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Comparison Methodology ü Meaning of a sample ü Confidence intervals • Making decisions and comparing alternatives • Special considerations in confidence intervals • Sample sizes © 1998, Geoff Kuenning

Estimating Confidence Intervals • Two formulas for confidence intervals – Over 30 samples from any distribution: z-distribution – Small sample from normally distributed population: t-distribution • Common error: using t-distribution for non-normal population – Central Limit Theorem often saves us © 1998, Geoff Kuenning

The z Distribution • Interval on either side of mean: • Significance level is small for large confidence levels • Tables of z are tricky: be careful! © 1998, Geoff Kuenning

The t Distribution • Formula is almost the same: • Usable only for normally distributed populations! • But works with small samples © 1998, Geoff Kuenning

Making Decisions • Why do we use confidence intervals? – Summarizes error in sample mean – Gives way to decide if measurement is meaningful – Allows comparisons in face of error • But remember: at 90% confidence, 10% of sample means do not include population mean © 1998, Geoff Kuenning

Testing for Zero Mean • Is population mean significantly nonzero? • If confidence interval includes 0, answer is no • Can test for any value (mean of sums is sum of means) • Example: our height samples are consistent with average height of 170 cm – Also consistent with 160 and 180! © 1998, Geoff Kuenning

Comparing Alternatives • Often need to find better system – Choose fastest computer to buy – Prove our algorithm runs faster • Different methods for paired/unpaired observations – Paired if ith test on each system was same – Unpaired otherwise © 1998, Geoff Kuenning

Comparing Paired Observations • Treat problem as 1 sample of n pairs • For each test calculate performance difference • Calculate confidence interval for differences • If interval includes zero, systems aren’t different – If not, sign indicates which is better © 1998, Geoff Kuenning

Example: Comparing Paired Observations • Do home baseball teams outscore visitors? • Sample from 9 -4 -96: © 1998, Geoff Kuenning

Example: Comparing Paired Observations • H-V 2 -2 -7 5 6 -1 -7 6 7 3 2 1 -1 6 • Mean 1. 4, 90% interval (-0. 75, 3. 6) – Can’t reject the hypothesis that difference is 0. – 70% interval is (0. 10, 2. 76) © 1998, Geoff Kuenning

Comparing Unpaired Observations • A sample of size na and nb for each alternative A and B • Start with confidence intervals – If no overlap: mean A B • Systems are different and higher mean is better (for HB metrics) – If overlap and each CI contains other mean: • Systems are not different at this level • If close call, could lower confidence level – If overlap and one mean isn’t in other CI • Must do t-test © 1998, Geoff Kuenning B mean A B A

The t-test (1) 1. Compute sample means and 2. Compute sample standard deviations sa and sb 3. Compute mean difference = 4. Compute standard deviation of difference: © 1998, Geoff Kuenning

The t-test (2) 5. Compute effective degrees of freedom: 6. Compute the confidence interval: ! 7. If interval includes zero, no difference © 1998, Geoff Kuenning

Comparing Proportions • If k of n trials give a certain result, then confidence interval is ! • If interval includes 0. 5, can’t say which outcome is statistically meaningful • Must have k>10 to get valid results © 1998, Geoff Kuenning

Special Considerations • Selecting a confidence level • Hypothesis testing • One-sided confidence intervals © 1998, Geoff Kuenning

Selecting a Confidence Level • Depends on cost of being wrong • 90%, 95% are common values for scientific papers • Generally, use highest value that lets you make a firm statement – But it’s better to be consistent throughout a given paper © 1998, Geoff Kuenning

Hypothesis Testing • The null hypothesis (H 0) is common in statistics – Confusing due to double negative – Gives less information than confidence interval – Often harder to compute • Should understand that rejecting null hypothesis implies result is meaningful © 1998, Geoff Kuenning

One-Sided Confidence Intervals • Two-sided intervals test for mean being outside a certain range (see “error bands” in previous graphs) • One-sided tests useful if only interested in one limit • Use z 1 - or t 1 - ; n instead of z 1 - /2 or t 1 - /2; n in formulas © 1998, Geoff Kuenning

Sample Sizes • Bigger sample sizes give narrower intervals – Smaller values of t, v as n increases – in formulas • But sample collection is often expensive – What is the minimum we can get away with? • Start with a small number of preliminary measurements to estimate variance. © 1998, Geoff Kuenning

