Implications of Interactions in Treatment Comparisons
Transportability of treatment effect estimates depends on the nature of interactions. In the absence of interactions, an effect estimated on a highly selected sample will apply to a much different population. In an observational study, the corresponding condition is that there need be no overlap in the baseline distribution of noninteracting factors. When there is an interaction, one can live with only a small to moderate amount of overlap in characteristics (between randomized vs. population target or between characteristics of treated vs. nontreated patients in an observational study) if the interaction is of a simple form. With more overlap, interactions can be complex (if modeled) and results from analysis of the sample will allow estimation of treatment effect in a different population. In the absence of significant overlap, confidence bands allowing for interaction properly inform the researcher about uncertainties in treatment effects.
Randomized clinical trials do not require representative patients; they require representative treatment effects. Generalizability of randomized trial findings for relative efficacy comes from one of three things:
 true absence of interactions,
 interacting factors have a similar distribution in the RCT as in the target population, or
 the RCT sample has enough representation of the distribution of interacting factors to allow them to be modeled and used to estimate treatment effects in target patients, and the researcher knows to include these interactions in her model.
Note: For shaded boxes marked with ⮩ click on the box to view the associated text.
Background
It is a commonly held belief that clinical trials, to provide treatment effects that are generalizable to a population, must use a sample that reflects that population’s characteristics. The confusion stems from the fact that if one were interested in estimating an average outcome for patients given treatment A, one would need a random sample from the target population. But clinical trials are not designed to estimate absolutes; they are designed to estimate differences as discussed further here. These differences, when measured on a scale for which treatment differences are allowed mathematically to be constant (e.g., difference in means, odds ratios, hazard ratios), show remarkable constancy as judged by a large number of published forest plots. What would make a treatment estimate (relative efficacy) not be transportable to another population? A requirement for nongeneralizability is the existence of interactions with treatment such that the interacting factors have a distribution in the sample that is much different from the distribution in the population.
A related problem is the issue of overlap in observational studies. Researchers are taught that nonoverlap makes observational treatment comparisons impossible. This is only true when the characteristic whose distributions don’t overlap between treatment groups interacts with treatment. The purpose of this article is to explore interactions in these contexts.
As a side note, if there is an interaction between treatment and a covariate, standard propensity score analysis will completely miss it.
For the following I assume that there is at most one variable (patient age) interacting with treatment effect. The data generating and analysis models are logistic models containing only age and treatment. I also assume that the goal is to assess efficacy isolated from net clinical benefit, e.g., we are not taking side effects into account.
Situations addressed below come from the following 2 × 2 × 2 × 3 combinations:
 randomized vs. observational study
 true data generating process has no interaction with age vs. has a linear interaction
 overlap in ages between RCT sample and target population (or between treatments in an observational study) is completely absent vs. partial overlap in the age distribution
 fitted model is linear with no interaction, has a linear interaction, or has a spline function of age interacting with treatment
I’ll explore what happens when interaction should be included in a model but is omitted, when an interaction is not needed but is included in the model, and when the interaction is simple but is allowed to be complex.
Generalizability of Randomized Clinical Trials
When the outcome generating process in a randomized clinical trial (RCT) is such that treatment does not interact with any baseline patient characteristic over the range of characteristics seen in the union of the RCT and the population, the estimate of relative efficacy (e.g., odds ratio) of treatment applies to every patient in the population. In addition to computing patientspecific odds ratios (ORs), one can easily compute patientspecific absolute risk reductions from treatment as shown here. If one allows for needless interaction terms, it is possible to get poor extrapolation of treatment benefit in unrepresented patient groups, unless the estimated interaction effects are very small (not guaranteed until the sample size is very large).
True Model Has No Treatment Interactions
Estimation of Treatment Effect in Population With No Overlap
As an example, consider a simulated study of about 2500 patients with a binary outcome where the model contains only age and treatment, the age effect is linear, and there is no interaction. The following R code simulates the data from a hypothetical population, fits a binary logistic model, and displays estimated log odds of outcome vs. age and treatment (left panel) and the estimated agespecific treatment effect (right panel), along with the true population values for both panels over the whole range of age. The RCT excluded those with age < 50 but the target population is exactly those patients. Tick marks on fitted curves display treatmentspecific raw data spike histograms.
The original age range is 13.3  88.1 in 5000 subjects, and the clinical trial includes 2461 of the subjects.
Age distributions in the target population as compared to the RCT sample are shown below.
Logistic Regression Model
lrm(formula = y ~ tx + age, data = d)
Frequencies of Missing Values Due to Each Variable
y tx age 2539 0 0
Model Likelihood Ratio Test 
Discrimination Indexes 
Rank Discrim. Indexes 


Obs 2461  LR χ^{2} 211.90  R^{2} 0.117  C 0.679 
0 1735  d.f. 2  R^{2}_{2,2461} 0.082  D_{xy} 0.357 
1 726  Pr(>χ^{2}) <0.0001  R^{2}_{2,1535.5} 0.128  γ 0.357 
max ∂log L/∂β 2×10^{7}  Brier 0.190  τ_{a} 0.149 
β  S.E.  Wald Z  Pr(>Z)  

Intercept  4.3508  0.4387  9.92  <0.0001 
tx=b  1.0992  0.0959  11.46  <0.0001 
age  0.0674  0.0074  9.05  <0.0001 
Wald Statistics for y


χ^{2}  d.f.  P  

tx  131.30  1  <0.0001 
age  81.95  1  <0.0001 
TOTAL  189.63  2  <0.0001 
Extrapolation to the younger population is fine even with no younger patients in the RCT. Confidence bands are computed under the assumption of an additive linear age effect.
This is the correct model.
Logistic Regression Model
lrm(formula = y ~ tx * age, data = d)
Frequencies of Missing Values Due to Each Variable
y tx age 2539 0 0
Model Likelihood Ratio Test 
Discrimination Indexes 
Rank Discrim. Indexes 


