The Affymetrix microarray is a device designed to simultaneously measure the expression levels of thousands of genes in a particular tissue or cell type of interest. The microarrays are microscopic slides that are printed with thousands of tiny spots in defined positions, with each spot containing known DNA sequences that correspond to particular genes. The DNA molecules attached to each slide act as probes to detect gene expression from incoming samples, which is also known as the transcriptome or the set of messenger RNA (mRNA) transcripts expressed by a group of genes. Specifically, for the U133A GeneChip used in the analysis, there are over 14,500 detectable genes. For each of the detectable genes in the array, there is a corresponding probe set containing 11 distinct 25-base pairs. The 11 distinct pairs include a perfect match (PM) probe and a complementary mismatch (MM) probe. The PM probe is designed to match a particular sequence of interest perfectly, while the MM probe is designed to measure the level of mis-hybridization. Together, they help provide a more accurate measurement of gene expression.

Before any initial analysis may begin, the raw microarray data must first be preprocessed as a set of intensities by usually following three steps, background correction, normalization, and summarization. Preprocessing procedures combine multiple probe signals into a single value. There are inherent physical differences between each microarray chip as well as differences in the handling of the chip and internal dye intensity effects that introduce extraneous noise to the data. Thus, a problem arises when true biological variation in the microarray chips becomes entangled with an unwanted systematic variation. Background correcting procedures adjust the values for the ambient background intensities in each probe. Since the probes are usually randomly scattered within the microarray, there should not be any particular spatial pattern in the intensities. It removes local artifacts and noise so that the measurements are not so affected by neighboring measurements. For robust multichip average (RMA) method, background correction involves modeling the observed Perfect Match (PM) signal of each probe as a sum of an exponentially distributed true signal (S) term and a normally distributed noise term (e):

where the index i represents the particular sample or microarray, index j represents the probe pair (within each probe set corresponding to a particular gene), and index k represents the particular gene within the microarray. From this, the expected value of the true signal intensity given the observed PM signal, E(S_ijk|PM_ijk), is estimated as the background-corrected intensity.

Normalization is required to correct and compensate for the unwanted variation and scale the distribution of the gene expression so that actual biological differences in the gene expression may be more appropriately detected. Quantile normalization is a standardizing technique to make similar distributions. As an example, consider three microarrays with 4 probes. Array 1 has values {3, 5, 6, 7} for 4 probes, array 2 has {9, 9, 3, 6} and array 3 has {6, 3, 7, 4}. The first step is to rank the values within each array. Array 1, 2, and 3 have ranks {1, 2, 3, 4}, {3.5, 3.5, 1, 2}, and {3, 1, 4, 2}, respectively. The second step is to obtain averages of each rank across arrays. Rank 1 average is (3+3+3)⁄ 3=3; rank 2 average is (5+6+4) ⁄ 3 =5; rank 3 average is (6+9+6) 3 = 7; rank 4 average is (7+9+7) ⁄3=7.67. The last step is to replace the average rank values with the rank position in each array. Thus, normalized array 1 has values {3, 5, 7, 7.67}. For normalized array 2, we use the average value of rank 3 average and rank 4 average due to a tie. Thus, normalized array 2 has {7.335, 7.335, 3, 5} values. Normalized array 3 has {7, 3, 7.67, 5} values.

Finally, the summarization step combines probe intensities across the probe set into a single value that may be considered the gene expression level. The summarization step involves using background-corrected and normalized intensities and their logged values. These log-transformed intensities denoted as Y_ijk, is then modeled as the following linear additive model:

where µ_ik represents the log scale expression level for microarray i for gene k, á jk is the probe affinity effect for probe pair j and gene k, and ϵijk is the independent identically distributed error term. Given the noisy nature of gene expression data, the median polish method developed by Tukey [11] is utilized to estimate the parameters on the right-hand side in the additive model. The median polish method involves iteratively extracting row and column medians to estimate the row and column effects that correspond to the microarray and probe pair effects, respectively. The estimate of µik gives the final RMA expression measure for microarray i and gene k.

By continuing with the previous example, the final RMA expression measures for the probe set via the median polish method is calculated as follows: for the normalized array 1, {3, 5, 7, 7.67}, the median is 6 across the probe set. For the normalized array 2, {7.335, 7.335, 3, 5}, the median is 6.1675. For the normalized array 3, {7, 3, 7.67, 5}, the median is 6. After removing the row medians obtained across the probe set, the array 1 has values {-3, -1, 1, 1.67}, the array 2 has {1.1675, 1.1675, -3.1675, -1.1675} and the array 3 has {1, -3, 1.67, -1}. Now we obtain column medians across arrays per each probe. Medians for probes 1, 2, 3, and 4 are 1, -1, 1, and -1, respectively. After removing the column medians, the residual values for arrays 1, 2 and 3 are {-4, 0, 0, 2.67}, {.1675, 2.1675, -4.1675, -.1675}, and {0, -2, .67, 0}. The median polish algorithm stops because all row medians and column medians are zero. To obtain the RMA expression measure for each of the three arrays, the residual values are subtracted from the original values. Thus, the resulting expression values for arrays 1, 2, and 3 are {7, 5, 7, 5}, {7.1675, 5.1675, 7.1675, 5.1675}, and {7, 5, 7, 5}. By taking the mean across each probe set, the RMA expression measures for the three arrays are found to be 6, 6.1675, and 6.

