Power and Sample Size Calculations for the 2-Sample Z-Statistic

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1 Powe and Sample Size Calculations fo the -Sample Z-Statistic James H. Steige ovembe 4, 004 Topics fo this Module. Reviewing Results fo the -Sample Z (a) Powe and Sample Size in Tems of a oncentality Paamete. The -Sample Independent Sample Z (a) Deiving the Distibution of Z i. The Mean ii. The Vaiance (b) Powe Calculation i. The oncentality Paamete (c) Sample Size Calculation 3. The Genealized Independent Sample Z-Statistic (a) Deiving the Distibution of Z i. The Mean ii. The Vaiance (b) Powe and Sample Size Calculation

2 Reviewing Results fo the -Sample Z-Test In ou pevious lectue, we found that powe, P, can be calculated fom the following equation. p P Es R whee ( 0 ) is the standadized e ect size, and R is the eection point fo the Z-statistic. We also found that sample size,, can be computed as ceiling. Powe in Tems of a oncentality Paamete On the way to deiving the above equations fo powe and, we also deived the mean and standad deviation of the Z-statistic, and found that they ae Z p and Z, espectively. Suppose we de ne a noncentality paamete as Z p Then P ( R) ote, howeve, that by collapsing p into in this way, we have lost, and it is no longe clea that thee is a simple equation fo that can be deived as a function of and R. Thee is, as we will see late, an advantage to de ning a noncentality paamete. It tuns out that, fo any Z test in a vey geneal family, powe is simply ( R) ( Z R). In what follows, we will assume equal vaiances in all populations, because ou goal is to use the esults fo Z-tests as an appoximation of the coesponding esults fo t-tests 3 The -Sample, Independent Sample Z-Statistic We will study the behavio of independent sample Z-tests. As you ecall, the test statistic fo the most basic -sample, independent sample Z-statistic when vaiances ae equal is

3 Z X X + otice that thee ae many ways to wite the above statistic. A key to being able to e-expess the fomula is to ealize the following simple identity: So anothe way to wite Z is Z X X p q X q X + 3. Deiving the Distibution of Z + X Once we have witten Z in this fom, it is easy to deive its mean and vaiance, using ou standad esults on linea combinations and linea tansfomations. 3.. The Mean otice that the only andom vaiables in the equation fo Z ae X and X. Recall that the mean of X X must be, and multiplication comes staight though in the mean, so Z + + whee in this case, is the di eence between the two means in standad deviation units, i.e., 3.. The Vaiance We can quickly show that the Z-statistic has a vaiance of. Recall that multiplying a vaiable by a constant multiples its vaiance by the squae of that constant. We will e-expess the equation fo the -sample Z as X X Z X X X

4 Recall fom ou linea combination theoy that the vaiance of the linea combination X X is X X But the Z statistic multiplies X X by some constants, so the vaiance of Z must be multiplied by the squae of these constants. Speci cally " # Z + X X " # In the q nal analysis, we have poven that the -sample Z-statistic has a mean of + and a standad deviation of. otice that, if the sample sizes ae equal, i.e.,, the mean of the Z-statistic educes to Z Once again, we nd that the noncentality paamete is simply the mean of the distibution of the Z-statistic. 3. Calculating Powe Since the standad deviation of the Z-statistic is still, powe is simply a function of how fa Z (o, if you pefe, ) is fom the eection point R. So, once again, fo a eection point R, powe (P ) is 3.. Calculating Sample Size P ( Z R) ote, howeve, that by leaving in the equation, we can also solve fo the sample size equied to assue a given powe. Speci cally, since! P ( Z R) R 4

5 we have, afte taking of both sides of the equation and doing a bit of eaanging (P ) R To guaatee powe at o above P, we use the ceiling function, and we have " # ceiling 3.. An Example Let s ty an example. Suppose :5 and 5 pe goup. With :05, and a -sided test, the eection point R is :96. In this case, powe is computed as! P R! 5 :5 :96! 5 :5 :96 [(3:5355)(:5) :96] ( :9) :43 The aea to the left of :9 in the standad nomal distibution is about :43. Clealy, sample size is inadequate in this case. What sample size would we need to achieve a powe of :80? Fist, we need (P ) (:80). Fom 5

6 the nomal cuve table, this is about :84. Fom the above equation, we get " # ceiling " # :84 + :96 ceiling :5 ceiling(6:7) 63 4 The Genealized Independent Sample Z- Test Entie books have been witten about t-tests that go beyond simple tests of equality fo two means. So how does one genealize to this new situation? The answe is that you poceed in exactly the same way you did fo the one sample and -sample t tests. You examine the distibution of the Z- statistic, obtain the mean and vaiance, and then wite a fomula fo P and fo. Let s take a simple special case st. Suppose you wished to test the hypothesis that 0. Assuming equal vaiances, the Z-statistic fo testing this hypothesis is Z X X + 4 We can ewite the above statistic as Z + 4 X X You can quickly deive, in the same manne as peviously, that the standad deviation of the statistic is and its mean is Z + 4 If the sample sizes ae both equal to, this educes to Z 5 5 6

7 ote that the in the denominato has been eplaced by a 5. In the sections that follow, we pesent a geneal fomula fo the mean and vaiance of the Z-statistic 4. Deiving the Distibution of Z Suppose the null hypothesis is H 0 : JX c 0 The Z-statistic fo testing this hypothesis, if vaiances ae assumed equal, is Z P J c X 0 PJ c With equal sample sizes, this can be witten Z P J c X 0 PJ c s P J P c X 0 c 4.. The Mean The mean of Z can be deived immediately fom the peceding fomula, since each X has a mean of. The esult is Z s P J P c 0 c s P E c s whee the c ae the linea weights used in the null hypothesis, and is de ned geneally as the amount by which the null hypothesis is wong in 7

8 standad deviation units, i.e.,. 4.. The Vaiance P J c 0 The vaiance standad deviation of the Z-statistic ae always. See if you can pove this esult fo youself. 4. Powe and Sample Size Calculations This allows us to wite vey geneal fomulae fo powe and sample size when the sample sizes ae equal to. Fo powe, we have s! P P E c s R Fo sample size, we manipulate the above equation s (P ) P E c s R s P c Squaing both sides and manipulating a bit moe, we end up with " X # ceiling c Once you become familia with a couple of key values fo the eection point R and the powe value (P ); you can deduce powe and sample size acoss a wide vaiety of situations. Fo example, suppose you ae doing a -tailed test with :0; and you need powe of P :90 to detect a standadized e ect size of :50.. You need to estimate the sample size pe goup equied to test the hypothesis H 0 : 3 4 8

9 o, equivalently In this case, the sum of squaed linea weights is X c () + ( ) + ( ) + () 4 The powe value (P ) is (90) :8: The eection point fo the -sided test is :576. So the equied sample size is " X # ceiling c " # :8 + :576 ceiling 4 :50 ceiling[38:5] 39 9

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