Obs 2461  LR χ^{2} 212.02  R^{2} 0.117  C 0.679 
0 1735  d.f. 3  R^{2}_{3,2461} 0.081  D_{xy} 0.357 
1 726  Pr(>χ^{2}) <0.0001  R^{2}_{3,1535.5} 0.127  γ 0.357 
max ∂log L/∂β 5×10^{6}  Brier 0.190  τ_{a} 0.149 
β  S.E.  Wald Z  Pr(>Z)  

Intercept  4.2193  0.5852  7.21  <0.0001 
tx=b  1.3998  0.8939  1.57  0.1174 
age  0.0652  0.0100  6.53  <0.0001 
tx=b × age  0.0051  0.0150  0.34  0.7351 
Wald Statistics for y


χ^{2}  d.f.  P  

tx (Factor+Higher Order Factors)  131.11  2  <0.0001 
All Interactions  0.11  1  0.7351 
age (Factor+Higher Order Factors)  82.19  2  <0.0001 
All Interactions  0.11  1  0.7351 
tx × age (Factor+Higher Order Factors)  0.11  1  0.7351 
TOTAL  188.87  3  <0.0001 
Note that the closetozero estimated interaction in the trial sample led to an appropriate extrapolation to the target population of age < 50, with the confidence bands getting wider.
Now see what happens when age is unnecessarily allowed to have a nonlinear nonadditive effect, by fitting a model that is quadratic in age interacting with treatment.
This model included an unnecessary linear interaction term.
Logistic Regression Model
lrm(formula = form, data = d, tol = 1e12)
Frequencies of Missing Values Due to Each Variable
y tx age 2539 0 0
Model Likelihood Ratio Test 
Discrimination Indexes 
Rank Discrim. Indexes 


Obs 2461  LR χ^{2} 214.13  R^{2} 0.119  C 0.679 
0 1735  d.f. 7  R^{2}_{7,2461} 0.081  D_{xy} 0.358 
1 726  Pr(>χ^{2}) <0.0001  R^{2}_{7,1535.5} 0.126  γ 0.358 
max ∂log L/∂β 2×10^{5}  Brier 0.190  τ_{a} 0.149 
β  S.E.  Wald Z  Pr(>Z)  

Intercept  3.0988  1.2779  2.42  0.0153 
tx=b  1.3609  2.0910  0.65  0.5151 
age  0.0445  0.0231  1.93  0.0542 
age’  0.2124  0.2213  0.96  0.3372 
age’’  0.4384  0.4906  0.89  0.3716 
tx=b × age  0.0043  0.0376  0.11  0.9087 
tx=b × age’  0.0067  0.3136  0.02  0.9829 
tx=b × age’’  0.0141  0.6574  0.02  0.9829 
Wald Statistics for y


χ^{2}  d.f.  P  

tx (Factor+Higher Order Factors)  131.33  4  <0.0001 
All Interactions  0.11  3  0.9910 
age (Factor+Higher Order Factors)  85.26  6  <0.0001 
All Interactions  0.11  3  0.9910 
Nonlinear (Factor+Higher Order Factors)  2.11  4  0.7151 
tx × age (Factor+Higher Order Factors)  0.11  3  0.9910 
Nonlinear  0.00  2  0.9998 
Nonlinear Interaction : f(A,B) vs. AB  0.00  2  0.9998 
TOTAL NONLINEAR  2.11  4  0.7151 
TOTAL NONLINEAR + INTERACTION  2.23  5  0.8162 
TOTAL  190.78  7  <0.0001 
By allowing not only an unnecessary interaction but also allowing the interaction to be nonlinear, when the predicted values are for a region completely outside the sample data, the extrapolation is still reasonable but the (honest) confidence intervals are much wider. We don’t know much at all about the relative treatment effect with age < 50 when we allowed the two treatments to have differently shaped age effects in the trial data. A Bayesian model that put a skeptical prior on either the nonlinear or the interaction effect would have credible intervals that are not so wide on the left.
This model included unnecessary linear and nonlinear interaction terms.
RCT Sample Partially Overlaps with Target Population
Instead of having nonoverlapping age distributions between the RCT sample and the target population, let’s include screened patients with probabilities that are functions of age as shown in the first graph below. Then again consider the simplest logistic model.
The original age range is 13.3  88.1 in 5000 subjects, and the clinical trial includes 3556 of the subjects.
Age distributions in the target population as compared to the RCT sample are shown below.
Logistic Regression Model
lrm(formula = y ~ tx + age, data = d)
Frequencies of Missing Values Due to Each Variable
y tx age 1444 0 0
Model Likelihood Ratio Test 
Discrimination Indexes 
Rank Discrim. Indexes 


Obs 3556  LR χ^{2} 329.69  R^{2} 0.131  C 0.697 
0 2658  d.f. 2  R^{2}_{2,3556} 0.088  D_{xy} 0.394 
1 898  Pr(>χ^{2}) <0.0001  R^{2}_{2,2013.7} 0.150  γ 0.394 
max ∂log L/∂β 3×10^{11}  Brier 0.171  τ_{a} 0.149 
β  S.E.  Wald Z  Pr(>Z)  

Intercept  4.4121  0.2941  15.00  <0.0001 
tx=b  1.0470  0.0846  12.38  <0.0001 
age  0.0682  0.0052  13.04  <0.0001 
Wald Statistics for y


χ^{2}  d.f.  P  

tx  153.17  1  <0.0001 
age  169.92  1  <0.0001 
TOTAL  286.93  2  <0.0001 
This is the correct model.
Logistic Regression Model
lrm(formula = y ~ tx * age, data = d)
Frequencies of Missing Values Due to Each Variable
y tx age 1444 0 0
Model Likelihood Ratio Test 
Discrimination Indexes 
Rank Discrim. Indexes 


