Effect of the number of removed lymph nodes on prostate cancer recurrence and survival: evidence from an observational study
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Abstract
Background
The aim of this article is to analyze the effect on biochemical recurrence and on overall survival of removing an extensive number of pelvic lymph nodes during prostate cancer surgery. The lack of evidence from randomized clinical trials to address this specific question has hampered the ability to determine the true effect of the number of nodes removed.
Results
Our analysis is based on a large observational study, and this can lead unadjusted estimates to be very sensitive to confounding bias due to the different prognosis of individuals. We assess the effect of the number of lymph nodes removed by means of an Inverse Probability Weighting adjustment based on a Poisson regression model, and by a Doublyrobust adjustment.
Conclusions
Our findings suggest that a large number of nodes removed is associated with a significant improvement in time to biochemical recurrence. However, it appears to have no impact on overall survival.
Keywords
Prostate cancer Retrospective study Doublyrobust estimation Survival analysisBackground
In prostate cancer (PCa) studies it is still debatable whether more extensive pelvic lymph node dissections are associated with better oncological outcomes. Nowadays, no prospective randomized study aimed at assessing the benefit on cancer control of anatomically defined extended pelvic lymph node dissection (PLND) as compared to limited or no PLND is indeed currently available.
Recent literature [1] reports that extended PLND is associated with higher rates of node positive findings, as the probability of detecting lymph node invaded by prostate cancer invasion is directly related to the extent of PLND. Another retrospective trial found a statistically significant association between the extent of PLND and cancer specific survival. The hazard of cancerrelated death was found to be significantly lower for higher numbers of nodes removed [2]. The main limitation of this multicentre study is the lack of a homogeneous and standardized pathologic assessment of the removed lymph nodes. Given the long natural history of treated PCa, it is likely that if a potential beneficial effect of extended PLND existed, this is more likely to be detected in men with more aggressive cancer characteristics. Such hypothesis has been tested in more recent retrospective studies focusing on men with adverse pathological features. For example, Moschini et al. analyzed a large cohort of 1586 patients with locally advanced PCa treated with RP plus ePLND and found that a higher number of removed LNs was an independent predictor of increased cancerspecific survival, suggesting a potential therapeutic role of a more extensive PLNDs in this subgroup of patients [3].
Taken together, all of these data show that the impact of PLND as a curative treatment remains an open question. After the withdrawal of the only available prospective study reported on this topic [4], the scientific community is eagerly awaiting the results from the two ongoing randomized clinical trials comparing extended and limited PLND during RP in intermediate and highrisk PCa to provide definitive evidence to this highly controversial topic.
Overall, a general agreement exists that whenever PLND is indicated, this should be anatomically extended at least for diagnostic purposes [5, 6]. Nevertheless, whether this approach improves patient outcomes is still unknown. In case of detection of positive nodes at PLND, patients may be offered different postoperative adjuvant treatments, such as: (i) radiotherapy, (ii) hormone therapy, (iii) a combination of both. Correct patient staging may help in tailoring the optimal postoperative patient management and thus indirectly improve patient outcomes. Therefore, the correct identification of men potentially harboring prostate cancer nodal metastases is crucial to optimize patient outcome. Moreover, assessing whether such extensive dissections are directly associated with improved cancer control rates using methodologically sound approaches, which can account for the effect of patient selection biases, is still an unmet need.
Methods
The aim of this article is to examine the effect on BCR and on allcauses tmonth survival of removing different numbers of nodes during prostate cancer surgery. Our analysis is based on nonrandomized observational study including 3046 PCa patients. In randomized studies the distributions of the patients characteristics are balanced across groups, so that groups are similar expect for the treatment. This constitutes a critical issue, since in nonrandomized (or observational) studies treatment exposure may be associated with covariates that are also associated with the survival function, and thus “treated” patients may differ from “untreated” ones. We specify that in the paper we refer to “treatment” meaning the removal of a given number of lymph nodes. In this framework the usual analysis based on unweighted KaplanMeier survival curves may be misleading due to confounding [7].
Interpreting differences in groups or effects of covariates in observational studies is a major debatable point, since it is tempting to attribute the differences to the lack of random treatment assignment. Indeed, the lack of randomization can lead observational effect estimates to be very sensitive to confounding bias due to the different prognosis of individuals between treatment groups. As a consequence, observational randomized discrepancies cannot be automatically attributed to randomization itself.
Thus, observational studies need adjustments for baseline confounders, for time varying confounding an for selection bias. Guidelines that rely on observational data due to the absence of randomized trials benefit when the analysis mimics the analysis of a hypothetical target trial. To this extent inverse probability weight (IPW) can be considered as an adjustment for pre or postrandomization variables in observational studies. In particular, in this article we use IPW to emulate the random allocation of patients to treatment groups (defined by the number of nodes removed), within a prostate cancer observational cohort study. We consider only pretreatment variables for the adjustment.
The IPW approach, although being one of the most popular methods used for adjusting for confounding, is subject to some criticism. In particular, it tends to be sensitive to misspecification of the treatment assignment model. To overcome this problem we also exploit the Doubly robust method proposed by [8, 9], which tends to be more robust against misspecification either of the treatment assignment model or of the survival model. In the Appendix we report on a simulation study aimed at illustrating and exploring such properties for large and moderate sample sizes.
The rest of this paper is structured as follows. We first describe the data, and show by means of some preliminary analysis how the number of nodes removed clearly affects the probability of missing a positivenode patient modelled as beta binomial distribution. We then analyze whether the removal of more lymph nodes during prostatectomy may improve patient time from surgery to BCR and OS through better treatment assignment due to a more precise nodal staging assessment. For that reason, our analysis is split into two parts: the first part of our “Results” section focuses on time to BCR, while the second part reports the results of the analyses on overall survival (OS). Concluding remarks are reported in the “Discussion and conclusion” section.
Results
Data and patient classification
Our analysis is based on a prostate cancer dataset collected at San Raffaele Hospital (HSR) in Milan, Italy, between years 1991 and 2012. We consider a sample of 3046 patients with localized prostate cancer treated with radical prostatectomy and pelvic lymph node dissection.
Descriptive statistics
Nodes group 110  Nodes group 1120  Nodes group 21+  Total  

