Chapter 16: Nonparametric Tests

Transcription

1 Chapter 16: Nonparametric Tests In Chapter 1-13 we discussed tests of hypotheses in a parametric statistics framework: which assumes that the functional form of the (population) probability distribution of the measurement X is known, except for a small number of unknown parameter(s), whose value(s) determine the underlying distribution, e.g., o X N(, ), or X Gamma(, ) : One or both the parameters are unknown. o X Poisson( ) : Value of is unknown. f is assumed to belong to a family of distributions defined by possible values of the parameter(s) in a defined set. Parameter is the basis for forming Hypothesis of interest As long as the assumption about f is correct, these tests are optimal, in some statistical sense, e.g., for a fixed level test, N-P lemma gives the most powerful test to detect a false null. The concept called Robustness studies the effect of violations of the underlying assumption, and suggests methods of estimation and hypothesis tests that do not require these parametric assumption(s).

2 Estimation - Suppose that the measurement X has a symmetric distribution. In that case, population Mean and Median are equal. o Sample mean is the MVUE, MLE for the population mean when the population is normally distributed. If the correct distribution of X is Cauchy (Symmetric, but heavier tails) o Its population mean and variance don t exist. o The sample mean is not a consistent estimator. o The distribution of X,for any n, is same as that of X. This doesn t mean that several samples from Cauchy distribution don t have more information than one observation. It simply says that the sample mean is not an efficient summary of the Data. o The sample median is not sensitive to outliers, has been proven to be consistent and asymptotically unbiased. In this sense, Median is a Robust Estimator of the population median (= population mean for symmetric distributions.) Other Robust estimators of location are Trimmed mean, Winsorized mean, weighted linear combination of ordered data.

3 Test of Hypothesis: Assuming that we have a random sample of size n from N(, ) distribution, which implies that X N(, ). n o We then use this distribution to calculate the cut off point for a level N( 0, ) n test, or the p-value (area under density in the tail(s) corresponding to the observed value x. Example: Let us say the there is a % chance that an observation has a recording error. o Then the sample is not from a normal population, and the distribution of X is not normal, even though moments exist. E( X ),and Var( X ) =, n if the population o Using the normal distribution for obtaining the cut-off point for thelevel test, or the p-value calculations would not give the correct inference. Exercise 13.30, (page 45) was discussed earlier. o 5 observations on Tar content of cigarettes. Assuming normality, H : was rejected at 5% level. o Just one observation recorded incorrectly, 16.0 (instead of 14.5). o Difference between the sample mean and 0 increased, but the Null was not rejected. o Incorrect observation increased the variance!

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5 Simple Robustness Experiment: Simulate data from various distributions C1: X ~ Normal(5,1) C: X ~.95*Normal(5,1)+.05*Normal(4.5,4) [Mixture of Normals] C9: X ~ Cauchy(5.0, 1.0) Sample size 5, Just 1 observation was contaminated by Chance in the mixture model. Descriptive Statistics: C1, C4, C9 Variable N Mean StDev Minimum Q1 Median Q3 Maximum C C C One-Sample Z: C1, C4, C9 Test of mu = 5 vs not = 5 The assumed standard deviation = 1 Variable N Mean StDev 95% CI Z P C (4.335, 5.119) C (4.45, 5.09) C (6.633, 7.417) One-Sample T: C1, C4, C9 Test and CI for Standard Deviation: C1, C4 Method Null hypothesis: Sigma = 1 Alternative hypothesis: Sigma not = 1 The standard method is only for the normal distribution. The adjusted method is for any continuous distribution with first and nd moments. 95% Confidence Intervals for SD Method Variable Standard Adjusted C1 (0.576, 1.06) (0.568, 1.051) C4 (0.73, 1.88) (0.658, 1.564) C9 (5.56, 9.90) (3.74, 41.98) Tests Variable Chi-Square DF P-Value C1 Standard Adjusted C4 Standard Adjusted C9 Standard Adjusted Test of mu = 5 vs not = 5 Variable N Mean StDev 95% CI T P C (4.43, 5.031) C (4.435, 5.00) C (4.090, 9.960)

