------------------------------------------------------------------------------ UnifyPow Workshop and SUGI 23 Presentation Mystic Michelle: Ordinary (pi = .50) or Gifted (pi = .80)? Testing a single proportion -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: PI .80 NULL .50 NTOTAL 20 40 ALPHA .01 .05 Testing Ho: pi = 0.5 NOTE: SETTING 2-TAILED CRITICAL REGIONS FOR THE BINOMIAL DISTRIBUTION. Denote the critical regions for a 2-tailed test as "major" and "minor" depending on which one is consistent with the true pi. Thus for Ho: pi = .35 with a conjecture of true pi = .20, the major critical region would be in the lower tail of the binomial(Ntotal, .35) distribution. Let alpha_major and alpha_minor be the Type I error rates in these tails. UnifyPow first finds the largest minor critical region such that alpha_minor LE alpha/2, where "LE" stands for "less than or equal." Then it finds the largest major region such that alpha_major LE alpha - alpha_minor. This ensures that alpha_minor + alpha_major LE alpha, and yet favors the major tail, thus increasing power. The critical values tabled below are in the rejection region. For example, the alpha = .05, two-tailed test of Ho: pi = .35 with NTotal = 40 gives lower and upper critical values of 8 and 21 if the conjectured true pi is less than .35. Thus, the major critical region is r = 0, 1, ..., 8 and the minor one is r = 21, 22, ..., 40. If the conjectured true pi exceeds .35, then the major region is r = 20, 21, ..., 40 and the minor one is r = 0, 1, ..., 7. ------------------------------------------------------------------------------ UnifyPow Workshop and SUGI 23 Presentation Mystic Michelle: Ordinary (pi = .50) or Gifted (pi = .80)? Testing a single proportion Scenario: pi .80 . Mystic Michelle claims 80% accuracy. ------------------------------------------------ | | ALPHA | | |-------------------| | | 0.01 | 0.05 | | |---------+---------| | | Total N | Total N | | |---------+---------| | | 20 | 40 | 20 | 40 | | |----+----+----+----| | |Pow-|Pow-|Pow-|Pow-| | | er | er | er | er | |--------------------------+----+----+----+----| |Method |Statistic | | | | | |------------+-------------| | | | | |Exact |2-tail bnml |.630|.912|.804|.981| |Binomial |-------------+----+----+----+----| | |1-tail bnml |.630|.957|.804|.992| ------------------------------------------------ ------------------------------------------------------------------------------ UnifyPow Workshop and SUGI 23 Presentation Mystic Michelle: Ordinary (pi = .50) or Gifted (pi = .80)? Testing a single proportion Critical values and actual alpha levels using binomial distribution. ---------------------------------------------------------------------- | | ALPHA | | |-----------------------------------------| | | 0.01 | 0.05 | | |--------------------+--------------------| | | |Lower |Upper | |Lower |Upper | | |Actual| Crit | Crit |Actual| Crit | Crit | | |Alpha |Value |Value |Alpha |Value |Value | |--------------------------+------+------+------+------+------+------| |Method |Total N |Type | | | | | | | |--------+--------+--------| | | | | | | |Exact |20 |2-tail | | | | | | | |Binomial| |bnml | 0.007| 3| 16| 0.041| 5| 15| | | |--------+------+------+------+------+------+------| | | |1-tail | | | | | | | | | |bnml | 0.006| .| 16| 0.021| .| 15| | |--------+--------+------+------+------+------+------+------| | |40 |2-tail | | | | | | | | | |bnml | 0.006| 11| 29| 0.038| 13| 27| | | |--------+------+------+------+------+------+------| | | |1-tail | | | | | | | | | |bnml | 0.008| .| 28| 0.040| .| 26| ---------------------------------------------------------------------- These critical values are part of the rejection region. The note above describes how they are set. ------------------------------------------------------------------------------ UnifyPow Workshop and SUGI 23 Presentation Mystic Michelle: Ordinary (pi = .50) or Gifted (pi = .80)? Testing a single proportion -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: PI .80 POWER .99 .995 ALPHA .005 TAILS 1 Testing Ho: pi = 0.5 NOTE: SETTING 2-TAILED CRITICAL REGIONS FOR THE BINOMIAL DISTRIBUTION. Denote the critical regions for a 2-tailed test as "major" and "minor" depending on which one is consistent with the true pi. Thus for Ho: pi = .35 with a conjecture of true pi = .20, the major critical region would be in the lower tail of the binomial(Ntotal, .35) distribution. Let alpha_major and alpha_minor be the Type I error rates in these tails. UnifyPow first finds the largest minor critical region such that alpha_minor LE alpha/2, where "LE" stands for "less than or equal." Then it finds the largest major region such that alpha_major LE alpha - alpha_minor. This ensures that alpha_minor + alpha_major LE alpha, and yet favors the major tail, thus increasing power. The critical values tabled below are in the rejection region. For example, the alpha = .05, two-tailed test of Ho: pi = .35 with NTotal = 40 gives lower and upper critical values of 8 and 21 if the conjectured true pi is less than .35. Thus, the major critical region is r = 0, 1, ..., 8 and the minor one is r = 21, 22, ..., 40. If the conjectured true pi exceeds .35, then the major region is r = 20, 21, ..., 40 and the minor one is r = 0, 1, ..., 7. ------------------------------------------------------------------------------ UnifyPow Workshop and SUGI 23 Presentation Mystic Michelle: Ordinary (pi = .50) or Gifted (pi = .80)? Testing a single proportion Same problem, but now find minimum N to achieve specified power at given alphas. ------------------------------------------------------------------------------ UnifyPow Workshop and SUGI 23 Presentation Mystic Michelle: Ordinary (pi = .50) or Gifted (pi = .80)? Testing a single proportion Scenario: pi .80 ------------------------------------------ | | ALPHA | | |-------------| | | 0.005 | | |-------------| | |Minimum Power| | |-------------| | | .990 | .995 | | |------+------| | |Total |Total | | | N | N | |--------------------------+------+------| |Method |Statistic | | | |------------+-------------| | | |Exact |1-tail bnml | | | |Binomial | | 61| 69| ------------------------------------------ ------------------------------------------------------------------------------ UnifyPow Workshop and SUGI 23 Presentation Mystic Michelle: Ordinary (pi = .50) or Gifted (pi = .80)? Testing a single proportion Critical values and actual alpha levels using binomial distribution. ------------------------------------------------- | | ALPHA | | |--------------------| | | 0.005 | | |--------------------| | | |Lower |Upper | | |Actual| Crit | Crit | | |Alpha |Value |Value | |--------------------------+------+------+------| |Method |Minimum |Type | | | | |--------|Power | | | | | |Exact |--------+--------| | | | |Binomial|.990 |1-tail | | | | | | |bnml | 0.005| .| 41| | |--------+--------+------+------+------| | |.995 |1-tail | | | | | | |bnml | 0.004| .| 46| ------------------------------------------------- These critical values are part of the rejection region. The note above describes how they are set. ------------------------------------------------------------------------------ UnifyPow Workshop DCA for Lactic Acidosis in Children with Severe Malaria Uncontrolled pilot study, comparing mortality to 28% -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: PI .14 NULL .28 ALPHA .05 .20 NTOTAL 40 60 100 Testing Ho: pi = 0.28 NOTE: SETTING 2-TAILED CRITICAL REGIONS FOR THE BINOMIAL DISTRIBUTION. Denote the critical regions for a 2-tailed test as "major" and "minor" depending on which one is consistent with the true pi. Thus for Ho: pi = .35 with a conjecture of true pi = .20, the major critical region would be in the lower tail of the binomial(Ntotal, .35) distribution. Let alpha_major and alpha_minor be the Type I error rates in these tails. UnifyPow first finds the largest minor critical region such that alpha_minor LE alpha/2, where "LE" stands for "less than or equal." Then it finds the largest major region such that alpha_major LE alpha - alpha_minor. This ensures that alpha_minor + alpha_major LE alpha, and yet favors the major tail, thus increasing power. The critical values tabled below are in the rejection region. For example, the alpha = .05, two-tailed test of Ho: pi = .35 with NTotal = 40 gives lower and upper critical values of 8 and 21 if the conjectured true pi is less than .35. Thus, the major critical region is r = 0, 1, ..., 8 and the minor one is r = 21, 22, ..., 40. If the conjectured true pi exceeds .35, then the major region is r = 20, 21, ..., 40 and the minor one is r = 0, 1, ..., 7. ------------------------------------------------------------------------------ UnifyPow Workshop DCA for Lactic Acidosis in Children with Severe Malaria Uncontrolled pilot study, comparing mortality to 28% Scenario: pi .14 . DCA cuts mortality 50%. ---------------------------------------------------------- | | ALPHA | | |-----------------------------| | | 0.05 | 0.2 | | |--------------+--------------| | | Total N | Total N | | |--------------+--------------| | | 40 | 60 |100 | 40 | 60 |100 | | |----+----+----+----+----+----| | |Pow-|Pow-|Pow-|Pow-|Pow-|Pow-| | | er | er | er | er | er | er | |--------------------------+----+----+----+----+----+----| |Method |Statistic | | | | | | | |------------+-------------| | | | | | | |Exact |2-tail bnml |.504|.788|.939|.811|.931|.990| |Binomial |-------------+----+----+----+----+----+----| | |1-tail bnml |.676|.788|.964|.902|.965|.995| ---------------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop DCA for Lactic Acidosis in Children with Severe Malaria Uncontrolled pilot study, comparing mortality to 28% Critical values and actual alpha levels using binomial distribution. ---------------------------------------------------------------------- | | ALPHA | | |-----------------------------------------| | | 0.05 | 0.2 | | |--------------------+--------------------| | | |Lower |Upper | |Lower |Upper | | |Actual| Crit | Crit |Actual| Crit | Crit | | |Alpha |Value |Value |Alpha |Value |Value | |--------------------------+------+------+------+------+------+------| |Method |Total N |Type | | | | | | | |--------+--------+--------| | | | | | | |Exact |40 |2-tail | | | | | | | |Binomial| |bnml | 0.033| 5| 18| 0.161| 7| 16| | | |--------+------+------+------+------+------+------| | | |1-tail | | | | | | | | | |bnml | 0.043| 6| .| 0.171| 8| .| | |--------+--------+------+------+------+------+------+------| | |60 |2-tail | | | | | | | | | |bnml | 0.046| 10| 25| 0.196| 12| 22| | | |--------+------+------+------+------+------+------| | | |1-tail | | | | | | | | | |bnml | 0.030| 10| .| 0.172| 13| .| | |--------+--------+------+------+------+------+------+------| | |100 |2-tail | | | | | | | | | |bnml | 0.045| 19| 38| 0.184| 22| 35| | | |--------+------+------+------+------+------+------| | | |1-tail | | | | | | | | | |bnml | 0.044| 20| .| 0.158| 23| .