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Evaluation of breast cancer service screening programme with a Bayesian approach: mortality analysis in a Finnish region

  • Epidemiology
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Abstract

Evaluation of long-term effectiveness of population-based breast cancer service screening program in a small geographic area may suffer from self-selection bias and small samples. Under a prospective cohort design with exposed and non-exposed groups classified by whether women attended the screen upon invitation, we proposed a Bayesian acyclic graphic model for correcting self-selection bias with or without incorporation of prior information derived from previous studies with an identical screening program in Sweden by chronological order and applied it to an organized breast cancer service screening program in Pirkanmaa center of Finland. The relative mortality rate of breast cancer was 0.27 (95% CI 0.12–0.61) for the exposed group versus the non-exposed group without adjusting for self-selection bias. With adjustment for selection-bias, the adjusted relative mortality rate without using previous data was 0.76 (95% CI 0.49–1.15), whereas a statistically significant result was achieved [0.73 (95% CI 0.57–0.93)] with incorporation of previous information. With the incorporation of external data sources from Sweden in chronological order, adjusted relative mortality rate was 0.67 (0.55–0.80). We demonstrated how to apply a Bayesian acyclic graphic model with self-selection bias adjustment to evaluating an organized but non-randomized breast cancer screening program in a small geographic area with a significant 27% mortality reduction that is consistent with the previous result but more precise. Around 33% mortality was estimated by taking previous randomized controlled data from Sweden.

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Acknowledgment

This research study was supported by the FiDiPro Research Project of Tampere School of Public Health Granted from Academy of Finland

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Correspondence to Tony Hsiu-Hsi Chen.

Appendix: Bayesian acyclic graphic model for adjusting for selection-bias

Appendix: Bayesian acyclic graphic model for adjusting for selection-bias

A Bayesian acyclic graphic model is proposed to estimate adjRR and its 95% CI. The detailed notations and method are elaborated as follows:

Let Y ij denote the observed numbers of breast cancer death in the ith study with the j detection mode (j = 1 for exposed, 2 = non-exposed, and 3 = uninvited). Hence, Y ij follows a Poisson distribution with the expected value of μ ij . Attendance rate (r i) in the ith study is treated as a random variable specified by a beta distribution, Beta(N i,exposed, N i,non-exposed). N i,exposed and N i,non-exposed are numbers of the exposed and the non-exposed groups, respectively, from the ith study (see Fig. 2).

Fig. 2
figure 2

A Bayesian acyclic graph model for self-selection bias adjustment

Following the framework of generalized linear model, the relationship between the outcome Y and detection modes was regressed through a logarithm link function like the following:

$$ \log \left( {{\frac{{\mu_{i,j} }}{{{\text{Person - years}}_{i,j} }}}} \right) = \alpha + \beta_{1} \times I_{S} + \beta_{2} \times I_{{\overline{S} }} + b_{i} $$
(1)

\( I_{S} \) and \( I_{{\overline{S} }} \) are two indicator variables for participant and non-participant, respectively, opposed to the uninvited group (baseline group). Two hyperparameters, β1 and β2, are the corresponding coefficients indicating the magnitudes of the risk for breast cancer death for the participant and the non-participant groups, respectively, compared with the uninvited group. Note that b i is a latent variable (random-effect) for capturing the heterogeneity across studies. We suppose b i follows a normal distribution with 0 and tau as the mean value and the inverse of variance (σ2).

Once β1 and β2 were estimated, we calculated the adjusted relative risk and its 95% CI using the following formula:

$$ {\text{adjRR}} = r\% \times \exp (\beta_{1} ) + \left( {1 - r\% } \right) \times \exp (\beta_{2} ) $$
(2)

Equation 2 is similar to previous formula used for adjusting self-selection bias [11, 12]: \( {{\left( {r\% \times P_{S} + (1 - r\% ) \times P_{{\overline{S} }} } \right)} \mathord{\left/ {\vphantom {{\left( {r\% \times P_{S} + (1 - r\% ) \times P_{{\overline{S} }} } \right)} {P_{{\overline{I} }} }}} \right. \kern-\nulldelimiterspace} {P_{{\overline{I} }} }} \), which is a function of attendance rate (r%) and the mortality ratios for the participant group (P s ) and the non-participant group \( \left( {P_{{\overline{S} }} } \right) \) compared to the uninvited group \( \left( {P_{{\overline{I} }} } \right) \). By comparing our Eq. 2 with the previous method, there are two equivalents: \( \exp (\beta_{1} ) = P_{S}/P_{\bar{I}}\) and \( \exp (\beta_{2} ) = P_{\bar {S}}/P_{\bar{I}} \).

