Presentation on theme: "Is it statistically significant?"— Presentation transcript:
1Is it statistically significant? Hypothesis testingIs it statistically significant?Most of the slides are adopted from:
2Example In a trial a jury must decide between two hypotheses. Null hypothesis H0: The defendant is innocentAlternative hypothesis H1: The defendant is guiltyThe jury must make a decision on the basis of evidence presented.In the language of statistics convicting the defendant is called rejecting the null hypothesis in favor of the alternative hypothesis. That is, the jury is saying that there is enough evidence to conclude that the defendant is guilty.If the jury acquits, then it is stating that there is not enough evidence to support the alternative hypothesis.
3GoalAssess the evidence provided by data in favor of some claim (null hypothesis) about the population.The procedure begins with the assumption that the null hypothesis is true.The goal is to determine whether there is enough evidence to infer that the alternative hypothesis is true, or the null is not likely to be true.Results of the test are expressed in terms of a probabilityThis probability measures how well the data and the hypothesis agree.
4Example usage in our work Usually we need to find if the performance of our proposed algorithm is statistically superior than that of a base line algorithm.General procedureFor i=1:nGenerate a random train/test dataset.Run both algorithms and get performance numbers.Record the difference between the performance of the proposed and base line algorithms. Let it be x.Null hypothesis H0 : x <= 0 i.e. base line better than proposed algorithm.Alternate hypothesis H1 : x > 0 Our algorithm is better.We want to reject H0 hypothesis in favor of H1 with certain confidence value.
5Statistical ExampleConsider mean demand for computers during assembly lead time. The operations manager wants to know whether the mean is different from 350 units.Our test statistics is over mean demand.Null hypothesis: H0 = 350Our alternative hypothesis becomes: H1 ≠ 350Assume σ = 75 and the sample size n = 25, and the sample mean is calculated to beA large value of (say, 600) provides enough evidence.If is close to 350 (say, 355), then this does not provide a great deal of evidence to infer that the population mean is different than 350.
6Is the Sample Mean in the Guts of the Sampling Distribution? Statistical ExampleAssume H0 : μ = 350 is true.Determine the sampling distribution of sample mean .Assume has meanVariance of is /sqrt(25) = 15Is the Sample Mean in the Guts of the Sampling Distribution?
7Hypothesis Test 1: Unstandardized test For the guts to be the center 95% of the distribution [α = 0.05].The critical values that define the guts will be 1.96 standard deviations of X-BarUCV = *15= 379.4LCV = 350 – 1.96*15= 320.6
8Hypothesis Test 2: Z-score Calculate z score of the sample mean.Z = ( )/ = ( – 350)/15 = 1.344Is this Z-Score in the guts of the sampling distribution?
9Hypothesis Test 3: p-value Increase “Rejection Region” until it “captures” sample mean.P( > ) = P(Z > 1.344) =p-value = double of this area for two tailed test =Since rejection region is defined to be 5% and our sample mean is in the 18.02% region, it in not in rejection region.
10Statistical Conclusions Unstandardized Test StatisticSince LCV (320.6) < (370.16) < UCV (379.4), we reject the null hypothesis at a 5% level of significance.Standardized Test StatisticSince -Z/2(-1.96) < Z(1.344) < Z/2 (1.96), we fail to reject the null hypothesis at a 5% level of significance.P-valueSince p-value (0.1802) > 0.05 , we fail to reject the null hypothesis at a 5% level of significance
12AssumptionsDistribution of test statistics follows standard deviation.Generally one appeals to the central limit theorem to justify assuming that a test statistic varies normally.Central limit theorem (CLT): mean (x bar) of a sufficiently large number of independent random variables (x), each with finite mean and variance, will be approximately normally distributed.If the variation of the test statistic is strongly non-normal, a Z-test should not be used.should be known or can be estimated reliably.when is same for each x and all x are independent from each other. [Follows from var(c.x) = c2 var(x) and var (x1 + x2) = var (x1)+ var (x2) if x1 and x2 are independent].A Z-test is appropriate when you are handling moderate to large samples (n > 30).
13Hypothesis Test 4: Student’s t-test The test statistic (x bar) follows a Student’s T-distribution if the null hypothesis is true.T-test is appropriate for small sample size (n < 30).Great if the populations’ standard deviation is unknown.s= sample standard deviation.Once a t value is determined, a p-value can be found using a table of values from Student's t-distributionThe Student t-distribution is symmetric and bell-shaped, like the normal distribution, but has heavier tails
14Student’s t-testTwo sample t-test can be used to compare means of two independent and identically distributed samples.
15Hypothesis Test 5: Wilcoxon Signed Test A non-parametric statistical hypothesis test.Alternative to the paired Student's t-test when the population cannot be assumed to be normally distributed.H0 : μ = 0Rank xis based on their magnitude. Let their rank be Ri.Wilcoxon signed rank statistics φi = I(xi > 0)ScoreFind the critical value for the given sample size n and the wanted confidence level.Compare S to the critical value, and reject H0 if S is less than or is equal to the critical value.
17Two types of Errors Type I error Type II error H0 T F I II When we reject a true null hypothesis. That is, jury convicts an innocent person.P(Type 1 error) = α [usually 0.05 or 0.01]Type II errorWhen we don’t reject a false null hypothesis. That is, when a guilty defendant is acquitted.P(Type 2 error) = βH0TFRejectIIIThe two probabilities are inversely related. Decreasing one increases the other, for a fixed sample size.