Obs 3556  LR χ^{2} 330.73  R^{2} 0.131  C 0.697 
0 2658  d.f. 3  R^{2}_{3,3556} 0.088  D_{xy} 0.394 
1 898  Pr(>χ^{2}) <0.0001  R^{2}_{3,2013.7} 0.150  γ 0.394 
max ∂log L/∂β 1×10^{11}  Brier 0.171  τ_{a} 0.149 
β  S.E.  Wald Z  Pr(>Z)  

Intercept  4.6606  0.3849  12.11  <0.0001 
tx=b  0.4360  0.6056  0.72  0.4715 
age  0.0726  0.0069  10.56  <0.0001 
tx=b × age  0.0108  0.0106  1.02  0.3089 
Wald Statistics for y


χ^{2}  d.f.  P  

tx (Factor+Higher Order Factors)  154.98  2  <0.0001 
All Interactions  1.04  1  0.3089 
age (Factor+Higher Order Factors)  169.90  2  <0.0001 
All Interactions  1.04  1  0.3089 
tx × age (Factor+Higher Order Factors)  1.04  1  0.3089 
TOTAL  292.05  3  <0.0001 
This model included an unnecessary linear interaction term.
Logistic Regression Model
lrm(formula = form, data = d, tol = 1e12)
Frequencies of Missing Values Due to Each Variable
y tx age 1444 0 0
Model Likelihood Ratio Test 
Discrimination Indexes 
Rank Discrim. Indexes 


Obs 3556  LR χ^{2} 333.43  R^{2} 0.132  C 0.697 
0 2658  d.f. 7  R^{2}_{7,3556} 0.088  D_{xy} 0.395 
1 898  Pr(>χ^{2}) <0.0001  R^{2}_{7,2013.7} 0.150  γ 0.395 
max ∂log L/∂β 5×10^{11}  Brier 0.171  τ_{a} 0.149 
β  S.E.  Wald Z  Pr(>Z)  

Intercept  4.7933  0.5949  8.06  <0.0001 
tx=b  0.6350  0.9700  0.65  0.5127 
age  0.0752  0.0113  6.67  <0.0001 
age’  0.0274  0.1673  0.16  0.8699 
age’’  0.0392  0.4011  0.10  0.9221 
tx=b × age  0.0319  0.0183  1.74  0.0820 
tx=b × age’  0.2854  0.2321  1.23  0.2188 
tx=b × age’’  0.5634  0.5254  1.07  0.2836 
Wald Statistics for y


χ^{2}  d.f.  P  

tx (Factor+Higher Order Factors)  155.86  4  <0.0001 
All Interactions  3.03  3  0.3873 
age (Factor+Higher Order Factors)  175.89  6  <0.0001 
All Interactions  3.03  3  0.3873 
Nonlinear (Factor+Higher Order Factors)  2.73  4  0.6041 
tx × age (Factor+Higher Order Factors)  3.03  3  0.3873 
Nonlinear  1.93  2  0.3813 
Nonlinear Interaction : f(A,B) vs. AB  1.93  2  0.3813 
TOTAL NONLINEAR  2.73  4  0.6041 
TOTAL NONLINEAR + INTERACTION  3.82  5  0.5755 
TOTAL  299.44  7  <0.0001 
This model included unnecessary linear and nonlinear interaction terms.
Case Where Treatment Truly Interacts with Age
Estimation of Treatment Effect in Population With No Overlap
Next turn to the case where the true data generating model has a linear treatment by age interaction which we may or may not include in our model. The true model has a treatment effect of 1.0 for age 50 patients, and each year below 50 results in a further reduction by 0.03 in the effect.
Logistic Regression Model
lrm(formula = y ~ tx + age, data = d)
Frequencies of Missing Values Due to Each Variable
y tx age 2539 0 0
Model Likelihood Ratio Test 
Discrimination Indexes 
Rank Discrim. Indexes 


Obs 2461  LR χ^{2} 257.32  R^{2} 0.144  C 0.701 
0 1787  d.f. 2  R^{2}_{2,2461} 0.099  D_{xy} 0.403 
1 674  Pr(>χ^{2}) <0.0001  R^{2}_{2,1468.2} 0.160  γ 0.403 
max ∂log L/∂β 9×10^{6}  Brier 0.178  τ_{a} 0.160 
β  S.E.  Wald Z  Pr(>Z)  

Intercept  3.8181  0.4509  8.47  <0.0001 
tx=b  1.4027  0.1020  13.76  <0.0001 
age  0.0583  0.0077  7.61  <0.0001 
Wald Statistics for y


χ^{2}  d.f.  P  

tx  189.24  1  <0.0001 
age  57.92  1  <0.0001 
TOTAL  223.45  2  <0.0001 
This model failed to include a needed interaction term.
Logistic Regression Model
lrm(formula = y ~ tx * age, data = d)
Frequencies of Missing Values Due to Each Variable
y tx age 2539 0 0
Model Likelihood Ratio Test 
Discrimination Indexes 
Rank Discrim. Indexes 


Obs 2461  LR χ^{2} 258.51  R^{2} 0.144  C 0.702 
0 1787  d.f. 3  R^{2}_{3,2461} 0.099  D_{xy} 0.403 
1 674  Pr(>χ^{2}) <0.0001  R^{2}_{3,1468.2} 0.160  γ 0.403 
max ∂log L/∂β 2×10^{12}  Brier 0.178  τ_{a} 0.160 
β  S.E.  Wald Z  Pr(>Z)  

Intercept  4.2193  0.5852  7.21  <0.0001 
tx=b  0.3879  0.9367  0.41  0.6788 
age  0.0652  0.0100  6.53  <0.0001 
tx=b × age  0.0171  0.0157  1.09  0.2765 
Wald Statistics for y


