Approximating a Sampling Distribution

Transcription

1 Chpter 3 Approximting Smpling Distribution Tble 3.1: Heights of the rectngles in the probbility histogrm of the smpling distribution of the test sttistic for Fisher s test for the Bllerin study. Height of rectngle x P (X = x) P (X = x)/ Totl Study Suggestions Chpter 1 introduced the CRD s device for conducting comprtive study of two tretments. Chpter 2 introduced hypothesis testing s technique for deciding whether the tretments hve n identicl effect. A hypothesis test yields number, the P-vlue, which quntifies the debte between the Skeptic nd the Advocte. A prcticl problem hs risen, however; the P-vlue cn be difficult to compute. In fct, if study hs lrge number of subjects, the P-vlue cn be impossible to compute, even with n electronic computer equipped with ny of the populr existing sttisticl softwre pckges. Thus, Chpter 3 ddresses the problem of finding n esy wy of obtining n pproximte P-vlue. The two pproximtion methods introduced in Chpter 3 re much esier to understnd if person hs lerned to represent smpling distribution with picture the probbility histogrm. Given smpling distribution, mke sure you cn drw its probbility histogrm by following the three steps in the key extrct on pge 80 of the text. In prticulr, prctice obtining the height of vlue s rectngle by dividing the probbility of the vlue by δ. (Remember, fter sorting, or ordering, the possible vlues of the test sttistic from smllest to lrgest, δ equls the difference between ny two successive vlues.) In ddition, given probbility histogrm, be certin tht you cn crete its smpling distribution. The centers of the bses of the rectngles of the probbility histogrm correspond to the possible vlues of the test sttistic, nd the re of rectngle equls the probbility of its vlue (center). Some exmples of these techniques follow. Consider the Bllerin study introduced in Chpter 1 of the text. The smpling distribution for the Bllerin study is presented in Tble 3.1. In order to obtin its probbility histogrm, first we must determine the vlue of δ, the (constnt) difference between ny two of the ordered possible vlues of the test sttistic. From Tble 3.1, clerly δ = (Alterntively, you cn remember tht δ = n/(n 1 n 2 ). For the current study, this formul gives δ = 50/[25(25)] = 0.08.) Second, we determine the height of ech rectngle. This is tedious, but bsiclly simple. Ech rectngle is centered t possible vlue of x; the height of the rectngle is the probbility of tht vlue divided by δ. The heights re given in Tble

2 24 CHAPTER 3. APPROXIMATING A SAMPLING DISTRIBUTION Figure 3.1: Probbility histogrm of the smpling distribution of the test sttistic for Fisher s test for the Bllerin study. Figure 3.2: Probbility histogrm of the smpling distribution of the test sttistic for Fisher s test for n unnmed unblnced study the P-vlue equls, 0.15(0.725) = (rounded). You do not need to verify ll the heights, but check enough to mke sure you know how to compute them. Finlly, we drw the probbility histogrm, s shown in Figure 3.1. Next, we consider the reverse of the bove exmple. In prticulr, if you re given probbility histogrm, the vlue of x, nd the lterntive, then you should be ble to compute the P-vlue. For exmple, Figure 3.2 is the probbility histogrm for n unblnced study. Note tht the height of ech rectngle is printed bove it. We shll use this hypotheticl exmple to illustrte severl computtions. 1. Suppose tht x = 0.25 nd the lterntive is p 1 > p 2. The P-vlue equls P (X x) = P (X 0.25). But 0.25 is the lrgest possible vlue of x; thus, P (X 0.25) = P (X = 0.25). This ltter probbility equls the re of the rectngle centered t Becuse the re of rectngle is its bse multiplied by its height, 2. Suppose tht x = 0.10 nd the lterntive is p 1 > p 2. The P-vlue equls P (X 0.10). This is the sum of two numbers: the res of the rectngles centered t 0.10 nd Thus, the P-vlue equls, 0.15(2.267) = = Suppose tht x = 0.20 nd the lterntive is p 1 < p 2. The P-vlue equls P (X 0.20). This is the sum of three numbers: the res of the rectngles centered t 0.20, 0.35, nd Thus, the P-vlue equls, 0.15( ) = 0.15(1.276) = The reminder of Chpter 3 focuses on two wys to pproximte smpling distribution, which will llow us to compute n pproximte P-vlue. The first method is simultion experiment. Consider gin the Bllerin study. It cn be shown tht there re over 126 trillion possible ssignments of subjects (spins) to tretments (directions).