Mean  Med  Freq  Mean  Med  Freq  Mean  Med  Freq  Mean  Med  Freq  
Nodes removed  7.1  7  15.3  15  27.8  26  16.6  15.0  
Positive nodes  0.1  0  0.3  0  0.8  0  0.4  0.0  
Age (years)  64.8  65.4  65.4  65.9  64.7  64.8  65.0  65.0  
Gleason score  6.1  6  6.2  6  6.4  6  6.2  6.0  
PSA  8.8  6.9  11.6  7.1  12.3  7.1  11.1  7.0  
T1 stage  485 (58%)  756 (55%)  425 (50%)  1666 (55%)  
T2 stage  281 (34%)  448 (33%)  288 (26%)  1017 (33%)  
T3 stage  67 (8%)  163 (11.9%)  133 (14.0%)  363 (11.9%)  
BCR (1 =yes)  93 (11%)  187 (14%)  79 (9%)  359 (12%)  
OS (1 =dead)  19 (2%)  49 (4%)  16 (2%)  84 (3%)  
N  833 (27%)  1367 (45%)  846 (28%)  3046 (100%) 
We note that almost half (45%) of the patients in the study belong to the second nodes group, with number of removed nodes between 11 and 20, and that the overall average number of nodes removed is 16.6. Table 1 also reveals that patients with larger numbers of nodes removed are more likely to have higher T stage, to be slightly younger, to have higher PSA values as well as higher Gleason scores than patients with fewer nodes removed. Therefore the distribution of the covariates does not seem to be independent from the treatment (the number of nodes removed). Moreover, the average number of positive nodes for all patients is 0.4, a value that is doubled for the groups of patients with more than 20 nodes removed. Hence, there seems to exist a positive association between the number of nodes removed and the number of positive nodes detected: the higher the number of nodes removed the higher is the probability of detecting a positive node.
For each ith nodepositive patient with number of removed nodes Nex_{i}, let X_{i} be the number of observed positive nodes among the given number Nex_{i} of removed nodes, and p_{i} be the unknown probability that an examined node is positive.
The idea is to estimate the probability of missing a positive node (thus classifying incorrectly a patient  false negative) for a nodepositive patient, based on the total number of removed nodes (Nex) and the number of nodes found positive (x).
This model relies on several assumptions which, although strong, are needed to ensure mathematical tractability. Among others, we are here assuming that our data do not contain incorrectly staged positive nodes, and that, within the same patient, all nodes have the same probability of being positive. For an extended discussion of the method and the reliability of those assumptions see Gönen et al. (2009).
Time to biochemical recurrence analysis
Unweighted survival analysis
We first perform an unweighted survival analysis, based on the Kaplan Meier estimation of the survival functions for each of the three node groups.
Logrank tests for the difference in survival estimates among node groups  BCR
Node groups  Log rank test pvalue 

110 vs 1120  0.116 
110 vs 21+  0.718 
1120 vs 21+  0.265 
However, since the study is observational as we have mentioned, the unweighted KaplanMeier survival curves may provide misleading results due to confounding: the apparent absence of differences across the groups might be in part explained by differences in the composition of the patient groups. If the experiment were randomized, then the number of nodes removed (that is, the treatment) would have been distributed in the same way across the patients, regardless of their covariates (PSA, Gleason score, age). In the previous section we have seen that this is actually not the case, since the surgeon’s decision was likely affected by these covariates. As more nodes are removed from patients with higher T stage, PSA and Gleason score, who also might be at higher risk of recurrence, the time to event distributions need to be adjusted for the effect of such confounding.
Inverse probability weighted analysis
A first possible method to adjust the analysis is based on the Inverse Probability Weights (IPW) adjustment, according to which each patient receives a weight that is inversely proportional to the estimated probability of having his number of nodes removed equal to the observed number; see, among others, [7, 11]. The IPW is a particular type of the propensity score method, which is aimed at adjusting for confounding by weighting observations by the inverse of the estimated propensity scores, that are probabilities of exposure to the treatment received conditionally on the covariates; see, among others, [12, 13, 14, 15]. We have estimated the weights by fitting a Poisson regression model to estimate each patient’s probability of receiving any number of nodes removed. As covariates (Z) we have included age, PSA score, Tstage and Gleason score. We have also truncated the weights to a maximum of 35 (similarly to [16]).