Choosing a Sample Size • To get a given percentage error ±r%: • Here, z represents either z or t as appropriate • For a proportion p = k/n: © 1998, Geoff Kuenning

Example of Choosing Sample Size • Five runs of a compilation took 22. 5, 19. 8, 21. 1, 26. 7, 20. 2 seconds • How many runs to get ± 5% confidence interval at 90% confidence level? • = 22. 1, s = 2. 8, t 0. 95; 4 = 2. 132 © 1998, Geoff Kuenning

Linear Regression Models ü What is a (good) model? ü Estimating model parameters • Allocating variation • Confidence intervals for regressions • Verifying assumptions visually © 1998, Geoff Kuenning

What Is a (Good) Model? • For correlated data, model predicts response given an input • Model should be equation that fits data • Standard definition of “fits” is least-squares – Minimize squared error – While keeping mean error zero – Minimizes variance of errors © 1998, Geoff Kuenning

Least-Squared Error N • If y then error in estimate for xi is N yi • Minimize Sum of Squared Errors (SSE) • Subject to the constraint © 1998, Geoff Kuenning

Estimating Model Parameters • Best regression parameters are where • Note error in book! © 1998, Geoff Kuenning

Parameter Estimation Example • Execution time of a script for various loop counts: • = 6. 8, = 2. 32, xy = 88. 54, x 2 = 264 • b 0 = 2. 32 (0. 29)(6. 8) = 0. 35 © 1998, Geoff Kuenning

Graph of Parameter Estimation Example © 1998, Geoff Kuenning

Variants of Linear Regression • Some non-linear relationships can be handled by transformations – For y = aebx take logarithm of y, do regression on log(y) = b 0+b 1 x, let b = b 1, – For y = a+b log(x), take log of x before fitting parameters, let b = b 1, a = b 0 – For y = axb, take log of both x and y, let b = b 1, © 1998, Geoff Kuenning

Allocating Variation • If no regression, best guess of y is • Observed values of y differ from , giving rise to errors (variance) • Regression gives better guess, but there are still errors • We can evaluate quality of regression by allocating sources of errors © 1998, Geoff Kuenning

The Total Sum of Squares • Without regression, squared error is © 1998, Geoff Kuenning

The Sum of Squares from Regression • Recall that regression error is • Error without regression is SST • So regression explains SSR = SST - SSE • Regression quality measured by coefficient of determination © 1998, Geoff Kuenning

Evaluating Coefficient of Determination • Compute © 1998, Geoff Kuenning

Example of Coefficient of Determination • For previous regression example – y = 11. 60, y 2 = 29. 79, xy = 88. 54, – – SSE = 29. 79 -(0. 35)(11. 60)-(0. 29)(88. 54) = 0. 05 SST = 29. 79 -26. 9 = 2. 89 SSR = 2. 89 -. 05 = 2. 84 R 2 = (2. 89 -0. 05)/2. 89 = 0. 98 © 1998, Geoff Kuenning

Standard Deviation of Errors • Variance of errors is SSE divided by degrees of freedom – DOF is n 2 because we’ve calculated 2 regression parameters from the data – So variance (mean squared error, MSE) is SSE/(n 2) • Standard deviation of errors is square root: © 1998, Geoff Kuenning

Checking Degrees of Freedom • Degrees of freedom always equate: – SS 0 has 1 (computed from ) – SST has n 1 (computed from data and , which uses up 1) – SSE has n 2 (needs 2 regression parameters) – So © 1998, Geoff Kuenning

Example of Standard Deviation of Errors • For our regression example, SSE was 0. 05, so MSE is 0. 05/3 = 0. 017 and se = 0. 13 • Note high quality of our regression: – R 2 = 0. 98 – se = 0. 13 – Why such a nice straight-line fit? © 1998, Geoff Kuenning

Confidence Intervals for Regressions • Regression is done from a single population sample (size n) – Different sample might give different results – True model is y = 0 + 1 x – Parameters b 0 and b 1 are really means taken from a population sample © 1998, Geoff Kuenning

Calculating Intervals for Regression Parameters • Standard deviations of parameters: • Confidence intervals are bi t sbi • where t has n - 2 degrees of freedom ! © 1998, Geoff Kuenning