3 3.1. STUDY SUGGESTIONS 25 Tble 3.2: Results of simultion experiment with 10,000 runs for the Bllerin study. Freq. Rel. Freq. x of x of x P (X = x) Totl 10,000 Tble 3.3: Results of simultion experiment with 10,000 runs for the Bllerin study. Rel. Freq. Rel. Freq. Rel. Freq. x of x of x of x Wheres the exct smpling distribution is obtined by considering ll the possible ssignments, the ide behind simultion experiment is tht we cn pproximte the smpling distribution by looking t only some of the ssignments. Exmining 10,000 ssignments usully gives n excellent pproximtion to the smpling distribution. Ech run of simultion experiment selects n ssignment t rndom from the collection of ll possible ssignments. Then the run computes the vlue of x tht would be given by the selected ssignment (remembering the ssumption tht the Skeptic is correct). I performed simulttion experiment for the Bllerin study with 10,000 runs; my results re in Tble 3.2. The first column of the tble lists the different vlues of x tht I obtined in my experiment. The second column presents the frequency of occurrence of ech vlue of x. Becuse there re 10,000 runs, ech frequency in the second column is divided by 10,000 to yield the reltive frequencies of occurrence, which re presented in the third column of the tble. The fourth column presents the exct probbilities. A creful inspection shows tht ech number in the third column is very close to the djcent number in the fourth column; thus, the reltive frequencies provide n excellent pproximtion to the smpling distribution. It will be convenient to hve cumultive sums of the reltive frequencies; these re presented in Tble 3.3. This tble cn be used to obtin n pproximte P-vlue. Recll for the Bllerin study tht x = 0.24 nd Julie chose the second lterntive. Thus, Julie s P-vlue is P (X 0.24) which cn be pproximted by the reltive frequency of x My pproximtion of Julie s P-vlue is , good pproximtion of the exct P-vlue, For second exmple, Tble 3.4 presents the results of simultion experiment for the Crohn s study. Recll tht x = 0.27 nd we chose the first lterntive. The exct P-vlue equls P (X 0.27) which cn be pproximted by the reltive frequency of x My pproximtion of the P-vlue is , n excellent pproximtion of the exct P- vlue, Section 3.3 of the text introduces the importnt ides of the center nd spred of smpling distribution. The four probbility histogrms on pge 89 of the text hve single pek (one or two rectngles wide) nd re symmetric or nerly symmetric. These histogrms differ most notbly in their mounts of spred. Of the myrid wys in which one might mesure spred, sttisticins choose the stndrd devition becuse, s demonstrted in Section 3.4, it helps us solve our problem, nmely, obtining n pproximtion to the smpling distribution.

