I n many cases, the SPRT will come to a decision with fewer samples than would have been required for a fixed size test.

Transcription

1 STATGRAPHICS Rev. 9/6/3 Sequential Saling Suary... Data Inut... 3 Analysis Otions... 3 Analysis Suary... 5 Cuulative Plot... 6 Decision Nubers... 9 Test Perforance... O. C. Curve... ASN Function... Forulas... Suary The Sequential Saling rocedure ileents various Sequential Probability Ratio Tests (SPRTs). Unlie statistical tests which have a fixed sale size, the nuber of sales required by sequential tests is not redeterined. Instead, after each sale is taen, one of 3 decisions is ade:. Sto the test and reject the null hyothesis.. Sto the test and accet the null hyothesis. 3. Continue saling. I n any cases, the SPRT will coe to a decision with fewer sales than would have been required for a fixed size test. Sale StatFolio: wald.sg 3 by StatPoint Technologies, Inc. Sequential Saling -

2 Sale Data: STATGRAPHICS Rev. 9/6/3 The file wald.sgd contains a set of observations resented in the classic boo by Wald (947). The data are shown in the following table: Sale Measureent It is assued that the data are rando sales fro a noral distribution with nown standard deviation = 5. It is desired to test the following hyotheses concerning the ean of the distribution: Null hyothesis: = 35 Alternative hyothesis: = 5 3 by StatPoint Technologies, Inc. Sequential Saling -

3 STATGRAPHICS Rev. 9/6/3 Data Inut The data inut dialog box requests the nae of the colun containing the easureents: Data variable: an otional colun containing the data that has been collected so far. If no data has yet been collected, this field ay be left blan. Select: subset selection. Analysis Otions After the data has been secified, the Analysis Otions dialog box is dislayed in order to secify the test to be conducted: 3 by StatPoint Technologies, Inc. Sequential Saling - 3

4 STATGRAPHICS Rev. 9/6/3 Test araeter: the araeter whose value is secified by the null and alternative hyotheses. Otions are: o Noral ean (siga nown): the ean of a noral distribution, assuing that the value of the standard deviation is nown. The value of siga ust be secified. o Noral ean (siga unnown): the ean of a noral distribution, assuing that the value of the standard deviation is not nown. The value of siga will be estiated fro the data. o Noral siga (ean nown): the standard deviation of a noral distribution, assuing that the value of the ean is nown. The value of the ean ust be secified. o Noral siga (ean unnown): the standard deviation of a noral distribution, assuing that the value of the ean is not nown. The value of the ean will be estiated fro the data. 3 by StatPoint Technologies, Inc. Sequential Saling - 4

5 STATGRAPHICS Rev. 9/6/3 o Binoial roortion : the robability of an event for the binoial distribution. If this otion is selected, all data values ust be either or. o Poisson rate : the rate araeter or ean of the Poisson distribution. If this otion is selected, all data values ust be non-negative integers. o Negative binoial ean: the ean of the negative binoial distribution. If this otion is selected, all data values ust be non-negative integers. This distribution is often used in lace of the Poisson distribution for overdisersed counts (counts for which the variance is greater than the ean). The value of the araeter, (an integer greater than or equal to ), ust be secified. Null hyothesis H: the value of the araeter secified by the null hyothesis. Null hyothesis alha ris: the -ris of the test. This is the robability that the test will end with an incorrect rejection of the null hyothesis. Alternative hyothesis H: the value of the araeter secified by the alternative hyothesis. Alternative hyothesis beta ris: the -ris of the test. This is the robability that the test will end with an incorrect accetance of the null hyothesis. Two-sided test: whether the test is two-sided or one-sided (available when testing the noral ean only). If a two-sided test is selected, the alternative hyothesis is assued to consist of two values equally saced around the null hyothesis. Analysis Suary The Analysis Suary dislays sale statistics for the data (if any), a suary of the hyothesis test, and the status of the test: Sequential Analysis (One Sale) Data variable: Measureent Count Average 33.5 Median 37.5 Standard deviation 6.49 Miniu 4. Maxiu 6. Stnd. sewness Stnd. Kurtosis Hyothesis Test Null hyothesis ean = 35. alha ris =. Alternative hyothesis ean = 5. beta ris =.3 Known siga = 5. Decision: reject null hyothesis at sale After each data value, a decision is ade to accet or reject the null hyothesis if there is sufficient inforation to sto the test. Otherwise, the rogra will indicate that continued 3 by StatPoint Technologies, Inc. Sequential Saling - 5

6 Cuulative su STATGRAPHICS Rev. 9/6/3 saling (ore data) is required. Tyically, observations are collected and added to the data colun one at a tie until enough data have been collected to sto the test. Cuulative Plot The Cuulative Plot illustrates the decision regions aroriate for a sequential saling lan. For the current exale, it lots the cuulative su of the data values after sales have been taen. Let X i be the observed data value for the i-th sale. Then the cuulative su is defined by X i i () In the lot below, these sus are lotted versus sale nuber: (X ) 4 Sequential Saling Plot for Measureent Sale Also included on the lot are three regions:. A rejection region, colored red. If the cuulative su enters the rejection region, saling is stoed and the null hyothesis is rejected.. An accetance region, colored green. If the cuulative su enters the accetance region, saling is stoed and the null hyothesis is acceted. 3. A white region. As long as the cuulative su reains in this region, additional sales ust be collected. The oint at which the statistic first enters the accetance or rejection region, at which further data collection would sto, is indicated by a red asteris. For the exale data, saling would 3 by StatPoint Technologies, Inc. Sequential Saling - 6