Example of Regression Confidence Intervals • Recall se = 0. 13, n = 5, x 2 = 264, = 6. 8 • So • Using a 90% confidence level, t 0. 95; 3 = 2. 353 © 1998, Geoff Kuenning

Regression Confidence Example, cont’d • Thus, b 0 interval is ! 0. 35 2. 353(0. 16) = (-0. 03, 0. 73) – Not significant at 90% • And b 1 is ! 0. 29 2. 353(0. 004) = (0. 28, 0. 30) – Significant at 90% (and would survive even 99. 9% test) © 1998, Geoff Kuenning

Confidence Intervals for Nonlinear Regressions • For nonlinear fits using exponential transformations: – Confidence intervals apply to transformed parameters – Not valid to perform inverse transformation on intervals © 1998, Geoff Kuenning

Confidence Intervals for Predictions • Previous confidence intervals are for parameters – How certain can we be that the parameters are correct? • Purpose of regression is prediction – How accurate are the predictions? – Regression gives mean of predicted response, based on sample we took © 1998, Geoff Kuenning

Predicting m Samples • Standard deviation for mean of future sample of m observations at xp is S N ymp • Note deviation drops as m • Variance minimal at x = • Use t-quantiles with n– 2 DOF for interval © 1998, Geoff Kuenning

Example of Confidence of Predictions • Using previous equation, what is predicted time for a single run of 8 loops? • Time = 0. 35 + 0. 29(8) = 2. 67 • Standard deviation of errors se = 0. 13 S N yp • 90% interval is then ! © 1998, Geoff Kuenning

Verifying Assumptions Visually • Regressions are based on assumptions: – Linear relationship between response y and predictor x • Or nonlinear relationship used in fitting – Predictor x nonstochastic and error-free – Model errors statistically independent • With distribution N(0, c) for constant c • If assumptions violated, model misleading or invalid © 1998, Geoff Kuenning

Testing Linearity • Scatter plot x vs. y to see basic curve type Linear Outlier © 1998, Geoff Kuenning Piecewise Linear Nonlinear (Power)

Testing Independence of Errors N • Scatter-plot i versus yi • Should be no visible trend • Example from our curve fit: © 1998, Geoff Kuenning

More on Testing Independence • May be useful to plot error residuals versus experiment number – In previous example, this gives same plot except for x scaling • No foolproof tests © 1998, Geoff Kuenning

Testing for Normal Errors • Prepare quantile-quantile plot • Example for our regression: © 1998, Geoff Kuenning

Testing for Constant Standard Deviation • • Tongue-twister: homoscedasticity Return to independence plot Look for trend in spread Example: © 1998, Geoff Kuenning

Linear Regression Can Be Misleading • Regression throws away some information about the data – To allow more compact summarization • Sometimes vital characteristics are thrown away – Often, looking at data plots can tell you whether you will have a problem © 1998, Geoff Kuenning

Example of Misleading Regression x 10 6. 58 8 5. 76 13 7. 71 9 8. 84 11 8. 47 14 7. 04 6 5. 25 4 12. 50 12 5. 56 © 1998, Geoff Kuenning 7 I y 8. 04 x 10 II y 9. 14 x 10 III y 7. 46 x 8 6. 95 8 8. 14 8 6. 77 8 7. 58 13 8. 74 13 12. 74 8 8. 81 9 8. 77 9 7. 11 8 8. 33 11 9. 26 11 7. 81 8 9. 96 14 8. 10 14 8. 84 8 7. 24 6 6. 13 6 6. 08 8 4. 26 4 3. 10 4 5. 39 19 10. 84 12 9. 13 12 8. 15 8 4. 82 7 7. 26 7 6. 42 8 IV y

What Does Regression Tell Us About These Data Sets? • • Exactly the same thing for each! N = 11 Mean of y = 7. 5 Y = 3 +. 5 X Standard error of regression is 0. 118 All the sums of squares are the same Correlation coefficient =. 82 R 2 =. 67 © 1998, Geoff Kuenning

Now Look at the Data Plots I II IV © 1998, Geoff Kuenning

For Discussion Today Project Proposal 1. Statement of hypothesis 2. Workload decisions 3. Metrics to be used 4. Method