Control charts for binary correlated variables Linda Lee Ho - PowerPoint PPT Presentation

Control charts for binary correlated variables Linda Lee Ho Airlane P Alencar USP - Brazil

Introduction • Automated manufacturing industries • Common practice - Inspections of items produced continuously • n binary random variables, 1 , , Z Z n   • ~ Z Bernoulli p i • =1 for a non-conforming item and 0 Z i otherwise

Introduction • Monitoring: the number (or the proportion) of non-conforming items in a sample of n units • np chart or p chart – used for SPC • Assumption: independent Z i

Introduction • However, in many production processes • A common component in the whole process may yield a correlation between the different items. • That is         , 0; Corr Z Z i j i j

Introduction • The consequences of this correlation n •  is inflated  0 Z i  1 i    n   • (1 )   Var Z np p (it is overdispersed). i    1 i

Assumptions • New production process • Phase I • For high quality processes: – the correlation of the binary variables is not easily identified. n  – High frequency of  is expected 0 Z i  1 i

Overdispersed binomial distribution n  •  T Z the number of non conforming items in the i  1 i sample of correlated n items.   n           0    I t   n t t   t b 1 1 P T t a a b        1 p                     ; ; ; ln (1 ) a e ln 1 p p b e   p   • Var(T)=np(1-p)[1+(n-1)  ]

Overdispersed binomial distribution

np x np  control charts

ML estimation • Maximum likelihood and method of moment estimation of the correlation parameter are presented • Comparison of performance: compared by simulation.

ML Estimation • Likelihood function   n          k n t  t     n    1 a b b ( , , , ) 1 1 0 i L T T T a a b   i 1 2 k     t ( 0, 1,..., ) t i k i i   n k           t  t n nk      0 k a 1 1   i 1  a a b b b 1 i   1    t ( 1.,,. ) i k i

ML Estimation  t i ˆ b  ˆ ˆ    0 t        1 2 1 ML n n ((1 ) (1 ) ... 1) b b i ˆ ML ML   n nk 1 (1 ) b 1 ML  k t i i   ˆ 1   a   ˆ ˆ ML n nk b k b ML ML

ML x MM

np x np  control charts        H : versus H : H : versus H : . 0 0 1 1 a a 0 1 0 1 • For a type error I equal 5%:    ˆ ˆ • Reject null hypothesis if .95 ML   | 0  ˆ ˆ • Or a a .05 ML  | a 1

Critical values – at 5%

Control chart np  • a Shewhart control chart named np  to monitor the non-conforming fraction when the binary variables are correlated. • The traditional np chart is a particular case of  np  the control chart when =0.

np  control charts  • Upward shifts: p p 1 0 • Given  , control limit (CL) is determined

Performance of np , np  control charts

Performance of np , np  control charts

np  x EWMA np  control charts

Performance of EWMA np  control charts

Conclusions • ML and MM estimation of the correlation parameter are similar. • ML estimator – low bias • the control chart needs at least to double np  the sample size - to have the similar performance of the traditional np control chart .

Thank you!