Lecture 16: Words as dataPivot: A step back•So far in lab you’ve looked at a couple ways to derive confidence intervals -- One involving the quantiles of the bootstrap distribution, one involving the +/- two standard error approach, and then the classical t-statistic technique involving quantiles from the t-distribution•We are going to take a step back briefly and recall the small miracle that is the t-distribution -- Considering the simple case of the sample mean and its relationship to the population mean will help us understand regression technology a little betterPivot: A step back•Recall that Gosset worked out the sampling distribution of a standardized statistic, the t-statistic, under the assumption that the data we’ve observed come form a population that is well described by a normal distribution•Under that assumption, he demonstrated that the quantity•has a sampling distribution that does not depend on anything but the number of observations that go into the sample mean and sample standard deviation (where is the unknown mean of the population)x − µs/√nµPivot: A step back•In technical parlance, we refer to •as a pivotal quantity -- Its distribution can be written down explicitly (although we won’t) and it depends only on sample size•We used this fact to help us form confidence intervals...x − µs/√nPivot•If we know that•has a specific sampling distribution, we can find upper and lower points that define where we typically expect to see our observations (again, “typical” referring to lots of repeated experiments)•In this case, the points are the quantiles of the Student’s t-distribution...x − µs/√nPivot•If we let define the upper and lower points which contain 95% of our statistics (95% referring to 95 out of 100 times we repeated our experiment), then •which we can simplify to give•or•which is the basic line of reasoning we followed for the t-based confidence intervalsProb!−qt≤x − µs/√n≤ qt"= 0.95Prob!−qts/√n ≤ x − µ ≤ qts/√n"= 0.95Prob!x − qts/√n ≤−µ ≤ x + qts/√n"= 0.95qtSome comments•In this case, we are using the t-distribution as our reference distribution for the test statistic for the null hypothesis that •It is known as a t-test (or a paired t-test since we’re working with differences of matched pairs)xs/√nµ = 0Testing and confidence intervals•Recall from last lecture that hypothesis tests and confidence intervals are looking for consistency between samples and population parameters, but they are coming at it from slightly different perspectives•Confidence intervals: Fix the (sample) statistic and ask what values of the population parameter are consistent with the fixed statistic•Hypothesis tests: Fix the population parameter value and ask what (sample) statistics are consistent with that fixed value•It turns out there is a one-to-one correspondence between tests and confidence intervals -- This dance with the t-distribution makes that clearTesting and confidence intervals•Let [lo,hi] be a 95% confidence interval, say, for a population parameter -- Then for any we can test the null hypothesis that , rejecting the null if is not contained in [lo,hi] •The resulting test has significance level 0.05 -- In general, any percent confidence interval is equivalent to a test with significance level •The logic works in reverse if we start with a hypothesis test and consider the set of values for which we would fail to reject the null hypothesis that -- These values form a confidence interval for the population parameterθH0: θ = θ0θ0θ0100(1 − α)αH0: θ = θ0θ0The standard error, classically•There are three parameters in our population model -- The two regression coefficients (slope and intercept) and the error standard deviation•We estimate the regression coefficients by least squares giving us and and we can assess sampling variability using the bootstrap•The classical tools, however, follow what we did for the mean -- The estimate of the error standard deviation is simply•and the classical estimate of the standard error of , say, is!σ ="#i(yi−!β0−!β1xi)2n − 2!σ"#i(xi−x)2β1,β0σ!β1!β0!β1The classics•Using the chart on the previous page, you see the link between testing and confidence intervals for the sample mean and testing and confidence intervals for the regression slope and intercept estimates•The bootstrap confidence intervals are providing a generalization of the classics when we’re not sure the assumptions Gosset required hold -- And the confidence intervals can be inverted to provide us with an example of a bootstrap hypothesis test•But before the bootstrap, we were testing hypotheses in a different way...A test•We could, of course, conduct a test along the lines of our earlier work on testing -- In particular, suppose we want to test the hypothesis that , or that the population parameter in a linear relationship relating Mercury levels and fish length is zero•Thinking back to the shuffling of labels that we started with at the beginning of the quarter, what special symmetry does this hypothesis suggest? What might we shuffle here?H0: β1= 0•This is a randomization test of the null hypothesis that the coefficient in a regression of Mercury content on fish length is zero, β1= 0Testing•I present this alternative test (alternative to working with the bootstrap confidence interval) because it demonstrates the richness of the tools you are learning -- There are several different ways to think about the same problem and, for the most part, the answers should agree•In general, if you are forming a test and there is something natural to permute (as in the case of randomized trials or this regression example), then you’re better off making use of that structure for a test•In the classical case, you’re reasoning from the t-distribution consistently, and so you don’t have as many choices to make -- You are also somewhat limited in the problems you can address (think about the median example from your lab)Back to the river•We’re going to next consider a second pair of variables -- It will introduce a small complication that will provide us with a richer view of regression analysis•Let’s have a look at the relationship between fish length and
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