χ^{2}  d.f.  P  

tx (Factor+Higher Order Factors)  191.47  2  <0.0001 
All Interactions  1.18  1  0.2765 
age (Factor+Higher Order Factors)  58.35  2  <0.0001 
All Interactions  1.18  1  0.2765 
tx × age (Factor+Higher Order Factors)  1.18  1  0.2765 
TOTAL  228.30  3  <0.0001 
Because of the more limited age range in the trial there was insufficient power to provide definitive statistical evidence for an interaction, but the point estimate for the interaction effect is not unreasonable. Though confidence bands are wide because of no overlap, the extrapolated treatment effects are reasonable as a result.
This is the correct model.
Logistic Regression Model
lrm(formula = form, data = d, tol = 1e12)
Frequencies of Missing Values Due to Each Variable
y tx age 2539 0 0
Model Likelihood Ratio Test 
Discrimination Indexes 
Rank Discrim. Indexes 


Obs 2461  LR χ^{2} 261.24  R^{2} 0.146  C 0.702 
0 1787  d.f. 7  R^{2}_{7,2461} 0.098  D_{xy} 0.403 
1 674  Pr(>χ^{2}) <0.0001  R^{2}_{7,1468.2} 0.159  γ 0.404 
max ∂log L/∂β 1×10^{12}  Brier 0.178  τ_{a} 0.161 
β  S.E.  Wald Z  Pr(>Z)  

Intercept  3.0988  1.2779  2.42  0.0153 
tx=b  0.1305  2.1946  0.06  0.9526 
age  0.0445  0.0231  1.93  0.0542 
age’  0.2124  0.2213  0.96  0.3372 
age’’  0.4384  0.4906  0.89  0.3716 
tx=b × age  0.0267  0.0396  0.68  0.4996 
tx=b × age’  0.0955  0.3313  0.29  0.7731 
tx=b × age’’  0.1934  0.6939  0.28  0.7805 
Wald Statistics for y


χ^{2}  d.f.  P  

tx (Factor+Higher Order Factors)  191.14  4  <0.0001 
All Interactions  1.19  3  0.7545 
age (Factor+Higher Order Factors)  61.60  6  <0.0001 
All Interactions  1.19  3  0.7545 
Nonlinear (Factor+Higher Order Factors)  2.67  4  0.6150 
tx × age (Factor+Higher Order Factors)  1.19  3  0.7545 
Nonlinear  0.08  2  0.9590 
Nonlinear Interaction : f(A,B) vs. AB  0.08  2  0.9590 
TOTAL NONLINEAR  2.67  4  0.6150 
TOTAL NONLINEAR + INTERACTION  3.78  5  0.5816 
TOTAL  229.68  7  <0.0001 
This model correctly captures linear interaction, but also allows for unnecessary nonlinear interaction.
RCT Sample Partially Overlaps with Target Population
Sticking with the true treatment effect interacting with age, we generate data with the same partial overlap as before, and repeat the same analyses. We start by fitting a model that is oblivious to interaction.
Logistic Regression Model
lrm(formula = y ~ tx + age, data = d)
Frequencies of Missing Values Due to Each Variable
y tx age 1444 0 0
Model Likelihood Ratio Test 
Discrimination Indexes 
Rank Discrim. Indexes 


Obs 3556  LR χ^{2} 323.95  R^{2} 0.130  C 0.699 
0 2702  d.f. 2  R^{2}_{2,3556} 0.087  D_{xy} 0.397 
1 854  Pr(>χ^{2}) <0.0001  R^{2}_{2,1946.7} 0.152  γ 0.397 
max ∂log L/∂β 8×10^{12}  Brier 0.165  τ_{a} 0.145 
β  S.E.  Wald Z  Pr(>Z)  

Intercept  3.7735  0.2939  12.84  <0.0001 
tx=b  1.2305  0.0876  14.04  <0.0001 
age  0.0566  0.0052  10.80  <0.0001 
Wald Statistics for y


χ^{2}  d.f.  P  

tx  197.20  1  <0.0001 
age  116.73  1  <0.0001 
TOTAL  281.79  2  <0.0001 
This model failed to include a needed interaction term.
Logistic Regression Model
lrm(formula = y ~ tx * age, data = d)
Frequencies of Missing Values Due to Each Variable
y tx age 1444 0 0
Model Likelihood Ratio Test 
Discrimination Indexes 
Rank Discrim. Indexes 


Obs 3556  LR χ^{2} 338.39  R^{2} 0.136  C 0.700 
0 2702  d.f. 3  R^{2}_{3,3556} 0.090  D_{xy} 0.400 
1 854  Pr(>χ^{2}) <0.0001  R^{2}_{3,1946.7} 0.158  γ 0.401 
max ∂log L/∂β 3×10^{12}  Brier 0.164  τ_{a} 0.146 
β  S.E.  Wald Z  Pr(>Z)  

Intercept  4.6606  0.3849  12.11  <0.0001 
tx=b  1.0820  0.6129  1.77  0.0775 
age  0.0726  0.0069  10.56  <0.0001 
tx=b × age  0.0412  0.0109  3.79  0.0002 
Wald Statistics for y


χ^{2}  d.f.  P  

tx (Factor+Higher Order Factors)  210.59  2  <0.0001 
All Interactions  14.36  1  0.0002 
age (Factor+Higher Order Factors)  125.47  2  <0.0001 
All Interactions  14.36  1  0.0002 
tx × age (Factor+Higher Order Factors)  14.36  1  0.0002 
TOTAL  310.44  3  <0.0001 
The amount of interaction estimated from the larger older sample extrapolated well to the younger target population.
This is the correct model.
Logistic Regression Model
lrm(formula = form, data = d, tol = 1e12)
Frequencies of Missing Values Due to Each Variable
y tx age 1444 0 0
Model Likelihood Ratio Test 
Discrimination Indexes 
Rank Discrim. Indexes 


