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 for a non-conforming item and 0
• therwise

1,

,

n

Z Z

 

~

i

Z Bernoulli p

i

Z

Introduction

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

i

Z

Introduction

• However, in many production processes
• A common component in the whole process

may yield a correlation between the different items.

• That is

 

 

, 0;

i j

Corr Z Z i j    

Introduction

• The consequences of this correlation
• is inflated
• (it is overdispersed).

1 n i i

Z

 



1

(1 )

n i i

Var Z np p

 

       

Assumptions

• New production process
• Phase I
• For high quality processes:

– the correlation of the binary variables is not easily identified. – High frequency of is expected

1 n i i

Z

 



Overdispersed binomial distribution

• the number of non conforming items in the

sample of correlated n items.

• Var(T)=np(1-p)[1+(n-1)]

1 n i i

T Z

 



    



 

1 1

I t n t t

n P T t a a b t b

 

    

     

   

ln 1

1 ; ; ; ln (1 )

b e

p a e p p p



 



 



 

         

     

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

   

 

     

1

1

1 2

( 1.,,. )

( 0, 1,..., )

( , , , ) 1 1

1 1 1

1

i i i i i

k

n k

k n nk k i k i

n t t t i k i t t

L T T T a a b

n a a b b b t

n a b b t

a

 

  

   

   

    

                    

ML Estimation

 

 

1

ˆ ˆ ˆ

i i ML n ML ML

t k a nk b k b



 

1 2 1 1

ˆ ˆ ˆ ((1 ) (1 ) ... 1) ˆ 1 (1 )

i

i t n n n

ML ML ML ML

t b b b nk b

   

        



ML x MM

• For a type error I equal 5%:
• Reject null hypothesis if
• Or

1 1

: : : : .

a a       

1 1

H versus H H versus H

|

.95

ˆ ˆ

ML



 





| 1

.05

ˆ ˆ

a

ML

a a





np x np control charts

Critical values – at 5%

Control chart

• a Shewhart control chart named

to monitor the non-conforming fraction when the binary variables are correlated.

• The traditional np chart is a particular case of

the control chart when =0.

np



np

np

np control charts

• Upward shifts:
• Given , control limit (CL) is determined

1

p p 

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 the sample size - to have the similar performance of the traditional np control chart .

np

Thank you!