This paper presents proportional odds cure models to allow spatial correlations by including spatial frailty in the interval censored data setting. Parametric cure rate models with independent and dependent spatial frailties are proposed and compared. Our approach enables different underlying activation mechanisms that lead to the event of interest; in addition, the number of competing causes which may be responsible for the occurrence of the event of interest follows a Geometric distribution. Markov chain Monte Carlo method is used in a Bayesian framework for inferential purposes. For model comparison some Bayesian criteria were used. An influence diagnostic analysis was conducted to detect possible influential or extreme observations that may cause distortions on the results of the analysis. Finally, the proposed models are applied for the analysis of a real data set on smoking cessation. The results of the application show that the parametric cure model with frailties under the first activation scheme has better findings.
Advances in medical and health science have led to several clinical studies that showed non-negligible proportions of patients who respond favorably to a treatment and are non-susceptible to the event of interest. Such a proportion of non-susceptible patients is considered cured or with prolonged disease-free survival which is usually referred to as the cured fraction. Survival models with cure fraction, also known as long-term survival models or cure rate models, have been widely developed to accommodate a cured fraction.
One of the most popular cure rate model is the mixture model introduced by Berkson and Gage (1952). It assumes that a certain proportion of the patients are cured, in the sense that they do not present the event of interest during a long period of time and can be seen as cured or immune from the causes of death under study (Rodrigues
Gu
In many clinical trial studies, it is very common for patients to be periodically examined for disease occurrence or progression. In such a situation, the exact failure time which patients cannot be observed exists, but can only be allowed to lie in an interval after a sequence of examination times that is known as interval-censored (Peto, 1973). The estimation methods available to the right-censored data, such as the Kaplan-Meier estimator, is not considered adequate to be applied in interval-censored data, because it can lead to biased estimation and invalid inferences. The information concerning interval censorship should be taken into account in modeling (Lindsey and Ryan, 1998; Rücker and Messerer, 1988; Sun and Chen, 2010).
The lifetimes data are sometimes collected from several regions, which can lead to different effects for each observation. To consider these different effects in the cure model for interval-censored data, Xiang
This paper analyzes smoking cessation data. In a smoking cessation study, all patients (smokers) were randomized both into a smoking intervention (SI) group, or into a usual care (UC) group that receives no special anti-SI. The SI program treatments were conducted in Rochester city, being located in the center of the map. The details concerning the programs can be found in Murray
In this paper, we present a proportional odds cure model (Gu
We also conduct influence diagnostics to examine the assumptions of the model and conduct studies on sensitivity to detect possible influential or extreme observations that can cause distortions in the results of the analysis. Here the influence of the diagnostics of case deletion are developed for the posterior joint distribution based on the
The remainder of the paper is organized as follows. In Section 2, the introduction of frailty models and some distributions for spatial frailties are presented and followed by the spatial distributional aspects of our modeling. The Bayesian inference for the proposed models is developed in the Section 3. In Section 4, the proposed models are fit into a read data set (smoking cessation study). Finally, Section 5 concludes with some general remarks.