This Poisson regression model was constructed by an acyclic graphic model as shown in Figure 2. Following Spiegelhalter et al. [13], the oval nodes stand for stochastic nodes and square nodes for constant observed values. The single solid arrow captures the stochastic relationships from parents’ nodes to their offsprings’ ones and the double-lined arrow represents the logic link without stochastic property. With the acyclic graphic model, one can establish the relationships among observed data, random variables, and unobserved parameters by specifying the joint probability. In this case, the joint probability is expressed as:

$$ \begin{array}{l} P\left( {\mu_{ij} ,\alpha ,\beta_{1} ,\beta_{2} ,b_{i} ,\tau ,{\text{adjRR}},r_{i} } \right) \propto \hfill \\ \quad \quad \prod\limits_{i} {\prod\limits_{j} {P\left( {\mu_{ij} |\alpha ,\beta_{1} ,\beta_{2} } \right) \times P\left( {{\text{adjRR}}|\alpha ,\beta_{1} ,\beta_{2} ,r_{i} } \right)} } \times P\left( \alpha \right) \times P\left( {\beta_{1} } \right) \times P\left( {\beta_{2} } \right) \times P\left( {r_{i} } \right) \times P\left( {b_{i} |\tau } \right)P\left( \tau \right) \hfill \\ \end{array} $$
(3)

The relationship between the observed deaths from breast cancer and detection modes (the participant, the non-participant, and the uninvited groups) is programmed in the right side of the Fig. 2, which is linked to the left side of attendance rate based on the observed data on the participant and the non-participant groups to estimate adjRR as mentioned above.

As mentioned in the methodological section, we used two types of prior distribution. The use of different prior distributions is specified in Table 3. The non-informative priors for alpha and beta follows a normal distribution denoted as N(0, 106). The second type is informative priors for α and βs, following normal distribution with parameters (mean and variance) borrowing prior information either from previous studies by Hakama et al. or from other relevant studies from literatures. For prior information using the Hakama study, the posterior distribution that combines the prior information with the likelihood based on data from the Pirkanmaa was computed and the corresponding adjusted RR was also calculated. The estimated results with non-informative and informative priors borrowing from the Hakama study only are shown in Table 3. Table 3 also shows the incorporation of prior distribution in a sequential way with chronological order for which the posterior distribution of earlier study was treated as the prior distribution for the next study. Namely, to obtain the abovementioned adjRR in a service screening program in Pirkanmaa, Finland, we retrieved data from three other breast cancer screening programs from Sweden. Data are listed in a chronological order in Table 4. The service screening program was conducted in Pirkanmaa since 1988. Three studies were Dalarna County [9], Gothenburg [10], and Hakama’s [3] conducted since 1977, 1982, and 1987, respectively. Therefore, we first estimated parameters starting from Dalarna County in Sweden. The estimated parameters of posterior distribution from Dalarna County in Sweden were used as the informative prior for the second study, the Gothenburg trial. The same procedure was applied to Hakama’s, and then to Pirkanmma. These estimated results are shown in Table 3. For parameter estimation, we used the Gibbs sampler to implement the MCMC method and obtain the marginal posterior distributions for parameters of interests following Spiegelhalter et al. [13] method.

Table 4 Number of participants, person-years, and breast cancer death of four studies

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Wu, J.CY., Anttila, A., Yen, A.MF. et al. Evaluation of breast cancer service screening programme with a Bayesian approach: mortality analysis in a Finnish region. Breast Cancer Res Treat 121, 671–678 (2010). https://doi.org/10.1007/s10549-009-0604-x

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