A Little Glimpse into the Hypothesis Testing


Are you watching closely? It is truth or just perception?

Sometimes it is really difficult to unveil the mask. For example, let’s enter a hospital. A new drug is introduced which claims it can reduce the high blood pressure. How can the hospital verify it? You can say immediately:

“Its easy, apply the drug on a patient and check whether it is effective or not!”

Wait!!! Is it the real end? What if the drug is actually not effective but somehow works for the patient? What if the other way round? You know that s**t happens!

Basically, it is not judicious to comment anything after applying the drug on just one patient. So the next idea is may be to apply the drug on 100 patients and then check how many of them are benefited by this. Now we are talking sense! But, we have to decide a cutoff, x i.e. if more than x people are benefited, then we will declare the drug as effective. Now what should be the value of x? Common sense is, x>50. But is it 55, 60 or 65? Now, we are entering in the realm of hypothesis testing. This basic framework can be thought as follows:

We have say data x_1, x_2, \cdots, x_n   (effect of drug on the patients measured somehow) and we want to test the following hypothesis:

          The new drug is really effective in reducing the blood pressure

How can we test that? With what confidence (obviously in terms of probability) we can claim that our test is true? Well, fortunately Statistics answer all of that. Let’s introduce the testing produce more formally.

Suppose we have x_1, x_2, \cdots \cdots x_n \sim \mathcal{N}(\mu,\sigma^2)

and for the time being we know \sigma^2 but we don’t know \mu. We have data in our hand and somehow we believe that \mu=0. Now we want to be confident about our belief based on the data.  Two immediate question arises. What is “confidence” mathematically? How should we achieve the confidence? It is clear that, we want to test \mu=0  vs  \mu \neq 0. In this lecture, we will just try to find some rationale approach to construct a test. Later, we will talk about confidence. There are mainly two types of test that are availed in the most of the situation. Out the them, the most common and well known technique is Likelihood Ratio Test, and the other one, also very useful is known as Union Intersection Test. Technically the test test is written as:

H_0\,\,\,: \,\,\,\,\,\,\,\,\,\,\,\mu =0

H_1\,\,\,: \,\,\,\,\,\,\,\,\,\,\,\mu \neq 0

Often H_0 is termed as null hypothesis and H_1 as alternative or research hypothesis. (The naming of “null” and “alternative/research” has a history. Earlier, in the field of biology, the testing is used to test the efficiency of new method or drugs. H_0 is generally considered to be the case whether the new method has no improvement, hence the name ‘null’. H_1 is considered to be the case where it has improvement, i.e. something new is found in the research, hence the name ‘research hypothesis’. )

  • Likelihood Ratio Test: 

In the previous example, we have data x_1,x_2, \cdots \cdots, x_n. Using the data we want to test whether \mu = 0 or not. Now what should the rational way to do it? One of the most intuitive method is to check the likelihood of the data under \mu = 0 and then check whether it is significant or not. How should we check the significance? One obvious way is to compare it with the maximum likelihood of the data. So the test becomes something like:

Reject the assumption \mu =0 if  \frac{\sup_{\mu =0} L(\mu | X)}{\sup_{\mu} L(\mu | X)} = \frac{\sup_{\mu =0} L(\mu | X)}{L(\hat{\mu}_{mle} | X)}is small. (As the small value means less significance!)

Now how small is small can be answered in the next lecture where we will talk about “confidence”. This is one of the main way to do testing. This method has many advantages which we will discuss later. In general if our parameter space is \Theta and we want to test

H_0\,\,\,\,\,\,: \,\,\,\,\, \theta \in \Theta_0

H_1\,\,\,\,\,\,: \,\,\,\,\, \theta \in \Theta_1

where \Theta_0 \cup\,\, \Theta_1 = \Theta  and \Theta_0 \cap \,\, \Theta_1 = \phi , the Likelihood Ratio test becomes:

Reject H_0 if \frac{\sup_{\theta \in \Theta_0}L(\theta | X)}{\sup_{\theta \in \Theta}L(\theta | X)} = \frac{\sup_{\theta \in \Theta_0}L(\theta | X)}{L(\hat{\theta}_{mle} | X)}is sufficiently small.

 

  • Union Intersection Test: 

The basic idea of UIT is to break a complicated hypothesis into several simpler hypothesis. For better understanding, let’s consider the previous example. We have x_1,x_2, \cdots \cdots, x_n \sim \mathcal{N}(\mu,\sigma^2) and we want to test:

H_0\,\,\,\, : \,\,\,\,\,\,\,\, \mu =0     H_1\,\,\,\, : \,\,\,\,\,\,\,\, \mu \neq 0

Now, let’s fix some new \mu' \neq 0. Lets consider the following test:

H_{0\mu'} \,\,\,\,\,\, : \,\,\,\,\,\,\,\,\,\,\,\, \mu = 0
H_{1\mu'} \,\,\,\,\,\, : \,\,\,\,\,\,\,\,\,\,\,\, \mu = \mu'

Of course it easier than the previous test as we don’t need to find the MLE as sometimes it is really hard ! We can test it just by comparing the densities i.e.

\frac{f(X|\mu')}{f(X|0)}

If it is large we can reject H_{0\mu'}  ( We will also show in subsequent lectures why this intuitive approach is best in some sense! ). Now suppose we reject   H_{0\mu'}. Then we can clearly conclude that we should reject  H_0 because, as it is beaten under the alternative  \mu = \mu', it will surely be beaten by the alternative  \mu \neq 0 !  So, calm down and try to understand that, if we reject  H_{0\mu'}   for at least one \mu' \neq 0 then we can reject H_0 ! So, ti accept H_0, we must accept H_{0\mu'} for all \mu' \neq 0.  So, the rejection region of H_0 is nothing but the union of the rejection region of all the tests of the form H_{0\mu'} for \mu' \neq 0 and consequently the acceptance region is the intersection of the acceptance region of all the test of these form, which motivates the naming! So here the advantage is, we can conclude about the test

H_{0} \,\,\,\,\,\, : \,\,\,\,\,\,\,\,\,\,\,\, \mu =0

H_{1} \,\,\,\,\,\, : \,\,\,\,\,\,\,\,\,\,\,\, \mu \neq 0using some simple tests described above.

 

Now, as for our blood pressure example we can have x_i  as the pressure difference of i^{th} patient before and after applying the drug. If we assume that the data follows Normal distribution with mean \mu (unknown) and variance say 1, then our H_0 will be \mu =0 as it the null hypothesis, i.e. no changes occurs on an average. The correct alternative or research hypothesis would be  \mu >0 as we want the blood pressure to be reduced. Using the above mentioned methods, one can test to find out which one is true !

 

In the next lecture we will more formally (rather mathematically ) introduce the hypothesis testing problem. We will also try evaluate the testing procedures using the “confidence” and “errors” of tests in mathematical way and try to figure which approach is better and why.

 

 

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