Step 1. Fit a Poisson regression model and compute the predicted probabilities of having the number of removed nodes be equal to the observed number given the covariates, P(Nex=xZ_{i}),i=1,…,n, so that n is the total number of patients. Then, sum over the three nodes groups to obtain the predicted probability of each node group g=1,2, or 3 and each covariate value:
$$\begin{array}{@{}rcl@{}} P_{1}(Z_{i}) &=&\sum_{j=1}^{10} P(Nex = j  Z_{i}) \\ P_{2}(Z_{i})&=&\sum_{j=11}^{20} P(Nex = j  Z_{i}) \\ P_{3}(Z_{i})&=&\sum_{j=21}^{\infty} P(Nex = j  Z_{i}) \end{array} $$  Step 2. Generate the Inverse Probability Weights (IPWs) for each patient as the inverse function of the corresponding predicted probability, which depends both on the number of nodes removed (g) and on the covariates:where g indicates the treatment group (1, 2, or 3).$$w_{g}(Z_{i})=\frac{1}{P_{g}(Z_{i})}, $$

Step 3. Fit three weighted Kaplan Meier estimators with weights w_{g}(Z_{i}), separately for each nodes group: S_{1}(t)=P(T≥tg=1), S_{2}(t)=P(T≥tg=2) and S_{3}(t)=P(T≥tg=3).
 Step 4. Estimate the treatment effects in terms of difference in survival probabilities at time t among the three groups:$${S}_{1}(t)  {S}_{2}(t); {S}_{1}(t)  {S}_{3}(t); {S}_{2}(t)  {S}_{3}(t). $$

Step 5. Estimate confidence intervals for the treatment differences using a bootstrap technique, for each time t.
Output of Poisson regression
Coefficient  Coefficient  

Intercept  2.576 ^{∗∗∗}  Gleason  0.045 ^{∗∗∗} 
T2 stage  0.038 ^{∗∗∗}  PSA  0.002 ^{∗∗∗} 
T3 stage  0.147 ^{∗∗∗}  Age  0.003 ^{∗∗∗} 
The graph reveals how the time to BCR curves would have looked like, had each of the node groups had the same covariates distribution. In particular, time to BCR seems to improve as the number of removed nodes increases.
Comparing Figs. 4 to 2 in the previous section we note that the estimated survival function of the group of patients with more than 20 nodes removed is higher in the IPWadjusted Kaplan Meier than in the unweighted analysis. Moreover, no difference emerges between the survival distribution of patients with fewer than 10 nodes and the distribution for patients with 1120 nodes removed.
Hence, once adjusting for confounding, we can conclude that there is a significant positive effect on time to BCR of removing a higher number of nodes.
Doubly robust analysis
An alternative way for adjusting for the nonrandomized nature of treatment is provided by the Doubly robust method; see, among others, [8, 9, 17]. The IPW method tends to be possibly sensitive to misspecification of the treatment assignment model, while the Doubly robust method tends to be more robust against misspecification either of the treatment assignment model P(Nex=xZ) or of the survival estimate model S(tZ,Nex,x).
Indeed, while one will never know whether the weights are correctly specified, if that is the case then the survival probabilities are estimated consistently by IPW, as long as the model for the outcome is also correctly specified. The parameter estimators of the model for outcome, on the other hand, are not guaranteed to be consistent if that corresponding model is not correctly specified. So, IPW methods require that both models be correctly specified. Doubly robust methods, on the other hand, are theoretically guaranteed to consistently estimate the true survival probabilities even if one of the two models is wrongly specified (hence the name “robust”). In our setting, the survival estimates based on the Doubly robust method are consistent as long as either the Cox model or the weight (or both) models are correctly specified.
As stated in [9], “In a causal inference model, an estimator is Doubly Robust (DR) if it remains consistent when either (but not necessarily both) a model for treatment assignment mechanism or a model for the distribution of the counterfactual data is correctly specified.” “With observational data one can never be sure that a model for the treatment assignment mechanism or a model for the counterfactual data is correct.” “DR estimators, in contrast with inverse probability weightedestimators, give the analyst two chances, instead of one, to make valid inference.” In that article, the authors also present the results of a simulation study that demonstrates the impressive finite sample performance of DR estimators in terms of finite sample efficiency and robustness. In [18], additional simulation based evidence is provided to quantify the theoretical properties of the Doubly Robust approach.

