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Date Created: 06/24/14
120C Final Review Probability of a Type 1 Error 1 PReject H0 H0 is true a is also called the level of the test Probability of a Type 2 Error 8 PAccept H0Ha is true Power of a Test 1 8 PReject H0 Ha is true Power is the probability that we have enough evidence against our null when the null is not true LRT H09 E 90 Ha9 90 max9e90H1f9xi maxeeel L1f9xi Decision Rule Reject H0 if 1 S ka with the significance level of max9E0P9 1 S ka S a Note that the value of ka depends on which a we select Note 00 is a subset of the whole parameter space 0 Remarks 0 The goal is to maximize the power of a fixed test level a 0 Minimizing the type II error is equivalent to maximizing the power of a test 0 The computation of type I err and power is through the critical region of the decision rule 0 The best test uniformly most powerful test has critical region C such that 0 PTX E CH0 true afixed 0 maximize PT X E C Ha true max power Deciding what method we should use for hypothesis tests Two Independent Samples Matched Pairs Sample Sizes Different Same Relation Non Existence Related Measurement Not on Same Subject On Same Subject Normal Assumption On Both Samples Only on Difference Nonparametric vs Parametric Tests 0 Assumptions Distributional form parametric 0 iid observations parametric independence nonparametric 0 Sensibility to outliers parametric 0 Power 0 Nonparametric test is conservative Trade offs Distributional exibility vs change in power 0 Sustainability to outliers vs partial information use signsranks vs full data Sign Test for One Example Outline Given n iid observations X 1 the goal is to conduct a test about the location of the distribution Hoili M0 Hal gtlt 10 OR letp PX gt 100T39 1 F010 p and H021 Hap gtlt so use M of times X gt i0K of times X lt 10 as a test statistic Rejection Region When Hazp gt 12 p valuePrM 2 m lt d When Hazp lt 12 p valuePrM S k lt d When Hazp at 12 p value2PrM S min mk lt a Note Normal approximation Wcontinuity correction works better for large samples ngt25 30 Normal ApproximationPr Z 2 W 7 Sign Test for a Matched Pairs Experiment Outline Given n independent pairs X 1 Y Let Fx be CDF of X and Gy be CDF of Y then location shift FxGx 9 The goal is to conduct a test about the equality of locations for distributions H09 OHa9 gtlt 0 OR Let pPrX gt Y then Hozp 5 Hazp gtlt 5 So use N of positive differences Di X 1 Yi as a test statistic and proceed as above Wilcoxon Signed Rank Test for a Matched Pairs Experiment Outline Given n independent paired observations X 1 Y the goal is to conduct a test about the equality of locations of the distributions H0 The distributions are the same Ha The location of one distribution is shifted Test Statistic T Sum of the ranks of the positive differences Tquot Sum of the ranks of the negative differences Rejection Region T 39 S To one sided T S To one sided T minT T S To two sided For T quot S T 0 we assume that the population relative frequency for the X s is shifted right of that of the Y s one tailed For T S T 0 we assume that the population relative frequency for the X s is shifted left of that of the Y s one tailed For relatively large samples gt25 normal approximation may be used Test Statistic Tquot39 nn14 nn12n124 Rejection Region Z 2 Z one sided gt Z S zaone sidedlt Z 2 zg two sided 2 U Statistic Definition 0 W Sum of the Ranks 0 U W smallest rank sum 0 U quot nlnz U Where n1 as observations from sample 1 and n2 observations from sample 2 0 U is the sum of the number of observations from sample 2 that are less than each of the observations from sample 1 RankSum Test for Independent Samples Nature of the test when rank sum for one sample is very large or correspondingly for the other sample is very small it indicates statistically significant difference between the locations of the two populations 0 The Mann Whitney U Test is carried out under this idea Mann Whitney U Test Nature of the test The statistic is obtained by ordering all n1 n2 observations according to their magnitude and counting the number of observations in sample I that precede each observation