The raw CEL files from both datasets were preprocessed and normalized altogether via the RMA algorithm to ensure consistency between the datasets using the “Affy” package in R [12]. The dataset contains over 23,000 probe sets which warranted an initial variable screening. Nonspecific gene filtering is a popular technique to help alleviate the high dimensional nature of genetic data by keeping only the probe sets that show high variability and remove low variability probe sets that are likely non-informative. This should help with potential overfitting with the removal of false positives and redundant covariates for more efficient analysis. Affymetrix microarrays contain control probe sets (denoted by the “AFFX” prefix) that do not correspond to any particular genes and are used to help normalize gene intensities. These control probe sets are thus removed before performing gene filtering. Nonspecific gene filtering is then applied in the analysis, keeping only the 10,000 probe set features with the highest variance in the training dataset. Thus, the features used to train the model are the remaining 10,000 probe sets in addition to the demographic and clinical covariates of age, sex, and stage.

Survival analysis models the expected duration of time until an event occurs, whether it is the death of a biological entity, recurrence of a disease, or failure of a mechanical system. Survival modeling involves examining the relationship between survival time data, which is commonly subject to censoring, and one or more covariates.

The Cox proportional hazards model is a popular choice to model covariates in relation to a survival time due to its relative simplicity and convenience in handling censored observations [13]. Given the survival data (ti, δi, Z i), i = 1, … n, where t_i is the observed survival or censored time, δi is the censoring indicator for the survival time, and Z_i = (Z_i1, Z _i2, …, Z_ip)T is a p -dimensional covariate vector for the ith individual, the Cox model is

where λ(t|Z) is the hazard rate at time t given the covariates Z, λ₀(t) is the baseline hazard rate, and β is a p-dimensional parameter vector associated with each of covariates. Cox [13] proposed the partial likelihood to estimate the parameters β through the following formula:

The estimated parameters for the cox model are given by minimizing the partial likelihood, which is given as:

where R(ti) = {j: tj ≥ ti} denotes the risk set at time t_i. Only uncensored event times contribute their own factor to the partial likelihood. However, both censored and uncensored observations appear in the denominator, where the sum over the risk set includes all individuals who are still at risk immediately prior to t_i. The proportional hazards condition states that covariates are multiplicatively related to the hazard; the model itself does not directly estimate survival times but rather estimates how the covariates affect the hazard rates. The logarithm of the partial likelihood is minimized due to the ease of calculation. This is given as:

The survival of two individuals can be compared with hazard ratios; given two individuals with covariates Z₁ and Z₂, the hazard ratio between the two individuals is given by

Because the number of covariates greatly outnumbers the number of observations in high dimensional microarray data, overfitting and high variance may be significant problems. Thus, additional consideration must be taken to select the most relevant covariates. Tibshirani proposed extending the lasso penalty in the least-squares regression models into the context of Cox models by minimizing the partial likelihood with the lasso [14]. The lasso estimates for the Cox model are

where γ is a specified penalty parameter and p is the number of covariates. The nature of the lasso constraint causes it to shrink irrelevant coefficients and produces coefficients that are exactly zero. As a result, it simultaneously reduces the estimation variance while providing a final interpretable model with a feasible set of variables. If too large of a penalty parameter λ is chosen, the estimated parameters maybe significantly biased; if too small, the model may not be sufficiently sparse. In some circumstances, one may not want to eliminate certain covariates with the lasso penalty from the model during training if the covariates hold some underlying importance or special weight in regard to the response. This may be done by modifying the minimization problem above:

By adding an individual penalty term v_j for each coefficient j, chosen coefficients may be left out of the regularization process even if the penalty term v_j is zero. Thus, this prevents the specified covariates from being removed by the lasso penalty in the final model.

