Integral, Measure and Martingale

2017年8月29日星期二

Highest density region

https://stats.stackexchange.com/a/148452/61705

2017年8月26日星期六

Different likelihoods

Maximum Likelihood

Find β and θ that maximizes L(β, θ|data).

Partial Likelihood

If we can write the likelihood function as:

L(β, θ|data) = L1(β|data) L2(θ|data)

Then we simply maximize L1(β|data).

Profile Likelihood

If we can express θ as a function of β then we replace θ with the corresponding function.

Say, θ = g(β). Then, we maximize:

L(β, g(β)|data)

Marginal Likelihood

We integrate out θ from the likelihood equation by exploiting the fact that we can identify the probability distribution of θ conditional on β.

2017年8月22日星期二

Calculate 10 year probability use Cox model

Also pay attention to time unit here.

2017年8月13日星期日

Transform or link?

https://ecommons.cornell.edu/bitstream/handle/1813/31620/BU-1049-MA.pdf?sequence=1

2017年8月12日星期六

2017年8月6日星期日

ranking and empirical distributions

In the absence of repeated values (ties), the cdf can be obtained computationally by sorting the observed data in ascending order, i.e., $X_{s} = \{x_{(1)}, x_{(2)}, \ldots , x_{(N)}\}$ . Then $F(x)=(n_x)/N$ , where $(n_x)$ represents the ascending rank of $x$ . Likewise, the p-value can be obtaining by sorting the data in descending order, and using a similar formula, $P(X \geqslant x) = (\tilde{n}_x)/N$ , where $(\tilde{n}_x)$ represents the descending rank of $x$ .

https://brainder.org/2012/11/28/competition-ranking-and-empirical-distributions/

2017年7月23日星期日

Spline regression

In regression modeling when we include a continuous predictor variable in our model, either as the main exposure of interest or as a confounder, we are making the assumption that the relationship between the predictor variable and the outcome is linear. In other words, a one unit increase in the predictor variable is associated with a fixed difference in the outcome. Thus, we make no distinction between a one unit increase in the predictor variable near the minimum value and a one unit increase in the predictor variable near the maximum value. This assumption of linearity may not always be true, and may lead to an incorrect conclusion about the relationship between the exposure and outcome, or in the case of a confounder that violates the linearity assumption, may lead to residual confounding. Spline regression is one method for testing non-linearity in the predictor variables and for modeling non-linear functions.