Obs 3556  LR χ^{2} 343.94  R^{2} 0.138  C 0.700 
0 2702  d.f. 7  R^{2}_{7,3556} 0.090  D_{xy} 0.401 
1 854  Pr(>χ^{2}) <0.0001  R^{2}_{7,1946.7} 0.159  γ 0.401 
max ∂log L/∂β 7×10^{12}  Brier 0.164  τ_{a} 0.146 
β  S.E.  Wald Z  Pr(>Z)  

Intercept  4.7933  0.5949  8.06  <0.0001 
tx=b  2.5181  0.9368  2.69  0.0072 
age  0.0752  0.0113  6.67  <0.0001 
age’  0.0274  0.1673  0.16  0.8699 
age’’  0.0392  0.4011  0.10  0.9221 
tx=b × age  0.0698  0.0179  3.91  <0.0001 
tx=b × age’  0.4148  0.2398  1.73  0.0836 
tx=b × age’’  0.8163  0.5469  1.49  0.1355 
Wald Statistics for y


χ^{2}  d.f.  P  

tx (Factor+Higher Order Factors)  211.50  4  <0.0001 
All Interactions  19.06  3  0.0003 
age (Factor+Higher Order Factors)  132.65  6  <0.0001 
All Interactions  19.06  3  0.0003 
Nonlinear (Factor+Higher Order Factors)  5.66  4  0.2263 
tx × age (Factor+Higher Order Factors)  19.06  3  0.0003 
Nonlinear  3.97  2  0.1374 
Nonlinear Interaction : f(A,B) vs. AB  3.97  2  0.1374 
TOTAL NONLINEAR  5.66  4  0.2263 
TOTAL NONLINEAR + INTERACTION  20.91  5  0.0008 
TOTAL  316.28  7  <0.0001 
The spline interaction estimates did not extrapolate well to the very young.
This model correctly captures linear interaction, but also allows for unnecessary nonlinear interaction.
Ability to Compare Treatments in Observational Studies
When a categorical baseline characteristic has all of the patients in one of its categories getting a single treatment, one cannot estimate efficacy in that category unless (1) there is no interaction between treatment and the variable or (2) the baseline variable does not relate to outcome. If one of these conditions is not satisfied, one would have to do a conditional analysis, i.e., to estimate efficacy in patients not in the offending category. If a continuous baseline variable has an interval for which every patient in that interval received only one treatment, then one cannot estimate the treatment effect in that interval unless (1) there is no interaction (linear or nonlinear) between the variable and treatment or (2) the variable is irrelevant to outcome. So when you hear that nonoverlap implies that only a conditional treatment comparison can be done, be aware of the true underlying assumptions. Note that nonoverlap on a variable implies nonoverlap for one or more values of a propensity score.
To illustrate these points, consider an observational study where the data generating process is from the same population logistic model as used above, but instead of considering generalizability to a population, consider the insample comparison of treatments a and b adjusted for age. We start with a sample in which all those with age ≥ 50 got treatment b and all those with age < 50 got treatment a.
True Model Has No Treatment Interactions
No Overlap Between Treatment Groups
All patients age < 50 had treatment a, and those ≥ 50 had b.
Age distributions in the observed treatment groups are shown below.
Logistic Regression Model
lrm(formula = y ~ tx + age, data = d)
Model Likelihood Ratio Test 
Discrimination Indexes 
Rank Discrim. Indexes 


Obs 2999  LR χ^{2} 87.42  R^{2} 0.046  C 0.616 
0 2431  d.f. 2  R^{2}_{2,2999} 0.028  D_{xy} 0.232 
1 568  Pr(>χ^{2}) <0.0001  R^{2}_{2,1381.3} 0.060  γ 0.232 
max ∂log L/∂β 8×10^{9}  Brier 0.149  τ_{a} 0.071 
β  S.E.  Wald Z  Pr(>Z)  

Intercept  4.5743  0.3532  12.95  <0.0001 
tx=b  1.0283  0.1608  6.40  <0.0001 
age  0.0714  0.0080  8.96  <0.0001 
Wald Statistics for y


χ^{2}  d.f.  P  

tx  40.90  1  <0.0001 
age  80.26  1  <0.0001 
TOTAL  82.33  2  <0.0001 
The modelbased covariate adjustment provided the correct estimate of the treatment effect even with no overlap, since the specified model is the correct model.
This is the correct model.
Logistic Regression Model
lrm(formula = y ~ tx * age, data = d)
Model Likelihood Ratio Test 
Discrimination Indexes 
Rank Discrim. Indexes 


Obs 2999  LR χ^{2} 88.00  R^{2} 0.047  C 0.616 
0 2431  d.f. 3  R^{2}_{3,2999} 0.028  D_{xy} 0.231 
1 568  Pr(>χ^{2}) <0.0001  R^{2}_{3,1381.3} 0.060  γ 0.232 
max ∂log L/∂β 5×10^{7}  Brier 0.149  τ_{a} 0.071 
β  S.E.  Wald Z  Pr(>Z)  

Intercept  4.2550  0.5399  7.88  <0.0001 
tx=b  1.6471  0.8237  2.00  0.0455 
age  0.0641  0.0124  5.19  <0.0001 
tx=b × age  0.0124  0.0161  0.77  0.4430 
Wald Statistics for y


χ^{2}  d.f.  P  

tx (Factor+Higher Order Factors)  40.98  2  <0.0001 
All Interactions  0.59  1  0.4430 
age (Factor+Higher Order Factors)  81.73  2  <0.0001 
All Interactions  0.59  1  0.4430 
tx × age (Factor+Higher Order Factors)  0.59  1  0.4430 
TOTAL  84.33  3  <0.0001 
With no age overlap, the treatment by age interaction is estimated very inefficiently and in this case incorrectly. Wide confidence bands correctly capture the difficulty of the task of estimating agespecific treatment effects.
This model included an unnecessary linear interaction term.
Logistic Regression Model
lrm(formula = form, data = d, tol = 1e12)
Model Likelihood Ratio Test 
Discrimination Indexes 
Rank Discrim. Indexes 


