# Using Longitudinal methods (via least squares) over Survival analysis in Point Estimation

Survival Analysis makes strong assumptions of the data and should be avoided with preference given to longitudinal data analysis (mixed or marginal methods).

Assumptions of Survival Analysis

Assumption 1: Non-informative censoring: Patient dropouts or lost to followup does not depend on the event of interest. Survival Analysis models time to event, if people who are more likely to have the event dropout earlier or later then Survival Analysis does not work.

If censoring is informative (assumption not holding) then it biases the Hazard Ratio which is the point estimate.

Assumption 2: The baselines hazard probability distribution has to be the same as the covariate hazard probability distribution. This is an assumption on probability distributions of the data, while not specifying a particular distribution this assumption still makes an assumption on probability distributions of the data.

Assumption 3: That the Hazard ratio is constant over time. This is a big assumption as it states the effect of the covariate is the same across the entire followup. Even common time varying covariate models assume proportional hazards.

Longitudinal Data Analysis and Assumption Testing

This can be done using least squares directly. This is because we are not concerned with the inflation of errors due to the lack of independence assumption, since we don’t need to calculate a P value.

Therefore the only assumption that is important is linearity as per simple linear regression. If the linearity assumption does not hold we can fit a more appropriate equation to model the effect of x (predictor) onto y (outcome) or to predict y (outcome).

Practical Concerns with replacing Survival Analysis with Longitudinal Data Analysis

Survival times consist of the followup time and a failure variable which is usually 1 for an event and 0 for no event.

Survival analysis despite its assumptions are advantageous for calculating point estimates in the presence of patients who were lost to followup. These patients did not have an event and their followup time was not as long as the window of the analysis (the minimum followup time).

This advantage of utilising lost to followup data of survival analysis can be replicated in longitudinal analysis with some effort.

Longitudinal analysis can be carried out with two outcome or dependent variables. One is followup time and the other is survival status. Formulas become more complicated for recurrent survival analysis, but this can be achieved.

A multivariate longitudinal least squares estimate is derived from the set of known predictors with 2 outcome variables of followup time and survival status. Dependence of error terms is not an issue as standard errors and probability are avoided in the “Statistics Without Probability” paradigm.

MultivariateRegression

MultivariateRegression.pdf

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