| ---------------------------------------------------------------------- These critical values are part of the rejection region. The note above describes how they are set. ------------------------------------------------------------------------------ UnifyPow Workshop Placebo vs. DCA for Lactic Acidosis in Children with Malaria 28% die untreated. What if DCA cuts this by 25%? -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: PI .28 .21 WEIGHT 1 2 ALPHA .01 .045 NTOTAL 750 999 1500 Testing Ho: pi1 - pi2 = 0 ------------------------------------------------------------------------------ UnifyPow Workshop Placebo vs. DCA for Lactic Acidosis in Children with Malaria 28% die untreated. What if DCA cuts this by 25%? o Actual study of children with _severe_ malaria. o 28% base mortality rate supported by meta-analysis of published surveys. o 25% reduction in mortality supported by animal-model study. o Small pilot study had 2/10 deaths in each group. o 2:1 randomization to DCA pleases local health officials. o One interim look using alpha = .01 & final analysis at .045. o Able to randomize 1500. Should interim look come at 750 or 999? ------------------------------------------------------------------------------ UnifyPow Workshop Placebo vs. DCA for Lactic Acidosis in Children with Malaria 28% die untreated. What if DCA cuts this by 25%? Scenario: pi .28 .21 ---------------------------------------------------------- | | ALPHA | | |-----------------------------| | | 0.01 | 0.045 | | |--------------+--------------| | | Total N | Total N | | |--------------+--------------| | |750 |999 |1500|750 |999 |1500| | |----+----+----+----+----+----| | |Pow-|Pow-|Pow-|Pow-|Pow-|Pow-| | | er | er | er | er | er | er | |--------------------------+----+----+----+----+----+----| |Method |Statistic | | | | | | | |------------+-------------| | | | | | | |Approximate |2-tld t apr |.331|.458|.674|.554|.680|.847| |Uncondit'l |-------------+----+----+----+----+----+----| |"chi^2*" |1-tld t apr |.426|.557|.759|.672|.781|.909| |------------+-------------+----+----+----+----+----+----| |Exact |2-tld t apr |.307|.427|.639|.527|.651|.823| |Uncondit'l**|-------------+----+----+----+----+----+----| | |1-tld t apr |.399|.526|.728|.647|.757|.892| |------------+-------------+----+----+----+----+----+----| |Fisher's |2-tld aprx |.288|.412|.633|.505|.636|.819| |exact |-------------+----+----+----+----+----+----| |conditional |1-tld aprx |.378|.511|.722|.626|.745|.889| |------------+-------------+----+----+----+----+----+----| |Likhd Ratio |2-tail Z |.323|.447|.661|.544|.669|.838| |for Log Odds|-------------+----+----+----+----+----+----| |Ratio |1-tail Z |.417|.546|.747|.663|.772|.903| ---------------------------------------------------------- *The Approximate Unconditional corresponds to the Ordinary Pearson chi-square test for a 2 x 2 table. Technically, the method here uses a regular t test with Y = 0 (no) or 1 (yes), which is known to offer more accurate p-levels and can be done with any standard t-test routine. See D'Agostino, Chase, and Belanger (1988), American Statistician, 1988, 42:198-202. **The Exact Unconditional corresponds to the test proposed by Suissa and Shuster (1985), J Royal Stat Soc A, 148:317-327). ------------------------------------------------------------------------------ UnifyPow Workshop Do the Biostat Boys use fair dice? -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: GOODNESSOFFIT .042 .067 .092 .116 .142 .166 .125 .100 .075 .050 .025 NULL .028 .056 .083 .111 .139 .166 .139 .111 .083 .056 .028 ALPHA .05 .10 POWER .80 .90 Testing goodness of fit: Ho: actual probability distribution has "null" form. ------------------------------------------------------------------------------ UnifyPow Workshop Do the Biostat Boys use fair dice? A trio of shady characters called the Biostat Boys run a craps game at lunch near the FDA. One of the regular patrons, Lucky Luke, has been losing lately and wonders if the dice are fair. The known distribution for a fair pair of dice is: X: 2 3 4 5 6 7 8 9 10 11 12 p(X): .028 .056 .083 .111 .139 .166 .139 .111 .083 .056 .028 Luke believes that one dice is weighted so that the "1" comes up 1/4 of the time, not 1/6. So he computes that the distribution may be: X: 2 3 4 5 6 7 8 9 10 11 12 p(X): .042 .067 .092 .116 .142 .166 .125 .100 .075 .050 .025 How many tosses must he observe in order to have 90% power (alpha = .05) to find such a discrepancy, based on the full distribution? ------------------------------------------------------------------------------ UnifyPow Workshop Do the Biostat Boys use fair dice? Scenario: {0.042 0.067 0.092 0.116 0.142 0.166 0.125 0.1 0.075 0.05 0.025} -------------------------------------------------------- | | ALPHA | | |---------------------------| | | 0.05 | 0.1 | | |-------------+-------------| | |Minimum Power|Minimum Power| | |-------------+-------------| | | .800 | .900 | .800 | .900 | | |------+------+------+------| | |Total |Total |Total |Total | | | N | N | N | N | |--------------------------+------+------+------+------| |Method |Statistic | | | | | |------------+-------------| | | | | |Ordinary |Chi-square | | | | | |Pearson | | 1109| 1402| 913| 1187| |------------+-------------+------+------+------+------| |Likhd Ratio |Chi-square | 1186| 1500| 976| 1270| -------------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Do the Biostat Boys use fair dice? -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: GOODNESSOFFIT .625 .333 .042 NULL .694 .278 .028 ALPHA .05 .10 POWER .80 .90 Testing goodness of fit: Ho: actual probability distribution has "null" form. ------------------------------------------------------------------------------ UnifyPow Workshop Do the Biostat Boys use fair dice? What if Lucky Luke only counts how many "1"s appear on a throw? The distributions are X: 0 1 2 fair p(X): .694 .278 .028 biased p(X): .625 .333 .042 ------------------------------------------------------------------------------ UnifyPow Workshop Do the Biostat Boys use fair dice? Scenario: {0.625 0.333 0.042} -------------------------------------------------------- | | ALPHA | | |---------------------------| | | 0.05 | 0.1 | | |-------------+-------------| | |Minimum Power|Minimum Power| | |-------------+-------------| | | .800 | .900 | .800 | .900 | | |------+------+------+------| | |Total |Total |Total |Total | | | N | N | N | N | |--------------------------+------+------+------+------| |Method |Statistic | | | | | |------------+-------------| | | | | |Ordinary |Chi-square | | | | | |Pearson | | 390| 512| 312| 424| |------------+-------------+------+------+------+------| |Likhd Ratio |Chi-square | 413| 542| 330| 448| -------------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Variation in sarcoma type by region -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: 2WAYCONTTABLE .50 .20 .30 > .60 .25 .15 > .35 .35 .30 > .45 .20 .35 WEIGHT .30 .20 .25 .25 ALPHA .01 .05 POWER .90 .95 Testing group x category independence: Ho: all groups have same probability distribution for outcome ------------------------------------------------------------------------------ UnifyPow Workshop Variation in sarcoma type by region There are 3 different types of soft-tissue sarcomas of the arms and legs: o Fibroid o Lipoid o Mixed (or other) Incidence data can be obtained from good cancer registries. The Question: Do the relative proportions of these 3 types differ among 4 geographic regions? Scenario- SARCOMA SOFT TISSUE TYPE % OF TOTAL ------------------------- SAMPLE REGION Fibroid Lipoid Mixed ---------- A .50 .20 .30 30% B .60 .25 .15 20% C .35 .35 .30 25% D .45 .20 .35 25% How large must Ntotal be to have 90% or 95% power? ------------------------------------------------------------------------------ UnifyPow Workshop Variation in sarcoma type by region Scenario: {0.5 0.2 0.3} v. {0.6 0.25 0.15} v. {0.35 0.35 0.3} v. {0.45 0.2 0.35} -------------------------------------------------------- | | ALPHA | | |---------------------------| | | 0.01 | 0.05 | | |-------------+-------------| | |Minimum Power|Minimum Power| | |-------------+-------------| | | .900 | .950 | .900 | .950 | | |------+------+------+------| | |Total |Total |Total |Total | | | N | N | N | N | |--------------------------+------+------+------+------| |Method |Statistic | | | | | |------------+-------------| | | | | |Ordinary |Chi-square | | | | | |Pearson | | 500| 580| 380| 440| |------------+-------------+------+------+------+------| |Likhd Ratio |Chi-square | 480| 560| 360| 420| -------------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Two-Group Design Traditional t-Tests -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: MU -.86 -.42 SD .45 .57 .65 ALPHA .05 .01 WEIGHT 2 1 NTOTAL 15 21 33 Testing location difference between 2 groups: Ho: mu1 - mu2 = 0 ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Two-Group Design Traditional t-Tests Scenario represents reductions of 45% vs. 25%. ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Two-Group Design Traditional t-Tests Scenario: mu -.86 -.42 Ordinary t test ------------------------------------------------------------------------- | | Standard Deviation | | |--------------------------------------------| | | 0.45 | 0.57 | 0.65 | | |--------------+--------------+--------------| | | Total N | Total N | Total N | | |--------------+--------------+--------------| | | 15 | 21 | 33 | 15 | 21 | 33 | 15 | 21 | 33 | | |----+----+----+----+----+----+----+----+----| | |Pow-|Pow-|Pow-|Pow-|Pow-|Pow-|Pow-|Pow-|Pow-| | | er | er | er | er | er | er | er | er | er | |--------------------------+----+----+----+----+----+----+----+----+----| |Alpha |Type | | | | | | | | | | |------------+-------------| | | | | | | | | | |0.05 |2-tail t |.380|.518|.727|.257|.353|.526|.209|.284|.427| | |-------------+----+----+----+----+----+----+----+----+----| | |1-tail t |.519|.652|.828|.379|.485|.655|.318|.407|.559| |------------+-------------+----+----+----+----+----+----+----+----+----| |0.01 |2-tail t |.156|.260|.472|.091|.147|.276|.068|.108|.201| | |-------------+----+----+----+----+----+----+----+----+----| | |1-tail t |.235|.358|.581|.145|.219|.372|.112|.167|.283| ------------------------------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Two-Group Design Traditional t-Tests -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: MU -.86 -.42 SD .45 .57 .65 ALPHA .05 .01 WEIGHT 2 1 POWER .80 .90 Testing location difference between 2 groups: Ho: mu1 - mu2 = 0 ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Two-Group Design Traditional t-Tests Scenario: mu -.86 -.42 Ordinary t test ---------------------------------------------------------------------- | | Standard Deviation | | |-----------------------------------------| | | 0.45 | 0.57 | 0.65 | | |-------------+-------------+-------------| | |Minimum Power|Minimum Power|Minimum Power| | |-------------+-------------+-------------| | | .800 | .900 | .800 | .900 | .800 | .900 | | |------+------+------+------+------+------| | |Total |Total |Total |Total |Total |Total | | | N | N | N | N | N | N | |--------------------------+------+------+------+------+------+------| |Alpha |Type | | | | | | | |------------+-------------| | | | | | | |0.05 |2-tail t | 39| 54| 63| 84| 81| 108| | |-------------+------+------+------+------+------+------| | |1-tail t | 33| 42| 51| 69| 63| 87| |------------+-------------+------+------+------+------+------+------| |0.01 |2-tail t | 60| 75| 93| 117| 120| 150| | |-------------+------+------+------+------+------+------| | |1-tail t | 51| 66| 81| 102| 102| 132| ---------------------------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Matched-Pairs Design Traditional t-Based Tests -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: PAIREDMU -.86 -.42 SD .60 .80 CORR .5 .6 SDMULT 1 1.2 ALPHA .05 .01 TOTALPAIRS 17 25 Testing difference of single pair of correlated measures: Ho: mu(Y1 - Y2) = 0 ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Matched-Pairs Design Traditional t-Based Tests ---- NOTE ---- UnifyPow requires: mu1 mu2 ........ at least one pair of means; see 2-group example below: AB/BA cross-over design SD1 SD2 ........ exactly one pair of SDs Corr(Y1,Y2) .... at least one correlation For each value of Corr(Y1,Y2), UnifyPow computes SD of diff = Y1 - Y2. Call this: SDdiff[Corr(Y1,Y2)]. The SDMult variable (m) allows the user to further vary SDdiff[Corr(Y1,Y2)], i.e. SDdiff[Corr(Y1,Y2), m] = m*SDdiff[Corr(Y1,Y2)]. Default is m = 1.00 only. ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Matched-Pairs Design Traditional t-Based Tests Scenario: PairedMu -.86 -.42 & SD 0.6 0.8 Matched-pairs t test -------------------------------------------------------------------- | | x SD (SD Multiplier) | | |---------------------------------------| | | 1 | 1.2 | | |-------------------+-------------------| | | Corr(Y1, Y2) | Corr(Y1, Y2) | | |-------------------+-------------------| | | 0.5 | 0.6 | 0.5 | 0.6 | | |---------+---------+---------+---------| | | Total | Total | Total | Total | | | Pairs | Pairs | Pairs | Pairs | | |---------+---------+---------+---------| | | 17 | 25 | 17 | 25 | 17 | 25 | 17 | 25 | | |----+----+----+----+----+----+----+----| | |Pow-|Pow-|Pow-|Pow-|Pow-|Pow-|Pow-|Pow-| | | er | er | er | er | er | er | er | er | |--------------------------+----+----+----+----+----+----+----+----| |Alpha |Type | | | | | | | | | |------------+-------------| | | | | | | | | |0.05 |2-tail t |.657|.833|.744|.900|.504|.684|.588|.771| | |-------------+----+----+----+----+----+----+----+----| | |1-tail t |.777|.906|.846|.949|.641|.795|.718|.862| |------------+-------------+----+----+----+----+----+----+----+----| |0.01 |2-tail t |.375|.604|.469|.715|.245|.418|.312|.518| | |-------------+----+----+----+----+----+----+----+----| | |1-tail t |.491|.709|.588|.805|.343|.529|.421|.629| -------------------------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Two-Group Design Wilcoxon Tests Based on mu & SD, Assuming Logistic Parent -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: MU -.86 -.42 SD .45 .57 .65 ALPHA .05 .01 WEIGHT 2 1 NTOTAL 15 21 33 WILCOXON PARENT LOGISTIC METHODS ALL Testing location difference between 2 groups: Ho: mu1 - mu2 = 0 Ho: p1 = .50 where p1 = Pr[Y{i,1} - Y{i',2} > 0 ] + .50*Pr[Y{i,1} - Y{i',2} = 0 ]. Y{i,g} = Y for case i in group g. Nonparametric Moments (if no ties possible) ------------------------------------------- Let Y{i,g} be the outcome score for case i in group g. (For the PairedMu problem, Y{i,g} is a difference score.) Then, p1 = Pr[Y{i,1} - Y{i',2} > Null] p2 = Pr[(Y{i,1} - Y{k,2} > Null) and (Y{i',1} - Y{k,2} > Null)] p3 = Pr[(Y{i,1} - Y{k,2} > Null) and (Y{i,1} - Y{k',2} > Null)] UnifyPow will reverse ordering relations in order to force p1 > .5 Ties are handled by partitioning probabilities appropriately, e.g., p1 = Pr[Y{i,1} - Y{i',2} > Null] + .50*Pr[Y{i,1} - Y{i',2} = Null] The (default) Lehmann-Hettsmansperger method uses p1, p2, and p3, whereas 'METHOD NOETHER' uses only p1. 'METHOD ARE' does not use the p-type moments, but rather uses the asymptotic relative efficiences of the Wilcoxon versus the t-test. This includes using ARE = .864, the theoretical minimum. Parent Distributions -------------------- Powers for the Wilcoxon will be approximated assuming Normal, Logistic, and Laplace parent distributions, thus giving a range of tail thicknesses (kurtoses) and asymptotic relative efficiencies (ARE): Parent Kurtosis ARE -------- -------- ----- Normal 0.0 0.955 Logistic 1.2 1.097 Laplace 3.0 1.500 <> 0.864 ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Two-Group Design Wilcoxon Tests Based on mu & SD, Assuming Logistic Parent Scenario: mu -.86 -.42 -------------------------------------------------------- | | Nonparametric Moments | | |--------------------------------------------| | | p1 | p2 | p3 | | |--------------+--------------+--------------| | | Std Dev | Std Dev | Std Dev | | |--------------+--------------+--------------| | |0.45|0.57|0.65|0.45|0.57|0.65|0.45|0.57|0.65| |---------+----+----+----+----+----+----+----+----+----| |Parent | | | | | | | | | | |---------| | | | | | | | | | |Normal |.755|.707|.684|.627|.566|.537|.627|.566|.537| |---------+----+----+----+----+----+----+----+----+----| |Logistic |.768|.719|.695|.645|.582|.551|.645|.582|.551| |---------+----+----+----+----+----+----+----+----+----| |Laplace |.788|.741|.716|.675|.612|.580|.675|.612|.580| |---------+----+----+----+----+----+----+----+----+----| |min ARE | N/A| N/A| N/A| N/A| N/A| N/A| N/A| N/A| N/A| -------------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Two-Group Design Wilcoxon Tests Based on mu & SD, Assuming Logistic Parent Scenario: mu -.86 -.42 AND Alpha: 0.05 ------------------------------------------------------------------------- | | Standard Deviation | | |--------------------------------------------| | | 0.45 | 0.57 | 0.65 | | |--------------+--------------+--------------| | | Total N | Total N | Total N | | |--------------+--------------+--------------| | | 15 | 21 | 33 | 15 | 21 | 33 | 15 | 21 | 33 | | |----+----+----+----+----+----+----+----+----| | |Pow-|Pow-|Pow-|Pow-|Pow-|Pow-|Pow-|Pow-|Pow-| | | er | er | er | er | er | er | er | er | er | |--------------------------+----+----+----+----+----+----+----+----+----| |Method |Type |Parent | | | | | | | | | | |--------+--------+--------| | | | | | | | | | |Wilcoxon|2-tail W|Normal |.291|.438|.677|.200|.295|.472|.164|.238|.380| |Mann- | |--------+----+----+----+----+----+----+----+----+----| |Whitney | |Logistic|.321|.481|.727|.221|.327|.521|.181|.264|.422| |[Lehmann| |--------+----+----+----+----+----+----+----+----+----| |(p1, p2,| |Laplace |.375|.553|.800|.265|.392|.610|.219|.321|.508| |p3) |--------+--------+----+----+----+----+----+----+----+----+----| |aprx] |1-tail W|Normal |.432|.588|.800|.312|.426|.612|.263|.356|.516| | | |--------+----+----+----+----+----+----+----+----+----| | | |Logistic|.468|.631|.839|.340|.464|.659|.286|.388|.560| | | |--------+----+----+----+----+----+----+----+----+----| | | |Laplace |.528|.699|.891|.395|.535|.740|.335|.455|.645| |--------+--------+--------+----+----+----+----+----+----+----+----+----| |Wilcoxon|2-tail W|Normal |.346|.463|.655|.245|.329|.483|.202|.269|.397| |Mann- | |--------+----+----+----+----+----+----+----+----+----| |Whitney | |Logistic|.374|.499|.697|.268|.360|.526|.222|.296|.437| |[Noether| |--------+----+----+----+----+----+----+----+----+----| |(p1) | |Laplace |.422|.557|.758|.313|.420|.604|.262|.352|.515| |aprx] |--------+--------+----+----+----+----+----+----+----+----+----| | |1-tail W|Normal |.468|.588|.763|.354|.449|.607|.302|.382|.522| | | |--------+----+----+----+----+----+----+----+----+----| | | |Logistic|.498|.623|.797|.381|.483|.648|.326|.413|.562| | | |--------+----+----+----+----+----+----+----+----+----| | | |Laplace |.547|.677|.845|.432|.545|.719|.374|.474|.638| |--------+--------+--------+----+----+----+----+----+----+----+----+----| |Wilcoxon|2-tail W|Normal |.363|.498|.706|.246|.339|.506|.200|.273|.410| |Mann- | |--------+----+----+----+----+----+----+----+----+----| |Whitney | |Logistic|.415|.560|.769|.281|.385|.566|.227|.309|.462| |[aprx | |--------+----+----+----+----+----+----+----+----+----| |via ARE | |Laplace |.549|.706|.888|.377|.506|.708|.303|.410|.595| |W vs. t]| |--------+----+----+----+----+----+----+----+----+----| | | |min ARE |.328|.455|.660|.224|.308|.465|.183|.249|.376| | |--------+--------+----+----+----+----+----+----+----+----+----| | |1-tail W|Normal |.501|.633|.811|.366|.469|.637|.307|.394|.542| | | |--------+----+----+----+----+----+----+----+----+----| | | |Logistic|.554|.690|.858|.406|.518|.692|.340|.435|.594| | | |--------+----+----+----+----+----+----+----+----+----| | | |Laplace |.680|.811|.940|.510|.637|.810|.428|.542|.715| ------------------------------------------------------------------------- (CONTINUED) ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Two-Group Design Wilcoxon Tests Based on mu & SD, Assuming Logistic Parent Scenario: mu -.86 -.42 AND Alpha: 0.05 ------------------------------------------------------------------------- | | Standard Deviation | | |--------------------------------------------| | | 0.45 | 0.57 | 0.65 | | |--------------+--------------+--------------| | | Total N | Total N | Total N | | |--------------+--------------+--------------| | | 15 | 21 | 33 | 15 | 21 | 33 | 15 | 21 | 33 | | |----+----+----+----+----+----+----+----+----| | |Pow-|Pow-|Pow-|Pow-|Pow-|Pow-|Pow-|Pow-|Pow-| | | er | er | er | er | er | er | er | er | er | |--------------------------+----+----+----+----+----+----+----+----+----| |Method |Type |Parent | | | | | | | | | | |--------+--------+--------| | | | | | | | | | |Wilcoxon|1-tail W|min ARE | | | | | | | | | | |Mann- | | | | | | | | | | | | |Whitney | | | | | | | | | | | | |[aprx | | | | | | | | | | | | |via ARE | | | | | | | | | | | | |W vs. t]| | |.465|.593|.774|.339|.436|.598|.285|.366|.506| |--------+--------+--------+----+----+----+----+----+----+----+----+----| |Ordinary|2-tail t|Normal |.380|.518|.727|.257|.353|.526|.209|.284|.427| |t test |--------+--------+----+----+----+----+----+----+----+----+----| | |1-tail t|Normal |.519|.652|.828|.379|.485|.655|.318|.407|.559| ------------------------------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Two-Group Design Wilcoxon Tests Based on mu & SD, Assuming Logistic Parent Scenario: mu -.86 -.42 AND Alpha: 0.01 ------------------------------------------------------------------------- | | Standard Deviation | | |--------------------------------------------| | | 0.45 | 0.57 | 0.65 | | |--------------+--------------+--------------| | | Total N | Total N | Total N | | |--------------+--------------+--------------| | | 15 | 21 | 33 | 15 | 21 | 33 | 15 | 21 | 33 | | |----+----+----+----+----+----+----+----+----| | |Pow-|Pow-|Pow-|Pow-|Pow-|Pow-|Pow-|Pow-|Pow-| | | er | er | er | er | er | er | er | er | er | |--------------------------+----+----+----+----+----+----+----+----+----| |Method |Type |Parent | | | | | | | | | | |--------+--------+--------| | | | | | | | | | |Wilcoxon|2-tail W|Normal |.098|.184|.388|.063|.109|.223|.049|.083|.164| |Mann- | |--------+----+----+----+----+----+----+----+----+----| |Whitney | |Logistic|.112|.212|.442|.071|.126|.259|.056|.095|.191| |[Lehmann| |--------+----+----+----+----+----+----+----+----+----| |(p1, p2,| |Laplace |.140|.265|.532|.091|.163|.335|.072|.125|.253| |p3) |--------+--------+----+----+----+----+----+----+----+----+----| |aprx] |1-tail W|Normal |.161|.275|.507|.105|.171|.315|.084|.133|.240| | | |--------+----+----+----+----+----+----+----+----+----| | | |Logistic|.181|.310|.563|.118|.194|.358|.094|.151|.274| | | |--------+----+----+----+----+----+----+----+----+----| | | |Laplace |.220|.374|.652|.147|.244|.444|.118|.192|.349| |--------+--------+--------+----+----+----+----+----+----+----+----+----| |Wilcoxon|2-tail W|Normal |.156|.239|.414|.096|.145|.255|.074|.109|.190| |Mann- | |--------+----+----+----+----+----+----+----+----+----| |Whitney | |Logistic|.175|.268|.460|.109|.165|.291|.083|.125|.219| |[Noether| |--------+----+----+----+----+----+----+----+----+----| |(p1) | |Laplace |.208|.319|.534|.135|.207|.362|.105|.160|.282| |aprx] |--------+--------+----+----+----+----+----+----+----+----+----| | |1-tail W|Normal |.223|.323|.513|.145|.209|.341|.115|.163|.265| | | |--------+----+----+----+----+----+----+----+----+----| | | |Logistic|.246|.356|.559|.162|.234|.382|.129|.184|.300| | | |--------+----+----+----+----+----+----+----+----+----| | | |Laplace |.286|.412|.631|.197|.285|.459|.158|.228|.371| |--------+--------+--------+----+----+----+----+----+----+----+----+----| |Wilcoxon|2-tail W|Normal |.145|.243|.447|.085|.138|.259|.064|.101|.188| |Mann- | |--------+----+----+----+----+----+----+----+----+----| |Whitney | |Logistic|.180|.296|.525|.103|.167|.311|.077|.123|.227| |[aprx | |--------+----+----+----+----+----+----+----+----+----| |via ARE | |Laplace |.286|.447|.710|.162|.259|.458|.119|.188|.341| |W vs. t]| |--------+----+----+----+----+----+----+----+----+----| | | |min ARE |.124|.209|.394|.073|.119|.226|.056|.088|.164| | |--------+--------+----+----+----+----+----+----+----+----+----| | |1-tail W|Normal |.221|.339|.557|.137|.207|.353|.106|.158|.269| | | |--------+----+----+----+----+----+----+----+----+----| | | |Logistic|.265|.399|.632|.162|.245|.411|.125|.186|.315| | | |--------+----+----+----+----+----+----+----+----+----| | | |Laplace |.388|.556|.795|.238|.353|.564|.181|.269|.442| ------------------------------------------------------------------------- (CONTINUED) ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Two-Group Design Wilcoxon Tests Based on mu & SD, Assuming Logistic Parent Scenario: mu -.86 -.42 AND Alpha: 0.01 ------------------------------------------------------------------------- | | Standard Deviation | | |--------------------------------------------| | | 0.45 | 0.57 | 0.65 | | |--------------+--------------+--------------| | | Total N | Total N | Total N | | |--------------+--------------+--------------| | | 15 | 21 | 33 | 15 | 21 | 33 | 15 | 21 | 33 | | |----+----+----+----+----+----+----+----+----| | |Pow-|Pow-|Pow-|Pow-|Pow-|Pow-|Pow-|Pow-|Pow-| | | er | er | er | er | er | er | er | er | er | |--------------------------+----+----+----+----+----+----+----+----+----| |Method |Type |Parent | | | | | | | | | | |--------+--------+--------| | | | | | | | | | |Wilcoxon|1-tail W|min ARE | | | | | | | | | | |Mann- | | | | | | | | | | | | |Whitney | | | | | | | | | | | | |[aprx | | | | | | | | | | | | |via ARE | | | | | | | | | | | | |W vs. t]| | |.193|.300|.504|.121|.183|.315|.095|.141|.239| |--------+--------+--------+----+----+----+----+----+----+----+----+----| |Ordinary|2-tail t|Normal |.156|.260|.472|.091|.147|.276|.068|.108|.201| |t test |--------+--------+----+----+----+----+----+----+----+----+----| | |1-tail t|Normal |.235|.358|.581|.145|.219|.372|.112|.167|.283| ------------------------------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Two-Group Design Wilcoxon Test, Specifying p1 Directly, Assuming Logistic Parent -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: 2WILCOXON .75 SD 1.00 1.25 1.40 WEIGHT 2 1 NTOTAL 15 21 33 METHODS NOETHER PARENT LOGISTIC NOTE: No ALPHA statement. Default is alpha = 0.05 only. Testing location difference between 2 groups: Ho: mu1 - mu2 = 0 Ho: p1 = .50 where p1 = Pr[Y{i,1} - Y{i',2} > 0 ] + .50*Pr[Y{i,1} - Y{i',2} = 0 ]. Y{i,g} = Y for case i in group g. Nonparametric Moments (if no ties possible) ------------------------------------------- Let Y{i,g} be the outcome score for case i in group g. (For the PairedMu problem, Y{i,g} is a difference score.) Then, p1 = Pr[Y{i,1} - Y{i',2} > Null] p2 = Pr[(Y{i,1} - Y{k,2} > Null) and (Y{i',1} - Y{k,2} > Null)] p3 = Pr[(Y{i,1} - Y{k,2} > Null) and (Y{i,1} - Y{k',2} > Null)] UnifyPow will reverse ordering relations in order to force p1 > .5 Ties are handled by partitioning probabilities appropriately, e.g., p1 = Pr[Y{i,1} - Y{i',2} > Null] + .50*Pr[Y{i,1} - Y{i',2} = Null] The (default) Lehmann-Hettsmansperger method uses p1, p2, and p3, whereas 'METHOD NOETHER' uses only p1. 'METHOD ARE' does not use the p-type moments, but rather uses the asymptotic relative efficiences of the Wilcoxon versus the t-test. This includes using ARE = .864, the theoretical minimum. Parent Distributions -------------------- Powers for the Wilcoxon will be approximated assuming Normal, Logistic, and Laplace parent distributions, thus giving a range of tail thicknesses (kurtoses) and asymptotic relative efficiencies (ARE): Parent Kurtosis ARE -------- -------- ----- Normal 0.0 0.955 Logistic 1.2 1.097 Laplace 3.0 1.500 <> 0.864 ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Two-Group Design Wilcoxon Test, Specifying p1 Directly, Assuming Logistic Parent You specified the LOGISTIC distribution and p1 = 0.750. This equates to psi = (mu2 - mu1 - NullValue)/SD = 0.8999. ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Two-Group Design Wilcoxon Test, Specifying p1 Directly, Assuming Logistic Parent Scenario: 2Wilcoxon .75 -------------------------------------------------------- | | Nonparametric Moments | | |--------------------------------------------| | | p1 | p2 | p3 | | |--------------+--------------+--------------| | | Relative Std | Relative Std | Relative Std | | | Dev | Dev | Dev | | |--------------+--------------+--------------| | | 1 |1.25|1.4 | 1 |1.25|1.4 | 1 |1.25|1.4 | |---------+----+----+----+----+----+----+----+----+----| |Parent | | | | | | | | | | |---------| | | | | | | | | | |Normal |.738|.695|.675|.604|.550|.527|.604|.550|.527| |---------+----+----+----+----+----+----+----+----+----| |Logistic |.750|.706|.686|.621|.565|.540|.621|.565|.540| |---------+----+----+----+----+----+----+----+----+----| |Laplace |.771|.727|.707|.652|.594|.568|.652|.594|.568| -------------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Two-Group Design Wilcoxon Test, Specifying p1 Directly, Assuming Logistic Parent Scenario: 2Wilcoxon .75 AND Alpha: 0.05 ------------------------------------------------------------------------- | | Relative Std Dev | | |--------------------------------------------| | | 1 | 1.25 | 1.4 | | |--------------+--------------+--------------| | | Total N | Total N | Total N | | |--------------+--------------+--------------| | | 15 | 21 | 33 | 15 | 21 | 33 | 15 | 21 | 33 | | |----+----+----+----+----+----+----+----+----| | |Pow-|Pow-|Pow-|Pow-|Pow-|Pow-|Pow-|Pow-|Pow-| | | er | er | er | er | er | er | er | er | er | |--------------------------+----+----+----+----+----+----+----+----+----| |Method |Type |Parent | | | | | | | | | | |--------+--------+--------| | | | | | | | | | |Wilcoxon|2-tail W|Normal |.254|.381|.601|.180|.263|.421|.153|.219|.348| |Mann- | |--------+----+----+----+----+----+----+----+----+----| |Whitney | |Logistic|.281|.421|.653|.199|.292|.466|.168|.243|.388| |[Lehmann| |--------+----+----+----+----+----+----+----+----+----| |(p1, p2,| |Laplace |.332|.492|.735|.239|.353|.555|.203|.297|.471| |p3) |--------+--------+----+----+----+----+----+----+----+----+----| |aprx] |1-tail W|Normal |.385|.526|.734|.285|.387|.559|.247|.332|.481| | | |--------+----+----+----+----+----+----+----+----+----| | | |Logistic|.418|.568|.778|.310|.422|.605|.268|.362|.524| | | |--------+----+----+----+----+----+----+----+----+----| | | |Laplace |.477|.640|.843|.362|.491|.689|.314|.426|.608| |--------+--------+--------+----+----+----+----+----+----+----+----+----| |Wilcoxon|2-tail W|Normal |.307|.412|.594|.221|.296|.436|.188|.249|.367| |Mann- | |--------+----+----+----+----+----+----+----+----+----| |Whitney | |Logistic|.334|.447|.637|.242|.325|.478|.206|.274|.405| |[Noether| |--------+----+----+----+----+----+----+----+----+----| |(p1) | |Laplace |.382|.508|.707|.285|.383|.556|.244|.328|.481| |aprx] |--------+--------+----+----+----+----+----+----+----+----+----| | |1-tail W|Normal |.425|.537|.710|.325|.412|.561|.284|.358|.490| | | |--------+----+----+----+----+----+----+----+----+----| | | |Logistic|.455|.573|.747|.351|.445|.602|.306|.388|.529| | | |--------+----+----+----+----+----+----+----+----+----| | | |Laplace |.505|.631|.805|.400|.507|.676|.353|.448|.606| |--------+--------+--------+----+----+----+----+----+----+----+----+----| |Ordinary|2-tail t|Normal |.331|.455|.656|.230|.315|.472|.193|.261|.393| |t test |--------+--------+----+----+----+----+----+----+----+----+----| | |1-tail t|Normal |.465|.591|.770|.345|.442|.603|.297|.380|.523| ------------------------------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Matched-Pairs Design One-Group Wilcoxon Tests, Assuming Normal Parent -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: PAIREDMU -.86 -.42 SD .60 .80 CORR .5 .6 SDMULT 1 1.2 TOTALPAIRS 17 25 WILCOXON NOTE: No ALPHA statement. Default is alpha = 0.05 only. Testing difference of single pair of correlated measures: Ho: mu(Y1 - Y2) = 0 Testing location of a single group: Ho: p1 = .50 where p1 = Pr[D{i} > 0 ] + .50*Pr[D{i} = 0 ]. D{i} = Y1{i} - Y2{i}. Nonparametric Moments (if no ties possible) ------------------------------------------- Let Y{i} be the outcome score for case i. (For the PairedMu problem, Y{i} is a difference score.) Then, p1 = Pr[Y{i} > Null] p2 = Pr[Y{i} + Y{i'} > 2*Null] p3 = (p2 + p1**2)/2 p4 = Pr[(Y{i} + Y{i'} > 2*Null) and (Y{i} + Y{i''} > 2*Null)] UnifyPow will reverse ordering relations in order to force p1 > .5 Ties are handled by partitioning probabilities appropriately, e.g., p1 = Pr[Y{i,1} - Y{i',2} > Null] + .50*Pr[Y{i,1} - Y{i',2} = Null] The (default) Lehmann-Hettsmansperger method uses p1, p2, and p4 (as well as p3), whereas 'METHOD NOETHER' uses only p1. 