Gu
where
Let
Given
Note that this survival function has a proportional odds structure when covariates
This CRPO model can also be obtained as an extension of the transformation-cure model (Zeng
The probability density function (pdf) and the hazard function associated to (
respectively, where
Note that, the survival function in (
Thus, the survival functions of uncured (susceptible) individuals can be expressed by
Now, if we assume another situation where the presence of all latent risks will ultimately lead to the occurrence of the event. In this case we are assuming the cause, which is later, not even knowing what it is, cause relapse into smoking. Thus, the time to the event of interest is defined by the random variable
The corresponding pdf and the hazard function are given by
and
respectively. The survival function (
Thus, the survival functions of susceptible individuals is given by
The first situation is also known as first activation (FA) scheme because, in this case, we assume the event of interest occurs when the first possible cause is activated. However, the second situation is known as the last activation (LA) scheme because the event of interest only takes place after all the latent causes have been activated (Cooner
where the superscript
Note that whichever the activation scheme, the density and hazard functions of the cure models are improper functions, since the survival functions are not proper. Its cure fraction is the same for all the activation schemes and can be obtained by
The cure fraction plays a key role in the survival models with cure fraction. So we consider the parametrization of the model in terms of the cure fraction. Since
where
The model in (
Henceforth, we assume the Weibull distribution for the latent variables
By following Banerjee and Carlin (2004), we introduce the frailties
Here, the frailties
The CAR model was originally developed by Besag (1974). These priors have become very popular in the Bayesian analysis of real data, especially in disease mapping. Let
Second, we assume that the spatial priors on (
We also assume the parameter
where
Let = {(
where
where
For the independent assumption, we employ the separate independent CAR prior on the random frailties
where
where
where
However, the full conditional distributions for parameters
where
Thus, the joint posterior density of
Now we assume the spatial priors of the parameters (
Further, we employ the parameter
Here, we consider the prior distributions for the parameters
where
To avoid range restrictions in parameters
The joint posterior density is given by
where
This joint posterior density is analytically intractable. Thus, we based our inference on MCMC simulation methods. We can observed that the full conditional distributions for the parameters
where
Thus, the full conditional distribution for
There are several Bayesian criteria to compare competing models for a given data set and to select the one that best fits the data. One of the most used in applied works is based on the posterior mean of deviance and called the DIC. For a model, the statistic DIC is defined as
where
Performing a sensitivity analysis is advisable since regression models are sensitive to underlying model assumptions. One of the most used ways of evaluating the influence of an observation in the fitted model is a case-deletion (Cook andWeisberg, 1982), where the effects are studied by completely removing cases from the analysis. This reasoning forms the basis of the Bayesian global influence methodology and makes it possible to determine which subjects might influence the analysis. The Bayesian case-deletion influence diagnostic measures for the joint posterior distribution based on the
Let
where
Let
where
Note that
where
It is not difficult to see for the divergence measures considered that
This section presents simulation studies for a cure rate model under FA with the dependent assumption of examining their performances. The interval-censored survival times (
We generate the latent Geometric variable
If
If
For
For
In this study, we consider
Random effects
For each generated sample, we simulate one chain of size 10,000 for each parameter, disregarding the first 1,000 iterations to eliminate the effect of the initial values and avoid correlation problems and thinning to every third iteration, thus obtaining an effective sample of size 3,000 upon which the posterior is based on. To evaluate the performance of the parameter estimates, the average bias (Bias), standard deviation (SD) of the estimate, average SD (SDs mean), and the mean square error (MSE) are calculated for a cure rate model under the FA (Table 1).
We can note that the bias and the MSE of the parameter Λ_{12} are larger than other parameters. The estimator of the Λ_{12} presents a negative biases; however, the biases and MSEs are always near zero. Moreover, the simulation results for the cure model considering the prior 1 are close to those obtained considering the prior 2.
A goal of this study is to show the need for robust models to deal with the presence of outliers in data. We consider two cases for perturbation with the parameter’s values and the setup the same as in the simulation studies; therefore, four data sets of size 100 were generated from the cure model under the FA with dependent spatial frailties.
We selected the cases 18 and 80 for perturbation. To create influential observations concerning the data set, we choose one or two of these selected cases and perturbed the response variable as:
Table 2 reports the posterior mean, the SD, the bias and the MSE of the parameters of WCRM-FA. We can note that the estimative of parameter Λ_{11} creasing in the perturbation cases when prior 1 is used. However, considering prior 2 for the parameters, the estimative of all parameters of cases B, C, and D are very closed the case A, which means that the parameters are not sensitive to perturbations.
For each simulated data set, the four divergence measures (
We note that all measures provide larger
To better show present the results, we plot the J-distance measure from the fitted models considering the prior 1 for the parameters. The Figure 1 presents the divergence measures before the perturbation (setup A), the model indicates the absence of outline observations, and after perturbation observations (setups B, C, and D). Note that outline observation 18 cannot be easy detected for the cure rate model under the FA.
Recall the interval-censored smoking cessation data briefly presented in the introduction section. In the smoking cessation study, all patients (smokers) were randomized into either a SI group, or a UC group that received no special anti-SI. The treatments of the SI program were realized in Rochester city localizing in the center of the maps. The program detail can be found in Murray
We fitted cure rate model under first and LA, considering the different spatial frailties in the models to the data set. The prior distributions for the parameters
Because of the high computational cost, we implement the MCMC algorithms in the C programming language; subsequently, the results were analyzed in the R language (R Development Core Team, 2010) through the “coda” package (Plummer
Table 4 provides the DIC scores for a variety of effects of the Weibull cure model under FA and LA. The DIC scores of the Model 1 and 5 show the best models despite DIC values that are close to each other. We also can note that the cure rate models under the FA are more adequate than models under the LA.