4 26 CHAPTER 3. APPROXIMATING A SAMPLING DISTRIBUTION Tble 3.4: Results of simultion experiment with 10,000 runs for the Crohn s study. Rel. Freq. Rel. Freq. Rel. Freq. x of x of x of x Throughout the book, test sttistics nd other rndom vribles will be stndrdized. Let X be rndom vrible with popultion men denoted by µ, nd popultion stndrd devition denoted by σ. The stndrdized version of X is denoted by Z, nd is given by the eqution Z = X µ. σ If X is the test sttistic for Fisher s test, the ppliction of interest to us in Chpter 3, then µ = 0, nd the stndrdized version of X is Z = X σ. Some of my students re confused by Z. Just remember tht X is rule tht tkes the outcome of chnce mechnism nd turns it into number, nd Z is rule tht tkes the number creted by X nd turns it into nother number. For exmple, in the Infidelity study, the chnce mechnism of ssigning subjects to tretments yields (fter responses re collected) the tble below. Version S F Totl Totl The test sttistic tkes this tble s input nd yields the number ˆp 1 ˆp 2 = = Since σ = for the Infidelity study, Z tkes the number produced by X, 0.30, nd uses it to obtin 0.30/ = As illustrted for the Infidelity study in Tbles 2.2 nd 2.3 on pges 93 nd 94 of the text, knowing the smpling distribution for X is equivlent to knowing the smpling distribution for Z. Thus, the P-vlue, which in Chpter 2 is expressed in terms of X, cn just s well be expressed in terms of Z. Crrying this process one step further, the P-vlue cn be obtined from the probbility histogrm for Z, s illustrted on pges of the text. It is importnt to relize tht t this point I hve chieved nothing of vlue. Computing probbilities for Z is no esier thn computing probbilities for X. The pictures on pges of the text revel, however, tht the probbility histogrms of Z for the four studies considered look very similr. This similrity suggests tht one picture might provide good pproximtion to the smpling distribution of Z for ny of these four studies, nd perhps for other studies s well. The one picture we use is the stndrd norml curve. The two-step lgorithm on pge 102 of the text provides simple nd quick wy to use the stndrd norml curve to obtin n pproximte P-vlue for Fisher s test. Pge 102 is milestone in the text; it took us 102 pges to introduce nd solve our first ctegory of rel problems. It ws difficult journey for the number of ides, their complexity, nd the sheer length of the trip. You now hve solid foundtion, however, for the reminder of the book, hving been introduced to mny of the importnt ides nd concepts of Sttistics.

5 3.3. EXAM QUESTIONS Solutions to Odd-Numbered Exercises Solutions for Section m 1 = 35, m 2 = 15, n 1 = 25, n 2 = 25, nd n = 50; thus, σ = m 1 = 50, m 2 = 15, n 1 = 34, n 2 = 31, nd n = 65; thus, σ = m 1 = 28, m 2 = 22, n 1 = 25, n 2 = 25, nd n = 50; thus, σ = Solutions for Section The vlues x = nd σ = yield z = 0.272/ = The lternte formul yields 43[14(14) 8(8)] z = = (22)(22)(22) (The two formuls will give the sme nswer, except for round-off error.) The pproximte P- vlue for the third lterntive is twice the re to the right of z ; for z = 1.79 the re equls 2(0.0367) = ; for z = 1.78 the re equls 2(0.0375) = The vlues x = nd σ = yield z = 0.237/ = The pproximte P-vlue for the third lterntive is twice the re to the right of 2.25 under the stndrd norml curve. This re equls 2(0.0122) = The vlues x = nd σ = yield z = 0.255/ = The pproximte P-vlue for the second lterntive is the re to the right of 2.82 under the stndrd norml curve. This re equls The vlues x = 0.08 nd σ = yield z = 0.08/ = The pproximte P-vlue for the third lterntive is twice the re to the right of 0.76 under the stndrd norml curve. This re is equl to 2(0.2236) = The stndrdized version of the test sttistic is 99[36(2) 14(48)] z = = (50)(84)(16) The pproximte P-vlue for the third lterntive is twice the re to the right of 3.26 under the stndrd norml curve. This re is equl to 2(0.0006) = Exm Questions 1. I performed simultion experiment with 1,000 runs. Ech run yielded n observed vlue of the test sttistic for Fisher s test. The results of the simultion experiment re in the tble below. Observed vlue of test sttistic Frequency Totl 1,000 Use the results of this simultion experiment to obtin n pproximte P-vlue for the first lterntive if x = The smpling distribution of the test sttistic for Fisher s test for n unblnced CRD is given by the following tble. x P (X = x) P (X x) P (X x) A resercher drws the probbility histogrm of this smpling distribution. Wht is the height of the rectngle centered t? 3. Refer to the previous question. A 1000 run simultion experiment ws conducted in which

