Docstoc

Type I and II error

Document Sample
Type I and II error Powered By Docstoc
					Type I and II error                                                                         http://cs.uni.edu/~campbell/stat/inf5.html




           Type I and II error
                  Type I error
                  Type II error
                  Conditional versus absolute probabilities
                  Remarks

           Type I error
           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 of 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.

           Type II error
           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.

           Conditional and absolute probabilities
           It is useful to distinguish between the probability that a healthy person is dignosed as diseased, and the
           probability that a person is healthy and diagnosed as diseased. The former may be rephrased as given
           that a person is healthy, the probability that he is diagnosed as diseased; or the probability that a person
           is diseased, conditioned on that he is healthy. The latter refers to the probability that a randomly chosen
           person is both healthy and diagnosed as diseased. Probabilities of type I and II error refer to the
           conditional probabilities. A technique for solving Bayes rule problems may be useful in this context.
           Examples:
           If the cholesterol level of healthy men is normally distributed with a mean of 180 and a standard
           deviation of 20, but men predisposed to heart disease have a mean cholesterol level of 300 with a
           standard deviation of 30, and the cholesterol level 225 is used to demarcate healthy from prediposed
           men; what fration of the population are healthy and diagnosed as predisposed? what fraction of the


1 of 3                                                                                                              15/4/08 9:33 PM
Type I and II error                                                                       http://cs.uni.edu/~campbell/stat/inf5.html


           population are predisposed and diagnosed as healthy? Assume 90% of the population are healthy (hence
           10% predisposed).
           Let A designate healthy, B designate predisposed, C designate cholesterol level below 225, D designate
           cholesterol level above 225. P(D|A) = .0122, the probability of a type I error calculated above. Hence
           P(AD)=P(D|A)P(A)=.0122 × .9 = .0110. P(C|B) = .0062, the probability of a type II error calculated
           above. Hence P(CD)=P(C|B)P(B)=.0062 × .1 = .00062.

           A problem requiring Bayes rule or the technique referenced above, is what is the probability that
           someone with a cholesterol level over 225 is predisposed to heart disease, i.e., P(B|D)=? This is
           P(BD)/P(D) by the definition of conditional probability. P(BD)=P(D|B)P(B). For P(D|B) we calculate
           the z-score (225-300)/30 = -2.5, the relevant tail area is .9938 for the heavier people; .9938 × .1 =
           .09938. P(D) = P(AD) + P(BD) = .0122 + .09938 = .11158 (the summands were calculated above).
           Inserting this into the definition of conditional probability we have .09938/.11158 = .89066 = P(B|D).

           Remarks
           If there is a diagnostic value demarcating the choice of two means, moving it to decrease type I error will
           increase type II error (and vice-versa).




           The power of a test is (1-*beta*), the probability of choosing the alternative hypothesis when the
           alternative hypothesis is correct.

           The effect of changing a diagnostic cutoff can be simulated.

           Applets: An applet by R. Todd Ogden also illustrates the relative magnitudes of type I and II error (and
           can be used to contrast one versus two tailed tests). [To interpret with our discussion of type I and II
           error, use n=1 and a one tailed test; alpha is shaded in red and beta is the unshaded portion of the blue
           curve. Because the applet uses the z-score rather than the raw data, it may be confusing to you. The
           allignment is also off a little.]

           Competencies: Assume that the weights of genuine coins are normally distributed with a mean of 480
           grains and a standard deviation of 5 grains, and the weights of counterfeit coins are normally distributed
           with a mean of 465 grains and a standard d eviation of 7 grains. Assume also that 90% of coins are
           genuine, hence 10% are counterfeit.
           What is the probability that a randomly chosen genuine coin weighs more than 475 grains?
           What is the probability that a randomly chosen counterfeit coin weighs more than 475 grains?
           What is the probability that a randomly chosen coin weighs more than 475 grains and is genuine?
           What is the probability that a randomly chosen coin weighs more than 475 grains and is counterfeit?
           What is the probability that a randomly chosen coin which weighs more than 475 grains is genuine?


2 of 3                                                                                                            15/4/08 9:33 PM
Type I and II error                                                                   http://cs.uni.edu/~campbell/stat/inf5.html


           Reflection: How can one address the problem of minimizing total error (Type I and Type II together)?

           return to index

           Questions?




3 of 3                                                                                                        15/4/08 9:33 PM

				
DOCUMENT INFO
Shared By:
Categories:
Tags: Type, error
Stats:
views:153
posted:3/28/2010
language:Swedish
pages:3
Description: Type I and II error