For the comparison with the models proposed by Carlin and Banerjee (2003), we consider the same prior distributions for the parameters
Table 6 presents the posterior summary of the parameters of the Model 15. We note that only the parameters
The survival model shows that the special intervention and the number of cigarettes smoked per day have negative effects on the hazard rate of the relapse time; therefore, individuals with special intervention do not present lower hazard rates for the relapse time when compared to those who attend UC. However, the individuals with a higher level of cigarette consumption do not present high hazard rates.
The estimated
The Figure 2 maps the posterior means of the frailties
The Figure 3 maps the posterior SD of the frailties
Figure 4 presents the estimates of
The posterior summaries of the parameters for the readjust Model 15 and RV for the posterior mean of the parameters are presented in Table 8. We can note that only the values of RV for the posterior means of the parameters Λ_{12} and ∑_{12} are more than one, but they still have posterior means near zero, and others parameters have posterior means near the obtained values for the completed data set. In this case, there are no inferential changes after removing the observations.
This paper described an approach to extend the proportional odds cure models to allow to spatial correlations by including spatial frailties in the interval-censored data setting. We used MCMC methods through Bayesian inference and the DIC for the model comparison. The results of the application show that the parametric cure model with frailties under the FA scheme have better fittings. A comparison of the proposed models with the model introduced by Carlin and Banerjee (2003) indicated that our model is more adequate. Moreover, the both models proposed are not sensitive to influence observations, which can be observed by the influence diagnostic analysis in the application. The interpretation of the covariates was easy due to the parametrization of the models considered in terms of the cure rate. The MCAR prior can also be used even if frailties do not present or do present low correlations. We assume
The research is partially supported by the Brazilian organizations CAPES (via Science without Borders-PDSE), CNPq and FAPESP.
Simulation results for the Weibull cure model under the first activation with depended spatial frailties
Parameter | True value | Estimate mean | SD of the estimate | Bias | MSE | SDs mean |
---|---|---|---|---|---|---|
Prior 1: | ||||||
−1.50 | −1.4812 | 0.0708 | 0.0188 | 0.0054 | 0.2685 | |
−0.50 | −0.5209 | 0.1280 | −0.0209 | 0.0168 | 0.2485 | |
−0.15 | −0.1360 | 0.0466 | 0.0140 | 0.0024 | 0.1915 | |
0.30 | 0.1930 | 0.0521 | −0.1070 | 0.0142 | 0.0677 | |
Λ_{11} | 4.00 | 4.0062 | 0.1597 | 0.0062 | 0.0255 | 2.4728 |
Λ_{22} | 4.00 | 4.0120 | 0.1918 | 0.0120 | 0.0369 | 2.6151 |
Λ_{12} | 0.00 | −0.4541 | 0.1343 | −0.4541 | 0.2242 | 1.9196 |
0.90 | 0.9001 | 0.0016 | 0.0001 | 0.0000 | 0.0653 | |
Prior 2: | ||||||
−1.50 | −1.4902 | 0.0701 | 0.0098 | 0.0050 | 0.2583 | |
−0.50 | −0.5376 | 0.1330 | −0.0376 | 0.0191 | 0.2227 | |
−0.15 | −0.1295 | 0.0493 | 0.0205 | 0.0028 | 0.1870 | |
0.30 | 0.1863 | 0.0443 | −0.1137 | 0.0149 | 0.0536 | |
Λ_{11} | 4.00 | 4.1638 | 0.1676 | 0.1638 | 0.0549 | 2.5070 |
Λ_{22} | 4.00 | 4.2657 | 0.1819 | 0.2657 | 0.1036 | 2.6919 |
Λ_{12} | 0.00 | −0.5809 | 0.1472 | −0.5809 | 0.3591 | 1.9647 |
0.90 | 0.8999 | 0.0015 | −0.0002 | 0.0000 | 0.0655 | |
0.90 | 0.9002 | 0.0015 | 0.0002 | 0.0000 | 0.0654 |
SD = standard deviation; Bias = average bias; MSE = mean square error; SDs mean = average SD; MCAR = multivariate conditionally autoregressive.
