Type I and II errors There are two kinds of errors that
can be made in significance testing: (1) a true null hypothesis can be incorrectly rejected and (2)
a false null hypothesis can fail to be rejected. The former error is called a
Type I error and the latter error is called a Type II error. These two types of
errors are defined in the table.
Statistical
Decision
True
State of the Null Hypothesis
H
True
H
False
Reject
H
Type
I error
Correct
Do
not Reject H
Correct
Type
II error
The probability of a Type I error is designated by the Greek
letter alpha (α) and is called the Type I error rate; the probability of a
Type II error (the Type II error rate) is designated by the Greek letter beta
(ß) . A Type II error is only an error in the sense that an opportunity to
reject the null hypothesis correctly was lost. It is not an error in the sense
that an incorrect conclusion was drawn since no conclusion is drawn when the null hypothesis is not rejected. A Type I error, on the
other hand, is an error in every sense of the word. A conclusion is drawn that
the null hypothesis is false when, in fact, it is true. Therefore, Type I
errors are generally considered more serious than Type II errors. The
probability of a Type I error (α) is called the significance level and is set by the experimenter. There is a tradeoff between
Type I and Type II errors. The more an experimenter protects himself or herself
against Type I errors by choosing a low level, the greater the chance of a Type
II error. Requiring very strong evidence to reject the null hypothesis makes it
very unlikely that a true null hypothesis will be rejected. However, it
increases the chance that a false null hypothesis will not be rejected, thus
lowering power. The Type I error rate is almost
always set at .05 or at .01, the latter being more conservative since it
requires stronger evidence to reject the null hypothesis at the .01 level then
at the .05 level. A type I error occurs when one rejects the
null hypothesis when it is true. The probability of a type I error is the level
of significance of the test of hypothesis, and is denoted by *alpha*. Usually a one-tailed test hypothesis is
is used when one talks about type I error. Examples: If
the cholesterol level of healthy men is normally distributed with a mean of 180
and a standard deviation of 20, and men with cholesterol levels over 225 are
diagnosed as not healthy, what is the probability of a type one error? z=(225-180)/20=2.25; the corresponding tail
area is .0122, which is the probability of a type I error. If
the cholesterol level of healthy men is normally distributed with a mean of 180
and a standard deviation of 20, at what level (in excess of 180) should men be
diagnosed as not healthy if you want the probability of a type one error to be
2%? 2% in the tail corresponds to a
z-score of 2.05; 2.05 × 20 = 41; 180 + 41 = 221. A type II error occurs when one rejects the
alternative hypothesis (fails to reject the null hypothesis) when the
alternative hypothesis is true. The probability of a type II error is denoted by
*beta*. One cannot evaluate the probability of a type II error when the
alternative hypothesis is of the form µ > 180, but often the alternative
hypothesis is a competing hypothesis of the form: the mean of the alternative
population is 300 with a standard deviation of 30, in which case one can
calculate the probability of a type II error. Examples: If men predisposed to heart disease have a mean
cholesterol level of 300 with a standard deviation of 30, but only men with a
cholesterol level over 225 are diagnosed as predisposed to heart disease, what
is the probability of a type II error (the null hypothesis is that a person is
not predisposed to heart disease). z=(225-300)/30=-2.5
which corresponds to a tail area of .0062, which is the probability of a type
II error (*beta*). If men predisposed to heart disease have a mean
cholesterol level of 300 with a standard deviation of 30, above what
cholesterol level should you diagnose men as predisposed to heart disease if
you want the probability of a type II error to be 1%? (The null hypothesis is
that a person is not predisposed to heart disease.) 1% in the tail corresponds to a z-score of
2.33 (or -2.33); -2.33 × 30 = -70; 300 - 70 = 230. |
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