in sample II 0 Population I is the population from which the smaller small was taken H0 The distributions of population I and II are identical Ha The distributions of population I and II have different locations two sided The distribution of population I is shifted to the right of population II The distribution of population I is shifted to the left of population II Test Statistic 1 U nlnz 1n12 n1 number of observations in sample I n2 number of observations in sample II W rank sum for sample I Mann Whitney U Test Continued Rejection Region 0 Two tailed test and given a reject H0 if U S U0 or U 2 nlnz U0 where PU S U0 3 0 Shifted right given Ct reject H0 if U S U0 where PU S U0 a 0 Shifted left given a reject H0 if U 2 nlnz U0where PU S U0 a Large Sample Situation Assumptionznl gt 10 and n2 gt 10 Test Statistic for Normal Approximation Z T jn1n2n1n21 12 Rejection Regions 1 two tailed test given Cl reject H0 if Z gt Z 2 or Z lt Za2 2 shifted right given 61 reject H0 if Z lt Za 3 shifted left given 61 reject H0 if Z gt Z0 Comparison between Mann Whitney test and two sample t test MannWhitney U Test Two Sample TTest Method Non Parametric Parametric Assumption No Assumption on Distribution Normal Distribution Information Not Complete Complete Power Less Large Conservative More Conservative Less Conservative Wording of a question that says significant increase need one sided test that tests whether the population of old is shifted to the left of the population of new Say Population I is old Population II is new Runs Test Nature 0 To determine whether a set of outcomes represents an independent sequence Definition 0 A run is a maximal subsequence of like elements 0 A very small positive association or very large negative association number of runs in a sequence indicates non randomness Test 0 H0 each occurrence is independence vs Ha there is some association 0 R is the number of runs in a sequence and denotes the test statistic 0 Rejection Region R S a or R S b 0 Level of test a PR S a PR 2 b 95 Confidence Interval 2 f1 bln 2 3951 Pnset a 005 X2 GOODNESS OF FIT TEST Tests how well the data fits the suggested distribution Definition and Test Nature 0 H0p1 pf 4H pk pf vs Ha some other set of probability 0bserved Expected2 Expected 0 2 test statistic has approximately a 2 distribution with k 1 df Odegrees of freedom number of estimates for p 0 test statistic 2 Z Contingency Table 0 Usage 0 Type of analysis count data with concerns of the independence of two methods subjects 0 Use to investigate a dependencycontingency between two classification criteria 0 Test Form T Shift A Total 1 a 2 a Total C 0 H0 Classifications are independent vs Ha Classifications are not independent 0 Formula for expected value Ei 1 1 with n as total counts 39 n 2 0ij Eij2 0 After computing expected value we calculate the us1ng the formula 2 1 1 ET i J39 0 Degrees of freedom of row 1 of column 1 0 Requirement that the expected value greater than the minimal of rows or columns 0 Rejection region 2 gt 12akC Hypothesis for Multiple Comparisons Fully Independent rct 0 EJk quotquotdegrees of freedom rtc r C t 2 Bayesian Priors Posteriors and Estimators SPSE yl 2 yn are random variables associated with sample size n 0 In the Bayesian approach the unknown parameter 9 is a rv with a probability distribution prior distribution of 9 0 The prior distribution is specified before data is collected and provides a theoretical description of info about 9 available before data collection 0 9 has a continuous distribution with density g9 f3 1 quotWyn Ly1I The marginal density of y1 p yn is my1 ynfoo LC1 r y6 g9 619 Therefore the posterior density of 9 y1 yn is 9 PM 00 Ly1ynq9 g lyl 339quot fooL3 13 n99d9 Bayesian estimate used to estimate risk obtain optimal estimate 0 Prior 709 expresses our opinion of 9 before data is factored in 0 Posterior f 9 x expresses the conditional probability after evidence is taken into account 0 Predictive Conditional probability of the new observation y given data X Calculation 1 Construct Prior 2 Find likelihood 3 Modify prior using likelihood to get posterior
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