Medical statistics part I, autumn 2010: One sample test of hypothesis

Transcription

1 Medical statistics part I, autumn 2010: One sample test of hypothesis Eirik Skogvoll Consultant/ Professor Faculty of Medicine Dept. of Anaesthesiology and Emergency Medicine 1

2 What is a hypothesis test? You have a hypothesis Observe something related to the hypothesis Decide ( test ) whether the hypothesis is correct or not 2

3 What is a hypothesis? A statement about one or more population parameters, e.g. μ 3 (the expected value) but also μ 1 = μ 2 σ 2 = 14 π = 0.4 π 1 = π 0 ψ = 1 β 1 = 0 ρ > 0 (two expected values) (the variance) (the probability in a binomial trial) (two probabilities) (odds ratio in a contingency table) (regression coefficient) (correlation coefficient) 3

4 Rosner tbl. 7.1 P (type II error) Decision Truth H 0 true H 0 false Accept H 0 1 α β Reject H 0 α 1 β P (type I error) Power Type I error: reject H 0 when H 0 is true Type II error: accept H 0 when H 0 is false 4

5 - general procedure State H 0 and H 1 (one- or two-sided) 5

6 - general procedure State H 0 and H 1 (one- or two-sided) Decide level of significance α = P (type I error) = P (reject H 0 H 0 true) 6

7 - general procedure State H 0 and H 1 (one- or two-sided) Decide level of significance α = P (type I error) = P (reject H 0 H 0 true) Find a test statistic whose probability distribution under H 0 is known 7

8 - general procedure State H 0 and H 1 (one- or two-sided) Decide level of significance α = P (type I error) = P (reject H 0 H 0 true) Find a test statistic whose probability distribution under H 0 is known Decide the rejection region of the test statistic, given H 0 and α 8

9 - general procedure State H 0 and H 1 (one- or two-sided) Decide level of significance α = P (type I error) = P (reject H 0 H 0 true) Find a test statistic whose probability distribution under H 0 is known Decide the rejection region of the test statistic, given H 0 and α Perform the experiment, and draw your conclusion based on the observed test statistic 9

10 - example Rosner expl. 7.2, p. 227 Question: Do mothers with low socioeconomic background deliver babies whose birthweight are lower than normal (120 oz, 1 oz 28 g, i.e g)? Birth weight is assumed to be Normally distributed. 10

11 - example State H 0 and H 1 (one- or two-sided) H 0 : μ μ 0 = 3360 g vs. H 1 : μ < μ 0 = 3360 g 11

12 - example Decide level of significance α = P (type I error) = P (reject H 0 H 0 true) Set α = 0,05 We only wish to reject H 0 5 % of the time when it in fact is true. 12

13 - example Find a test statistic whose probability distribution under H 0 is known 2 Birthweight is assumed N ( μ = 3360 g, σ ) 2 But σ is unknown, so we choose X μ T as test statistic, T = ~ T = T s Under H ET ( ) = 0 0 n n

14 - example Decide the rejection region of the test statistic, given H 0 and α The T distribution is symmetrical about 0, and under H : ET ( ) = 0 99, Need to find t so that 99, 0.05 Pr ( T t ) = 0.05 (one sided test) From table 5 we find t , 0.95 Thus, reject H if t obs p = 0.05 t 99, 0.05 =

15 - example Perform the experiment, and draw your conclusion based on the observed test statistic We observed x = 115 ( = 3220 g), s= 24 (=672 g), thus X μ Tobs = = = = 2,08 s ,2 n 100 P( T ) < 0.05 Conclusion: We reject H and accept H ( μ < μ )

16 P-value: one tailed The p value is the probability of obtaining a test statistic (T n-1 ) under H 0 which is less than or equal to the observed value t: Graphically, the area to the left of t: Birth weight example: P (T n-1 H 0 t obs ) = P (T 99 H ) =

17 - one tailed vs. two tailed So far we have assumed that H 1 has some direction relative to H 0 If no such assumption exists, a two tailed test is required Sign of H 1 > < Critical region One area Right tail One tailed test Two areas One in each tail Two tailed test One area Left tail One tailed test t-distribution 17

18 - two tailed rejection region Rejection regions: beyond (more extreme than) the critical values As the t distribution is symmetrical about 0, we may use the absolute value of t instead: Reject H 0 if t obs > t n-1, 1- α/2 Reject H 0 if t obs t n-1, α/2 Reject H 0 if t obs > t n-1,1-α/2 18

19 P value: two-tailed test The p value is the sum of probabilities of obtaining a test statistic (T n-1 ) less than or equal to, OR larger than, the observed value t The T distribution is symmetric: 2 P (T n-1 > t obs ) Graphically, the sum of the areas to the left and to the right: 19

20 Some software output (based on similar (not identical) data as in Rosner expl. 7.2) SPSS/ PASW One Sample t-test t = , df = 99, p-value = alternative hypothesis: true mean is less than percent confidence interval: -Inf sample estimates: mean of x R One-sample t test Variable Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] v mean = mean(v1) t = Ho: mean = 3360 degrees of freedom = 99 STATA Ha: mean < 3360 Ha: mean!= 3360 Ha: mean > 3360 Pr(T < t) = Pr( T > t ) = Pr(T > t) =