6 28 CHAPTER 3. APPROXIMATING A SAMPLING DISTRIBUTION ech run yielded vlue of the test sttistic for Fisher s test. The frequencies for the six different vlues of the test sttistic re: Mtch these frequencies with the six vlues of the test sttistic. (Hint: The frequencies nd test sttistic vlues mtch exctly s one would expect bsed on the long-run reltive frequency interprettion of probbility.) 4. Figure 3.3 on pge 85 of the text describes single run of simultion experiment for the Colloquium study. Use the informtion in the top box of the figure to determine (ccording to the Skeptic) the vlue of x if ll odd-numbered subjects (1, 3, 5,... ) re ssigned to tretment 1, nd ll even-numbered subjects (2, 4, 6,... ) re ssigned to tretment Bob enjoys plying Yhtzee ( gme in which person tosses five dice). Bob htes it when ll five dice hve different numbers showing, nd wnts to compute the probbility tht this hted event occurs. Unfortuntely, Bob is not very good t computing probbilities. Thus, Bob progrms his computer to simulte 10,000 tosses of five dice. In exctly 932 of his simulted tosses, the hted event occurs. Wht should Bob conclude bout the probbility tht the hted event occurs? Explin your nswer. 6. Eline performs blnced CRD with 50 subjects. She obtins totl of 28 successes, with 18 of the successes occurring on the second tretment. Use the stndrd norml curve to obtin the pproximte P-vlue for Fisher s test nd the second lterntive (<) for Eline s dt. 7. Refer to the previous question. () Use the stndrd norml curve to obtin the pproximte P-vlue for Fisher s test nd the first lterntive (>) for Eline s dt. (b) Use the stndrd norml curve to obtin the pproximte P-vlue for Fisher s test nd the third lterntive ( ) for Eline s dt. 8. Find the re under the stndrd norml curve between 0.72 nd Al performs CRD with 50 subjects on the first tretment, 25 subjects on the second tretment, nd obtins totl of 12 successes 10 on the first tretment, nd two on the second tretment. Use the stndrd norml curve to pproximte the P-vlue for the first lterntive (>). 10. Use the following probbility histogrm to compute P (X = 0.50) A rndom vrible X hs the following smpling distribution: x P (X = x) Totl 1.0 Construct the probbility histogrm for X. (Remember to lbel the possible vlues of X on the horizontl xis nd to specify the heights of the rectngles.) 12. Cliff performs CRD nd obtins the probbility histogrm, pictured below, of the test sttistic for Fisher s test. Given tht X = 0.5, find the exct P-vlue for the second lterntive (<)

7 3.3. EXAM QUESTIONS For n unblnced CRD, the possible vlues of the test sttistic for Fisher s test re: 0.24, 0.10, 0.04, 0.18, 0.32, 6, nd A resercher drws the probbility histogrm of the smpling distribution. Given P (X = 0.04) = , find the height of the rectngle centered t Sm performs n unblnced CRD with dichotomous response nd two tretments on 20 subjects nd obtins totl of 10 successes. True or flse? The probbility histogrm of the test sttistic for Fisher s test is symmetric bout the point A simultion experiment with 100 runs is performed. Thirty-five runs yield vlue of the rndom vrible X tht is lrger thn Wht is the simultion pproximtion of P (X 0.25)? 16. Bob performs CRD nd obtins the following dt Totl He then decides to perform simultion experiment to obtin n pproximte P-vlue. The first run of the experiment yields the following tble: Totl Wht should Bob do next? 17. Find the re under the stndrd norml curve to the right of Cliff performs CRD nd obtins the probbility histogrm, pictured below, of the test sttistic for Fisher s test. Given tht X = 1, find the exct P-vlue for the third lterntive ( ) A blnced CRD with two tretments nd dichotomous response yields 73 successes nd 47 filures. Compute the stndrd devition of the test sttistic for Fisher s test. 20. The possible vlues of the rndom vrible X re 3, 4, 5, 6, nd 7. The smpling distribution of X hs men of 5 nd stndrd devition of 0.5. Wht re the possible vlues of the stndrdized version of X? 21. A blnced CRD with two tretments nd dichotomous response yields 55 successes nd 35 filures. If 25 of the successes occur on the first tretment, compute the stndrd norml curve pproximtion to the P-vlue for Fisher s test for the second lterntive (<). 22. A blnced CRD with two tretments nd dichotomous response yields 28 successes nd 52 filures. If 18 of the successes occur on the first tretment, compute the stndrd norml curve pproximtion to the P-vlue for Fisher s test for the third lterntive ( ). 23. A controlled comprtive study yields the dt presented in the tble below Totl A 1,000 run simultion experiment yields the following results: x Frequency Totl 1,000

