Making Decisions with
Probability Distributions
[email protected]
BioPhia Consulting, Inc.
1
David LeBlond, PhD 2/2017
Definitions
Decision:
An action taken to minimize risk.
Probability distribution (PD):
A tool for quantifying risk.
x
a b c d e f
x
g h
Bad thing happens
2
David LeBlond, PhD 2/2017
Relationship to Quality
Knowing the…
means
knowing the…
quality
risk
risk
probability
probability
distribution.
x
a b c d e f
x
g h
Bad thing happens
3
David LeBlond, PhD 2/2017
Role of PDs in science & technology
Probability Theory
Probability Tools: 350+ years old
Statistical Tools: 110 years old
Statistics
Need PDs for model building
Inference:
Estimate model parameters from data
Deduction:
Predict future values, given estimates.
4
David LeBlond, PhD 2/2017
Some applications of PDs
•
•
•
•
•
•
•
•
risk assessment (FMEA, ICH Q9)
acceptance sampling
sample size estimation
Monte-Carlo predictions
process capability
statistical tests
interval estimation
Bayesian methods
x
a b c d e f
x
g h
Bad thing happens
5
David LeBlond, PhD 2/2017
Why calculate PDs MS Excel?
Cons
• Manual
• Formulas must be ‘verified’
• Many complex analyses cannot be done in EXCEL
Pros
• Availability
• No black box
• Empowering
• Provide templates to clients
• Complex formulas duplicated by ‘cut and paste’
• ‘Accessible’ modeling environment
• Common distributions relatively easy to model
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David LeBlond, PhD 2/2017
Outline
• Why PDs? Why Excel?
• pdf, pmf, cdf, qf
• Excel formulas for 5 count and 10 continous PDs
• Generating random draws from a PD
• Histograms in Excel
• Skewness and Kurtosis
• Testing for normality
• Transformations to normality
• Bonus: Matrix calculations in Excel
Approach:
minimize algebra, Greek letters
include a little historical trivia
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David LeBlond, PhD 2/2017
probability
density
function
cdf
cumulative
distribution
function
0.5
0.4
0.3
0.2
0.1
0
-4
-3
-2
-1
0
1
2
3
4
x
Cumulative distribution
pdf
Probability density
PDs for continuous variates
1
qf
0.8
0.6
0.4
0.2
0
-4
-3
-2
-1
0
1
2
3
4
quantile
function
x
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David LeBlond, PhD 2/2017
PDs for count variates
0.3
pmf
probability
mass
function
probability mass
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
8
9
10
x
cdf
cumulative
distribution
function
Cumulative distribution
1.2
1
qf
0.8
0.6
quantile
function
0.4
0.2
0
0
1
2
3
4
5
6
7
8
9
10
x
9
David LeBlond, PhD 2/2017
Parameters of a PD
X is a random variate, x is a specific value of X
Continuous pdf: = NORMDIST(x, m, s, FALSE)
True mean
m
True sigma
s
Count pmf: = BINOMDIST(x, n, p, FALSE)
Repetitions
n
Success rate
p
In Excel formula, substitute the cell range for x, m, s, n, p, …
10
David LeBlond, PhD 2/2017
Relationships
among
distributions
Neg.Binomial
n, p
David LeBlond, PhD 2/2017
Beta
a, b
Normal
m, s
Std Normal
LS Student t
v, m, s
Binomial
n, p
Poisson
m
Lognormal
m, s
Student t
v
Hypergeometric
n, M, N
F
v1, v2
Bernoulli
p
Std Uniform
Gamma
a, b
Chisquare
v
Exponential
b
Uniform
h, d
Weibull
c,b 11
Jacob Bernoulli,
1713, Art of Conjecture
Swiss Mathematician
Binomial
Story: A jar contains a very large (i.e., infinite) number of widgets. A known
proportion, p, are defective. We randomly select n widgets and measure x, the
number in this sample that are defective.
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
n=10, p=0.1
n=1  Bernoulli PD
remember: cdf is really a step function.
