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I have a failure time data set. I want to fit Cox PH model with quadratic term like

$h(t;x)=h_0(t)*exp(x\beta_0+x^2\beta_1)$

What will be the mathematical formula for variance of quadratic term? Could anybody give the details form?

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You can think of $x^2$ as just a second covariate (i.e. $x_1 = x, x_2 = x^2$) that just happens to be the first covariate squared.

As such, the standard errors will estimated in the standard fashion: by inverting the negative Hessian matrix. To answer what I believe is your misunderstanding, there will be no closed form solution to variance (i.e. can't just plug into a simple formula), BUT it will still have an estimated standard error produced by any standard software.

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