Deductive and Inductive Reasoning
In mathematics, we usually apply deductive rules to prove theorems. The deductive reasoning starts from a general truth (rule) and deduce a specific statement. Since general statement leads to specific statements, there is no chance of having uncertainty in the status of specific statement, because status of specific statement totally depends on the general statements. Examples :
(1)- An angle A is acute, if 0 < A < 90; B = 30; then B is acute.
(2)- If it rains, then there are clouds in the sky. If we reason that there are no clouds in the sky that means there is no rain. This is reasoning by making first statement into the contra-positive statement. Abstractly if statement A implies B is true then complement of B implies complement of A will also be true.
(3)- A implies B, and B implies C, then A implies C.
All the above three are deductive reasoning. There is no concept of uncertainty in deductive reasoning because we are going from general to specific, also called top-down approach.
Inductive Reasoning
On the contrary, inductive reasoning means reasoning from specific observations to broader generalizations and theories. Testing of Hypothesis based on drawn sample from a population (target universe of sampling process) is an inductive statistics science. Similarly estimating a parameter of population based on sample observations is also a science of inductive reasoning. Aim is to know about unknown universe, after observing a part of it. Uncertainty plays a part here. This uncertainty is measured by level of significance (α) , or confidence level (1-α). The confidence interval is also calculated within which the parameter of universe is thought to lie inside it with certain probability (1-α).
Null hypothesis H0 and Alternative hypothesis H1
In inductive reasoning, if we decide something about universe say by two competing statements - the H0 against an alternative statement H1, then there are chances that we can commit either of two errors. Here H0 and H1 are mutually exclusive statements about the Universe. Mutually exclusive means that they are non-overlapping. Occurence of H0 precludes Occurence of H1 and vice-versa. H0 and H1 may be exhaustive or may not be. Exhaustive means nothing else can occur about universe beyond the union of H0 and H1 or in other words,there is no other situation(s) available other than H0 or H1 for universe under consideration.
H1 is the statement to which we may be concerned most. H1 is an inequality statement, like medicine A is more effective than medicine B. It is termed as an alternative hypothesis. We choose a competing statement H0, such that if it gets rejected then the alternative hypothesis H1 will get supported for acceptance. Such H0 is called null hypothesis. Null hypothesis is to be chosen meticulously. Specifically, the null hypothesis involves the absence of a difference or the absence of an association between or amongst the two or more than two indicators. Under the null hypothesis, a distribution of test statistic should be available. The null hypothesis can never be that, where there is a difference available or there are association. Thus H0 : p = 0.05, H0: p1 = p2, H0: p1-p2 =0, H0 ρ = 0.01, or H0 ρ1 = ρ2 are valid examples of null hypothesis. Whereas H0 : p ≥ 0.05, H0: p1 ≠ p2, H0: p1-p2 ≠0, H0 ρ ≤ 0.01, or H0 ρ1 ≠ ρ2 are invalid example of null hypothesis. These inequalities conjectures and associative conjectures can very well be example of alternative hypothesis.
Some other time, we might be concerned most about H0, which is a point hypothesis, like average height of adults μ in a city is μ0. In that case, we find out a competing hypothesis H1, as H1: μ ≠ μ0
The errors, in the decision which we take is not correct, can be of two types.
Type I Error
One is that we have not chosen decision H0 but actually H0 is true. This is type one error. This is an Event (reject H0 । H0 true) means accepting the “not H0", but fact is H0 is true or “Rejecting H0 when H0 is true.” This may also be called false rejection. It is measured probabilistically by α.
Alternatively, it is a mistaken rejection of null hypothesis. It is also called as false positive because when we test for some disease, we take H0 as person of normal health. When it happens that the test points out positive to the disease, but actually that disease is not present in the person, the situation is false positive. Thus, though test asserts health of person as diseased person, yet person is not actually suffering from the disease is an example of rejecting H0 ( person is healthy) while it is true. This is a type 1 error.
Intuitively, type I errors can be thought of as errors of commission, i.e. the experimenter concludes that a statement is really a fact. For instance, consider a study where he compares a drug with a placebo. H0 would be that drug and placebo has no difference in their effects or their effects are same. If the patients who are given the drug get better than the patients given the placebo by chance, it may appear that the drug is effective. Researcher based on data will reject H0. And if the fact is that there is no difference in the effects of drug and placebo, this conclusion is incorrect and it will be errors of commission. Appearance of drug behaviour had made the experimenter to commit such an error.
Type II Error
On the other hand, we chose the H0 to be correct, but actually it is not correct or we can say H1 is correct. Accepting H0 when H0 is false is mistaken acceptance of H0. This is an Event ( accept H0 । H0 false) or in words accepting H0 when H0 is not true. This is false acceptance. This is type 2 error. Also termed as false negative. It is measured probabilistically by β.
Alternatively, it is a mistaken acceptance of null hypothesis. It is also called as false negative because when we test for some disease, we take H0 as person of normal health. When it happens that the test points out negative to the disease, but actually that disease is present in the person, the situation is false negative. Thus, though test negates health of person as diseased person, but person is actually suffering from the disease is an example of accepting H0 ( person is healthy) while it is not true. This is a type II error.
Intuitively, type II errors can be thought of as errors of omission, i.e. the experimenter concludes that a statement is really not a fact. For instance, consider a study where he compares a drug with a placebo. H0 would be that drug and placebo has no difference in their effects or their effects are same. If the patients who are given the drug did not get better more often than the patients given the placebo, it may appear that the drug is not effective. Researcher based on data will accept H0. And if the fact is that there is effect of drug with respect to placebo, this conclusion is incorrect and it will be errors of omission. Appearance of drug behaviour had made the experimenter to come to such conclusion, where he perhaps omitted the size and direction of the missed determination and the circumstances. The type II errors are random fluke and depends on size and direction of missed determination / observations and the variabilities of circumstances.
Probability of Type I and Type II errors
Probability of type one error is called alpha (α),while probability of type 2 error is called beta (β).
Pr ( wrongly rejecting null hypothesis or errors of commission or mistaken rejection or false positive) = α
Pr ( wrongly retaining null hypothesis or errors of omissions or mistaken acceptance or false negative) = β