Steps 1. and 2. Proceed analogously as in Step 1 and Step 2 of the IPW method.

Step 3. Fit three weighted Cox models, with weights w_{g}(Z_{i}). In particular, we fit one model for each treatment group g=1,2,3, where group 1 refers to 110 nodes, group 2 refers to 1120 nodes, and group 3 to 21+ nodes. For detailed notation see also Section 2.2 of Bang and Robins [9].

Step 4. Compute three predicted survival functions at month t for every patient one for each treatment group: \(\tilde {S}_{i}(t  Group=g, Z)\). In this way each patient is assigned three estimated survivals, one factual and the other two counterfactual.
 Step 5. For each group g, compute the average survival function at month t over all n patients:$$\tilde{S}_{g}(t)=\frac{1}{n}\sum_{i=1}^{n} \tilde{S}_{i}(t Group={g}, Z). $$
 Step 6. Estimate the effects in terms of differences in survival at time t among groups:$$\tilde{S}_{1}(t)  \tilde{S}_{2}(t); \tilde{S}_{1}(t)  \tilde{S}_{3}(t);\tilde{S}_{2}(t)  \tilde{S}_{3}(t) $$

Step 7. Estimate confidence intervals using bootstrap, for each time t.
Time to BCR
Nodes group  t=60 months  t=90 months  t=120 months  

Mean  95% C.I.  Mean  95% C.I.  Mean  95% C.I.  
Unweighted KM  1 (110)  0.866  0.836  0.898  0.764  0.710  0.822  0.692  0.612  0.782 
2 (1120)  0.841  0.815  0.867  0.748  0.710  0.789  0.679  0.627  0.736  
3 (21+)  0.855  0.820  0.892  0.795  0.729  0.868  0.795  0.729  0.868  
1 vs 2  0.026  0.015  0.066  0.016  0.051  0.083  0.031  0.048  0.111  
1 vs 3  0.011  0.038  0.061  0.031  0.119  0.057  0.047  0.141  0.048  
2 vs 3  0.014  0.061  0.032  0.047  0.124  0.030  0.078  0.160  0.004  
Weighted KM  1 (110)  0.847  0.839  0.855  0.717  0.701  0.732  0.627  0.605  0.650 
2 (1120)  0.835  0.812  0.858  0.744  0.711  0.778  0.681  0.637  0.728  
3 (21+)  0.883  0.871  0.895  0.846  0.828  0.863  0.846  0.828  0.863  
1 vs 2  0.013  0.032  0.058  0.027  0.102  0.047  0.031  0.119  0.057  
1 vs 3  0.036  0.092  0.020  0.129  0.232  0.026  0.161  0.271  0.050  
2 vs 3  0.049  0.104  0.006  0.102  0.197  0.007  0.130  0.229  0.031  
Doubly Robust  1 (110)  0.839  0.791  0.887  0.705  0.605  0.805  0.592  0.432  0.752 
2 (1120)  0.812  0.773  0.850  0.693  0.625  0.762  0.614  0.516  0.713  
3 (21+)  0.852  0.809  0.896  0.819  0.764  0.873  0.819  0.747  0.890  
1 vs 2  0.027  0.035  0.089  0.012  0.098  0.122  0.035  0.112  0.182  
1 vs 3  0.014  0.080  0.052  0.113  0.222  0.005  0.141  0.275  0.007  
2 vs 3  0.041  0.103  0.021  0.125  0.225  0.025  0.176  0.298  0.054 
Overall survival analysis
Logrank tests for the difference in survival estimates among node groups  OS
Node groups  Log rank test pvalue 

110 vs 1120  0.13 
110 vs 21+  0.797 
1120 vs 21+  0.255 
Also for time to OS, both the graph and the tests show that there are no differences across the three survival curves, thus revealing that the number of nodes removed (Nex) does not seem to affect OS.
Overall survival
Nodes group  t=60 month  t=90 month  t=110 month  