'METHOD ARE' does not use the p-type moments, but rather uses the asymptotic relative efficiences of the Wilcoxon versus the t-test. This includes using ARE = .864, the theoretical minimum. Parent Distributions -------------------- Powers for the Wilcoxon will be approximated assuming Normal, Logistic, and Laplace parent distributions, thus giving a range of tail thicknesses (kurtoses) and asymptotic relative efficiencies (ARE): Parent Kurtosis ARE -------- -------- ----- Normal 0.0 0.955 Logistic 1.2 1.097 ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Matched-Pairs Design One-Group Wilcoxon Tests, Assuming Normal Parent Laplace 3.0 1.500 <> 0.864 ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Matched-Pairs Design One-Group Wilcoxon Tests, Assuming Normal Parent Scenario: PairedMu -.86 -.42 & SD 0.6 0.8 ----------------------------------------------------------------------- | | x SD Multiplier | | |-----------------------------------------------------------| | | 1 | 1.2 | | |-----------------------------+-----------------------------| | | Corr(Y1, Y2) | Corr(Y1, Y2) | | |-----------------------------+-----------------------------| | | 0.5 | 0.6 | 0.5 | 0.6 | | |--------------+--------------+--------------+--------------| | |Nonparametric |Nonparametric |Nonparametric |Nonparametric | | | Moments | Moments | Moments | Moments | | |--------------+--------------+--------------+--------------| | | p1 | p2 | p4 | p1 | p2 | p4 | p1 | p2 | p4 | p1 | p2 | p4 | |---------+----+----+----+----+----+----+----+----+----+----+----+----| |Parent | | | | | | | | | | | | | |---------| | | | | | | | | | | | | |Normal |.729|.806|.695|.750|.830|.730|.694|.764|.639|.713|.787|.670| |---------+----+----+----+----+----+----+----+----+----+----+----+----| |Logistic |.752|.818|.713|.773|.841|.747|.716|.776|.657|.735|.799|.688| |---------+----+----+----+----+----+----+----+----+----+----+----+----| |Laplace |.789|.834|.740|.808|.855|.771|.756|.796|.686|.775|.817|.716| ----------------------------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Matched-Pairs Design One-Group Wilcoxon Tests, Assuming Normal Parent Scenario: PairedMu -.86 -.42 & SD 0.6 0.8 AND Alpha 0.05 -------------------------------------------------------------------- | | x SD (SD Multiplier) | | |---------------------------------------| | | 1 | 1.2 | | |-------------------+-------------------| | | Corr(Y1, Y2) | Corr(Y1, Y2) | | |-------------------+-------------------| | | 0.5 | 0.6 | 0.5 | 0.6 | | |---------+---------+---------+---------| | | Total | Total | Total | Total | | | Pairs | Pairs | Pairs | Pairs | | |---------+---------+---------+---------| | | 17 | 25 | 17 | 25 | 17 | 25 | 17 | 25 | | |----+----+----+----+----+----+----+----| | |Pow-|Pow-|Pow-|Pow-|Pow-|Pow-|Pow-|Pow-| | | er | er | er | er | er | er | er | er | |--------------------------+----+----+----+----+----+----+----+----| |Method |Type |Parent | | | | | | | | | |--------+--------+--------| | | | | | | | | |Wilcoxon|2-tail W|Normal |.589|.806|.686|.889|.438|.637|.518|.733| |Signed | |--------+----+----+----+----+----+----+----+----| |Rank | |Logistic|.639|.847|.732|.917|.485|.690|.568|.781| |[Lehmann| |--------+----+----+----+----+----+----+----+----| |(p1, p2,| |Laplace |.708|.896|.788|.946|.564|.770|.643|.845| |p3, p4) |--------+--------+----+----+----+----+----+----+----+----| |aprx] |1-tail W|Normal |.739|.900|.823|.952|.590|.769|.672|.847| | | |--------+----+----+----+----+----+----+----+----| | | |Logistic|.782|.927|.858|.967|.637|.812|.718|.882| | | |--------+----+----+----+----+----+----+----+----| | | |Laplace |.836|.954|.896|.980|.711|.871|.782|.923| |--------+--------+--------+----+----+----+----+----+----+----+----| |Matched-|2-tail t|Normal |.657|.833|.744|.900|.504|.684|.588|.771| |pairs t |--------+--------+----+----+----+----+----+----+----+----| |test |1-tail t|Normal |.777|.906|.846|.949|.641|.795|.718|.862| -------------------------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's 2-Group Design 7-pt Likert Scale for Improvement in Headache -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: 2WILCOXON .03 .04 .08 .10 .27 .28 .20 .10 .10 .15 .25 .20 .15 .05 WEIGHT 2 1 POWER .90 .95 ALPHA .05 .01 Testing location difference between 2 groups: Ho: p1 = .50 where p1 = Pr[Y{i,1} - Y{i',2} > 0 ] + .50*Pr[Y{i,1} - Y{i',2} = 0 ]. Y{i,g} = Y for case i in group g. Nonparametric Moments (if no ties possible) ------------------------------------------- Let Y{i,g} be the outcome score for case i in group g. (For the PairedMu problem, Y{i,g} is a difference score.) Then, p1 = Pr[Y{i,1} - Y{i',2} > Null] p2 = Pr[(Y{i,1} - Y{k,2} > Null) and (Y{i',1} - Y{k,2} > Null)] p3 = Pr[(Y{i,1} - Y{k,2} > Null) and (Y{i,1} - Y{k',2} > Null)] UnifyPow will reverse ordering relations in order to force p1 > .5 Ties are handled by partitioning probabilities appropriately, e.g., p1 = Pr[Y{i,1} - Y{i',2} > Null] + .50*Pr[Y{i,1} - Y{i',2} = Null] The (default) Lehmann-Hettsmansperger method uses p1, p2, and p3, whereas 'METHOD NOETHER' uses only p1. ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's 2-Group Design 7-pt Likert Scale for Improvement in Headache Scenario: 2Wilcoxon -------------------------- | | Nonparametic | | | Moments | | |--------------| | | p1 | p2 | p3 | |---------+----+----+----| |Parent | | | | |---------| | | | |Custom |.706|.560|.566| -------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's 2-Group Design 7-pt Likert Scale for Improvement in Headache Scenario: 2Wilcoxon, {.03 .04 .08 .10 .27 .28 .20}, {.10 .10 .15 .25 .20 .15 .05} -------------------------------------------------------- | | Alpha | | |---------------------------| | | 0.05 | 0.01 | | |-------------+-------------| | |Minimum Power|Minimum Power| | |-------------+-------------| | | .900 | .950 | .900 | .950 | | |------+------+------+------| | |Total |Total |Total |Total | | | N | N | N | N | |--------------------------+------+------+------+------| |Method |Type | | | | | |------------+-------------| | | | | |Wilcoxon- |2-tail W | 87| 105| 126| 147| |Mann-Whitney|-------------+------+------+------+------| |[Lehmann |1-tail W | | | | | |(p1, p2, p3)| | | | | | |aprx] | | 72| 87| 108| 129| -------------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Matched-Pairs Design/ 7-pt Likert scale Ordered Categorical Outcome, 1-Group Wilcoxon Test -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: 1WILCOXON .02 .03 .05 .05 .05 .06 .09 .14 .14 .14 .09 .08 .06 LIMITS -6 6 NULL 0 NTOTAL 50 75 100 ALPHA .05 METHOD NOETHER Testing location of a single group: Ho: p1 = .50 where p1 = Pr[Y{i} > 0 ] + .50*Pr[Y{i} = 0 ]. Category values (interval scale): -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Nonparametric Moments (if no ties possible) ------------------------------------------- Let Y{i} be the outcome score for case i. (For the PairedMu problem, Y{i} is a difference score.) Then, p1 = Pr[Y{i} > Null] p2 = Pr[Y{i} + Y{i'} > 2*Null] p3 = (p2 + p1**2)/2 p4 = Pr[(Y{i} + Y{i'} > 2*Null) and (Y{i} + Y{i''} > 2*Null)] UnifyPow will reverse ordering relations in order to force p1 > .5 Ties are handled by partitioning probabilities appropriately, e.g., p1 = Pr[Y{i,1} - Y{i',2} > Null] + .50*Pr[Y{i,1} - Y{i',2} = Null] The (default) Lehmann-Hettsmansperger method uses p1, p2, and p4 (as well as p3), whereas 'METHOD NOETHER' uses only p1. ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Matched-Pairs Design/ 7-pt Likert scale Ordered Categorical Outcome, 1-Group Wilcoxon Test Scenario: 1Wilcoxon -------------------------- | | Nonparametic | | | Moments | | |--------------| | | p1 | p2 | p4 | |---------+----+----+----| |Parent | | | | |---------| | | | |Custom |.695|.713|.574| -------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Matched-Pairs Design/ 7-pt Likert scale Ordered Categorical Outcome, 1-Group Wilcoxon Test Scenario: 1Wilcoxon, {.02 .03 .05 .05 .05 .06 .09 .14 .14 .14 .09 .08 .06}, AND NULL: 0 ------------------------------------------- | | Alpha | | |--------------| | | 0.05 | | |--------------| | | Total N | | |--------------| | | 50 | 75 |100 | | |----+----+----| | |Pow-|Pow-|Pow-| | | er | er | er | |--------------------------+----+----+----| |Method |Type | | | | |------------+-------------| | | | |Wilcoxon |2-tail W |.765|.916|.974| |Signed Rank |-------------+----+----+----| |[Lehmann |1-tail W | | | | |(p1, p2, p3,| | | | | |p4) aprx] | |.859|.959|.989| |------------+-------------+----+----+----| |Wilcoxon |2-tail W |.743|.892|.958| |Signed Rank |-------------+----+----+----| |[Noether |1-tail W | | | | |(p1) aprx] | |.833|.940|.980| ------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Mindy Bowdy: 4-Group Design Example 3 in O'Brien and Muller (1993, Section 8.2.3) (1) Dominators, (2) Ordinaries, (3) Loners, (4) Friendlies -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: MU .35 .50 .52 .60 WEIGHT .20 .50 .10 .20 SD .16 .19 ALPHA .05 NTOTAL 60 80 100 ANOVA testing on 4 means. CONTRASTS Ord vs Loners 0 1 -1 0 Almost Overall (2 DF) 1 -.833 -.167 0 > 0 -.833 -.167 1 ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Mindy Bowdy: 4-Group Design (1) Dominators, (2) Ordinaries, (3) Loners, (4) Friendlies -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: MU .35 .50 .52 .60 WEIGHT .20 .50 .10 .20 SD .16 .19 ALPHA .0167 .05 NTOTAL 60 80 100 NOOVERALL ANOVA testing on 4 means. CONTRASTS {Ord & Loners} vs Fr 0 -.833 -.167 1 Dom vs {Ord & Loners} 1 -.833 -.167 0 Fr vs Dom -1 0 0 1 ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Mindy Bowdy: 4-Group Design (1) Dominators, (2) Ordinaries, (3) Loners, (4) Friendlies Scenario: mu .35 .50 .52 .60 ---------------------------------------------------------- | | Standard Deviation | | |-----------------------------| | | 0.16 | 0.19 | | |--------------+--------------| | | Total N | Total N | | |--------------+--------------| | | 60 | 80 |100 | 60 | 80 |100 | | |----+----+----+----+----+----| | |Pow-|Pow-|Pow-|Pow-|Pow-|Pow-| | | er | er | er | er | er | er | |--------------------------+----+----+----+----+----+----| |Test |Alpha |Type | | | | | | | |--------+--------+--------| | | | | | | |Overall |0.05 |Regular | | | | | | | |test | |F |.899|.970|.992|.763|.887|.951| |--------+--------+--------+----+----+----+----+----+----| |Ord vs |0.05 |2-tail t|.059|.062|.065|.056|.058|.060| |Loners | |--------+----+----+----+----+----+----| | | |1-tail t|.086|.093|.099|.079|.084|.090| |--------+--------+--------+----+----+----+----+----+----| |Almost |0.05 |Regular | | | | | | | |Overall | |F | | | | | | | |(2 DF) | | |.933|.982|.996|.821|.923|.969| |--------+--------+--------+----+----+----+----+----+----| |{Ord & |0.05 |2-tail t|.429|.542|.639|.323|.413|.496| |Loners} | |--------+----+----+----+----+----+----| |vs Fr | |1-tail t|.558|.666|.751|.445|.541|.622| | |--------+--------+----+----+----+----+----+----| | |0.0167 |2-tail t|.265|.367|.465|.183|.253|.326| | | |--------+----+----+----+----+----+----| | | |1-tail