Obs 2999  LR χ^{2} 89.34  R^{2} 0.047  C 0.617 
0 2431  d.f. 5  R^{2}_{5,2999} 0.028  D_{xy} 0.233 
1 568  Pr(>χ^{2}) <0.0001  R^{2}_{5,1381.3} 0.059  γ 0.234 
max ∂log L/∂β 7×10^{12}  Brier 0.149  τ_{a} 0.072 
β  S.E.  Wald Z  Pr(>Z)  

Intercept  5.0833  0.9349  5.44  <0.0001 
tx=b  0.4348  7.1719  0.06  0.9517 
age  0.0868  0.0241  3.60  0.0003 
age’  0.1233  0.1090  1.13  0.2580 
tx=b × age  0.0361  0.1482  0.24  0.8073 
tx=b × age’  0.1420  0.1520  0.93  0.3502 
Wald Statistics for y


χ^{2}  d.f.  P  

tx (Factor+Higher Order Factors)  40.98  3  <0.0001 
All Interactions  1.50  2  0.4734 
age (Factor+Higher Order Factors)  80.92  4  <0.0001 
All Interactions  1.50  2  0.4734 
Nonlinear (Factor+Higher Order Factors)  1.31  2  0.5193 
tx × age (Factor+Higher Order Factors)  1.50  2  0.4734 
Nonlinear  0.87  1  0.3502 
Nonlinear Interaction : f(A,B) vs. AB  0.87  1  0.3502 
TOTAL NONLINEAR  1.31  2  0.5193 
TOTAL NONLINEAR + INTERACTION  1.90  3  0.5925 
TOTAL  83.33  5  <0.0001 
This model included unnecessary linear and nonlinear interaction terms.
Some Overlap in Ages Between Treatments
Now consider an observational study, again with confounding by indication, related to age. Partial overlap is defined by specifying separate study inclusion probability functions. Given the age and treatment these functions specify the probability that a patient would be included in the observational study. The first plot shows the inclusion probabilities separately for treatments a and b. Then the result of a linear nointeraction model are shown. As before, blue curves depict true datagenerating model log odds.
The graph below shows the probability of selection into each of the observed treatment groups as a function of age.
Age distributions in the observed treatment groups are shown below.
Logistic Regression Model
lrm(formula = y ~ tx + age, data = d)
Model Likelihood Ratio Test 
Discrimination Indexes 
Rank Discrim. Indexes 


Obs 3031  LR χ^{2} 474.99  R^{2} 0.212  C 0.744 
0 2236  d.f. 2  R^{2}_{2,3031} 0.144  D_{xy} 0.488 
1 795  Pr(>χ^{2}) <0.0001  R^{2}_{2,1759.4} 0.236  γ 0.489 
max ∂log L/∂β 9×10^{6}  Brier 0.166  τ_{a} 0.189 
β  S.E.  Wald Z  Pr(>Z)  

Intercept  4.8959  0.3443  14.22  <0.0001 
tx=b  1.0104  0.1787  5.65  <0.0001 
age  0.0769  0.0061  12.53  <0.0001 
Wald Statistics for y


χ^{2}  d.f.  P  

tx  31.98  1  <0.0001 
age  156.93  1  <0.0001 
TOTAL  335.25  2  <0.0001 
This is the correct model.
Logistic Regression Model
lrm(formula = y ~ tx * age, data = d)
Model Likelihood Ratio Test 
Discrimination Indexes 
Rank Discrim. Indexes 


Obs 3031  LR χ^{2} 475.20  R^{2} 0.212  C 0.744 
0 2236  d.f. 3  R^{2}_{3,3031} 0.144  D_{xy} 0.488 
1 795  Pr(>χ^{2}) <0.0001  R^{2}_{3,1759.4} 0.235  γ 0.489 
max ∂log L/∂β 1×10^{8}  Brier 0.166  τ_{a} 0.189 
β  S.E.  Wald Z  Pr(>Z)  

Intercept  4.8638  0.3512  13.85  <0.0001 
tx=b  1.6044  1.3432  1.19  0.2323 
age  0.0763  0.0063  12.18  <0.0001 
tx=b × age  0.0141  0.0314  0.45  0.6546 
Wald Statistics for y


χ^{2}  d.f.  P  

tx (Factor+Higher Order Factors)  32.33  2  <0.0001 
All Interactions  0.20  1  0.6546 
age (Factor+Higher Order Factors)  157.04  2  <0.0001 
All Interactions  0.20  1  0.6546 
tx × age (Factor+Higher Order Factors)  0.20  1  0.6546 
TOTAL  330.68  3  <0.0001 
The confidence bands are correctly registering that there is little information about the treatment effect outside of the heavier overlap region.
This model included an unnecessary linear interaction term.
Logistic Regression Model
lrm(formula = form, data = d, tol = 1e12)
Model Likelihood Ratio Test 
Discrimination Indexes 
Rank Discrim. Indexes 


Obs 3031  LR χ^{2} 476.54  R^{2} 0.213  C 0.744 
0 2236  d.f. 5  R^{2}_{5,3031} 0.144  D_{xy} 0.489 
1 795  Pr(>χ^{2}) <0.0001  R^{2}_{5,1759.4} 0.235  γ 0.489 
max ∂log L/∂β 2×10^{7}  Brier 0.166  τ_{a} 0.189 
β  S.E.  Wald Z  Pr(>Z)  

