Statistical Inference and Modelling of Breast Invasive Ductal Carcinoma Treatments

Final Paper

Goals and Objectives

Breast invasive ductal carcinoma (IDC) is the most common type of breast cancer, with about 80% of all forms of breast cancer being IDC, according to the American Cancer Society (DePolo, 2024). There are numerous nonsurgical treatments of IDC, such as radiotherapy, chemotherapy, and hormone therapy (Wright, 2023), and each of their effectiveness is partly determined by the patient’s condition, such as age and tumor stage. For instance, due to interactions between other treatments or conditions as a consequence of aging (e.g. Diabetes, liver disease, metabolism), more optimal doses of chemotherapy are generally discouraged for older patients due to potentially toxic side effects, implying a smaller difference in survival between older patients with or without chemotherapy (Given, Given, 2008). However, it is unclear how combinations of treatments can interact in a model to predict a patient’s survival until death.

For our research project, we have selected a dataset of approximately 1900 primary breast cancer samples, obtained from the Molecular Taxonomy of Breast Cancer International Consortium (METABRIC) database through cBioPortal for Cancer Genomics.

Our project question is: “How do radiotherapy, chemotherapy, and hormone therapy influence the length of time a patient with IDC will survive, given control variables age, surgery type, tumor stage, and their present survival status?”

Our analysis will involve the inference of covariates within linear models, as we seek to determine the interaction of cancer therapies on allowing patients to survive longer from IDC.

  • Variables from columns 32 to 693 consist of genetic attributes containing m-RNA levels z-score for 331 genes, and mutation for 175 genes; they have been omitted due to being difficult to interpret.
  • Due to the distribution of “cancer_type_detailed” categories and for ease of computation, we will be filtering the dataset for IDC patients that are either alive or dead from disease, as IDC consists of the majority of the dataset, and the patients that have died from other causes are irrelevant to the project question and not specific enough (e.g. Accident, non-cancer diseases).

Process and Approach

  • Developed a team contract with my peers to summarise our data overview and reach an understanding of how we would allocate our later work.
  • Selected a dataset from METABRIC for IDC patients, and demonstrated that the datasets can be read from the web into R.
  • Cleaned and wrangled the data into a tidy format.
  • Modelled residual and QQ-plots, boxplots, scatterplots, histograms, and VIF of covariates to perform exploratory data analysis on the cleaned dataset to glean a better understanding of what results are to be expected.
  • Compared 2 prediction models that differ on the presence of an interaction term age_at_diagnosis:chemotherapy1 to determine if the term led to both models providing different statistically significant coefficients.

Model selection

  • As our goal is to determine how the various forms of therapy influence a patient’s survival until death, interaction terms regarding all combinations of therapies are considered, since it is unclear if the effect of one therapy will influence the effect of another (e.g. Chemotherapy and hormone therapy in similar timeframe).

  • Interaction terms for age and each type of therapy are included, as prior studies have indicated varying degrees of influence between age and treatment method (Given, Given 2008; Cleveland Clinic, 2024; U.S. National Library of Medicine; Steinfeld, Diamond, Hanks, Coia, Kramer, 1989).

  • Interaction terms for tumor_stage and each type of therapy are also included since prior studies have indicated that the type of treatment a patient receives is influenced by the stage, size of tumor and the spread of cancer cells. (“Invasive Ductal Carcinoma”, 2024; “Treatment of breast cancer”, 2024)

  • Due to not being significant parts of the question of interest and the indeterminate interaction between them and the therapies, interaction terms for type_of_breast_surgery and death_from_cancer are ignored for the full model.

  • We had attempted to fit linear models without the log transformation, and the resulting residual plots had demonstrated heteroscedasticity; this is discouraged as it would lead to inconsistent results.

  • Full model for finalmod_both: log(overall_survival_months) \~ age_at_diagnosis + chemotherapy \* hormone_therapy \* radio_therapy + age_at_diagnosis:chemotherapy + age_at_diagnosis:hormone_therapy + age_at_diagnosis:radio_therapy + tumor_stage + tumor_stage:chemotherapy + tumor_stage:hormone_therapy + tumor_stage:radio_therapy + type_of_breast_surgery + death_from_cancer

  • Forward and backward AIC selection with the same full model was attempted, but only backward selection produced the same model, while forward selection produced a less optimal model.

Variable Estimate Std. Error t value Pr(|t|)
(Intercept) 5.947000 0.214458 27.730 2e-16
age_at_diagnosis -0.014682 0.003806 -3.857 0.000124
chemotherapy1 -0.701659 0.294312 -2.384 0.017348
hormone_therapy1 -0.622362 0.266825 -2.332 0.019915
tumor_stage2 -0.238917 0.093835 -2.546 0.011073
tumor_stage3 -0.815091 0.166351 -4.900 1.15e-06
death_from_cancerDied of Disease -0.857453 0.054094 -15.851 < 2e-16
age_at_diagnosis:chemotherapy1 0.009678 0.005384 1.797 0.072624
age_at_diagnosis:hormone_therapy1 0.009458 0.004510 2.097 0.036273
hormone_therapy1:tumor_stage2 0.216084 0.117380 1.841 0.065997
hormone_therapy1:tumor_stage3 0.711055 0.203734 3.490 0.000508

  • Full model for finalmod_both_1: log(overall_survival_months) ~ age_at_diagnosis + chemotherapy + hormone_therapy + tumor_stage + death_from_cancer + age_at_diagnosis:hormone_therapy + hormone_therapy:tumor_stage

  • Version of the model finalmod_both without the interaction term age_at_diagnosis:chemotherapy1

  • Has all terms to be statistically significant on the 5% level.