Simulation results of the perturbed cases for the Weibull cure rate models under the first activation
Setup | Perturbed case | Prior 1 | Prior 2 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Parameters | Mean | SD | Bias | MSE | Parameters | Mean | SD | Bias | MSE | ||
A | None | −1.482 | 0.278 | 0.018 | 0.000 | −1.341 | 0.277 | 0.159 | 0.025 | ||
−0.859 | 0.259 | −0.359 | 0.129 | −0.554 | 0.248 | −0.054 | 0.003 | ||||
−0.036 | 0.193 | 0.114 | 0.013 | −0.105 | 0.197 | 0.045 | 0.002 | ||||
0.228 | 0.068 | −0.072 | 0.005 | 0.112 | 0.072 | −0.188 | 0.035 | ||||
Λ_{11} | 4.172 | 2.533 | 0.172 | 0.030 | Λ_{11} | 4.052 | 2.486 | 0.052 | 0.003 | ||
Λ_{22} | 4.152 | 2.693 | 0.152 | 0.023 | Λ_{22} | 4.012 | 2.636 | 0.012 | 0.000 | ||
Λ_{12} | −0.502 | 1.945 | −0.502 | 0.252 | Λ_{12} | −0.540 | 1.905 | −0.540 | 0.291 | ||
0.902 | 0.064 | 0.002 | 0.000 | 0.899 | 0.068 | −0.001 | 0.000 | ||||
0.899 | 0.067 | −0.001 | 0.000 | ||||||||
B | {18} | −1.526 | 0.263 | −0.026 | 0.001 | −1.572 | 0.275 | −0.072 | 0.005 | ||
−0.439 | 0.253 | 0.061 | 0.004 | −0.622 | 0.260 | −0.122 | 0.015 | ||||
−0.155 | 0.189 | −0.005 | 0.000 | −0.099 | 0.190 | 0.051 | 0.003 | ||||
0.231 | 0.061 | −0.069 | 0.005 | 0.332 | 0.073 | 0.032 | 0.001 | ||||
Λ_{11} | 4.057 | 2.433 | 0.057 | 0.003 | Λ_{11} | 4.056 | 2.494 | 0.056 | 0.003 | ||
Λ_{22} | 3.947 | 2.630 | −0.053 | 0.003 | Λ_{22} | 3.913 | 2.561 | −0.087 | 0.008 | ||
Λ_{12} | −0.497 | 1.904 | −0.497 | 0.247 | Λ_{12} | −0.406 | 1.991 | −0.406 | 0.164 | ||
0.900 | 0.065 | 0.000 | 0.000 | 0.899 | 0.065 | −0.001 | 0.000 | ||||
0.900 | 0.065 | 0.000 | 0.000 | ||||||||
C | {80} | −1.510 | 0.275 | −0.010 | 0.000 | −1.524 | 0.263 | −0.024 | 0.001 | ||
−0.586 | 0.257 | −0.086 | 0.007 | −0.679 | 0.264 | −0.179 | 0.032 | ||||
−0.139 | 0.194 | 0.011 | 0.000 | −0.083 | 0.193 | 0.067 | 0.005 | ||||
0.255 | 0.082 | −0.045 | 0.002 | 0.235 | 0.078 | −0.065 | 0.004 | ||||
Λ_{11} | 3.832 | 2.406 | −0.168 | 0.028 | Λ_{11} | 4.018 | 2.386 | 0.018 | 0.000 | ||
Λ_{22} | 3.535 | 2.423 | −0.465 | 0.216 | Λ_{22} | 3.919 | 2.649 | −0.081 | 0.007 | ||
Λ_{12} | −0.115 | 1.835 | −0.115 | 0.013 | Λ_{12} | −0.302 | 1.969 | −0.302 | 0.091 | ||
0.900 | 0.063 | 0.000 | 0.000 | 0.899 | 0.064 | −0.001 | 0.000 | ||||
0.900 | 0.068 | 0.000 | 0.000 | ||||||||
D | {18, 80} | −1.460 | 0.259 | 0.040 | 0.002 | −1.599 | 0.266 | −0.099 | 0.010 | ||
−0.348 | 0.248 | 0.152 | 0.023 | −0.316 | 0.246 | 0.184 | 0.034 | ||||
−0.203 | 0.189 | −0.053 | 0.003 | −0.208 | 0.191 | −0.058 | 0.003 | ||||
0.187 | 0.059 | −0.113 | 0.013 | 0.253 | 0.061 | −0.047 | 0.002 | ||||
Λ_{11} | 4.205 | 2.569 | 0.205 | 0.042 | Λ_{11} | 4.259 | 2.541 | 0.259 | 0.067 | ||
Λ_{22} | 4.086 | 2.634 | 0.086 | 0.007 | Λ_{22} | 4.002 | 2.549 | 0.002 | 0.000 | ||
Λ_{12} | −0.515 | 1.971 | −0.515 | 0.265 | Λ_{12} | −0.588 | 1.946 | −0.588 | 0.346 | ||
0.900 | 0.064 | 0.000 | 0.000 | 0.901 | 0.065 | 0.001 | 0.000 | ||||
0.898 | 0.068 | −0.002 | 0.000 |
SD = standard deviation; Bias = average bias; MSE = mean square error.