Mean  95% C.I.  Mean  95% C.I.  Mean  95% C.I.  
Unweighted KM  1 (110)  0.969  0.953  0.985  0.947  0.918  0.977  0.934  0.895  0.974 
2 (1120)  0.963  0.948  0.977  0.934  0.910  0.959  0.900  0.863  0.939  
3 (21+)  0.970  0.947  0.993  0.945  0.905  0.987  0.906  0.842  0.975  
1 vs 2  0.006  0.016  0.029  0.013  0.025  0.051  0.033  0.019  0.086  
1 vs 3  0.001  0.032  0.030  0.002  0.049  0.053  0.027  0.048  0.102  
2 vs 3  0.007  0.037  0.022  0.011  0.059  0.037  0.006  0.079  0.067  
Weighted KM  1 (110)  0.963  0.959  0.968  0.945  0.938  0.952  0.935  0.926  0.944 
2 (1120)  0.964  0.952  0.976  0.920  0.897  0.943  0.889  0.856  0.922  
3 (21+)  0.979  0.973  0.986  0.954  0.942  0.967  0.923  0.906  0.941  
1 vs 2  0.001  0.027  0.025  0.024  0.019  0.067  0.046  0.013  0.104  
1 vs 3  0.016  0.053  0.021  0.009  0.069  0.051  0.012  0.073  0.096  
2 vs 3  0.016  0.050  0.019  0.033  0.090  0.023  0.034  0.118  0.050  
Doubly Robust  1 (110)  0.974  0.956  0.991  0.963  0.930  0.996  0.952  0.896  1.008 
2 (1120)  0.972  0.955  0.988  0.936  0.892  0.980  0.908  0.842  0.974  
3 (21+)  0.985  0.972  0.998  0.975  0.941  1.008  0.937  0.839  1.035  
1 vs 2  0.008  0.069  0.085  0.038  0.060  0.136  0.060  0.068  0.188  
1 vs 3  0.011  0.090  0.067  0.010  0.121  0.100  0.020  0.160  0.200  
2 vs 3  0.020  0.071  0.032  0.048  0.141  0.045  0.040  0.202  0.122 
After adjusting, no significant impact of the number of removed nodes on OS emerges for the three time points.
Discussion and conclusion
Extended pelvic lymph node dissection (PLND) has certainly a key staging role in Prostate Cancer (PCa). Assessing the relative benefits and burden of PLND for oncological and nononcological outcomes in patients undergoing radical prostatectomy for PCa is still debatable. The issue of whether PLND may affect prostate cancer outcomes such as progression and survival has been an argument of extreme interest in the urologic community over the last decades. Indeed, the impact of PLND on cancer outcomes remains controversial [1] due to the lack of prospective, randomized trials. The authors of [19] found a statistically significant negative association between the number of removed lymph nodes and BCRfree survival in patients with no positive nodes found. However, there is currently no available study supporting its role with regards to oncological outcomes in men with clinically localized disease. Thus, our approach represents a novel perspective to evaluate – within an observational setting – the effect on biochemical recurrence and on allcauses tmonth survival of removing different numbers of nodes, providing important evidence also from non randomized trials.
We have shown how it is possible to assess the effect of the number of lymph nodes removed by means of an Inverse Probability Weighting adjustment (here, based on Poisson regression) and by a Doublyrobust adjustment method.
As a first analysis we had run traditional Cox models both for BCR and OS, with the number of nodes removed being classified in the three groups, and with the logtransformed original variable. The results of such standard analyses do not show an effect of the number of nodes removed (results not shown).
Root Mean Squared Error (MSE) and Bias for the estimation of the marginal survival probability S(1000) for the three groups defined by the number of nodes received
Generation  Estimation  Unadjusted  Weighted  Doubly Robust  