Variable Estimate Std. Error t value Pr(|t|)
(Intercept) 5.773782 0.191845 30.096 2e-16
age_at_diagnosis -0.011469 0.003365 -3.408 0.000685
chemotherapy1 -0.189144 0.073044 -2.589 0.009782
hormone_therapy1 -0.570923 0.265644 -2.149 0.031908
tumor_stage2 -0.271223 0.092222 -2.941 0.003363
tumor_stage3 -0.830862 0.166344 -4.995 7.19e-07
death_from_cancerDied of Disease -0.857409 0.054167 -15.829 < 2e-16
age_at_diagnosis:hormone_therapy1 0.008276 0.004467 1.852 0.064316
hormone_therapy1:tumor_stage2 0.240445 0.116753 2.059 0.039766
hormone_therapy1:tumor_stage3 0.726981 0.203816 3.567 0.000382

Results

The final regression model includes the following key variables: age, chemotherapy, hormone therapy, and tumor stage (specifically stages 2 and 3). It also accounts for the outcome of death from cancer (death from cancer and death from other causes) and incorporates interaction terms between age and hormone therapy, as well as hormone therapy and tumor stage (for stages 2 and 3).

The results from bidirectional selection and backward selection consistently provided the same set of covariates, but the results from forward selection did not, possibly due to it missing covariates that are only significant in combination with other covariates. The covariates of the final regression model are all statistically significant on the 5% significance level, with the exception of the interaction term between age and hormone therapy. This consistency between bidirectional and backward selection enhances the credibility of the findings, indicating that the observed relationships are relatively robust and not due to random chance.

In the model selection, we discarded radiotherapy, indicating that it doesn’t provide a significant influence on the log survival period of patients. For chemotherapy, regardless of tumor stage, the log survival period is decreased by 0.189144. This indicates that chemotherapy consistently reduces survival periods, and its negative effects on survival do not change significantly at higher tumor stages.

For hormone therapy, it initially reduces the log survival period by 0.570923 at tumor stage 1, but its effect decreases to a reduction of 0.330478 in log survival at stage 2, and it even leads to an increase in log survival by 0.156058 in stage 3. From tumor stage 1 to stage 3, the impact of hormone therapy changes from harmful to beneficial, ultimately improving survival chances. This highlights the importance of carefully considering the use of hormone therapy based on tumor stage.

The model shows a negative correlation between age_at_diagnosis and overall_survival_months, indicating that for each year of age (at the time of diagnosis), the patient’s log months of survival is decreased by 0.011469. This indicates that younger cancer patients will have a higher chance of recovering from IDC (higher overall survival months). The presence of hormone therapy increases change in survival to 0.003193, implying that the impact of age on survival months becomes less severe with hormone therapy.

The final model shows that the intercept for patients who have died from disease is lower by 0.857409 compared to living patients. This is consistent with the scatterplot between age/living condition from the exploratory analysis, where the overall survival was significantly lower for deceased patients compared to living ones. The scatterplot does not show the two slopes to be parallel, which would have been accounted for by the interaction term between death_from_cancer and age_at_diagnosis; however, this term was removed during model selection due to low significance.

To further validate the reliability of the model, the plot of fitted values versus residuals shows a good fit with residuals centered around zero, demonstrating homoscedasticity. Additionally, the residuals are randomly scattered without any discernible pattern beyond being centered around 0, suggesting that the errors are independent and that the relationship between the independent variables and the dependent variable is assumed to be linear. Furthermore, the points in the QQ plot align closely along the diagonal line, indicating that the residuals are approximately normally distributed. These observations confirm that the regression model does not violate the key assumptions of regression analysis, thereby minimizing the risk of drawing incorrect conclusions.

Impact and Reflection

To address our initial research question, our model suggests that chemotherapy and hormone therapy have varying influences on survival time before death, depending on the initial tumor stage, while radiotherapy does not have a significant association with survival time before death. In general, our model suggests that chemotherapy has a negative association with survival time before death, regardless of tumor stage leve,l while hormone therapy has a negative association for patients with an initial tumor stage of 1 and 2, while it has a positive association when the patient’s original tumor stage was 3. Our model does not align with prior research, as prior research suggests that chemotherapy is generally effective and increases the survival time before death. Similarly, prior research suggests that hormone therapy and radiotherapy are generally effective at treating patients with IDC. However, our model aligns with prior research suggesting chemotherapy is most effective for later stages of cancer (Penn Medicine, n.d.).

Our model likely differs from prior research due to several limitations in our model. One limitation is our variable selection method, a bidirectional stepwise selection starting with a full model. While bidirectional model selection allows for better flexibility and model fit to the data, there is also the risk of the final model being overfitted to the existing data. Our response variable, being the log of overall_survival_months to maintain homoscedasticity, also makes our model estimates rather difficult to interpret. Moreover, our results may have been skewed by using tumor stage 1 as our baseline, since we did not have enough data points for tumor stage 0. Lastly, our model could not account for the context of the patients undergoing such treatments, such as the patients that took the treatment likely already being in poorer health than those that did not, or the side effects of such treatments impacting health, such as organ damage from chemotherapy (“Side effects of chemotherapy”, Canadian Cancer Society, 2024), osteoporosis from hormone therapy (“Side effects of hormone therapy”, Canadian Cancer Society, 2017) and low blood cell counts from radiation therapy (“Side effects of radiation therapy”, Canadian Cancer Society, 2017). Overall, the limitations of our model are reflected by the relatively low adjusted R2 of 0.3103432 (7 s.f.), which indicates our model only accounts for around 31.03% (2 d.p.) of the variability in survival time before death for IDC patients.

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