
QBIPDFName:
with p, , and m denoting the shape parameters. The quasibinomial type I distribution is used to model Bernoulli trials. The parameter p denotes the initial probability of success, m denotes the number of Bernoulli trials, and denotes how the probability of success increases or decreases with the number of successes. Specificially, when = 0, the quasibinomial type I distribution reduces to the binomial distribution. When ≠ 0, the probability of success in the xth trial becomes
<SUBSET/EXCEPT/FOR qualification> where <x> is a positive integer variable, number, or parameter; <p> is a number, parameter, or variable in the range (0,1) that specifies the first shape parameter; <phi> is a number, parameter, or variable that specifies the second shape parameter; <m> is a number, parameter, or variable that specifies the third shape parameter; <y> is a variable or a parameter (depending on what <x> is) where the computed quasi binomial type I pdf value is stored; and where the <SUBSET/EXCEPT/FOR qualification> is optional.
LET Y = QBIPDF(X,0.7,0.01,20) PLOT QBIPDF(X,0.3,0.005,20) FOR X = 0 1 20
LET P = <value> LET PHI = <value> LET Y = QUASI BINOMIAL TYPE I RANDOM NUMBERS FOR I = 1 1 N
QUASI BINOMIAL TYPE I PROBABILITY PLOT Y
QUASI BINOMIAL TYPE I CHISQUARE GOODNESS OF FIT Y In fitting the quasibinomial type I distribution to data, we typically assume that the number of trials, m, is fixed and known and we then estimate p and . To obtain the maximum likelihood estimates of p and , enter the command
QUASI BINOMIAL TYPE I MAXIMUM LIKELIHOOD Y QUASI BINOMIAL TYPE I MAXIMUM LIKELIHOOD Y2 X2 For unbinned data, the maximum likelihood estimates are the solutions to the equations:
For binned data, the equations become
with k, n, and n_{x} denoting the number of classes, the total sample size, and the frequency of the xth class, respectively. These equations are known to have multiple solutions, so good starting values are required. By default, we use the starting values recommended by Consul and Famoye
with f_{0} denoting the frequency of the class x = 0 and denoting the sample mean. Alternatively, you can specify your own starting values by entering the commands
LET PHISTART = <value> Consul and Famoye give formulas for the Fisher information matrix (the inverse of the parameter variancecovariance matrix). They also give simplified formulas for the special cases m = 1, 2, or 3. You can generate estimates of p and based on the maximum ppcc value or the minimum chisquare goodness of fit with the commands
LET P1 = <value> LET P2 = <value> LET PHI1 = <value> LET PHI2 = <value> QUASI BINOMIAL TYPE I KS PLOT Y QUASI BINOMIAL TYPE I KS PLOT Y2 X2 QUASI BINOMIAL TYPE I KS PLOT Y3 XLOW XHIGH QUASI BINOMIAL TYPE I PPCC PLOT Y QUASI BINOMIAL TYPE I PPCC PLOT Y2 X2 QUASI BINOMIAL TYPE I PPCC PLOT Y3 XLOW XHIGH The default values of p1 and p2 are 0.05 and 0.95, respectively. The default values for phi1 and phi2 are phi1 = p1/m and phi2 = (1p1)/m. Due to the discrete nature of the percent point function for discrete distributions, the ppcc plot will not be smooth. For that reason, if there is sufficient sample size the KS PLOT (i.e., the minimum chisquare value) is typically preferred. However, it may sometimes be useful to perform one iteration of the PPCC PLOT to obtain a rough idea of an appropriate neighborhood for the shape parameters since the minimum chisquare statistic can generate extremely large values for nonoptimal values of the shape parameters. Also, since the data is integer values, one of the binned forms is preferred for these commands.