PM  SM  PM  SM  n  Nodes  S(1000)  \(\sqrt {\text {MSE}}\)  Bias  \(\sqrt {\text {MSE}}\)  Bias  \(\sqrt {\text {MSE}}\)  Bias 
1  1  1  1  1000  110  0.4808  0.0908  0.0287  0.0987  0.0060  0.0845  0.0093 
1120  0.4808  0.0299  0.0071  0.0292  0.0004  0.0262  0.0011  
>20  0.4808  0.1113  0.0728  0.1002  0.0001  0.0935  0.0183  
1  1  1  1  5000  110  0.4769  0.0459  0.0259  0.0436  0.0002  0.0373  0.0027 
1120  0.4769  0.0147  0.0073  0.0128  0.0005  0.0113  0.0001  
>20  0.4769  0.0836  0.0756  0.0463  0.0001  0.0417  0.0029  
1  1  1  1  10000  110  0.4775  0.0356  0.0235  0.0306  0.0005  0.0261  0.0017 
1120  0.4775  0.0116  0.0071  0.0093  0.0002  0.0080  0.0000  
>20  0.4775  0.0786  0.0742  0.0336  0.0007  0.0298  0.0013  
[5pt] 2  2  2  2  1000  110  0.4108  0.0948  0.0329  0.1067  0.0060  0.0951  0.0102 
1120  0.4108  0.0293  0.0070  0.0288  0.0005  0.0265  0.0005  
>20  0.4108  0.1091  0.0728  0.0968  0.0006  0.0927  0.0160  
2  2  2  2  5000  110  0.4067  0.0498  0.0312  0.0463  0.0003  0.0414  0.0031 
1120  0.4067  0.0141  0.0068  0.0125  0.0002  0.0116  0.0001  
>20  0.4067  0.0778  0.0697  0.0439  0.0001  0.0410  0.0018  
2  2  2  2  10000  110  0.4069  0.0391  0.0289  0.0320  0.0012  0.0292  0.0002 
1120  0.4069  0.0111  0.0064  0.0092  0.0002  0.0084  0.0002  
>20  0.4069  0.0739  0.0695  0.0322  0.0006  0.0300  0.0016  
[5pt] 2  3  2  3  1000  110  0.4108  0.0948  0.0329  0.1067  0.0060  0.0951  0.0102 
1120  0.2971  0.0283  0.0044  0.0284  0.0001  0.0269  0.0014  
>20  0.5514  0.1056  0.0727  0.0978  0.0009  0.0895  0.0186  
2  3  2  3  5000  110  0.4067  0.0498  0.0312  0.0463  0.0003  0.0414  0.0031 
1120  0.2941  0.0131  0.0046  0.0125  0.0000  0.0120  0.0001  
>20  0.5461  0.0764  0.0687  0.0443  0.0002  0.0390  0.0027  
2  3  2  3  10000  110  0.4069  0.0391  0.0289  0.0320  0.0012  0.0292  0.0002 
1120  0.2936  0.0099  0.0045  0.0090  0.0000  0.0086  0.0001  
>20  0.5473  0.0708  0.0666  0.0317  0.0001  0.0281  0.0015  
[5pt] 3  2  3  2  1000  110  0.4108  0.1562  0.1401  0.1061  0.0082  0.0949  0.0100 
1120  0.4108  0.0292  0.0011  0.0292  0.0001  0.0264  0.0009  
>20  0.4108  0.1930  0.1710  0.1110  0.0099  0.0969  0.0241  
3  2  3  2  5000  110  0.4067  0.1429  0.1399  0.0455  0.0023  0.0416  0.0008 
1120  0.4067  0.0133  0.0010  0.0133  0.0000  0.0122  0.0002  
>20  0.4067  0.1754  0.1711  0.0490  0.0010  0.0414  0.0020  
3  2  3  2  10000  110  0.4069  0.1362  0.1345  0.0324  0.0006  0.0300  0.0007 
1120  0.4069  0.0095  0.0011  0.0094  0.0000  0.0086  0.0001  
>20  0.4069  0.1721  0.1698  0.0349  0.0002  0.0297  0.0008 
Root Mean Squared Error (MSE) and Bias for the estimation of the marginal survival probability S(1000) for the three groups defined by the number of nodes received
Generation  Estimation  Unadjusted  Weighted  Doubly Robust  

PM  SM  PM  SM  n  Nodes  S(1000)  \(\sqrt {\text {MSE}}\)  Bias  \(\sqrt {\text {MSE}}\)  Bias  \(\sqrt {\text {MSE}}\)  Bias 
2  2  3  2  1000  110  0.4108  0.0948  0.0329  0.1144  0.0423  0.0929  0.0095 
1120  0.4108  0.0293  0.0070  0.0287  0.0027  0.0265  0.0005  
>20  0.4108  0.1091  0.0728  0.1038  0.0715  0.0895  0.0159  
2  2  3  2  5000  110  0.4067  0.0498  0.0312  0.0668  0.0483  0.0398  0.0029 
1120  0.4067  0.0141  0.0068  0.0126  0.0023  0.0116  0.0001  
>20  0.4067  0.0778  0.0697  0.0813  0.0747  0.0377  0.0011  
2  2  3  2  10000  110  0.4069  0.0394  0.0294  0.0569  0.0471  0.0283  0.0000 
1120  0.4069  0.0112  0.0064  0.0094  0.0021  0.0085  0.0002  
>20  0.4069  0.0735  0.0692  0.0809  0.0776  0.0275  0.0006  
[5pt] 2  3  3  3  1000  110  0.4108  0.0948  0.0329  0.1144  0.0423  0.0929  0.0095 
1120  0.2971  0.0283  0.0044  0.0282  0.0010  0.0269  0.0014  
>20  0.5514  0.1056  0.0727  0.1250  0.0866  0.0882  0.0173  
2  3  3  3  5000  110  0.4067  0.0507  0.0325  0.0657  0.0468  0.0397  0.0020 
1120  0.2941  0.0131  0.0044  0.0125  0.0008  0.0121  0.0003  
>20  0.5461  0.0769  0.0690  0.1004  0.0923  0.0373  0.0022  
2  3  3  3  10000  110  0.4069  0.0391  0.0289  0.0575  0.0478  0.0279  0.0001 
1120  0.2936  0.0099  0.0045  0.0090  0.0010  0.0086  0.0001  
>20  0.5473  0.0708  0.0666  0.1008  0.0969  0.0270  0.0013  
[5pt] 2  3  2  4  1000  110  0.4108  0.0948  0.0329  0.1067  0.0060  0.0920  0.0186 
1120  0.2971  0.0283  0.0044  0.0284  0.0001  0.0270  0.0082  
>20  0.5514  0.1056  0.0727  0.0978  0.0009  0.1013  0.0114  
2  3  2  4  5000  110  0.4067  0.0498  0.0312  0.0463  0.0003  0.0413  0.0056 
1120  0.2941  0.0131  0.0046  0.0125  0.0000  0.0134  0.0069  
3  0.5461  0.0764  0.0687  0.0443  0.0002  0.0470  0.0029  
2  3  2  4  10000  110  0.4069  0.0391  0.0289  0.0320  0.0012  0.0290  0.0019 
1120  0.2936  0.0099  0.0045  0.0090  0.0000  0.0105  0.0065  
>20  0.5473  0.0708  0.0666  0.0317  0.0001  0.0335  0.0024 
Our use of a Poisson regression model was motivated by the very nature of the treatment of interest here, clearly a counting variable. Using a multinomial distribution would have forced (at most ordinal) categories onto that variable. That would be a natural choice in presence of multiple (unordered) treatments, when a fully categorical analysis is appropriate (see, e.g., Feng et al., [20]). In our setting, such a classification would also have led to an increase in the number of parameters to be able to model all odds ratios with respect to the covariates potentially influencing the number of nodes extracted. Discretizing the problem to just a binary treatment outcome would have forced an even stronger classification of the treatment variable. So our choice was to allow for a parsimonious model for the number of nodes through the Poisson regression model, and to then consider (ordinal) groups for the estimation process. The reasons for such a twostep approach are the need to allow for easy interpretability and description of the results, while still keeping a moderate number of patients in each group for the estimation, while maintaining a modeling approach coherent with the nature of the underlying treatment variable. Here, “moderate number” is meant as a number of patients that allowed us to obtain informative 95% confidence intervals for the reported differences in survival probabilities across treatment groups. As a last comment, note that IPW could be very sensitive to the presence of extreme weights [16]. We truncated the weights to a maximum of 35, but we also performed the estimation with a truncation value of 20, with no apparent changes in the results.
Our findings suggest that a large number of nodes removed may be associated with a significant improvement in the time to BCR but there is no detectable impact on OS. However, the lack of any benefit of the extent of pelvic lymph node dissection on patient OS could also be related to the relatively short median followup of our population.
Appendix
Simulation study
In this Appendix we report on a simulation experiment designed to illustrate and explore the finitesample and the largesample properties of the estimation procedures that we have used.