7 Cuulative deviation STATGRAPHICS Rev. 9/6/3 continue until = sales have been collected, at which tie the null hyothesis would be acceted. If a two-sided test of were selected, the lot would show instead the cuulative su of deviations fro the null hyothesis, defined by X i i () The rejection region then consist of two sections, one for alternative hyotheses greater than the null and another for alternative hyotheses less than the null. A tyical exale is shown below: Sequential Saling Plot for Measureent Sale In the lot above, saling would continue beyond =, since all of the oints are within the white zone. If the value of siga was not nown, the SPRT erfored would be a sequential t-test. In such a test, the quantity lotted is the log lielihood ratio defined by t t f, (3) f, where t is the standard t-statistic calculated fro the sale ean and sale standard deviation x t (4) s / 3 by StatPoint Technologies, Inc. Sequential Saling - 7

8 Log lielihood ratio STATGRAPHICS Rev. 9/6/3 and f(t,) is the robability density function for the non-central t-distribution with degrees of freedo and non-centrality araeter (5) s / n The decision liits are then two arallel lines as shown below: Sequential Saling Plot - Measureent Sale In the lot above, the null hyothesis would have been acceted after 6 sales were collected, since the log lielihood ratio enters the green accetance zone at that oint. Pane Otions Miniu n to dislay: The lot can be extended beyond the end of the data by secifying a larger sale size. 3 by StatPoint Technologies, Inc. Sequential Saling - 8

9 STATGRAPHICS Rev. 9/6/3 Decision Nubers This table dislays the value of the lotted statistic and corresonding accetance and rejection values after each sale is collected: Decision Nubers Accetance Rejection Sale Cuulative su Nuber Nuber If the data leads to accetance or rejection of the null hyothesis, the value of the statistic at which that occurs is shown in red. Pane Otions Miniu n to dislay: The table can be extended beyond the end of the data by secifying a larger sale size. 3 by StatPoint Technologies, Inc. Sequential Saling - 9

10 STATGRAPHICS Rev. 9/6/3 Test Perforance The Test Perforance table dislays the roerties of the sequential test: Test Perforance Mean Prob(accet null) Average sale nuber Null hy Alt. hy Sale size for fixed test n = 5 The table has rows for various values of the araeter being tested. Two iortant quantities are dislayed:. Prob(accet null) This is the robability that the test will lead to accetance of the null hyothesis. The alha and beta ris set this robability when the ean exactly equals the null and alternative hyotheses, resectively. At values between the null and the alternative, the robability will vary.. Average sale nuber The average nuber of sales that ust be taen before the test either accets or rejects the null hyothesis. The table also shows the sale size required by a test with redeterined sale size. In ost cases, the average sale size for the sequential test will be saller than for the fixed size test. O. C. Curve The O. C. Curve or Oerating Characteristic Curve lots the robability that the null hyothesis will be acceted versus various values of the araeter being tested: 3 by StatPoint Technologies, Inc. Sequential Saling -

11 Average sale nuber Prob(accet null) STATGRAPHICS Rev. 9/6/3 OC Curve Mean This curve asses through the oints (, ) and (,). ASN Function The ASN Function or Average Sale Nuber Function lots the average sale size required before the null hyothesis is either acceted or rejected as a function of the true value of the araeter being tested: ASN Function Mean A horizontal line is also dislayed at the sale size required for a fixed size test. 3 by StatPoint Technologies, Inc. Sequential Saling -

12 Forulas STATGRAPHICS Rev. 9/6/3 Test of Noral Mean (One-Sided, Siga Known) Test statistic: X i i (6) Accetance nuber: (7) Rejection nuber: (8) Test of Noral Mean (Two-Sided, Siga Known) Test statistic: X i i (9) Accetance nuber: / () Rejection nuber: / () where () Test of Noral Mean (One-Sided, Siga Unnown) Test statistic: t t f, (3) f, Accetance nuber: T (4) Rejection nuber: T (5) 3 by StatPoint Technologies, Inc. Sequential Saling -

13 Test of Noral Mean (Two-Sided, Siga Unnown) STATGRAPHICS Rev. 9/6/3 Test statistic: t t f, (6) f, Accetance nuber: T (7) / Rejection nuber: T (8) / Test of Noral Siga (Mean Known) Test statistic: X i i (9) Accetance nuber: () Rejection nuber: () Test of Noral Siga (Mean Unnown) This case uses the sae statistic as the test which assues that the ean is nown. The accetance and rejection nubers after sales have been collected corresond to the accetance and rejection nubers after - sales when the ean is nown. Test of Binoial Proortion Test statistic: X i i () 3 by StatPoint Technologies, Inc. Sequential Saling - 3

14 STATGRAPHICS Rev. 9/6/3 3 by StatPoint Technologies, Inc. Sequential Saling - 4 Accetance nuber: (3) Rejection nuber: (4) Test of Poisson Rate Test statistic: i X i (5) Accetance nuber: (6) Rejection nuber: (7) Test of Negative Binoial Mean Test statistic: i X i (8) Accetance nuber: (9)

15 STATGRAPHICS Rev. 9/6/3 3 by StatPoint Technologies, Inc. Sequential Saling - 5 Rejection nuber: (3) Oerating Characteristic Curve h h h L ) ( (3) where h() is a quantity that deends on the value of the unnown araeter (see Wald, 947). Average Sale Nuber,, ) ( ) ( ) ( x f x f E L L n E (3) where f(x,) is the robability density or ass function for the data variable.