21 - one tailed or two tailed? Altman, 1991: One-sided tests are rarely appropriate. Even when we have strong prior expectations, for example that a new treatment cannot be worse than an old one, we cannot be sure that we are right. If we could be sure we would not need to do an experiment! 21

22 - one tailed or two tailed? Zar, 2010: However, in many instances, our interest lies only in whether is signifinatly larger (or significantly smaller) than μ 0 as grounds for rejecting H 0. ( ) X For example, one might be testing a drug hypothezised to cause weight X reduction in humans. The investigator is interested only in whether a weight loss occurs after the drug is taken. If there is either weight gain or no weight change, the drug will be considered a failure. ( ) It is important to appreciate that the decision ( ) must be based on the scientific question being addressed, before data are collected. 22

23 - one tailed or two tailed? Rosner, 2000: Generally, a two-sided test is always appropriate, because there can be no question about the conclusions. ( ) However, in certain situations only alternatives on one side of the null mean are of interest or are possible, and in this case a one-sided test is better because it has more power. 23

24 - one tailed or two tailed? Armitage, Berry & Matthews, 2002: If, for some reason, we decided that we were interested in possible departures only in one specified direction, say that a new treatment was superior to an old treatment, it would be reasonable to count as significant only those samples that differed sufficiently from the null hypothesis in that direction. ( ) Before the data are examined, one should decide to use a one-sided test only if it is quite certain that departures in one direction will always be ascribed to chance, and therefore regarded as non-significant however large they are. 24

25 - example: two-tailed z-test Find a test statistic whose probability distribution under H 0 is known 2 Birthweight is assumed N ( μ = 3360 g, σ ) 2 Suppose σ is known; then we may use as test statistic, X E( X) X μ In general: Z = = Var( X ) σ X E( X) X μ Here: Z = = N(0,1) SE( X ) σ n Under H EZ ( ) = 0 0 Z 25

26 - example: two-tailed z-test Decide the rejection region of the test statistic, given H 0 and α The Z distribution (standard Normal) is symmetrical about 0, and under H, EZ ( ) = 0 Need to find z so that Pr (Z z ) = (two-tailed test) From table 3 we find z 1.96 Thus, reject H if z 0 obs > 1.96 tha is, if z 1.96 or z > 1.96 obs obs 26

27 - example: two-tailed z-test Perform the experiment, and draw your conclusion based on the observed test statistic We observed x = 115 ( = 3220 g), σ = 672 g, thus X μ Z = = = = 2.08 s n 100 As 2.08 < 1.96: We reject H and accept H ( μ μ ) P-value P( Z > 2.08 ) = 2 P( Z 2.08) = =

28 Example (nausea in pregnancy) Rosner Problem 7.23: Nausea affects 30 % of pregnant women between 24 and 28 weeks If 110 out of 200 women taking erythromycin complain of nausea, is this compatible with a prevalence of 0.3? H0 : p= p0 = 0.3 vs. H1: p p0 Observed X = {number of women with nausea} n = total number of pregnant women X ~ bin( n, p0) EX ( ) = n p0 = = 60 Var( X ) = np0(1 p0) = = 42 X E( X) Z = N(0,1) var( X ) 28

29 Example (nausea in pregnancy, cont d) X n p0 Z = np0(1 p0) 110 Obs. X = 110, n= 200, pˆ = = npˆ(1 pˆ) = 200 = = OK to use normal approximation Z = = 7.72 np (1 p ) PZ ( > 7.72 = 2 PZ ( > 7.72) = Conclusion: reject H 0 29

30 Example (nausea in pregnancy) Rosner Problem 7.23 (modified): Nausea affects 30 % of pregnant women between 24 and 28 weeks If 11 out of 20 women taking erythromycin complain of nausea, is this compatible with a prevalence of 0.3? H : p= p = 0.3 vs. H : p p Observed X = {number of women with nausea} n = total number of pregnant women X ~ bin( n, p ) 0 EX ( ) = n p= = 6 0 Var( X ) = np (1 p ) = =

31 Example (nausea in pregnancy, cont d) 11 Obs. X = 11, n= 20, pˆ = = npˆ(1 pˆ) = 20 = = not OK to use normal approximation. We must resort to the binomial distribution. PX ( 11 = Rosner, Table ) = As H is two-sided, we muliply by 2; thus the p value = Conclusion: reject H 0 k P(X=k)

32 Summary: properties of a hypothesis test Rosner tbl. 7.1 P (type II error) Decision Truth H 0 true H 0 false Accept H 0 1 α β Reject H 0 α 1 β P (type I error) Power Type I error: reject H 0 when H 0 is true Type II error: accept H 0 when H 0 is false 32

33 Power or significance? We wish to reduce the probability of both a type I error (α) and a type II error (β), because A small α means that it is diffcult to reject H 0 A small β means that it is easier to reject H 0 (and accept H 1 ) But minimizing α and β at the same time is problematic, because α increases as β decreases, and vice versa Strategy: Usually α is kept at a comfortable level (0.10, 0.05, 0.01); the maximum acceptable probability of making a type I error; i.e. rejecting H 0 when it is true Then use a test that minimizes β, i.e. maximizes power ( =1 - β). Beware of H 1 : tests may have different properties! 33