Linear regression is a linear approach for modeling the relationship between a scalar dependent variable and one or more independent variables. Multiple linear regression takes a vector of independent variables and find each independent variables’ associations with the dependent variable. Here for each patient, let Y denotes the dependent variable (RFS) for each patient, β denote a vector of associations between dependent and independent variables, and let X denotes the vector of independent variable (expressions of genes):

X i = ( 1 , x i 1 , ⋯ , x i p ) (1)

β = ( β 0 , β 1 , ⋯ , β p ) (2)

So a multiple linear model is fit under the assumption:

y i = β 0 + x i 1 β 1 + x i 2 β 2 + ⋯ + x i p β p + ϵ (3)

Where ε is natural noise with following distribution:

ϵ ~ N ( 0 , σ )   i .i .d . (4)

where N stands for normal distribution, the distribution that is assumed for real random values, and i.i.d. stands for “independently and identically distributed”, as natural noises have the same probability distribution and are mutually independent.

When β is positive, x and y has a positive relationship, and as x increases y increases; vice versa. Note that the explanatory variable is also given an intercept β₀. β and β₀ are found by Least Square Estimation, rigorously defined by:

min β ∑ i = 1 n ( y i − X i t β ) 2 (5)

Under such assumption, β is found using the closed form solution:

β = ( X t X ) − 1 X t y (6)

Then a linear model is fit to describe the relationship between dependent variable and a vector of independent variables. A p-value is calculated for each β to test its statistical significance. More analysis of p-values will be discussed in Section 2.2.

Logistic regression is a regression model dealing with categorical dependent variables, which can only take two values, “0” or “1”. It estimates the probability of the binary response based on one or more explanatory variables. There are only two possible outcomes: having (y = 1) or not having (y = 0) a new tumor. The sum of the two probabilities is assumed to be 1, as stated:

p ( y i = 1 ) = p i , p ( y i = 0 ) = 1 − p i (7)

Multiple logistic regression takes a vector of independent variables and find each independent variables’ contribution to the classification result of the dependent variable. A linear model could still be used for logistic regression except one problem: the probabilities would be allowed to go smaller than 0 and bigger than 1. So the joint probability takes a logistic transformation:

p i = exp ( x i t β ) 1 + exp ( x i t β ) , (8)

where X denotes the vector of independent variables (expressions of genes) and β denotes the contribution an independent variable makes in determining the classification of the dependent variable:

X i = ( 1 , x i 1 , ⋯ , x i p ) (9)

β = ( β 0 , β 1 , ⋯ , β p ) (10)

Unlike linear regression, a Least Square Estimation cannot be found in logistic regression, so instead a Maximum Likelihood Estimation is sought. The following likelihood function is maximized:

∏ i = 1 n p i y i ( 1 − p i ) 1 − y i (11)

Also unlike linear regression, there is no closed form solution for the optimized β, so the β that maximizes the likelihood function can only be found using iterative algorithm such as the Newton Raphson algorithm [11] . A logistic model is then fit to describe the contribution each independent variable makes in deciding the category of the dependent binary variable. A p-value is then calculated for each β to test its statistical significance. More analysis of p-values will be discussed in Section 2.3.

Multiple testing is when a set of statistical inferences is tested simultaneously. In this study, each gene is given a p-value through multiple testing of all models fit. When a model is fit for each gene, a hypothesis test is performed. Null hypothesis (H0) states that the real coefficient (β) is zero (the value of this specific independent variable does not have association with the value of the dependent variable), and alternative hypothesis (H1) states the opposite.

From each null hypothesis, an T population distribution is assumed. Then by using a t-test, a p-value―how likely the T statistic, resulted from the alternative hypothesis, is to occur under F distribution―is calculated. In other words, p-value measures the probability of obtaining a result at least as extreme as the sample data, assuming the null hypothesis is true. So when the p-value is small, it is reasonable to reject the null hypothesis. In this study, p-value is used as the metric for selecting top genes and pathways.

In the case of multiple testing, a general rule of 0.05 as cutoff for p-values for statistical significance is not sufficient. Consider a large test number such as 10,000. If 0.05 is used as cutoff, it is expected that 500 true hypothesis to be rejected even though all null hypotheses are true. Also, because p-value cannot be larger than 1, using 0.05 cutoff would yield many “statistical significant” cases just by chance. As a result, an accommodated cutoff would be applied for multiple testing.

There are multiple ways to calculate the cutoff for p-value. For example, Bonferroni cutoff is a rigid correction for multiple comparison that controls the overall probability of a single false discovery. It adjusts the significance cutoff to α/m where α is the normal 0.05 cutoff and m is the number of hypothesis. In this study, a Bonferroni cutoff would be extremely small because there are tens of thousands of genes, thus an m value of tens of thousands [12] .

Another correction cutoff is False Discovery Rate (FDR), designed to control the expected proportion of rejected null hypotheses that are false. It is less stringent than the Bonferroni cutoff [13] . According to Benjamini-Hochberg (BH)’s method, the cutoff is calculated as:

p B H = max { P i : P i ≤ α i m }

Both corrections are used in this study, and the results will be discussed in Section 3.

With the complexity of traits of genes, it is sometimes difficult to interpret the p-values of all genes. Therefore, gene set pathway enrichment analysis is also used. It is useful for finding out if the differentially expressed genes are associated with a certain biological process or molecular function, so that an association between cancer and certain gene pathways can be drawn. Here, a list of genes and their p-values is analyzed using Gene Set Enrichment Analysis (GSEA) through KEGG pathway.

The sorted p-values (ascending order) for both regressions are plotted to help visualization. There were three to four thousand p-values that passed the rule-of-thumb 0.05 cutoff in both regressions, so we proceed to use more rigorous corrections of cutoff: FDR corrections and Bonferroni cutoff.

Results of analyses of the two variables appear to be very similar in terms of the top pathways and genes. Five pathways appear in the top six of both results. (Table 3).

After studying existing researches, we find some medical evidence for correlation between some pathways above and cancer.

Drug metabolism―cytochrome P450-Homo sapiens (human), id hsa00982, belongs to class of Xenobiotics biodegradation, Metabolism [33] . A 1991 research related drug metabolism with cancer predictions, and proposed to use this pathway of genes to predict individual risk of cancer [34] .

Metabolism of xenobiotics by cytochrome P450-Homo sapiens (human), id hsa00980, belongs to class of Xenobiotics biodegradation, Metabolism [33] . A research to find a new way of cancer diagnosis found about how a specific expression of cytochrome P450 is related to multiple cancers including lung cancer. The team found that a cloned dioxin-inducible form of the cytochrome P450 family, CYP1B1, expresses at a high frequency in a wide range of cancers, so a test on this pathway can provide a new way of diagnosis [35] .

Fatty acid degradation―Homo sapiens (human), id hsa00071, belongs to class of Lipid metabolism, Metabolism [33] . A study of cancer from lipid perspective suggests that through control of fatty acid pathway, cancer cell proliferation can be controlled since it requires fatty acids for synthesis of membranes and signaling molecules [36] .

The other two of the five pathways on the lists, though appeared top, have not been researched for cancer association. There is a great chance that new perspectives of cancer will be found through these two pathways.