\(\sqrt {\text {PSA}} \sim N(2,2)\)

Age ∼N(60,9)

Stage ∼ Multinomial(1,(.22,.33,.45)), i.e. taking the three values 1, 2, and 3

(Number of nodes examined  1) ∼ Poisson(λ_{PM}(z_{i})). Treatment group (Tx) was then set equal to 1 (110 Nodes), 2 (1120 Nodes) or 3 (>20 Nodes).
The vector z_{i} indicates the covariate vector for subject i, excluding the treatment group Tx_{i}, and λ_{PM}(z_{i}) indicates the mean parameter defined as
The propensity score models PM were defined as follows (in parentheses the values of the regression parameters):
and covariates absent from a specific model were multiplied by a coefficient equal to zero in the generation step, and not estimated in the estimation process.
Conditionally on the covariates (including now the number of nodes examined, as classified in Tx), we generated survival times according to an exponential Cox proportional hazards model. The parameter λ_{SM}(tz_{i}) of the exponential distribution for subject i was defined as
We considered the following four survival models SM (in parentheses the values of the regression parameters):
Here, too, covariates absent from a specific model were multiplied by a coefficient equal to zero in the generation step, and not estimated in the estimation process. The generated survival times were right censored with an independent censoring variable distributed as a negative exponential random variable with parameter 1/500.
Root Mean Squared Error (MSE) and Bias for the estimation of the marginal survival probability S(1000) for the three groups defined by the number of nodes received
Generation  Estimation  Unadjusted  Weighted  Doubly Robust  

PM  SM  PM  SM  n  Nodes  S(1000)  \(\sqrt {\text {MSE}}\)  Bias  \(\sqrt {\text {MSE}}\)  Bias  \(\sqrt {\text {MSE}}\)  Bias 
2  3  3  4  1000  110  0.4108  0.0948  0.0329  0.1144  0.0423  0.1175  0.0671 
1120  0.2971  0.0283  0.0044  0.0282  0.0010  0.0258  0.0051  
3  0.5514  0.1056  0.0727  0.1250  0.0866  0.1015  0.0641  
2  3  3  4  5000  110  0.4067  0.0498  0.0312  0.0668  0.0483  0.0733  0.0589 
1120  0.2941  0.0131  0.0046  0.0124  0.0010  0.0120  0.0039  
3  0.5461  0.0764  0.0687  0.1005  0.0926  0.0823  0.0751  
2  3  3  4  10000  110  0.4069  0.0391  0.0289  0.0575  0.0478  0.0643  0.0565 
1120  0.2936  0.0099  0.0045  0.0090  0.0010  0.0088  0.0034  
3  0.5473  0.0708  0.0666  0.1008  0.0969  0.0840  0.0804 
Estimated marginal survival probability S(1000) for thee three groups defined by the number of nodes received
Generation  Estimation  Unadjusted  Weighted  Doubly Robust  