8 30 CHAPTER 3. APPROXIMATING A SAMPLING DISTRIBUTION () Use the bove dt nd simultion results to pproximte the P-vlue for Fisher s test with the first lterntive (>). (b) Use the bove dt nd simultion results to pproximte the P-vlue for Fisher s test with the third lterntive ( ). (c) Remember tht smpling distributions re computed nd simultion experiments re performed on the ssumption tht the Skeptic is correct. With this in mind, you cn see tht the simultion experiment did not yield ll possible vlues of the test sttistic. List ll possible vlues of the test sttistic (ccording to the Skeptic) tht re not represented in the simultion experiment. 3.4 Solutions to Exm Questions 1. The P-vlue equls P (X 0.33). This probbility is pproximted by the reltive frequency of simulted vlues tht re greter thn or equl to The reltive frequency equls δ = 0.3, so the height equls /0.3 = The correspondence is: Vlue Freq The 2 2 tble is below Totl Thus, x = 6/14 2/14 = 4/14 = 2/7 = The reltive frequency of occurrence of the hted event is Thus, the probbility of the hted event is pproximtely (not exctly!) equl to by n ppliction of the longrun reltive frequency interprettion of probbility. 6. Eline s 2 2 tble is below Totl Thus, x = 0.32 nd σ = , giving z = 0.32/ = The pproximte P- vlue for the second lterntive is () (b) The re equls 9. Al s tbles re below = Totl By subtrction, x = Further, σ = 12(63) 50(25)(74) = Thus, z = 0.12/ = 1.33 nd the pproximte P-vlue equls () = The probbility histogrm for X is below (0.2) + 0.5() = 0.3.

9 3.5. MORE MATHEMATICS /0.14 = True Bob should check his method of simultion. It is obviously flwed since every simultion run must yield totl of 21 successes nd 19 filures, just s the ctul study did (0.5)(0.2) = σ = The possible vlues re 4, 2, 0, 2, nd The success rtes re ˆp 1 = nd ˆp 2 = 0.667, yielding x = In ddition, 55(35) σ = 45(45)(89) = Thus, z = 0.111/ = 1.07, nd the pproximte P-vlue is The success rtes re ˆp 1 = 50 nd ˆp 2 = 0.250, yielding x = In ddition, σ = 28(52) 40(40)(79) = Thus, z = 0.200/ = 1.86, nd the pproximte P-vlue for the third lterntive is 2(0.0314) = () The dt yield ˆp 1 = 0.35 nd ˆp 2 = Thus, x = The P-vlue equls P (X 0.25) which is pproximted by the reltive frequency of (X 0.25). This reltive frequency equls ( )/1000 = (b) For the third lterntive, equls the P-vlue P (X 0.25) + P (X 0.25) = P (X 0.35) + P (X 0.25), which is pproximted by Rel. freq. (X 0.35)+ Rel. freq. (X 0.25), This sum of reltive frequencies equls ( )/1000 = (c) There re nine possible vlues of the test sttistic becuse the position in the contingency tble cn be filled with 0, 1, 2, 3, 4, 5, 6, 7, or 8. It is esy to check tht if = 0, then x = 0.80 nd if = 1, then x = The other seven possible vlues re represented in the simultion experiment. 3.5 More Mthemtics I will continue the presenttion of Section 2.5 of this guide. Recll tht we hve derived the hypergeometric formul which gives probbilities for the rndom vrible A. Of course, 1 = P (A = ) = C(m 1, )C(n m 1, n 1 ). C(n, n 1 ) It is importnt to understnd this identity in the following wy. The function C hs two rguments nd ppers three times in this formul. The rguments tht vry with the summtion re the second rguments in the numertor terms. The sum of the first rguments in the numertor equls the first rgument in the denomintor, nd the sum of the second rguments in the numertor equls the second rgument in the denomintor. Section 3.6 of the text defines the men of the smpling distribution of A s µ = P (A = ). In order to obtin the vlue of µ, it is helpful to note tht for t 1, C(s, t) = s C(s 1, t 1). t