Intercept  6.2165  1.3037  4.77  <0.0001 
tx=b  0.2424  2.3347  0.10  0.9173 
age  0.1058  0.0280  3.78  0.0002 
age’  0.0262  0.0240  1.09  0.2761 
tx=b × age  0.0291  0.0579  0.50  0.6155 
tx=b × age’  0.1247  0.2974  0.42  0.6749 
Wald Statistics for y


χ^{2}  d.f.  P  

tx (Factor+Higher Order Factors)  20.25  3  0.0002 
All Interactions  0.26  2  0.8794 
age (Factor+Higher Order Factors)  154.87  4  <0.0001 
All Interactions  0.26  2  0.8794 
Nonlinear (Factor+Higher Order Factors)  1.30  2  0.5229 
tx × age (Factor+Higher Order Factors)  0.26  2  0.8794 
Nonlinear  0.18  1  0.6749 
Nonlinear Interaction : f(A,B) vs. AB  0.18  1  0.6749 
TOTAL NONLINEAR  1.30  2  0.5229 
TOTAL NONLINEAR + INTERACTION  1.51  3  0.6793 
TOTAL  330.06  5  <0.0001 
This model included unnecessary linear and nonlinear interaction terms.
Case Where Treatment Truly Interacts with Age
No Overlap Between Treatment Groups
Return to the data generating model used in the RCT simulation, for which there is truly a linear interaction with treatment, and start with the no overlap case.
Logistic Regression Model
lrm(formula = y ~ tx + age, data = d)
Model Likelihood Ratio Test 
Discrimination Indexes 
Rank Discrim. Indexes 


Obs 2999  LR χ^{2} 39.09  R^{2} 0.022  C 0.591 
0 2493  d.f. 2  R^{2}_{2,2999} 0.012  D_{xy} 0.181 
1 506  Pr(>χ^{2}) <0.0001  R^{2}_{2,1261.9} 0.029  γ 0.181 
max ∂log L/∂β 4×10^{11}  Brier 0.139  τ_{a} 0.051 
β  S.E.  Wald Z  Pr(>Z)  

Intercept  3.5659  0.3538  10.08  <0.0001 
tx=b  0.9499  0.1661  5.72  <0.0001 
age  0.0481  0.0081  5.96  <0.0001 
Wald Statistics for y


χ^{2}  d.f.  P  

tx  32.71  1  <0.0001 
age  35.51  1  <0.0001 
TOTAL  37.94  2  <0.0001 
The treatment effect is incorrect example at age=50.
This model failed to include a needed interaction term.
Logistic Regression Model
lrm(formula = y ~ tx * age, data = d)
Model Likelihood Ratio Test 
Discrimination Indexes 
Rank Discrim. Indexes 


Obs 2999  LR χ^{2} 42.28  R^{2} 0.023  C 0.592 
0 2493  d.f. 3  R^{2}_{3,2999} 0.013  D_{xy} 0.184 
1 506  Pr(>χ^{2}) <0.0001  R^{2}_{3,1261.9} 0.031  γ 0.185 
max ∂log L/∂β 3×10^{8}  Brier 0.138  τ_{a} 0.052 
β  S.E.  Wald Z  Pr(>Z)  

Intercept  4.2550  0.5399  7.88  <0.0001 
tx=b  0.5441  0.8578  0.63  0.5259 
age  0.0641  0.0124  5.19  <0.0001 
tx=b × age  0.0295  0.0167  1.77  0.0767 
Wald Statistics for y


χ^{2}  d.f.  P  

tx (Factor+Higher Order Factors)  36.26  2  <0.0001 
All Interactions  3.13  1  0.0767 
age (Factor+Higher Order Factors)  36.45  2  <0.0001 
All Interactions  3.13  1  0.0767 
tx × age (Factor+Higher Order Factors)  3.13  1  0.0767 
TOTAL  40.05  3  <0.0001 
The extrapolation is excellent. Now try the overkill model.
This is the correct model.
Logistic Regression Model
lrm(formula = form, data = d, tol = 1e12)
Model Likelihood Ratio Test 
Discrimination Indexes 
Rank Discrim. Indexes 


Obs 2999  LR χ^{2} 44.02  R^{2} 0.024  C 0.592 
0 2493  d.f. 5  R^{2}_{5,2999} 0.013  D_{xy} 0.184 
1 506  Pr(>χ^{2}) <0.0001  R^{2}_{5,1261.9} 0.030  γ 0.185 
max ∂log L/∂β 2×10^{5}  Brier 0.138  τ_{a} 0.052 
β  S.E.  Wald Z  Pr(>Z)  

Intercept  5.0833  0.9349  5.44  <0.0001 
tx=b  3.5002  7.5833  0.46  0.6444 
age  0.0868  0.0241  3.60  0.0003 
age’  0.1233  0.1090  1.13  0.2580 
tx=b × age  0.0483  0.1568  0.31  0.7581 
tx=b × age’  0.0498  0.1569  0.32  0.7508 
Wald Statistics for y


χ^{2}  d.f.  P  

tx (Factor+Higher Order Factors)  32.65  3  <0.0001 
All Interactions  0.97  2  0.6143 
age (Factor+Higher Order Factors)  35.52  4  <0.0001 
All Interactions  0.97  2  0.6143 
Nonlinear (Factor+Higher Order Factors)  1.70  2  0.4269 
tx × age (Factor+Higher Order Factors)  0.97  2  0.6143 
Nonlinear  0.10  1  0.7508 
Nonlinear Interaction : f(A,B) vs. AB  0.10  1  0.7508 
TOTAL NONLINEAR  1.70  2  0.4269 
TOTAL NONLINEAR + INTERACTION  4.59  3  0.2041 
TOTAL  39.24  5  <0.0001 
Extrapolation failed, and the failure was thankfully signaled by very wide confidence bands for agespecific treatment effects.
This model correctly captures linear interaction, but also allows for unnecessary nonlinear interaction.
Some Overlap in Ages Between Treatments
Logistic Regression Model
lrm(formula = y ~ tx + age, data = d)
Model Likelihood Ratio Test 
Discrimination Indexes 
Rank Discrim. Indexes 


