-Meier survival plots

1.00 1.00 1.00 Hygiene Age Smoking good or fair less than 50 not smoker

poor 50 or older smoker

0.98 0.98 0.98

0.96 0.96 0.96

Survival Probability Survival

0.94 0.94 0.94

0.92 0.92 0.92

0 5 10 15 0 5 10 15 0 5 10 15

Time (years) Time (years) Time (years) Kaplan-Meier plots present estimates of the survival function, S(t). S(t) = Probability of surviving to time t Cox proportional hazards regression

• If some simplifying assumptions hold, then one can compare survival probabilities using a regression framework. • Cox proportional hazards regression assumes that the hazards in different groups are proportional. • The hazard is the instantaneous probability of failure, and is related to S(t) • The hazard at time t can be thought of as the probability of failure right after t, given that one has survived up until time t. Cox regression results • The output from a Cox Variable Hazard 95% regression is the hazard ratio confidence ratio interval* • The hazard ratio can be Poor 1.9 (0.7, 4.9) interpreted as an Hygiene instantaneous relative risk. • A value of 1 means no Age > 50 1.3 (0.7, 2.4) association • Since the “smoker” hazard Smoker 2.1 (1.1, 3.9) ratio is significantly different from 1, we can say the teeth from the smokers are at significantly higher risk.

*Corrected for correlation between teeth within same patient Count outcome – “Poisson” regression • Children randomly assigned to intensive application of fluoride varnish (3 times in 2 weeks) versus standard application applied semiannually* • Outcome was number of new primary caries per time at risk Treatment Surface-years Surfaces of new group n at risk decay per person P-value1 Mean ± SD Mean ± SD Standard 264 168 ± 60 7.4 ± 7.7 <.001 Intensive 251 161 ± 57 9.8 ± 8.6

• 1”Possion” regression (with robust standard errors) indicates a higher caries rate for the intensive treatment.

*P Weinstein, C Spiekerman & P Milgrom. Caries Research 2009 •There are some minor (?) differences between the two treatment groups at baseline, which we may want to adjust for when comparing the two treatment group.

Treatment Group Standard (n=306) Intensive (n=294) Age at baseline, months Mean ± SD 54.7 ± 4.5 55.8 ± 4.7 Baseline dmfs, n (%) 0 dmfs 159 (52%) 96 (33%) 1-7 dmfs 81 (26%) 100 (34%) >7 dmfs 66 (22%) 98 (33%) Poisson Regression • If counts have a Poisson distribution, most common model is the log-linear regression model log (outcome) = log (time) + explanatory variables log (time) is often called an offset • If time is included in the model (as an offset), the exponentiated regression coefficients have a relative risk or rate ratio interpretation • Because the Poisson distribution is somewhat restrictive (e.g., requires the mean and variance to be the same), the Poisson model may not be ideal for every count outcome. (E.g., the Standard group mean was 168 and variance was 602 = 3600)

• Other distributions and regression methods are available. For example, a negative binomial regression can be used as an alternative to the Poisson regression to account for variance > mean (i.e., over-dispersion) • Another option is to fit a log-linear model, but use robust standard estimates to compute p-values & confidence intervals (i.e., “Poisson” regression) • The interpretation of the regression coefficients are similar to Poisson regression. • Other options are zero-inflated Poisson and zero-inflated negative binominal regression. – account for extra “zeros” in the data. – methods are appropriate if the outcomes are generated by two processes (one producing only zero counts [e.g., will never get caries] and one that can produce nonzero counts [e.g., may get caries]). – regression coefficients have a different interpretation. “Poisson” regression results (with robust standard errors*) Adjusted Covariate Rate Ratio 95% CI P-value Treatment group .20 Standard 1 --- Intensive 1.13 0.94, 1.37 Caries at baseline <.0001 0 dmfs 1 --- 1-7 dmfs 1.93 1.52, 2.45 >7 dmfs 3.24 2.54, 4.13 (Also adjusted for enrollment cohort & site, gender and age) *Robust standard errors used to compute 95% CI and P-value; does not require mean = variance assumption. “Poisson” regression results (with robust standard errors*) Adjusted Covariate Rate Ratio 95% CI P-value Treatment group .20 Standard 1 --- Intensive 1.13 0.94, 1.37 Caries at baseline <.0001 0 dmfs 1 --- 1-7 dmfs 1.93 1.52, 2.45 >7 dmfs 3.24 2.54, 4.13 (Also adjusted for enrollment cohort & site, gender and age) • Adjusted rate ratio indicates the caries rate is only 13% higher in intensive group (after adjusting for caries at baseline, etc.) • The 95% CI and P-value indicates this difference is not statistically significant Regression methods when outcomes are not independent