10 32 CHAPTER 3. APPROXIMATING A SAMPLING DISTRIBUTION For lter use note tht for t 2, Thus, C(s, t) = µ = s(s 1) C(s 2, t 2). t(t 1) P (A = ) = C(m 1, )C(n m 1, n 1 ). C(n, n 1 ) The summnd equls 0 if = 0. For 1 use the bove identity to write this lst sum s m 1 n 1 n C(m 1 1, 1)C(n m 1, n 1 ) C(n 1, n 1 1) m 1 n 1 n. Next, the vrince of A equls E[(A µ) 2 ] = E(A 2 2µA + µ 2 ). Since the opertion of expecttion is simply specil kind of summtion, it inherits the linerity properties of summtion. In prticulr, this lst sum equls E(A 2 ) 2µE(A) + µ 2 = E(A 2 ) 2µ 2 + µ 2 = Note lso tht Thus, Finlly, E(A 2 ) µ 2. E[A(A 1)] = E(A 2 ) µ. Vr(A) = E[A(A 1)] + µ µ 2. E[A(A 1)] = ( 1)P (A = ) = ( 1) C(m 1, )C(n m 1, n 1 ). C(n, n 1 ) The summnd equls 0 if = 0 or = 1. For 2 use the erlier identity to write this lst sum s ( 1) m 1(m 1 1) n 1 (n 1 1) ( 1) n(n 1) = C(m 1 2, 2)C(n m 1, n 1 ) C(n 2, n 1 2) m 1 (m 1 1)n 1 (n 1 1). n(n 1) Substituting the bove vlues into Vr(A) = E[A(A 1)] + µ µ 2 nd simplifying yields (detils re left to the reder) Vr(A) = n 1n 2 m 1 m 2 n 2 (n 1). All tht remins is to trnslte these results for A to the results in the textbook for X. To this end, we need two results. Let Y be ny rndom vrible nd let c 1 nd c 2 be ny numbers. E(c 1 + c 2 Y ) = c 1 + c 2 E(Y ). Proof: The left hnd side of the eqution equls (c 1 + c 2 yp (Y = y)). y The result follows from the linerity of summtion. Let Y be ny rndom vrible nd let c 1 nd c 2 be ny numbers. Vr(c 1 + c 2 Y ) = c 2 2Vr(Y ). Proof: Let W = c 1 +c 2 Y, nd let µ denote the men of Y. By the definition of vrince nd the preceding result, Vr(W ) = E[(c 1 + c 2 Y c 1 c 2 µ) 2 ] = E[(c 2 Y c 2 µ) 2 ] = c 2 2Vr(Y ), s desired. As stted erlier, X = c 1 + c 2 A, with c 1 = m 1, nd c 2 = n. n 2 n 1 n 2 Thus, E(X) = m 1 + n n 1 m 1 n 2 n 1 n 2 n = 0. Vr(X) = n2 n 1 n 2 m 1 m 2 n 2 1 n2 2 n 2 (n 1) = m 1 m 2 n 1 n 2 (n 1). These vlues for the men nd vrince of X gree with the nswers given in the text. =