MORE ON THE EQUAL-VARIANCE, TWO-SAMPLE T-TEST

Transcription

1 1 MORE ON THE EQUAL-VARIANCE TWO-SAMPLE T-TEST Robustess: As poited out i the quote from Box (i the otes for Jauary 19 ad as the discussio (whe the example was itroduced of the example about comparig two computer packages suggests we ca't expect the assumptios of a iferece procedure to apply exactly. A procedure is said to be robust to departures from a model assumptio if the results are still reasoably accurate whe the assumptio is relaxed to some degree. Robustess is sometimes determied by theory sometimes by computer simulatios. For example i the two-sample t-test above if our samples are large eough the Cetral Limit Theorem tells us that eve if ad are ot ormally distributed the distributio of - is approximately ormal so that the test statistic will still have a distributio that is approximately t with m - degrees of freedom. Of course just how large is large eough will deped o the distributios of ad. Computer simulatios have show that moderate departures of ad froormality have little effect o the distributio of the t-statistic. Simulatios have also show that the equal-variace twosample t-test is relatively robust to departures from the equal variace assumptio provided the two sample sizes are equal or early equal. However lack of idepedece ca cause serious problems -- the results of a t-test may be very misleadig. Aother perspective o the two-sample equal-variace t-test. Those of you who have had regressio have see that a certai t-test is equivalet to a certai F-test. The same is true here. The F-test perspective the allows us to geeralize the method to situatios where we are comparig more tha two meas ad to some samplig methods other tha simple radom samples. First we eed more detail o t distributios: A t-distributio with k degrees of freedom is Z defied as the distributio of a radom variable of the form where U k Z~N(01 U~ χ (k (Chi-squared with k degrees of freedom. Z ad U are idepedet. A chi-squared distributio with k degrees of freedom is defied as the distributio of a radom variable that is a sum of squares of k idepedet stadard ormal radom variables. The proof that our test statistic T for the equal-variace two-sample t-test has a t- distributio follows from these facts:

2 T = ( µ = ( µ ( m S (algebra S S σ σ σ ( m Z = ( µ is stadard ormal (see earlier σ σ U = ( m S is chi-squared with m - degrees of freedom. (Ca be proved σ usig model assumptios U ad Z are idepedet (Ca be proved usig model assumptios. A F-distributio F(ν 1 ν with ν 1 degrees of freedom i the umerator ad ν degrees W ν1 of freedom i the deomiator is the distributio of a radom variable of the form U ν where W ~ χ (ν 1 U ~ χ (ν ad U ad W are idepedet. If we have a t radom variable of the form T = Z U k where U ad Z are as i the defiitio of t-distributio the T = Z U k. Now Z is a chi-squared radom variable with 1 degree of freedom ad U is chi-squared with k degrees of freedom so T is a F-distributio with 1 degree of freedom i the umerator ad k degrees of freedom i the deomiator. So we could do ay t-test (with two-sided alterative as a F-test by usig the square of the t-statistic. Lookig at the square of the t-statistic for the two-sample equal-variace t-test i the case of equal sample sizes will give us some isight ito geeralizig the F-test to work for more tha oe sample ad evetually to some other samplig desigs as well. Uder the ull hypothesis µ = µ the t-statistic is

3 3 T = 1 1 S Our additioal restrictio of equal sample sizes meas m =. So ad S = ( 1 S ( x 1 S ( 1 ( 1 = S S x T = ( S x S =. ( S x S The our F statistic is ( T = ( Sx S which is equivalet to ( 1 ( Sx S. With our assumptio that m = the deomiator i this re-expressio is just our pooled estimator of σ the commo variace of the two populatios. If the ull hypothesis is true the the two distributios (of ad are the same -- so we may cosider our two samples to be two samples of size from the same N(µσ distributio. But we kow that the sample meas of samples of size from a N(µσ distributio have a N(µσ / distributio (the samplig distributio. Now the sample variace of a distributio is a ubiased estimator of the populatio variace of that distributio. Applyig this to our N(µσ / samplig distributio we coclude that the radom variable S b = 1

4 4 is a ubiased estimator of σ /. (The b stads for "betwee sample" Usig algebra S b = = 1 (. ( of T is a ubiased Thus if the ull hypothesis is true the umerator estimator of σ so we expect the quotiet i T to be close to 1. It ca be proved that if the ull hypothes is false the the ratio T is greater tha 1. So the F-test (equivalet to the t-test ca be iterpreted as a test for the ratio of two estimates of σ. This idea ca be geeralized to more tha two samples: We form the sample variace for each sample take the mea of these sample variaces as oe estimate of the commo populatio variace σ ad compare with a "betwee sample" estimate of σ. With suitable modificatios this works ad is the idea behid the method of Aalysis of Variace. However we may as above multiply the umerator ad deomiator i the F- statistic by costats to make iterpretatios ad/or formulas easier. I the otatio used i the textbook for the special case = m cosidered here we would express the F- statistic as SST SSE ( where SST (the sum of squares for treatmets or treatmet sum of squares is SST = ad SSE (the sum of squares for error or error sum of squares is ( m i i= 1 i i= 1 SSE = ( ( is sometimes called the grad mea abbreviated GM. Exercises: i. Go through the algebra to check that the two expressios for the F-statistic are equivalet. ii. Express GM SST ad SSE usig the followig otatio:

5 The sample from the first radom variable is ad the sample from the secod radom variable is 1. (I other words the radom variable represetig the t th observatio from the i th populatio for i = 1 is it. We will eed to use double subscripts whe we go to more tha populatios; typically populatios i this class will be defied by treatmets. 5