# Does anyone know what is beta in Cox-nnet paper?

I was going through the methods in this paper: Cox-nnet: An artificial neural network method for prognosis prediction of high-throughput omics data [1]

There they define (Equation 4)

$$\theta_i=\boldsymbol{G \circ (Wx_i+b)^T\boldsymbol\beta}$$

$$\boldsymbol{W}_{H \times J}$$ are the layer coefficients, $$\boldsymbol b$$ are the biases and $$\boldsymbol G$$ is the activation function.

What does $$\boldsymbol\beta$$ mean in the context of Equation 4?

[1] Ching T, Zhu X, Garmire LX. Cox-nnet: An artificial neural network method for prognosis prediction of high-throughput omics data. PLoS Comput Biol. 2018;14(4):e1006076. Published 2018 Apr 10. doi:10.1371/journal.pcbi.1006076

• Seeing the question has an answer, I don't see reason for closing it. Commented Dec 31, 2019 at 17:06

Look back at Equation (2) in the linked paper:

$$\theta_i = \boldsymbol{x_i^t \beta}$$

"where $$\theta_i$$ is the log hazard ratio for patient $$i$$."

Regression coefficients are typically represented by the symbol $$\boldsymbol{\beta}$$. In a standard Cox model as in the above equation, $$\boldsymbol{\beta}$$ would represent the vector of coefficients that translate the covariate values $$\boldsymbol{x_i}$$ for patient $$i$$ into the estimated log hazard ratio for that patient, via the indicated dot product $$\boldsymbol{x_i^t \beta}$$.

For the neural net developed in that paper, the $$\boldsymbol{\beta}$$ in the equations that you show would then represent the regression coefficients for the covariate values $$\boldsymbol{x_i}$$ as transformed by the weight matrix, the node bias coefficients, and the specified activation function.

• Thank you so much for the kind reply. If I understood that correctly, you mean, that if we were to distill the neural net into a non-linear regression model, ..each covariate would have a coefficient $\beta$ (that would be a function of other variables and not constant).. and that is what they show here? Commented Dec 21, 2019 at 5:57
• @AasthaDua not quite. The nonlinearities are all encompassed in the activation function applied to the weighted and biased covariate values in $\boldsymbol{x}$. The $\beta$ coefficient values would be constants, although determining their values would involve using information from all of the variables. Technically this would still be considered a linear regression model as it is linear in terms of the coefficients. A nonlinear transformation of a predictor variable (as, say, with the frequently used log transform) does not by itself make a model “nonlinear” as that term is used in statistics.
– EdM
Commented Dec 21, 2019 at 10:52
• Thanks again for your reply! That makes it super clear. Can I think of this equation as a 2 layer neural net? First has params W and b. This is followed by a layer of non-linearity G. And that is followed by $\beta$ - the weights of the last layer of this neural net - which give out the output of the net. Commented Dec 23, 2019 at 21:25
• @AasthaDua that would be my understanding: the hidden layer applies the activation function to each predictor after weighting and biasing, the output from each predictor is multiplied by the corresponding $\beta$ value, and those contributions from the predictors are added to provide the combined linear predictor as an output. Note, however, that I have no substantive experience with neural nets and their hidden layers, just with Cox models.
– EdM
Commented Dec 23, 2019 at 22:03