Activation | Prior | Setup | Perctausrebed | No. of case | DIC | ||||
---|---|---|---|---|---|---|---|---|---|
First | 1 | A | None | 18 | 0.006 | 0.011 | 0.042 | 0.012 | 142.754 |
80 | 0.030 | 0.060 | 0.097 | 0.067 | |||||
B | {18} | 18 | 0.080 | 0.171 | 0.159 | 0.238 | 164.476 | ||
C | {80} | 80 | 0.246 | 0.602 | 0.277 | 2.128 | 153.082 | ||
D | {18,80} | 18 | 0.069 | 0.143 | 0.147 | 0.181 | 185.709 | ||
80 | 0.282 | 0.677 | 0.304 | 1.742 | |||||
2 | A | None | 18 | 0.007 | 0.014 | 0.046 | 0.014 | 140.186 | |
80 | 0.033 | 0.067 | 0.102 | 0.075 | |||||
B | {18} | 18 | 0.036 | 0.075 | 0.106 | 0.084 | 164.446 | ||
C | {80} | 80 | 0.294 | 0.940 | 0.288 | 14.544 | 149.760 | ||
D | {18,80} | 18 | 0.062 | 0.131 | 0.138 | 0.176 | 184.934 | ||
80 | 0.120 | 0.269 | 0.190 | 0.498 |
DIC = deviance information criterion.
Real dataset: Bayesian criteria for the fitted models
Cure rate model under first activation | Criteria | ||
---|---|---|---|
Model | Priors | DIC | pd |
1 | 416.3917 | 11.5105 | |
2 | 416.9691 | 11.6971 | |
3 | 417.4942 | 12.1639 | |
4 | 417.1560 | 12.6639 | |
5 | 416.6700 | 11.7184 | |
Cure rate model under last activation | Criteria | ||
Model | Priors | DIC | pd |
6 | 419.2723 | 11.8914 | |
7 | 419.1585 | 12.2811 | |
8 | 419.4509 | 12.6852 | |
9 | 417.8917 | 13.6477 | |
10 | 418.6432 | 13.9228 |
DIC = deviance information criterion.
Real dataset: Bayesian criteria for the fitted models
Cure rate model under first activation | Criteria | ||
---|---|---|---|
Model | Priors | DIC | pd |
11 | 414.3751 | 8.2277 | |
12 | 414.7620 | 10.9257 | |
13 | 414.7484 | 10.8353 | |
14 | 414.3483 | 10.9108 | |
15 | 414.4925 | 10.9244 | |
Cure rate model under last activation | Criteria | ||
Model | Priors | DIC | pd |
16 | 418.1336 | 9.4144 | |
17 | 417.2717 | 11.7087 | |
18 | 416.8620 | 11.4699 | |
19 | 416.8979 | 11.6343 | |
20 | 416.8782 | 11.5676 |
DIC = deviance information criterion.
