Download Statistical hypothesis testing – Inferential statistics I.

Statistical hypothesis testing –
Inferential statistics I.
What is hypothesis testing?
• Hypothesis: a theoretical statement concerning a certain feature
of the studied statistical population.
We want to know if our hypotheses are true or not by doing
research.
• Hypothesis testing (or significance test): a procedure of
assessing whether sample data is consistent with statements
(hypotheses) made about the statistical population.
Briefly, we make a decision about the hypothesis on the basis of our
sample data.
We want to get answers to questions starting typically like these:
– „Is there a difference between…”
– „Is there a relationship between…”
• Types of hypotheses:
– There are two kinds of hypothesis:
• H1: the statement we actually want to test; usually
postulates a non-zero difference or relationship
(called ‘alternative hypothesis’)
E.g: „The mean weight of males and females are different.”
• H0: a statement which usually claims a zero
difference or relationship against the H1
(called ‘null hypothesis’).
E.g: „The mean weight of males and females are not different.”
• Test statistic:
– It is a numerical value calculated from our sample which
forms a link between our sample and the null hypothesis.
• Null distribution:
– The probability distribution of a test statistic when the
null hypothesis is true.
– Null distribution of the test statistic is known by e.g.
statistical computer programs.
• p-value:
– This is a probability indicating how likely to get a
sample with such a test statistic like ours or with a
more extreme one provided that the H0 is true.
– p-value comes from the null distribution by contrasting
the value of our test statistic with the null distribution.
– The smaller the p-value the more unlikely the null
hypothesis is true.
• Significance level (α alpha):
– It is an arbitrarily and a priori declared probability
threshold.
– If the p-value of the hypothesis test is less than or
equals to alpha, then it is agreed that the null
hypothesis will be rejected.
– The value of alpha in the most biological research is
0.05.
• Principle of hypothesis testing:
– We have a link between the sample and the null
hypothesis, this is the test statistic.
– We know the probability distribution of the test
statistic when the null hypothesis is true.
– Contrasting our test statistic with the null distribution
we will get a probability showing how typical this
value of the test statistic of the null distribution.
– If the probability we got is less than a threshold
declared in advance, we will reject the null hypothesis
and accept the alternative hypothesis, otherwise we
accept the null hypothesis.
Errors in hypothesis testing
• Type I error:
– we reject H0 although that is true.
– Denoted by α. Occurs only when H0 is true.
– Pr(type I error) = p-value
• Type II error:
– we accept H0, although that is false.
– Denoted by β. Occurs only when H0 is false.
One- and two-tailed tests
(or One- and two-sided tests)
• Two-tailed tests: a test in which H0 can be rejected by
large deviations from expected in either direction.
E.g:
H0: the two population means are equal: μ1 = μ2
This can be rejected if either population has a greater mean than the
other.
• One-tailed test: a test in which H0 is tested in a more
specific way, it can be rejected by deviation only in one
direction.
E.g:
H0: the mean of population 1 is greater or equal to the mean of
population 2: μ1 >= μ2
It would be rejected only if the mean of population 1 was significantly
less than that of population 2.
Steps of hypothesis testing
1. Formulate the hypotheses of the test (H0
and H1).
2. Collect data (i.e. take a random sample).
3. Declare your significance level (alpha).
4. Compute your test statistic and p-value.
5. Make a decision on the H0.
Assumptions of statistical tests
• Most of the statistical tests have clear
assumptions on the data.
• If these assumptions are not met the test
can not be done, because it will give an
incorrect result.
• In this case you have to try an other test
that is appropriate for your study design.
• To get detailed knowledge on the concrete
assumptions of the a certain test, consult a
statistical text book.