Question

In the study, the authors state the following: “Because ventilator-free days and days free of multiple organ dysfunction syndromes are known to have a bimodal distribution, the data were initially analyzed by means of Student’s t-test, with between-group differences presented as means and 95% confidence intervals. A secondary analysis of these outcome measures involving a bootstrapped t-test was also conducted to support the results of the primary analysis…”

Try to Justify the logic of these two sentences.

i. why did they initially use a method based on Student’s t-test and why did they do a secondary analysis with a “bootstrapped t-test”. Try to explain briefly what the latter means.

This is an open question, please feel free to answer this question. If you know something or even just some points, I will grade it as helpful. Thanks!
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Answer 1

Before getting into the details of the explanation , let us first of try to understand as to why is there a need to have a bootstapped t test ?

Why is bootstapping required ?

bootstrapping is any test that relies on random sampling with replacement/ resampling.. It is the practice of estimating properties of an estimator (such as its variance) by measuring those properties when sampling from an approximating distribution.is done. For example, empirical distribution function

but How does it help ?

A Bootstrapped test allows us to assign various measures of accuracy like bias, variance, confidence intervals, etc to sample estimates.

It allows estimation of the sampling distribution of almost any statistic using random sampling methods for almost all kinds of distributions.

Such being the case, It may also be used for most of the statisitical analysis including constructing hypothesis tests.

It is often used as an alternative to drawing inferences statistically based on the assumption of a parametric model when

1. Our assumption is in doubt,
2. Where parametric inference is impossible or
3. When it requires complicated formulas for the calculation of standard errors..

So what is the reason and logic behind usage of bootstrapped test?

The basic idea of bootstrapping is that inference about a population from sample data (sample → population) can be modelled by resampling the sample data and performing inference about a sample from resampled data (resampled → sample).

As the population is unknown, the true error in a sample statistic against its population value is unknown.

Now in a bootstrap-resampling, the 'population' is in fact the sample itself, and this is known; hence the quality of inference of the 'true' sample from resampled data (resampled → sample) is measurable and accurate.

Now that we have good understanding of the idea behind application of bootstrapping , let us approach our problem.

We have an experiment done where the normal t test of difference of means was conducted and for the same data set ,the bootstrapped t test was also conducted.

The experiment had the following features :-

1. It is bimodal distribution
2. t test (normal and bootstrapped ) conducted
3. 95 % confidence interval was considered for significance

Now think !

Why was the simple t test done in first instance?

The primary t test was conducted as it gave us an understanding about the level of significance for difference between means of the samples between ventilator case and no organ failure cases.

Post that analysis now, why was the boot strapped t test conducted ?

The bootstrapped t test which is a secondary t test was done on the bi-modal distribution to re-validate the result accuracy as it is applicable for all kind of distributions since it does resampling.

This test acted as an inferencial representation for the entire population which is now represented by the sample itself.

This further helped us with plugging the inference inaccuracy due to true error in sample statistic owing to non availability of population value since now the sample is the population itself.