We employ the modified covariate method by Tian et al. [15]. A set of q covariates (both clinical and genetic data) Z, and a binary treatment variable T, which takes on values of -1 or +1 corresponding to observation or treatment, respectively, are given. Let function W be p dimensional functions of covariate Z including an intercept. To identify the subgroup of patients who may or may not benefit from the treatment, we consider a treatment interaction term. Thus, the modified covariate method begins with

where W^*i = W(Z)∙ T2. For simplicity in this analysis, W(Zi) is chosen as the identity function, W(Zi) = Zi. For survival data, the following Cox regression model can be used:

Under the modified covariate approach, the linear combination γTZ can be thought of as a treatment score and thus used to stratify patients according to how they would respond to a treatment. For example, if given T = + 1 as ACT treatment, higher treatment scores γ T Z correspond to a higher hazard, and thus ACT treatment is not recommended. On the other hand, lower treatment scores correspond to a relatively lower hazard, and thus ACT may be recommended. In the randomized trials, we assume P(T=1) = P (T= - 1) = meaning that the treatments are randomly assigned to each patient.

The features are transformed to the modified covariate approach, where each feature is multiplied by the corresponding treatment (either T/2 = + 0.5 or T/2 = - 0.5). The modified covariate Cox model is then trained using the preprocessed JBR.10 training set. We note that the coefficients associated with the modified covariate measure the interaction strength between treatment and subgroup of patients who may or may not benefit from the treatment. To further reduce the dimensionality and alleviate high potential variance, the model is trained using the lasso regularization penalties to select prominent predictive genomic markers. Given the potential importance of the clinical covariates, the model trained with the lasso is modified to keep the three clinical covariates in the model. We implement this approach using the glmnet package in R, which uses the coordinate descent algorithm to approximate the penalized coefficient estimates. Then, the results from both regularization methods are compared. Due to the nature of the relatively small training set, a Leave-One-Out Cross-Validation (LOOCV) scheme was used to tune the lasso penalty parameter. The LOOCV is a popular variant of k-fold cross-validation involving the splitting of samples randomly into k groups or folds. The standard approach for the LOOCV is made by setting aside one observation as the testing sample and fitting the model on the remaining n - 1 observations in the training sample. Then, the model evaluation is made on the left-out testing sample. This process is repeated for each of the observation, where each fold is served as the testing set once. Ultimately, the hyperparameters that maximize the average of the cross-validation metrics are chosen. This helps determine the optimal model hyperparameters so that the model is not overfitted on the training data.

In the Cox regression framework where there is censoring, maximizing the partial likelihood may lead to a problem when the testing sample used for cross-validation is too small. If the number of testing samples is too small, there may not be enough samples to build up an appropriate risk set R(t_i) present in the partial likelihood. This could potentially lead to undefined or unstable partial likelihoods. Verweij and Van Houwelingen [16] proposed a method to calculate a shrinkage factor by measuring CV score, mentioned as below:

where refers to the partial likelihood calculated on the entire dataset, refers to the partial likelihood on the dataset which excludes the i-th observation (out of n observations), and ^-i (λ) refers to the estimated parameters fitted using the training set, which excludes the left-out observation and the using candidate λ parameter. This avoids calculating the partial log likelihood directly on the held-out testing set and ensures a sufficient number of samples to be defined for the partial likelihood.

After training, the estimated parameters maybe used to find the risk scores (risk with respect to ACT treatment) for the training set given by 'Z. A threshold must then be determined to stratify the patients into a low-risk and high-risk group. Patients who have a risk score higher than the threshold are classified as high-risk patients and thus are not recommended chemotherapy treatment; conversely, patients whose risk scores are lower than the threshold are classified as low-risk patients and thus are recommended chemotherapy treatment.

One possible and natural choice is to set the decision threshold at zero. By using the modified covariate approach, the hazard ratio of the two treatments on an individual is given by:

If the patient risk score β'Z_i> 0, then the hazard rate is greater if the patient undergoes T = + 1 (which corresponds to ACT treatment) than if the patient does not. Conversely, if the risk score β' Z_i< 0, then the hazard rate for the patient is lower had they undergone ACT treatment compared to without ACT treatment. Setting the threshold to zero is a sensible choice under the assumption that the modified covariate method is indeed the actual model of interaction between treatment and the other covariates. Even though setting the decision threshold at zero may only capture the relative direction and not the size of the effect, we considered both zero risk score and the median of the risk scores as thresholds in order to make predicted treatment recommendations by classifying patients into a high or low-risk group. To evaluate the classifier, survival curves are used based on treatment types and classified risk groups. For example, for the patients classified in the high-risk group, those who underwent ACT should be expected to survive just as short or shorter than those who did not. Conversely, in the low-risk group, patients that underwent ACT are expected to survive longer or just as long as those who did not.

Additionally, patients may further be separated into the following two groups: one where the actual treatment they underwent corresponded to the treatment recommended according to their predicted risk group and the other where the two do not correspond. For a particular patient i, if TX_i represents the actual treatment that patient i received and R_i is the recommended treatment based on the previous risk scores, then the variable F_i indicating whether the patient i followed the recommended treatment is given by

A successful classifier, in principle, should split the patients such that those who follow the recommendation exhibit higher survival than those who do not. Finally, the modified covariate model trained using the JBR.10 dataset is validated using the DCC dataset.