Medical Statistics Medical Statistics Tao Yuchun Tao Yuchun 6 Statistical inference Statistical inference 2. Hypothesis testing ( t test ) The idea and steps of Hypothesis testing (1) The idea of Hypothesis testing See Example 6-1:See Example 6-1: For most healthy male adults, their pulse is 72 times/min averagely. A doctor randomly drew 25 healthy male adults from a mountainous area, measured everyones pulse, got sample mean and SD: mean=74.2 times/min, SD=6.0 times /min. QuestionWhether the population mean ofQuestion: Whether the population mean of pulse for healthy male adults in this region is pulse for healthy male adults in this region is different from 72times/min ? different from 72times/min ? In this example the population mean of pulse for the healthy mountainous male adults denotes m for the general denotes 0 0 =72 times/min General Pop. m= m= Mountain Pop. n=25 The relation between m and 0 has only two possibilities : m 0 m0 m0 m0m0 m0m0 For reasons sampling error m 0 same sampling error essential difference m 0 not same essential difference Question : Which is the truth? -- problem of hypothesis testing !Question : Which is the truth? -- problem of hypothesis testing ! Basic logic:Basic logic: set two hypotheses First, set two hypotheses for two possibilities. Null hypothesis: Null hypothesis : H 0 : m 0 Alternative hypothesis : Alternative hypothesis : H 1 : m 0 Second, How possible to occur the current situation and even more unfavorable situation Calculate a to H 0 under the null hypothesis ? Calculate a probability (P-value) probability (P-value). Last, If it is less possible to occur the current situation and even more unfavorable situation to H 0, then reject H 0 ; otherwise, not reject H 0. Given a small , compare P and , then draw the conclusion the conclusion. ( is called the level of a test) (2) The steps of Hypothesis testing I.Set hypotheses and the level of a test For Example 6-1: H 0 : m 0 H 1 : m > 0 One side = 0.05 Question: Why is one side for this exp. ? What is a one side test or two sides test?Question: Why is one side for this exp. ? What is a one side test or two sides test? one-sided test: For Alternative hypothesis H 1, only one only one probably exists from two possibilities. For Exp. 6-1: H 1 : m > 0 or H 1 : m < 0 two-sided test: For Alternative hypothesis H 1, any one any one probably exists from two possibilities. For Exp. 6-1: H 1 : m 0 You must choose it according to professional before knowledge before a test ! Question: Why did we choose this oneQuestion: Why did we choose this one () not another one () from ( m > 0 ) not another one ( m < 0 ) from two possibilities for this example ? two possibilities for this example ? According to medical professional knowledge, anyones pulse in a mountainous area is not possible less than it in general area. We may eliminate one possibility ( m < 0 ) from H 1. The final decision will be more certain, either ( m > 0 ) or ( m = 0 ) ! Be careful and considerate choosing one-sided test ! It is safe and stable choosing two-sided test anytime. is the level of a test, it is often chosen as the a small-probability event standard of a small-probability event, is also the type I error probability for type I error (relate later) a small-probability event: indicates the probability of an event that will happen to be very small, if this thinknot kind of event appears, we think it will not occur ! Its standard often chooses P0.05 or P0.01. is a limit to infer H 0 to be rejected or not to be rejected in hypothesis testing. II.Select an appropriate test and calculate the test statistic the test statistic According to the type of data, methods of design, special suitable conditions, etc, you may select a proper test that has existed, and calculate the value of the test statistic for this test from sample data. Question: Why must we choose a testQuestion: Why must we choose a test and calculate the test statistic ? and calculate the test statistic ? Because it can give me the P-value for this sample data ! Question: What is the P-value ? Is itQuestion: What is the P-value ? Is it really important ? really important ? I will explain it in next step. Yes, it is very important ! Because we can draw a conclusion just by it. For Example 6-1: We have known: If X ~, Then Pulse value ~, then When H 0 : m 0 72 times/min holds, Based on the current sample, you can get: Question: Where is the P-value ?Question: Where is the P-value ? See next step. III. Determine P-value, and make decision P-value is the area of the tail(s) in the distribution of the test statistic beyond the value(s) of the test statistic calculated based on the sample. See figure below. t = =24 P H 0 : m 0 =72 Two methods: You can get exact the P-value by statistical software; You can contrast the P-value with a frame of reference (i.e. the level of a test). For Example 6-1: When H 0 holds, the probability of the current situation (sample mean=72) and even more unfavorable situation (sample mean>72) to H 0 is the P-value. Question: How much is the P-value ?Question: How much is the P-value ? Question: How do I draw a conclusionQuestion: How do I draw a conclusion by the P-value ? by the P-value ? You will animatedly see how to get the P-value for this example from a animation. click me If P then reject H 0 at significance level If P then reject H 0 at significance level =0.05. =0.05. If P then not reject H 0 at significance If P then not reject H 0 at significance level =0.05. level =0.05. The P-value also indicates the probability of H 0 holding. Question: Why is H 0 rejected if P ?Question: Why is H 0 rejected if P ? H 0 Because the probability of H 0 holding (i.e. P- small-probability event value) is a small-probability event ! We may thinknot H 0 think it will not occur, so we rejected H 0. H 0 H 0 Otherwise, we cant reject H 0 (accepted H 0 ), because the P-value has exceeded the limit of frame of reference (i.e. the level of this test, =0.05 =0.05 ). The conclusion of Hypothesis testing The conclusion of Hypothesis testing bases on the size of the probability ! bases on the size of the probability ! For Example 6-1: The conclusion is: = 24 Checked one side t , = t 0.05,24 =1.711 now t = then P 0.05 reject H 0 at significance level=0.05. We may consider the population mean of pulse for the mountainous healthy male adults to be higher than the generals. The entire process of Hypothesis testing for Example 6-1 is: H 0 : m 0 H 1 : m > 0 One side = 0.05 = 24 Checked one side t , = t 0.05,24 =1.711 now t = then P 0.05 reject H 0 at significance level=0.05. We may consider the population mean of pulse for the mountainous healthy male adults to be higher than the generals. t tests (1) Comparing to a given population mean ( One-sample t test) Example 6-1 belongs to this kind of t test. Example 6-2: Example 6-2: For most healthy male adults, their blood sugar is 4.70mmol/L averagely. A researcher randomly drew 26 healthy male adults from a manager population, measured everyones blood sugar, got sample mean and Please check SD: mean= 4.84mmol/L, SD=0.85mmol/L. Please check whether the of blood sugar for the managers is 4.70 whether the of blood sugar for the managers is 4.70 ? H 0 : 0 = 4.70mmol/L H 1 : 0 = 4.70mmol/L = 0.05 = 25 Checked two sides t , = t 0.05,25 =2.060 now t = then P 0.05 no reason to reject H 0 at significance level=0.05. We may consider the of blood sugar for the managers to be same as the generals. C You will animatedly see how to get the P-value for this example from a animation. click me (http://en.wikipedia.org/wiki/Forbidden_City) =24 H 0 : m 0 One side: t , = t 0.05,24 = Accepted area m0m0 Rejected area m0m0 t = P =0.05 P 0.05 P 0.05 =25 H 0 : 0 =0.05 t /2, = t 0.05/2,25 = Accepted area 00 Rejected area 0 Rejected area 00 t = 0.840t = P P 0.05