t|.363|.474|.573|.264|.348|.429| |--------+--------+--------+----+----+----+----+----+----| |Dom vs |0.05 |2-tail t|.807|.906|.957|.663|.788|.872| |{Ord & | |--------+----+----+----+----+----+----| |Loners} | |1-tail t|.884|.950|.979|.772|.870|.928| | |--------+--------+----+----+----+----+----+----| | |0.0167 |2-tail t|.658|.806|.897|.486|.637|.754| | | |--------+----+----+----+----+----+----| | | |1-tail t|.755|.873|.938|.597|.735|.832| |--------+--------+--------+----+----+----+----+----+----| |Fr vs |0.05 |2-tail t|.964|.992|.998|.886|.957|.985| |Dom | |--------+----+----+----+----+----+----| | | |1-tail t|.984|.997|.999|.938|.979|.994| | |--------+--------+----+----+----+----+----+----| | |0.0167 |2-tail t|.909|.974|.993|.772|.896|.956| | | |--------+----+----+----+----+----+----| | | |1-tail t|.948|.987|.997|.849|.938|.976| ---------------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Asthma (Active) Treaments A vs. B, Outcome is FEV1 Two-Group (AB/BA) Cross-Over Design with Continuous Outcome -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: PAIREDMU 1.6 1.7 > 1.9 2.3 SD .60 .75 CORR .6 .7 .8 SDMULT 1.0 1.2 SIDES 2 TOTALPAIRS 50 NOOVERALL NOTE: No ALPHA statement. Default is alpha = 0.05 only. NOTE: No WEIGHTS statement. Default is balanced design. CONTRAST Drug x Order 1 -1 Drug A vs. Drug B 1 1 WARNING: Contrast coefficients do not sum to 0.0. Row: 1 Effect title: Drug A vs. Drug B ------------------------------------------------------------------------------ UnifyPow Workshop Asthma (Active) Treaments A vs. B, Outcome is FEV1 Two-Group (AB/BA) Cross-Over Design with Continuous Outcome Scenario: PairedMu 1.6 1.7; 1.9 2.3 & SD 0.6 0.75 Drug x Order ---------------------------------------------------------- | | x SD (SD Multiplier) | | |-----------------------------| | | 1 | 1.2 | | |--------------+--------------| | | Corr(Y1, Y2) | Corr(Y1, Y2) | | |--------------+--------------| | |0.6 |0.7 |0.8 |0.6 |0.7 |0.8 | | |----+----+----+----+----+----| | |Tot-|Tot-|Tot-|Tot-|Tot-|Tot-| | | al | al | al | al | al | al | | |Pai-|Pai-|Pai-|Pai-|Pai-|Pai-| | | rs | rs | rs | rs | rs | rs | | |----+----+----+----+----+----| | | 50 | 50 | 50 | 50 | 50 | 50 | | |----+----+----+----+----+----| | |Pow-|Pow-|Pow-|Pow-|Pow-|Pow-| | | er | er | er | er | er | er | |--------------------------+----+----+----+----+----+----| |Alpha |Type | | | | | | | |------------+-------------| | | | | | | |0.05 |2-tail t |.390|.485|.637|.288|.360|.486| ---------------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Asthma (Active) Treaments A vs. B, Outcome is FEV1 Two-Group (AB/BA) Cross-Over Design with Continuous Outcome Scenario: PairedMu 1.6 1.7; 1.9 2.3 & SD 0.6 0.75 Drug A vs. Drug B ---------------------------------------------------------- | | x SD (SD Multiplier) | | |-----------------------------| | | 1 | 1.2 | | |--------------+--------------| | | Corr(Y1, Y2) | Corr(Y1, Y2) | | |--------------+--------------| | |0.6 |0.7 |0.8 |0.6 |0.7 |0.8 | | |----+----+----+----+----+----| | |Tot-|Tot-|Tot-|Tot-|Tot-|Tot-| | | al | al | al | al | al | al | | |Pai-|Pai-|Pai-|Pai-|Pai-|Pai-| | | rs | rs | rs | rs | rs | rs | | |----+----+----+----+----+----| | | 50 | 50 | 50 | 50 | 50 | 50 | | |----+----+----+----+----+----| | |Pow-|Pow-|Pow-|Pow-|Pow-|Pow-| | | er | er | er | er | er | er | |--------------------------+----+----+----+----+----+----| |Alpha |Type | | | | | | | |------------+-------------| | | | | | | |0.05 |2-tail t |.800|.893|.971|.646|.761|.894| ---------------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Blackwelder's one-sided 'equivalency' testing strategy -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: PI .052 .052 WEIGHT 2 1 NULL .012 NTOTAL 8001 9999 12000 14001 TAILS 1 NOTE: No ALPHA statement. Default is alpha = 0.05 only. Testing Ho: pi1 - pi2 = 0.012 ------------------------------------------------------------------------------ UnifyPow Workshop Blackwelder's one-sided 'equivalency' testing strategy Abciximab plus a low-dose of heparin reduces complications and increases 30-day survival rates in patients undergoing high-risk coronary angioplasty or other kinds of revascularization (EPILOG Invetigators, 1997 [N Engl J Med, 336:1689]). But abciximab is expensive. Giving bivalirudin (Bittl et. al., 1995 [N Engl J Med, 333:764-9]) albeit with provisional ("bail-out") use of abciximab may give equivalent efficacy and safety at lower costs. Design. Bivalirudin plus provisional abciximab (B+provA) vs. abciximab plus a low-dose of heparin (A+lowH). Randomization ratio is 2:1, that is, 2/3 get B+provA. Primary end-point. As per EPILOG trial: "death by any cause, myocardial infarction, or urgent revascularization within 30 days." Statistical considerations. This equivalency trial will be handled using Blackwelder's method (1982 [Controlled Clinical Trials, 3:345-53]), which tests Ho: pi(B+provA) - pi(A+lowH) GE delta [B+provA appreciably worse] Ha: pi(B+provA) - pi(A+lowH) < delta [B+provA not appreciably worse; could be better] Delta sets the region of equivalency, which we take to be within 25% of the 30-day event rate of 0.052 found in the EPILOG trial. Thus, take delta = 0.012 < 0.013 = 0.25*0.052. Our sample-size analysis will assume that pi(B+provA) = pi(A+lowH) = 0.052. We seek a power of 0.90 for an alpha = .05 test of this one-tailed hypothesis. We shall use either the traditional Pearson ("approximate") unconditional test or its exact counterpart (Suissa & Shuster, 1985 [J Royal Stat Soc A, 48:317-27]), whichever is more powerful. [Even though their true alpha-levels are practically identical in large studies, the two methods will have different powers when the group pi conjectures are unequal and the sample-sizes are not balanced. See O'Brien and Muller (1993).] ------------------------------------------------------------------------------ UnifyPow Workshop Blackwelder's one-sided 'equivalency' testing strategy -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: PI .0494 .052 WEIGHT 2 1 NULL .012 NTOTAL 8001 9999 12000 14001 TAILS 1 NOTE: No ALPHA statement. Default is alpha = 0.05 only. Testing Ho: pi1 - pi2 = 0.012 ------------------------------------------------------------------------------ UnifyPow Workshop Blackwelder's one-sided 'equivalency' testing strategy Sponsor actually believes that bivaliridin may cut primary events by at least 5%. That is, they think that pi(B+provA) = .95*.052 = .0494. ------------------------------------------------------------------------------ UnifyPow Workshop Blackwelder's one-sided 'equivalency' testing strategy ------------------------------------------------------------------------- | | Alpha | | |---------------------------| | | 0.05 | | |---------------------------| | | Total N | | |---------------------------| | | 8001 | 9999 |12000 |14001 | | |------+------+------+------| | |Power |Power |Power |Power | |-------------------------------------------+------+------+------+------| |True State |Delta |Method | | | | | |-------------+-------------+---------------| | | | | |pi .052 .052 |0.012 |Approximate | | | | | | | |Uncondit'l | | | | | | | |"chi^2*" | 0.737| 0.817| 0.874| 0.915| | | |---------------+------+------+------+------| | | |Exact | | | | | | | |Uncondit'l** | 0.737| 0.817| 0.874| 0.915| |-------------+-------------+---------------+------+------+------+------| |pi .0494 .052|0.012 |Approximate | | | | | | | |Uncondit'l | | | | | | | |"chi^2*" | 0.880| 0.934| 0.965| 0.981| | | |---------------+------+------+------+------| | | |Exact | | | | | | | |Uncondit'l** | 0.875| 0.930| 0.962| 0.980| ------------------------------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Thromboembolism (case/control) & The Pill ('outcome') Matched Pairs, Binary Outcome -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: MCNEMAR .32 .08 NTOTAL 50 to 150 by 25 NOTE: No ALPHA statement. Default is alpha = 0.05 only. Testing Ho: pi12 - pi21 = 0 using McNemar's test. ------------------------------------------------------------------------------ UnifyPow Workshop Thromboembolism (case/control) & The Pill ('outcome') Matched Pairs, Binary Outcome Scenario: McNemar .32 .08 ----------------------------------------------------- | | ALPHA | | |------------------------| | | 0.05 | | |------------------------| | | Pairs | | |------------------------| | | 50 | 75 |100 |125 |150 | | |----+----+----+----+----| | |Pow-|Pow-|Pow-|Pow-|Pow-| | | er | er | er | er | er | |--------------------------+----+----+----+----+----| |Method |Statistic | | | | | | |------------+-------------| | | | | | |McNemar |2-tailed |.798|.931|.980|.995|.999| |(virtually |-------------+----+----+----+----+----| |exact) |1-tailed |.839|.954|.988|.997|.999| ----------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Jean Netticks: ARX Enzyme Levels in Father and Son Long been assumed that rho[ARX(dad), ARX(son)] < .55. New genetic theory predicts a correlation of rho > .70. -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: RHO .70 NULL .55 NTOTAL 50 75 100 TAILS 1 NOTE: No ALPHA statement. Default is alpha = 0.05 only. Testing single correlation coefficient using Fisher's r-to-Z: Ho: Z(rho) = Z(0.55 ) ------------------------------------------------------------------------------ UnifyPow Workshop Jean Netticks: ARX Enzyme Levels in Father and Son Long been assumed that rho[ARX(dad), ARX(son)] < .55. New genetic theory predicts a correlation of rho > .70. Scenario: rho .70 ------------------------------------------- | | ALPHA | | |--------------| | | 0.05 | | |--------------| | | Total N | | |--------------| | | 50 | 75 |100 | | |----+----+----| | |Pow-|Pow-|Pow-| | | er | er | er | |--------------------------+----+----+----| |Method |Statistic | | | | |------------+-------------| | | | |Fisher's |1-tail Z | | | | |r-to-Z test | | | | | |of one rho | |.525|.680|.790| ------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop rho[ARX(dad), ARX(son)]: Is there a difference between obese and normal fathers? Obese: rho = .55 Normal: rho = .70 -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: RHO .55 .70 WEIGHT 1 3 NTOTAL 400 to 1600 by 400 TAILS 2 NOTE: No ALPHA statement. Default is alpha = 0.05 only. Testing correlations using Fisher's r-to-Z: Ho: Z(rho1) - Z(rho2) = 0 ------------------------------------------------------------------------------ UnifyPow Workshop rho[ARX(dad), ARX(son)]: Is there a difference between obese and normal fathers? Obese: rho = .55 Normal: rho = .70 Scenario: rho .55 .70 ------------------------------------------------ | | ALPHA | | |-------------------| | | 0.05 | | |-------------------| | | Total N | | |-------------------| | |400 |800 |1200|1600| | |----+----+----+----| | |Pow-|Pow-|Pow-|Pow-| | | er | er | er | er | |--------------------------+----+----+----+----| |Method |Statistic | | | | | |------------+-------------| | | | | |Comparing |2-tail Z | | | | | |two | | | | | | |correlations| | | | | | |(r-to-Z) | |.567|.858|.961|.990| ------------------------------------------------ ------------------------------------------------------------------------------ UnifyPow Workshop Corey Latour: OLS Modeling to Predict Job Satisfaction Adding a 4-level nominal predictor (3 dummy variables). -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: R**2 .45 .50 NUMPARMS 5 8 NTOTAL 100 125 150 ALPHA .05 .01 Testing Ho: Beta_6 = Beta_7 = Beta_8 = 0. ------------------------------------------------------------------------------ UnifyPow Workshop Corey Latour: OLS Modeling to Predict Job Satisfaction Adding a 4-level nominal predictor (3 dummy variables). Scenario: R**2 .45 .50 ---------------------------------------------------------- | | ALPHA | | |-----------------------------| | | 0.05 | 0.01 | | |--------------+--------------| | | Total N | Total N | | |--------------+--------------| | |100 |125 |150 |100 |125 |150 | | |----+----+----+----+----+----| | |Pow-|Pow-|Pow-|Pow-|Pow-|Pow-| | | er | er | er | er | er | er | |--------------------------+----+----+----+----+----+----| |Method |Statistic | | | | | | | |------------+-------------| | | | | | | |Comparing |Regular F | | | | | | | |nested R**2 | | | | | | | | |values | |.741|.843|.909|.506|.648|.762| ---------------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Corey Latour: Predicting Salaries: Will the TeamValu scale signicantly improve the model? -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: 1BETAOLS 1.5 2.0 SDX .6 .8 TOLERANCE .6 .7 SD 3 3.5 NUMPARMS 4 NTOTAL 50 75 TAILS 1 NOTE: No ALPHA statement. Default is alpha = 0.05 only. Testing Ho: Beta_j = 0 in OLS model with 4 parameters. ------------------------------------------------------------------------------ UnifyPow Workshop Corey Latour: Predicting Salaries: Will the TeamValu scale signicantly improve the model? Alpha: 0.05 AND Beta Coefficient: 1.5 ------------------------------------------------ | | SD(Resid) | | |-------------------| | | 3 | 3.5 | | |---------+---------| | | Total N | Total N | | |---------+---------| | | 50 | 75 | 50 | 75 | | |----+----+----+----| | |Pow-|Pow-|Pow-|Pow-| | | er | er | er | er | |--------------------------+----+----+----+----| |Tol(X) |SD(X) |Type | | | | | |--------+--------+--------| | | | | |0.6 |0.6 |1-tail t|.490|.636|.399|.525| | |--------+--------+----+----+----+----| | |0.8 |1-tail t|.696|.844|.581|.737| |--------+--------+--------+----+----+----+----| |0.7 |0.6 |1-tail t|.541|.694|.442|.579| | |--------+--------+----+----+----+----| | |0.8 |1-tail t|.754|.890|.638|.793| ------------------------------------------------ ------------------------------------------------------------------------------ UnifyPow Workshop Corey Latour: Predicting Salaries: Will the TeamValu scale signicantly improve the model? Alpha: 0.05 AND Beta Coefficient: 2 ------------------------------------------------ | | SD(Resid) | | |-------------------| | | 3 | 3.5 | | |---------+---------| | | Total N | Total N | | |---------+---------| | | 50 | 75 | 50 | 75 | | |----+----+----+----| | |Pow-|Pow-|Pow-|Pow-| | | er | er | er | er | |--------------------------+----+----+----+----| |Tol(X) |SD(X) |Type | | | | | |--------+--------+--------| | | | | |0.6 |0.6 |1-tail t|.696|.844|.581|.737| | |--------+--------+----+----+----+----| | |0.8 |1-tail t|.891|.971|.794|.918| |--------+--------+--------+----+----+----+----| |0.7 |0.6 |1-tail t|.754|.890|.638|.793| | |--------+--------+----+----+----+----| | |0.8 |1-tail t|.928|.985|.846|.949| ------------------------------------------------ ------------------------------------------------------------------------------ UnifyPow Workshop Corey Latour: Predicting Salaries: Will the TeamValu scale signicantly improve the model? Alpha: 0.05 ------------------------------------------------------- | | SD(Resid) | | |-------------------| | | 3 | 3.5 | | |---------+---------| | | Total N | Total N | | |---------+---------| | | 50 | 75 | 50 | 75 | | |----+----+----+----| | |Pow-|Pow-|Pow-|Pow-| | | er | er | er | er | |---------------------------------+----+----+----+----| |Beta |Tol(X) |SD(X) |Type | | | | | |for | | | | | | | | |TeamVa-| | | | | | | | |lu | | | | | | | | |-------+-------+-------+---------| | | | | |1.5 |0.6 |0.6 |1-tail t |.490|.636|.399|.525| | | |-------+---------+----+----+----+----| | | |0.8 |1-tail t |.696|.844|.581|.737| | |-------+-------+---------+----+----+----+----| | |0.7 |0.6 |1-tail t |.541|.694|.442|.579| | | |-------+---------+----+----+----+----| | | |0.8 |1-tail t |.754|.890|.638|.793| |-------+-------+-------+---------+----+----+----+----| |2 |0.6 |0.6 |1-tail t |.696|.844|.581|.737| | | |-------+---------+----+----+----+----| | | |0.8 |1-tail t |.891|.971|.794|.918| | |-------+-------+---------+----+----+----+----| | |0.7 |0.6 |1-tail t |.754|.890|.638|.793| | | |-------+---------+----+----+----+----| | | |0.8 |1-tail t |.928|.985|.846|.949| ------------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Two-Group Design (revisited) Simple Example to Introduce EXEMPLARY SSH problem, OBS FEEDBACK VHCHANGE N_EXEMP 1 ET -0.86 20 2 SP -0.42 10 ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Two-Group Design (revisited) Simple Example to Introduce EXEMPLARY SSH problem, General Linear Models Procedure Class Level Information Class Levels Values FEEDBACK 2 ET SP Number of observations in data set = 30 ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Two-Group Design (revisited) Simple Example to Introduce EXEMPLARY SSH problem, General Linear Models Procedure Dependent Variable: VHCHANGE Frequency: N_EXEMP Sum of Mean Source DF Squares Square F Value Pr > F Model 1 1.29066667 1.29066667 99999.99 0.0001 Error 28 0.00000000 0.00000000 Corrected Total 29 1.29066667 R-Square C.V. Root MSE VHCHANGE Mean 1.000000 0 0 -0.713333 Source DF Type I SS Mean Square F Value Pr > F FEEDBACK 1 1.29066667 1.29066667 99999.99 0.0001 Source DF Type III SS Mean Square F Value Pr > F FEEDBACK 1 1.29066667 1.29066667 99999.99 0.0001 ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Two-Group Design (revisited) Simple Example to Introduce EXEMPLARY SSH problem, -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: EXEMPLARY SSH NUMPARMS 2 NEXEMPLARY 30 SD .45 .57 .65 ALPHA .05 .01 NTOTAL 15 21 33 EFFECTS ET vs SP on VHchange 1 1.29066667 ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Two-Group Design (revisited) Simple Example to Introduce EXEMPLARY SSH problem, Nobody should use UnifyPow this way to do a traditional t test! It is only done here to introduce the use of the EXEMPLARY SSH method to compute and table power or sample sizes for complex general linear models. ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Two-Group Design (revisited) Simple Example to Introduce EXEMPLARY SSH problem, Scenario: exemplary SSH ------------------------------------------------------------------------- | | Standard Deviation | | |--------------------------------------------| | | 0.45 | 0.57 | 0.65 | | |--------------+--------------+--------------| | | Total N | Total N | Total N | | |--------------+--------------+--------------| | | 15 | 21 | 33 | 15 | 21 | 33 | 15 | 21 | 33 | | |----+----+----+----+----+----+----+----+----| | |Pow-|Pow-|Pow-|Pow-|Pow-|Pow-|Pow-|Pow-|Pow-| | | er | er | er | er | er | er | er | er | er | |--------------------------+----+----+----+----+----+----+----+----+----| |Test |Alpha |Type | | | | | | | | | | |--------+--------+--------| | | | | | | | | | |ET vs SP|0.05 |2-tail t|.380|.518|.727|.257|.353|.526|.209|.284|.427| |on | |--------+----+----+----+----+----+----+----+----+----| |VHchange| |1-tail t|.519|.652|.828|.379|.485|.655|.318|.407|.559| | |--------+--------+----+----+----+----+----+----+----+----+----| | |0.01 |2-tail t|.156|.260|.472|.091|.147|.276|.068|.108|.201| | | |--------+----+----+----+----+----+----+----+----+----| | | |1-tail t|.235|.358|.581|.145|.219|.372|.112|.167|.283| ------------------------------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Two-Group Design (revisited) Simple Example to Introduce EXEMPLARY SSH problem, -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: EXEMPLARY SSH NUMPARMS 2 NEXEMPLARY 30 SD .45 .57 .65 ALPHA .05 .01 POWER .80 .90 EFFECTS ET vs SP on VHchange 1 1.29066667 ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Alalgia's Two-Group Design (revisited) Simple Example to Introduce EXEMPLARY SSH problem, Scenario: exemplary SSH ---------------------------------------------------------------------- | | Standard Deviation | | |-----------------------------------------| | | 0.45 | 0.57 | 0.65 | | |-------------+-------------+-------------| | |Minimum Power|Minimum Power|Minimum Power| | |-------------+-------------+-------------| | | .800 | .900 | .800 | .900 | .800 | .900 | | |------+------+------+------+------+------| | |Total |Total |Total |Total |Total |Total | | | N | N | N | N | N | N | |--------------------------+------+------+------+------+------+------| |Test |Alpha |Type | | | | | | | |--------+--------+--------| | | | | | | |ET vs SP|0.05 |2-tail t| 39| 52| 62| 82| 80| 106| |on | |--------+------+------+------+------+------+------| |VHchange| |1-tail t| 31| 42| 49| 67| 63| 86| | |--------+--------+------+------+------+------+------+------| | |0.01 |2-tail t| 59| 74| 92| 116| 119| 150| | | |--------+------+------+------+------+------+------| | | |1-tail t| 51| 65| 79| 102| 102| 131| ---------------------------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Mindy Bowdy: 4-Group Design + Covariate Example 4 in O'Brien and Muller (1993, Section 8.3) (D) Dominators, (R) Regulars {Ordinary or Loner}, (F) Friendlies OBS LYSIS DRFGRP LESI N_EXEMP 1 0.3950 D -2 2 2 0.3650 D -1 3 3 0.3350 D 0 4 4 0.3050 D 1 5 5 0.2750 D 2 6 6 0.5233 R -2 12 7 0.5133 R -1 12 8 0.5033 R 0 12 9 0.4933 R 1 12 10 0.4833 R 2 12 11 0.6000 F -2 4 12 0.6000 F -1 4 13 0.6000 F 0 4 14 0.6000 F 1 4 15 0.6000 F 2 4 ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Mindy Bowdy: 4-Group Design + Covariate Example 4 in O'Brien and Muller (1993, Section 8.3) (D) Dominators, (R) Regulars {Ordinary or Loner}, (F) Friendlies General Linear Models Procedure Class Level Information Class Levels Values DRFGRP 3 D R F Number of observations in data set = 100 ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Mindy Bowdy: 4-Group Design + Covariate Example 4 in O'Brien and Muller (1993, Section 8.3) (D) Dominators, (R) Regulars {Ordinary or Loner}, (F) Friendlies General Linear Models Procedure Dependent Variable: LYSIS Frequency: N_EXEMP Sum of Mean Source DF Squares Square F Value Pr > F Model 6 24.49015340 4.08169223 99999.99 0.0001 Error 94 0.00000000 0.00000000 Uncorrected Total 100 24.49015340 R-Square C.V. Root MSE LYSIS Mean 1.000000 0 0 0.485980 Source DF Type I SS Mean Square F Value Pr > F DRFGRP 3 24.44665340 8.14888447 99999.99 0.0001 LESI*DRFGRP 