PM  SM  PM  SM  Nodes  S(1000)  Est Surv  Est Surv  Est Surv 
1  1  1  1  110  0.4740  0.4522  0.4749  0.4751 
1120  0.4740  0.4669  0.4739  0.4738  
>20  0.4740  0.5476  0.4743  0.4740  
2  2  2  2  110  0.4037  0.3767  0.4042  0.4040 
1120  0.4037  0.3974  0.4041  0.4041  
>20  0.4037  0.4734  0.4035  0.4035  
2  3  2  3  110  0.4037  0.3767  0.4042  0.4040 
1120  0.2911  0.2864  0.2910  0.2909  
>20  0.5438  0.6100  0.5427  0.5427  
3  2  3  2  110  0.4037  0.2693  0.4014  0.4015 
1120  0.4037  0.4047  0.4037  0.4037  
>20  0.4037  0.5756  0.4031  0.4032  
2  2  3  2  110  0.4037  0.3767  0.4536  0.4040 
1120  0.4037  0.3974  0.4017  0.4041  
>20  0.4037  0.4734  0.3261  0.4038  
2  3  3  3  110  0.4037  0.3767  0.4536  0.4040 
1120  0.2911  0.2864  0.2899  0.2909  
>20  0.5438  0.6100  0.4454  0.5430  
2  3  2  4  110  0.4037  0.3767  0.4042  0.4046 
1120  0.2911  0.2864  0.2910  0.2972  
>20  0.5438  0.6100  0.5427  0.5422  
2  3  3  4  110  0.4037  0.3767  0.4536  0.4605 
1120  0.2911  0.2864  0.2899  0.2941  
>20  0.5438  0.6100  0.4454  0.4613 
Propensity scores for the three treatments (110, 1120, and >20 Nodes examined) were computed as described in the main text, i.e. by adding the appropriate terms of the Poisson probability mass function. The theoretical survival probabilities were computed as the average over the observed covariate distribution (fixed for each simulation) of the quantities
We estimated the survival probability at t^{∗}=500,1000, and 1500 days. For simplicity, below we only report the results for t^{∗}=1000 days. The conclusions drawn from for the other two time points were analogous.
Tables 7, 8 and 9 show the empirical rootMSE and Bias in estimating S(1000) with the naive KaplanMeier estimator (“Unadjusted”), with the weighted nonparametric estimator (“Weighted”), and with the Doubly robust estimator (“Doubly Robust”). Simulations were based on 3000 samples, and for sample sizes 1000, 3000, and 10000. Data were simulated and estimated by using different combinations of models PM and SM for the generation (“Generation”) and for the estimation (“Estimation”) processes.
Results show that the naive estimator is always biased. On the other hand, when both models used for estimation match the models used to estimate the data (i.e. they are the correct models), then both the weighted and the Doubly robust methods estimate the survival probabilities properly, with Bias and MSEs that tend to zero as the sample size increases. When only the propensity score is correctly specified, then both estimators are known to be consistent, and this is confirmed by the results. If, on the other hand, only the survival model is correctly specified, then the weighted estimator is clearly biased (as expected), while the Doubly robust estimator still shows its consistency. Lastly, when both models are wrongly specified, both estimators are clearly shown to be biased.
Interestingly, for the smaller sample size (1000) the MSE of the (biased) naive estimator is quite similar to that of the adjusted estimators, even when the latter two are known to be consistent. This implies that the former estimator has misleadingly smaller variance that the adjusted estimators. As sample size increases, as expected, the unadjusted estimator maintains a larger MSE than the other two estimators. Note that in the limit the MSE should become equal to the squared bias term (since the variance term tends to zero – recall that the MSE is equal to the sum of the variance of the estimator and the squared bias term). Indeed, for the n =10000 cases one can see that the absolute value of the Bias term of the unadjusted estimator approaches the value of the square root of the MSE.
For completeness, Table 10 shows the results of the three estimation procedures when they are applied, for each combination of propensity score and survival models, to just one sample of size n =10 million. That table again confirms the impressions gained from Tables 7, 8 and 9, and described above.
Note: Simulations were performed using the seed 13794 in R version 3.3.2.
Notes
Acknowledgements
We thank three anonymous referees for their helpful comments and suggestions on earlier versions of the manuscript. We are also grateful to the participants at the 27th International Biometric Conference and at the 13th CIBB Conference.
Funding
Publication charges for this article have been funded by Università VitaSalute San Raffaele and by Bocconi University.
About this supplement
This article has been published as part of BMC Bioinformatics Volume 19 Supplement 7, 2018: 12th and 13th International Meeting on Computational Intelligence Methods for Bioinformatics and Biostatistics (CIBB 2015/16). The full contents of the supplement are available online at https://bmcbioinformatics.biomedcentral.com/articles/supplements/volume19supplement7.
Authors’ contributions
All authors of this research paper have directly participated in the planning, execution, or analysis of the study. All authors of this paper have read and approved the final version submitted.
Ethics approval and consent to participate
The study received Institutional Review Board approval.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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