Posterior summaries of the parameter of the Model 15 for the smoking cessation data
Parameter | Survival model | Cure rate | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
Mean | SD | 2.5% | 97.5% | Mean | SD | 2.5% | 97.5% | |||
Intercept | 1.3736 | 0.5789 | 0.2600 | 2.5442 | ||||||
Sex (male = 0) | −0.1562 | 0.4551 | −1.0626 | 0.6992 | −0.5096 | 0.3718 | −1.2711 | 0.2052 | ||
SI/UC (UC = 0) | 0.8427 | 0.5191 | −0.1659 | 1.8703 | 0.8601 | 0.4310 | 0.0671 | 1.7192 | ||
Cigarettes per day | −0.1148 | 0.0378 | −0.1809 | −0.0322 | −0.0728 | 0.0290 | −0.1345 | −0.0201 | ||
Duration as smoker | −0.0246 | 0.0343 | −0.1003 | 0.0345 | 0.0197 | 0.0230 | −0.0264 | 0.0651 | ||
2.4097 | 0.3073 | 1.8113 | 3.0065 | |||||||
0.8968 | 0.0676 | 0.7261 | 0.9874 | |||||||
0.8994 | 0.0670 | 0.7370 | 0.9881 | |||||||
Λ_{11} | 2.6769 | 0.6377 | 1.5543 | 4.0673 | ||||||
Λ_{22} | 2.5743 | 0.6413 | 1.4924 | 3.9718 | ||||||
Λ_{12} | −0.0104 | 0.4625 | −0.9321 | 0.8919 | ||||||
∑_{11} | 0.4113 | 0.1075 | 0.2531 | 0.6754 | ||||||
∑_{22} | 0.4298 | 0.1236 | 0.2561 | 0.7298 | ||||||
∑_{12}/(∑_{11}∑_{22})^{1/2} | 0.0035 | 0.1828 | −0.3655 | 0.3559 |
Where Λ
Possible influential observations are detected by four divergence measures
Observations | Sex | Duration | Intervention | No. of cigarettes | Relapse | Time interval | Zip |
---|---|---|---|---|---|---|---|
72 | 0 | 20 | 1 | 25 | 1 | (3.159, 3.929) | 55987 |
138 | 1 | 25 | 1 | 20 | 1 | (2.998, 3.992) | 55021 |
151 | 0 | 39 | 1 | 10 | 1 | (0.923, 3.962) | 55057 |
199 | 0 | 22 | 0 | 20 | 1 | (3.885, 5.013) | 55904 |
Posterior summaries of the parameter of the Model 15 and relative variations adjusted for the smoking cessation data without detected individuals 72, 138, 151, and 199
Parameter | Survival model | Cure rate | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
Mean | SD | 2.5% | 97.5% | Mean | SD | 2.5% | 97.5% | |||
Intercept | 1.3697 | 0.5981 | 0.2041 | 2.5265 | ||||||
Sex (male = 0) | −0.2101 (−0.3455) | 0.4326 | −1.0848 | 0.6344 | −0.5657 (0.1101) | 0.3425 | −1.2511 | 0.0902 | ||
SI/UC (UC = 0) | 1.0138 (0.2031) | 0.5071 | 0.0486 | 2.0473 | 0.9023 | 0.4002 | 0.1406 | 1.7522 | ||
Cigarettes per day | −0.1048 (−0.0873) | 0.0377 | −0.1800 | −0.0304 | −0.0611 | 0.0257 | −0.1221 | −0.0183 | ||
Duration as smoker | −0.0308 (0.2518) | 0.0361 | −0.1073 | 0.0375 | 0.0184 (−0.0656) | 0.0230 | −0.0280 | 0.0629 | ||
2.6474 (0.0987) | 0.3515 | 1.9792 | 3.3582 | |||||||
0.9006 (0.0043) | 0.0665 | 0.7388 | 0.9880 | |||||||
0.9002 (0.0009) | 0.0650 | 0.7374 | 0.9864 | |||||||
Λ_{11} | 2.6749 (−0.0008) | 0.6497 | 1.5769 | 4.0686 | ||||||
Λ_{22} | 2.5717 (−0.0010) | 0.6364 | 1.5178 | 3.9480 | ||||||
Λ_{12} | 0.0113 (−2.0812) | 0.4709 | −0.9257 | 0.9130 | ||||||
∑_{11} | 0.4122 (0.0023) | 0.1086 | 0.2515 | 0.6701 | ||||||
∑_{22} | 0.4307 (0.0021) | 0.1208 | 0.2575 | 0.7207 | ||||||
∑_{12} | −0.0054 (−2.5601) | 0.1839 | −0.3637 | 0.3568 |
SD = standard deviation; SI = smoking intervention group; UC = usual care group.