Question

Why would the researcher want to test multiple groups multiple times. Explain. Share your example

Answer 1

Some things show the behaviour over time and these processes are called time series processes. In time series a researcher collects data over a fixed time period basis and then tries to capture patterns in data.

Multiple comparisons:

In statistics, the multiple comparisons, multiplicity or multiple testing problem occurs when one considers a set of statistical inferences simultaneously or infers a subset of parameters selected based on the observed values. In certain fields it is known as the look-elsewhere effect.

The more inferences are made, the more likely erroneous inferences are to occur. Several statistical techniques have been developed to prevent this from happening, allowing significance levels for single and multiple comparisons to be directly compared. These techniques generally require a stricter significance threshold for individual comparisons, so as to compensate for the number of inferences being made.

Multiple comparisons arise when a statistical analysis involves multiple simultaneous statistical tests, each of which has a potential to produce a "discovery." A stated confidence level generally applies only to each test considered individually, but often it is desirable to have a confidence level for the whole family of simultaneous tests.^[4] Failure to compensate for multiple comparisons can have important real-world consequences, as illustrated by the following examples:

• Suppose the treatment is a new way of teaching writing to students, and the control is the standard way of teaching writing. Students in the two groups can be compared in terms of grammar, spelling, organization, content, and so on. As more attributes are compared, it becomes increasingly likely that the treatment and control groups will appear to differ on at least one attribute due to random sampling error alone.
• Suppose we consider the efficacy of a drug in terms of the reduction of any one of a number of disease symptoms. As more symptoms are considered, it becomes increasingly likely that the drug will appear to be an improvement over existing drugs in terms of at least one symptom.

In both examples, as the number of comparisons increases, it becomes more likely that the groups being compared will appear to differ in terms of at least one attribute. Our confidence that a result will generalize to independent data should generally be weaker if it is observed as part of an analysis that involves multiple comparisons, rather than an analysis that involves only a single comparison.

For example, if one test is performed at the 5% level and the corresponding null hypothesis is true, there is only a 5% chance of incorrectly rejecting the null hypothesis. However, if 100 tests are conducted and all corresponding null hypotheses are true, the expected number of incorrect rejections (also known as false positives or Type I errors) is 5. If the tests are statistically independent from each other, the probability of at least one incorrect rejection is 99.4%.

The multiple comparisons problem also applies to confidence intervals. A single confidence interval with a 95% coverage probability level will contain the population parameter in 95% of experiments. However, if one considers 100 confidence intervals simultaneously, each with 95% coverage probability, the expected number of non-covering intervals is 5. If the intervals are statistically independent from each other, the probability that at least one interval does not contain the population parameter is 99.4%.

Techniques have been developed to prevent the inflation of false positive rates and non-coverage rates that occur with multiple statistical tests

Classification of multiple hypothesis tests :

The following table defines the possible outcomes when testing multiple null hypotheses. Suppose we have a number m of null hypotheses, denoted by: H₁, H₂, ..., H_m. Using a statistical test, we reject the null hypothesis if the test is declared significant. We do not reject the null hypothesis if the test is non-significant. Summing each type of outcome over all H_i yields the following random variables:

	Null hypothesis is true (H0)	Alternative hypothesis is true (HA)	Total
Test is declared significant	V	S	R
Test is declared non-significant	U	T	{\displaystyle m-R}
Total	{\displaystyle m_{0}}	{\displaystyle m-m_{0}}	m

• m is the total number hypotheses tested
• {\displaystyle m_{0}} is the number of true null hypotheses, an unknown parameter
• {\displaystyle m-m_{0}} is the number of true alternative hypotheses
• V is the number of false positives (Type I error) (also called "false discoveries")
• S is the number of true positives (also called "true discoveries")
• T is the number of false negatives (Type II error)
• U is the number of true negatives
• {\displaystyle R=V+S} is the number of rejected null hypotheses (also called "discoveries", either true or false)

In m hypothesis tests of which {\displaystyle m_{0}} are true null hypotheses, R is an observable random variable, and S, T, U, and V are unobservable random variables.