Tests of Hypotheses
Probabilistic scrutiny of the claims made against established knowledge
Knowledge In Statistics
True knowledge in statistics is not some metaphysical discourse on epistemology rather it's a statement made about a parameter of the population that has a statistical basis.
Null Hypothesis (H0)
In statistics established facts about a population parameter are treated as Null Hypotheses. But what is the source of these facts or accepted parameters in the first place, well there are at least three:
- Firstly, from the result of some prior experimentation.
- Secondly, determined by a statistical model of the process under study.
- Thirdly, from external considerations, such as design or engineering specifications.
What Is Population?
In statistics, Population is used in a bit different sense than the ordinary usage of the word. It does not only refer to a group of individual animate objects only even though it may, but it refers to inanimate objects as well. To be precise Population refers to the entire set of similar entities which are of interest to a particular statistical experiment.
Alternative Hypothesis (H1)
An alternative hypothesis is a claim made by someone with the intention of rejecting the established knowledge about some specific parameter of the population or the Null Hypothesis. In order to scrutinize this new claim, statisticians developed a probabilistic examination known as hypothesis testing. It is this test that determines the fate of our established knowledge in statistics and essentially leads to new insights about the data at hand.
A Simple Example
Suppose from previous experiments or trials AstraZeneca has established that its vaccine is 80% effective in curing covid-19. But now some unknown entity challenges this claim by stating that the average effectiveness of the said vaccine is lesser than 80%. In this case, our Null hypothesis H0 becomes, H0:average(effectiveness) = 0.8 and the alternative hypothesis H1 becomes, H1:average(effectiveness)<0.8.
Type I and Type II errors
There are three inevitable scenarios that arise when dealing with the null hypothesis. For ease of understanding we can depict this using a confusion matrix shown below:
I think the confusion matrix pretty much sums up the type of errors or simply put the type of mistakes one can be guilty of while trying to form an opinion about the Null Hypothesis. Let me make it even simpler, what can basically go wrong here is that you can either reject something that was true in the first place or fail to reject something that was actually false. As simple as that, now go and take a look at the confusion matrix again!
Significance Levels and P-values
In hypothesis testing what we attempt to do is obtain a sample from the population of interest find a test statistic like an average or a variance or a proportion of the sample and check the probability of the occurrence of this test statistic given that the Null Hypothesis H0 is true. The probability of this test statistic is what is called a p-value in short. This p-value is then checked against a significance level that is predetermined, in case it is lesser than the significance level then we claim to have successfully rejected the Null hypothesis, and for vice-versa we claim to have failed to reject the Null hypothesis.
What we are essentially doing is checking whether our test statistic occurred purely out of chance or not. And it has occurred purely out of chance if and only if the probability of its occurrence exceeded the significance level. In case it occurred with a probability that was very small and below a significant level then evidence for alternative hypothesis gains ground. This is because after all if the test statistic had a very minuscule chance of occurring then why did it occur at all?
Types of hypothesis testing
If we are hell-bent on proving the null hypothesis wrong then there are three obvious possibilities based on what our alternative hypothesis is. These three cases are discussed below:
- Right tail: If the alternative hypothesis claims that the parameter from the Null hypothesis is greater than that which is stated in the Null Hypothesis then we are doing a right tail test. The right tail here refers to the tail of the standard distributions like Normal, t, or chi-squared distribution.
- Left tail: In this case, the Alternative Hypothesis ascertains that the parameter is lesser than that which is stated by the Null hypothesis.
- Two tail: If the parameter is simply assumed to be not equal to that which is stated by the Null hypothesis, we go for a two-tail test
Test Statistic
Test Statistic is something that we encountered before. The calculation of a test statistic is solely dependent on the population parameter in question.
The test statistic is a value used in making a decision about the null hypothesis and is found by converting the sample statistic to a score with the assumption that the null hypothesis is true. There are three population parameters and three corresponding test statistics we consider in general:
- Test Statistic for Mean
- Test Statistic for Variance
- Test Statistic for Proportion
Conclusion
So whether you are questioning a stock’s average return or your country’s rate of unemployment or any other statistical parameter for that matter, hypothesis testing is a tool that will help you frame a counter hypothesis and form probabilistic evidence to establish your new claim or falsify an existing one.