6 In many practical problems, we are not willing to make strong assumption, like normal distribution, as a basis for test procedure, since we are not sure that the assumption is valid. Nonparametric Tests do not require assumption of a known density. This class of procedures include distribution free tests (except possibly that the distribution is continuous, and/or symmetric. Alternatives to one sample or paired t-test: o continuous distribution : PX 0 ( ) 0; o symmetric : ( Median ), or just test for the population median Section 16. Sign Test: H : against H : ; ; o (One sided or two sided alternatives). If x >, replace it by +; If x <, replace it by -; i 0 0 i 0 x =,discard the observation, and reduce the sample size by 1. i Paired-sample sign test: H0 : 1 against H1: 1 ; 1 ; 1 If x -x >0, replace it by +; If x -x < 0, replace it by -; 1i 1i 1i 1i If they are equal, discard the pair, and reduce the sample size by 1.

7 Each observation results in binary {+, -} outcomes, and we have independent Bernoulli trials. Under H 0, P(+) = = ½. o y= # of + (Heads, Successes) out of m un-discarded trials. o Under the null, Y~ Binomial (m, ½). o For one sided alternative, 0 calculate the p- value = P( Y y) m bi( x; m,.5) using Binomial x y Table I. Reject the null if p value. If m is large, we can use the normal approximation with Z y m m(1 ) o For one sided alternative, 0 calculate the p- value = ( ) y P Y y bi ( x ; m,.5) using Binomial x0 Table I. Reject the null if p value. o For the two-sided alternatives, p-value is defined by P( Y y) m bi( x; m,.5). xy o Exercises: 13.30, 13.44

8 Section 16.3 Wilcoxon Signed-Rank Test Sign test is easy to use, but uses very little info in the data. o Just sign of the difference between the observed and hypothesized values. More information is used when we also take into account the magnitude of differences. Consider the absolute values of the differences xi 0 ( ) or paired differences x1 i x ( ) i, discarding zero differences. Rank these numbers from smallest to largest. If two or more absolute differences are equal, each one is assigned the mean rank of the positions occupied by these observations. Now define, T + = Sum of the ranks of Positive differences T - = Sum of the ranks of Negative differences T=min(T +, T - ) Note that T + + T - = mm ( 1) One of these is used as a T-statistic for the level- T-test. The choice depends on the alternative hypothesis.

9 Alternative Hypothesis Reject the Null if:, T T 0, 0 T T, 0 T T Critical Values of the signed-rank test are given in Table 10, page 584 for n=4(1)5, and =.01,.0,.05,.10 Example: Exercise 13.30, Nonparametric Alternative to -sample t-test Section 16.4 Rank-Sum Tests - The U test (Wilcoxon-Mann-Whitney) Instead of normal distribution, assume that the two distributions are continuous with same shape, but with different location o Independent random samples n1 o Independent random samples n X,, ~ ( ) 1 X f x Y,, ~ ( ) 1 Y g y o The same shape means that for some, g( x) f ( x ) for all x. Null Hypothesis H0 1 : or 0. ( 1 ) continuous distributions are identical] [Two

10 Alternative: H :,i.e., <0;, i. e., 0;, i. e., Pool the two samples as if they were one common sample, and rank the observations from smallest to largest in the pooled sample, ties are given the mean ranks of the positions occupied by the tied observations. Calculate W 1 Sum of ranks of first sample, W Sum of ranks of the second sample Since there are a total of n n1 nobservations in the two nn ( 1) samples, their ranks add up to W1 W If the samples are from exactly the same distribution, they will be evenly distributed in the pooled sample. n ( n 1) n ( n 1) U1 W 1 ; U W ; U min( U1, U ) Define 1 1 The tests based on U s or W s are equivalent, since they differ by specified constants. The critical values of the U-test are given in Table XI, page for sample sizes up to 15. Just like, signed-rank test, one of these is used as a U- statistic for the level- U-test. The choice depends on the alternative hypothesis..

11 Alternative Hypothesis Reject the Null if: 0 U U 0 U U 0 U1 Example: Exercise 13.4 U