3 0.04350000 0.01450000 99999.99 0.0001 Source DF Type III SS Mean Square F Value Pr > F DRFGRP 3 24.36259090 8.12086363 99999.99 0.0001 LESI*DRFGRP 3 0.04350000 0.01450000 99999.99 0.0001 Contrast DF Contrast SS Mean Square F Value Pr > F DvRvF main | LESI=0 2 0.67221490 0.33610745 99999.99 0.0001 Ave LESI slope 1 0.02584615 0.02584615 99999.99 0.0001 DvRvF x LESI 2 0.01753846 0.00876923 99999.99 0.0001 DvR | LESI=0 1 0.38375657 0.38375657 99999.99 0.0001 RvF | LESI=0 1 0.14026335 0.14026335 99999.99 0.0001 Slopes: DvR 1 0.01083871 0.01083871 99999.99 0.0001 Slopes: RvF 1 0.00300000 0.00300000 99999.99 0.0001 T for H0: Pr > |T| Std Error of Parameter Estimate Parameter=0 Estimate DRFGRP D 0.3350000000 9999.99 0.0001 0 R 0.5033000000 9999.99 0.0001 0 F 0.6000000000 9999.99 0.0001 0 LESI*DRFGRP D -.0300000000 -9999.99 0.0001 0 R -.0100000000 -9999.99 0.0001 0 F 0.0000000000 9999.99 0.0001 0 ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Mindy Bowdy: 4-Group Design + Covariate Example 4 in O'Brien and Muller (1993, Section 8.3) (D) Dominators, (R) Regulars {Ordinary or Loner}, (F) Friendlies -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: EXEMPLARY SSH NEXEMPLARY 100 ALPHA .05 SD .12 .15 NTOTAL 200 300 500 NUMPARMS 6 TAILS 2 EFFECTS DvRvF main | LESI=0 2 0.6722149 Ave LESI slope 1 0.0258462 DvRvF x LESI 2 0.0175385 ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Mindy Bowdy: 4-Group Design + Covariate Example 4 in O'Brien and Muller (1993, Section 8.3) (D) Dominators, (R) Regulars {Ordinary or Loner}, (F) Friendlies -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: EXEMPLARY SSH NEXEMPLARY 100 ALPHA .025 SD .12 .15 NTOTAL 200 300 500 NUMPARMS 6 TAILS 2 EFFECTS DvR | LESI=0 1 0.3837566 RvF | LESI=0 1 0.1402634 Slopes: DvR 1 0.0108387 Slopes: RvF 1 0.0030000 ------------------------------------------------------------------------------ UnifyPow Workshop Dr. Mindy Bowdy: 4-Group Design + Covariate Example 4 in O'Brien and Muller (1993, Section 8.3) (D) Dominators, (R) Regulars {Ordinary or Loner}, (F) Friendlies Scenario: Exemplary SSH ---------------------------------------------------------- | | Standard Deviation | | |-----------------------------| | | 0.12 | 0.15 | | |--------------+--------------| | | Total N | Total N | | |--------------+--------------| | |200 |300 |500 |200 |300 |500 | | |----+----+----+----+----+----| | |Pow-|Pow-|Pow-|Pow-|Pow-|Pow-| | | er | er | er | er | er | er | |--------------------------+----+----+----+----+----+----| |Test |Alpha |Type | | | | | | | |--------+--------+--------| | | | | | | |DvRvF |0.05 |Regular | | | | | | | |main | | |F | | | | | | | |LESI=0 | | |.999|.999|.999|.999|.999|.999| |--------+--------+--------+----+----+----+----+----+----| |Ave LESI|0.05 |2-tail t| | | | | | | |slope | | |.470|.638|.848|.326|.456|.667| |--------+--------+--------+----+----+----+----+----+----| |DvRvF x |0.05 |Regular | | | | | | | |LESI | |F |.264|.380|.588|.182|.256|.404| |--------+--------+--------+----+----+----+----+----+----| |DvR | |0.025 |2-tail t| | | | | | | |LESI=0 | | |.999|.999|.999|.999|.999|.999| |--------+--------+--------+----+----+----+----+----+----| |RvF | |0.025 |2-tail t| | | | | | | |LESI=0 | | |.984|.999|.999|.897|.981|.999| |--------+--------+--------+----+----+----+----+----+----| |Slopes: |0.025 |2-tail t| | | | | | | |DvR | | |.154|.228|.380|.103|.148|.244| |--------+--------+--------+----+----+----+----+----+----| |Slopes: |0.025 |2-tail t| | | | | | | |RvF | | |.057|.074|.111|.045|.056|.078| ---------------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Lactic Acidosis in Children with Malaria (revisited) 28% die untreated. What if DCA cuts this by 25%? OBS TRTMENT DCA DIED PRDEATH N_EXEMP EXEMPFRQ 1 Placebo 0 1 0.28 100 28 2 Placebo 0 0 0.28 100 72 3 DCA 1 1 0.21 200 42 4 DCA 1 0 0.21 200 158 ------------------------------------------------------------------------------ UnifyPow Workshop Lactic Acidosis in Children with Malaria (revisited) 28% die untreated. What if DCA cuts this by 25%? The LOGISTIC Procedure Data Set: WORK.DATA3 Response Variable: DIED Response Levels: 2 Number of Observations: 4 Weight Variable: EXEMPFRQ Sum of Weights: 300 Link Function: Logit Response Profile Ordered Total Value DIED Count Weight 1 0 2 230.00000 2 1 2 70.00000 Model Fitting Information and Testing Global Null Hypothesis BETA=0 Intercept Intercept and Criterion Only Covariates Chi-Square for Covariates AIC 327.964 328.173 . SC 327.350 326.946 . -2 LOG L 325.964 324.173 1.790 with 1 DF (p=0.1809) Score . . 1.826 with 1 DF (p=0.1766) Analysis of Maximum Likelihood Estimates Parameter Standard Wald Pr > Standardized Odds Variable DF Estimate Error Chi-Square Chi-Square Estimate Ratio INTERCPT 1 0.9445 0.2227 17.9829 0.0001 . . DCA 1 0.3805 0.2824 1.8153 0.1779 0.988821 1.463 Association of Predicted Probabilities and Observed Responses Concordant = 25.0% Somers' D = 0.000 Discordant = 25.0% Gamma = 0.000 Tied = 50.0% Tau-a = 0.000 (4 pairs) c = 0.500 ------------------------------------------------------------------------------ UnifyPow Workshop Lactic Acidosis in Children with Malaria (revisited) 28% die untreated. What if DCA cuts this by 25%? -------------------------------------------------------------------- | UnifyPow 98.08.23 1998 Copyright (c) by Ralph G. O'Brien | | For information, see http://www.bio.ri.ccf.org/power.html | -------------------------------------------------------------------- Specifications processed: EXEMPLARY CHI**2 NEXEMPLARY 300 ALPHA .01 .045 NTOTAL 750 999 1500 EFFECTS DCA vs. Placebo 1 1.790 ------------------------------------------------------------------------------ UnifyPow Workshop Lactic Acidosis in Children with Malaria (revisited) 28% die untreated. What if DCA cuts this by 25%? Nobody should use UnifyPow this way to test two independent proportions. It is only done here to introduce the use of the EXEMPLARY CHI**2 method to compute and table powers or sample sizes for complex logistic regression or log-linear models, or similar methods using -2lnL chi**2 test statistics. ------------------------------------------------------------------------------ UnifyPow Workshop Lactic Acidosis in Children with Malaria (revisited) 28% die untreated. What if DCA cuts this by 25%? Scenario: exemplary chi**2 ---------------------------------------------------------- | | ALPHA | | |-----------------------------| | | 0.01 | 0.045 | | |--------------+--------------| | | Total N | Total N | | |--------------+--------------| | |750 |999 |1500|750 |999 |1500| | |----+----+----+----+----+----| | |Pow-|Pow-|Pow-|Pow-|Pow-|Pow-| | | er | er | er | er | er | er | |--------------------------+----+----+----+----+----+----| |Effect |Statistic | | | | | | | |------------+-------------| | | | | | | |DCA vs. |2-tail Z |.323|.447|.661|.544|.669|.838| |Placebo |-------------+----+----+----+----+----+----| | |1-tail Z |.416|.546|.747|.663|.772|.903| ---------------------------------------------------------- ------------------------------------------------------------------------------ UnifyPow Workshop Get Exemplary D values for Study of Lyon's Nonstress Test See O'Brien (1986, SUGI-11 Proceedings, 778-784) Summarized in Agresti (1990, Analysis of Categorical Data, Wiley, p 243). OBS STANDARD LYONS CELLPROB PRNRCARE NONROUTN EXEMPFRQ 1 1 1 0.04 0.40 0 24.0 2 1 1 0.04 0.40 1 16.0 3 1 2 0.08 0.32 0 54.4 4 1 2 0.08 0.32 1 25.6 5 1 3 0.04 0.27 0 29.2 6 1 3 0.04 0.27 1 10.8 7 2 1 0.02 0.30 0 14.0 8 2 1 0.02 0.30 1 6.0 9 2 2 0.18 0.22 0 140.4 10 2 2 0.18 0.22 1 39.6 11 2 3 0.64 0.15 0 544.0 12 2 3 0.64 0.15 1 96.0 ------------------------------------------------------------------------------ UnifyPow Workshop Get Exemplary D values for Study of Lyon's Nonstress Test See O'Brien (1986, SUGI-11 Proceedings, 778-784) Summarized in Agresti (1990, Analysis of Categorical Data, Wiley, p 243). The LOGISTIC Procedure Data Set: WORK.BABIES Response Variable: NONROUTN Response Levels: 2 Number of Observations: 12 Weight Variable: EXEMPFRQ Sum of Weights: 1000 Link Function: Logit Response Profile Ordered Total Value NONROUTN Count Weight 1 0 6 806.00000 2 1 6 194.00000 Model Fitting Information and Testing Global Null Hypothesis BETA=0 Intercept Intercept and Criterion Only Covariates Chi-Square for Covariates AIC 985.943 968.437 . SC 986.428 969.407 . -2 LOG L 983.943 964.437 19.505 with 1 DF (p=0.0001) Score . . 21.710 with 1 DF (p=0.0001) Analysis of Maximum Likelihood Estimates Parameter Standard Wald Pr > Standardized Odds Variable DF Estimate Error Chi-Square Chi-Square Estimate Ratio INTERCPT 1 1.1576 0.0960 145.3859 0.0001 . . XSTD 1 0.4381 0.0960 20.8250 0.0001 1.688701 1.550 Association of Predicted Probabilities and Observed Responses Concordant = 25.0% Somers' D = 0.000 Discordant = 25.0% Gamma = 0.000 Tied = 50.0% Tau-a = 0.000 (36 pairs) c = 0.500 ------------------------------------------------------------------------------ UnifyPow Workshop Get Exemplary D values for Study of Lyon's Nonstress Test See O'Brien (1986, SUGI-11 Proceedings, 778-784) Summarized in Agresti (1990, Analysis of Categorical Data, Wiley, p 243). The LOGISTIC Procedure Data Set: WORK.BABIES Response Variable: NONROUTN Response Levels: 2 Number of Observations: 12 Weight Variable: EXEMPFRQ Sum of Weights: 1000 Link Function: Logit Response Profile Ordered Total Value NONROUTN Count Weight 1 0 6 806.00000 2 1 6 194.00000 Model Fitting Information and Testing Global Null Hypothesis BETA=0 Intercept Intercept and Criterion Only Covariates Chi-Square for Covariates AIC 985.943 962.319 . SC 986.428 963.774 . -2 LOG L 983.943 956.319 27.623 with 2 DF (p=0.0001) Score . . 30.513 with 2 DF (p=0.0001) Analysis of Maximum Likelihood Estimates Parameter Standard Wald Pr > Standardized Odds Variable DF Estimate Error Chi-Square Chi-Square Estimate Ratio INTERCPT 1 1.0219 0.1056 93.6660 0.0001 . . XSTD 1 0.2883 0.1098 6.8935 0.0087 1.111257 1.334 XLYONS1 1 -0.4049 0.1402 8.3440 0.0039 -1.269382 0.667 Association of Predicted Probabilities and Observed Responses Concordant = 41.7% Somers' D = 0.000 Discordant = 41.7% Gamma = 0.000 Tied = 16.7% Tau-a = 0.000 (36 pairs) c = 0.500 ------------------------------------------------------------------------------ UnifyPow Workshop Get Exemplary D values for Study of Lyon's Nonstress Test See O'Brien (1986, SUGI-11 Proceedings, 778-784) Summarized in Agresti (1990, Analysis of Categorical Data, Wiley, p 243). The LOGISTIC Procedure Data Set: WORK.BABIES Response Variable: NONROUTN Response Levels: 2 Number of Observations: 12 Weight Variable: EXEMPFRQ Sum of Weights: 1000 Link Function: Logit Response Profile Ordered Total Value NONROUTN Count Weight 1 0 6 806.00000 2 1 6 194.00000 Model Fitting Information and Testing Global Null Hypothesis BETA=0 Intercept Intercept and Criterion Only Covariates Chi-Square for Covariates AIC 985.943 964.277 . SC 986.428 966.217 .