title size 3 tic label size 3 label size 3 legend size 3 height 3 x1label displacement 12 y1label displacement 15 . multiplot corner coordinates 0 0 100 95 multiplot scale factor 2 label case asis title case asis case asis tic offset units screen tic offset 3 3 title displacement 2 y1label Probability Mass x1label X . ylimits 0 1 major ytic mark number 6 minor ytic mark number 3 xlimits 0 20 line blank spike on . multiplot 2 2 . title P = 0.3, Phi = 0.01, M = 20 plot qbipdf(x,0.3,0.01,20) for x = 1 1 20 . title P = 0.3, Phi = 0.01, M = 20 let phi = 0.01 plot qbipdf(x,0.3,phi,20) for x = 1 1 20 . title P = 0.7, Phi = 0.01, M = 20 plot qbipdf(x,0.7,0.01,20) for x = 1 1 20 . title P = 0.7, Phi = 0.01, M = 20 let phi = 0.01 plot qbipdf(x,0.7,phi,20) for x = 1 1 20 . end of multiplot . justification center move 50 97 text Probability Mass Functions for Quasi Binomial Type I Program 2: let p = 0.7 let phi = 0.01 let m = 20 let psave = p let phisave = phi let y = quasi binomial type I rand numb for i = 1 1 500 . let y3 xlow xhigh = integer frequency table y class lower 0.5 class width 1 class upper 20.5 let y2 x2 = binned y . relative hist y2 x2 multiplot 2 2 4 limits freeze preerase off line color blue plot qbipdf(x,p,phi,m) for x = 0 1 20 limits preerase on line color black . quasi binomial type I mle y let p = pml let phi = phiml . quasi binomial type I chisquare goodness of fit y3 xlow xhigh . char x line bl quasi binomial type I probability plot y3 xlow xhigh . char bl line so let p1 = 0.5 let p2 = 0.9 let a1 = (1  p1)/m let a2 = (1  p2)/m let phi1 = 0 let phi2 = max(a1,a2) quasi binomial type I ks plot y3 xlow xhigh let p = shape1 let phi = shape2 quasi binomial type I chisquare goodness of fit y3 xlow xhigh QUASI BINOMIAL TYPE I PARAMETER ESTIMATION: NUMBER OF OBSERVATIONS = 500 SAMPLE MEAN = 17.26600 SAMPLE STANDARD DEVIATION = 1.901177 SAMPLE MINIMUM = 7.000000 SAMPLE MAXIMUM = 20.00000 ZEROCLASS FREQUENCY: = 0.000000 USERSPECIFIED VALUE FOR M = 20.00000 MAXIMUM LIKELIHOOD: ESTIMATE OF P = 0.7056712 ESTIMATE OF PHI = 0.9720844E02 THE COMPUTED VALUE OF THE CONSTANT P = 0.7056712E+00 THE COMPUTED VALUE OF THE CONSTANT PHI = 0.9720844E02 CHISQUARED GOODNESSOFFIT TEST NULL HYPOTHESIS H0: DISTRIBUTION FITS THE DATA ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA DISTRIBUTION: QUASI BINOMIAL TYPE I SAMPLE: NUMBER OF OBSERVATIONS = 500 NUMBER OF NONEMPTY CELLS = 9 NUMBER OF PARAMETERS USED = 3 TEST: CHISQUARED TEST STATISTIC = 1.331433 DEGREES OF FREEDOM = 5 CHISQUARED CDF VALUE = 0.068336 ALPHA LEVEL CUTOFF CONCLUSION 10% 9.23636 ACCEPT H0 5% 11.07050 ACCEPT H0 1% 15.08628 ACCEPT H0 CHISQUARED GOODNESSOFFIT TEST NULL HYPOTHESIS H0: DISTRIBUTION FITS THE DATA ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA DISTRIBUTION: QUASI BINOMIAL TYPE I SAMPLE: NUMBER OF OBSERVATIONS = 500 NUMBER OF NONEMPTY CELLS = 9 NUMBER OF PARAMETERS USED = 3 TEST: CHISQUARED TEST STATISTIC = 2.357771 DEGREES OF FREEDOM = 5 CHISQUARED CDF VALUE = 0.202254 ALPHA LEVEL CUTOFF CONCLUSION 10% 9.23636 ACCEPT H0 5% 11.07050 ACCEPT H0 1% 15.08628 ACCEPT H0
Date created: 8/23/2006 