Non-independent outcomes can by due to: • Multiple outcomes within a mouth – outcomes & explanatory variables measured on multiple teeth or surfaces within the same patient • Multiple outcomes over time – outcomes & explanatory variables measured on same patient at multiple visits over time • Cluster sampling – National surveys (e.g., NHANES) involve cluster sampling • Cluster randomization – Intervention randomized by school, classroom or dental practice

When the outcomes are not independent, but we assume they are: • Regression coefficient estimates typically still valid as long as sample size is sufficiently large. • Standard errors (SEs) for regression coefficient estimates may not be valid. • Hence, p-values and confidence intervals may not be valid • For explanatory variables that do not vary within a cluster, p- values tend to be too small and confidence intervals too narrow.

Generalized Estimating Equations (GEE) •One commonly used regression method that takes into the account the dependence or correlation between observations from the same cluster or individual is generalized estimating equations or GEE. •You can use the GEE method to fit a linear, logistic, Poisson, multinomial logistic and a variety of other regression models. •GEE-like methods also available for time to event / survival outcomes. •The GEE method doesn’t require the same number of outcomes per individual or cluster. •Regression coefficient estimates are interpreted the same way as ordinary linear regression, logistic regression coefficients, Poisson regression, etc.

Radiographic bone loss study • RCT of non-steroidal anti-inflammatory drug on radiographic bone loss • Outcome was binary, >2.5% annual bone loss, measured on each tooth • Explanatory variables were treatment (drug vs placebo) randomly assigned to a subject and prior disease activity (yes vs no) measured on each tooth

Coefficient Standard Odds 95 CI% P-value Estimate error ratio Logistic regression* Treatment -0.381 0.130 0.68 0.53, 0.88 0.0035 Prior disease 0.391 0.144 1.38 1.04, 1.83 0.027

*Logistic regression assumes the outcomes are all independent Radiographic bone loss study Coefficient Standard Odds 95 CI% P-value Estimate error ratio Logistic regression Treatment -0.381 0.130 0.68 0.53, 0.88 0.0035 Prior disease 0.391 0.144 1.38 1.04, 1.83 0.027 GEE logistic regression* Treatment -0.381 0.327 0.68 0.35, 1.30 0.25 Prior disease 0.391 0.205 1.38 0.92, 2.06 0.12 *GEE logistic regression using an Independence working correlation; robust standard errors account correlation within subjects. Radiographic bone loss study Coefficient Standard Odds 95 CI% P-value Estimate error ratio Logistic regression Treatment -0.381 0.130 0.68 0.53, 0.88 0.0035 Prior disease 0.391 0.144 1.38 1.04, 1.83 0.027 GEE logistic regression* Treatment -0.381 0.327 0.68 0.35, 1.30 0.25 Prior disease 0.391 0.205 1.38 0.92, 2.06 0.12 *GEE logistic regression using an Independence working correlation; robust standard errors account correlation within subjects. Final Thoughts •Regression modeling is an “art” –All models are “wrong”, but some are useful –Often several plausible models can fit the data equally well –Should validate model (e.g., check model assumptions, outliers?) McCullagh & Nelder (1989) •Limitations of multiple regression methods – Multiple regression can be used to control for confounding, but best way to avoid confounding is to do a controlled experiment. – Can only control for known & measured confounders, and need sufficient variation among the variables to “adjust” for confounding.