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Hypothesis testing I


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Hypothesis testing
A method of deciding whether empirical data contradicts an assumed statistical behaviour or not.
Confirmatory data analysis
An alternative term for statistical hypothesis testing, possibly slightly more general.
Test statistic
A function that assigns a small number of numerical values to sample data such that hypothesis testing can be applied.
Null hypothesis (H_0)
The statement which is to be proven wrong, usually those distributions in the model which correspond to a default position or pure chance.
Alternative hypothesis (H_1)
The statement which is complementary to the null hypothesis.
(Statistical) test
A function which assigns to each possible outcome of a random experiment a decision for the null or alternative hypothesis.
Statistically significant
The adjective characterizing a result that is unlikely to have occurred by chance alone in a hypothesis test.
Critical region / Region of rejection
The set of those samples for which the null hypothesis is to be rejected.
"Region of acceptance"
The set of those samples for which the null hypothesis is _not_ to be rejected.
Critical value
A threshold t such that the null hypothesis is to be rejected for all samples greater or equal to t (given that the sample set is ordered).
Type I error / False positive error
The error to incorrectly reject the null hypothesis, i.e. to misinterpret a random result as evidence for an assumed theory or effect.
Type II error / False negative error
The error to incorrectly accept the null hypothesis, i.e. to misinterpret a result supporting the assumed theory as pure chance.
Simple hypothesis
A hypothesis consisting of a single distribution.
Composite hypothesis
A hypothesis consisting of a family of distributions with at least two members.
Size / Significance level (α)
An upper bound for the rejection probabilities with respect to all distributions belonging to the null hypothesis.
Specificity
A term in biostatistics for the complement 1-α of the significance level α.
False negative rate (β)
The probability of a type II error with respect to a specific distribution of the alternative hypothesis.
Power
The rejection probability with respect to a specific distribution of the alternative hypothesis.
Most powerful test (of size α)
A test with the lowest probability of a type II error among all tests of a fixed size and a given simple alternative hypothesis.
Uniformly most powerful test
A test with composite alternative H_1 such that its critical region defines a most powerful test of size α against every single alternative in H_1.
p-value
The lowest size that would lead to a null hypothesis rejection for a _specific data sample_.
One-tailed test
A hypothesis test in which the alternative hypothesis is defined by a half-bounded interval.
Two-tailed test
A hypothesis test in which the alternative hypothesis is defined by a bounded interval.
Paired (two-sample) test
A two-sample test reduced to a one-sample test by subtracting each value of one sample from an associated value of the other sample.
Exact test
A test in which critical values can be computed without any approximation involved.
Conservative test
A test defined in such a way that the false positive rate is not greater than the size _for any possible sample_.
Non-parametric test
A test which is independent from special distribution assumptions.