Obs 3031  LR χ^{2} 421.49  R^{2} 0.189  C 0.732 
0 2221  d.f. 2  R^{2}_{2,3031} 0.129  D_{xy} 0.464 
1 810  Pr(>χ^{2}) <0.0001  R^{2}_{2,1780.6} 0.210  γ 0.464 
max ∂log L/∂β 1×10^{7}  Brier 0.170  τ_{a} 0.182 
β  S.E.  Wald Z  Pr(>Z)  

Intercept  4.7976  0.3406  14.09  <0.0001 
tx=b  0.7412  0.1640  4.52  <0.0001 
age  0.0751  0.0061  12.37  <0.0001 
Wald Statistics for y


χ^{2}  d.f.  P  

tx  20.41  1  <0.0001 
age  153.01  1  <0.0001 
TOTAL  319.21  2  <0.0001 
The nointeraction model missed the boat. Now fit the correct linear interaction model.
This model failed to include a needed interaction term.
Logistic Regression Model
lrm(formula = y ~ tx * age, data = d)
Model Likelihood Ratio Test 
Discrimination Indexes 
Rank Discrim. Indexes 


Obs 3031  LR χ^{2} 422.14  R^{2} 0.189  C 0.732 
0 2221  d.f. 3  R^{2}_{3,3031} 0.129  D_{xy} 0.464 
1 810  Pr(>χ^{2}) <0.0001  R^{2}_{3,1780.6} 0.210  γ 0.464 
max ∂log L/∂β 1×10^{5}  Brier 0.170  τ_{a} 0.182 
β  S.E.  Wald Z  Pr(>Z)  

Intercept  4.8638  0.3512  13.85  <0.0001 
tx=b  0.1333  1.0813  0.12  0.9019 
age  0.0763  0.0063  12.18  <0.0001 
tx=b × age  0.0208  0.0255  0.81  0.4151 
Wald Statistics for y


χ^{2}  d.f.  P  

tx (Factor+Higher Order Factors)  20.56  2  <0.0001 
All Interactions  0.66  1  0.4151 
age (Factor+Higher Order Factors)  153.46  2  <0.0001 
All Interactions  0.66  1  0.4151 
tx × age (Factor+Higher Order Factors)  0.66  1  0.4151 
TOTAL  326.02  3  <0.0001 
There is reasonable extrapolation. Now for the overkill model.
This is the correct model.
Logistic Regression Model
lrm(formula = form, data = d, tol = 1e12)
Model Likelihood Ratio Test 
Discrimination Indexes 
Rank Discrim. Indexes 


Obs 3031  LR χ^{2} 423.48  R^{2} 0.190  C 0.732 
0 2221  d.f. 5  R^{2}_{5,3031} 0.129  D_{xy} 0.464 
1 810  Pr(>χ^{2}) <0.0001  R^{2}_{5,1780.6} 0.209  γ 0.464 
max ∂log L/∂β 3×10^{5}  Brier 0.170  τ_{a} 0.182 
β  S.E.  Wald Z  Pr(>Z)  

Intercept  6.2165  1.3037  4.77  <0.0001 
tx=b  1.8369  1.9527  0.94  0.3468 
age  0.1058  0.0280  3.78  0.0002 
age’  0.0262  0.0240  1.09  0.2761 
tx=b × age  0.0602  0.0477  1.26  0.2067 
tx=b × age’  0.1114  0.2592  0.43  0.6672 
Wald Statistics for y


χ^{2}  d.f.  P  

tx (Factor+Higher Order Factors)  16.32  3  0.0010 
All Interactions  1.94  2  0.3784 
age (Factor+Higher Order Factors)  151.01  4  <0.0001 
All Interactions  1.94  2  0.3784 
Nonlinear (Factor+Higher Order Factors)  1.30  2  0.5233 
tx × age (Factor+Higher Order Factors)  1.94  2  0.3784 
Nonlinear  0.18  1  0.6672 
Nonlinear Interaction : f(A,B) vs. AB  0.18  1  0.6672 
TOTAL NONLINEAR  1.30  2  0.5233 
TOTAL NONLINEAR + INTERACTION  1.98  3  0.5774 
TOTAL  325.31  5  <0.0001 
Extrapolated treatment effect estimates are soso.
This model correctly captures linear interaction, but also allows for unnecessary nonlinear interaction.
Summary
With respect to relative efficacy (or both relative and absolute efficacy for continuous repsonse variables), clinical trials do not require having a representative sample of the target clinical population if there are no interactions with treatment. If there are interactions, let M denote the levels of interacting factors that are well represented in the target population. In our examples, M represents younger patients. Even with interaction present, the randomized trial still does not need to be on a representative patient sample if either (1) the trial sample is representative with respect to M (in which case omitting interactions from the model is not fatal), or (2) there is just enough representation in the sample with respect to M, and those interacting factors are appropriately modeled in the randomized trial. Unless M is richly represented in the trial, using statistical testing to decide on which interactions to include in the model is not advised, due to low power of such interaction tests. Suspected interactions, even statistically weak ones, should be included in the model when some degree of extrapolation is sought. When M is very poorly represented in the trial, extrapolation to the target population makes strong model assumptions. Thankfully confidence intervals for extrapolated efficacy estimates will be properly wide to reflect the weak basis for such extrapolation.
In a similar vein, observational treatment comparisons can be appropriate if factors that do not overlap between treatment groups do not interact with treatment. So a key to understanding both overlap and clinical trial generalizability is interactions.
Further Reading
 Why representativeness should be avoided by Rothman, Gallacher, and Hatch
 Treatment effects may remain the same even when trial participants differed from the target population by MJ Bradburn et al.
Questions and Discussion
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