Cumulative distribution
Probability mass
pmf: M = BINOMDIST(x,n,p,FALSE)
cdf: P = BINOMDIST(x,n,p,TRUE)
qf: x = CRITBINOM(n,p,P)
n=10, p=0.5
n=10, p=0.9
1.2
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
x
David LeBlond, PhD 2/2017
6
7
8
9 10
0
1
2
3
4
5
x
6
7
8
9 10
12
Negative Binomial
Blaize Pascal,
1654, France
mathematician, theologian, gambler
AKA “Pascal PD”
Story: A process has a known probability of success, p, on each run. We
run the process until we make n good lots and measure x, the number of
failed lots produced.
pmf: M=NEGBINOMDIST(X, n, p)
Probability mass
0.3
0.25
n=2, p=0.25
0.2
n=2, p=0.5
0.15
n=5, p=0.5
0.1
0.05
18
15
12
9
6
3
0
0
x
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David LeBlond, PhD 2/2017
Negative Binomial
Cumulative distribution
cdf: create pmf for 0,…,x then SUM(pmf range)
qf: create cdf range for -1 to x, then LOOKUP(P,cdf range,quantile range)+1
1.2
1
0.8
0.6
n=2, p=0.25
0.4
n=2, p=0.5
0.2
n=5, p=0.5
18
15
12
9
6
3
0
0
x
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David LeBlond, PhD 2/2017
Poisson
Simeon Poisson
1838, France
mathematician, astronomer
Story: A white ointment is contaminated with randomly dispersed black
specs. The true mean number of black specs per unit volume is m. We
randomly sample a unit volume and measure x, the number of black specs
in the sample.
pmf: M = POISSON(X, m, FALSE)
cdf: P = POISSON(X, m, TRUE)
qf: create cdf range for -1 to x, then LOOKUP(P,cdf range,quantile range)+1
m=1
m=2
m=4
m=8
0.35
0.3
0.25
0.2
0.15
0.1
1
0.8
0.6
m=1
m=2
m=4
m=8
0.4
0.2
x
12
10
8
6
4
2
0
12
10
8
6
4
0
2
0
0.05
0
Cumulative distribution
Probability mass
0.4
x
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David LeBlond, PhD 2/2017
Hypergeometric
Abraham de Moivre
1711, France-England
“Doctrine of Chances” prized by gamblers
Story: K of the N widgets in a small batch are known to be defective. We
randomly select n widgets and measure x, the number in this sample that are
defective. (Note if n-x > N-K or if x >K or x>n, pmf must = 0).
pmf: M =IF(AND(MAX(0,n+K-N)<=X,X<=MIN(K,n)),HYPGEOMDIST(X,n,K,N),0)
cdf: create pmf for 0,…,x then SUM(pmf range)
qf: create cdf range for -1 to x, then LOOKUP(P,cdf range,quantile range)+1
n=1, K=10, N=20
n=2, K=10, N=20
n=4, K=10, N=20
n=8, K=10, N=20
0.5
0.4
0.3
0.2
0.1
0
Cumulative distribution
1
0.8
0.6
n=1, K=10, N=20
n=2, K=10, N=20
n=4, K=10, N=20
n=8, K=10, N=20
0.4
0.2
x
12
10
8
6
4
2
12
10
8
6
4
2
0
0
0
Probability mass
0.6
x
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David LeBlond, PhD 2/2017
Normal
Karl Friedrich Gauss
1809, German
rigorously justified
Story: x is the sum of the effects of many small, independent factors. No
single process has a dominant effect. The true mean and sigma of x are m
and s.
Abraham de Moivre
1711, France
First described
Cumulative distribution
Probability density
pdf: D = NORMDIST(x, m, s, FALSE)
cdf: P = NORMDIST(x, m, s, TRUE)
qf: x = NORMINV(P, m, s) = m + s*NORMINV(P,0,1)
0.5
A
1

 0.61
0.4
B
e
0.3
A
0.2
B
0.1
0
-4
-3
-2
-1
0
x
1
2
3
4
1
0.8
0.6
0.9973
0.4
0.68
0.955
0.2
0
-4
-3
-2
-1
0
1
2
3
4
x
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David LeBlond, PhD 2/2017
LogNormal
Story: x is the multiplicative product of the effects of many
small, independent factors. The natural log of x follows a
normal distribution with true mean m and true standard
deviation s.
pdf:D = EXP(0-(LN(x)-m)^2/2/s^2)/SQRT(2*PI())/s/x
cdf: P=LOGNORMDIST(X, m , s)
Sir Francis Galton
qf: X=LOGINV(P,m, s)
1879, Darwin’s cousin
1
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Cumulative distribution
Probability density
developed regression, correlation
m=0, s=0.25
m=0, s=0.5
m=0, s=1
m=0, s=2
0.8
0.6
m=0, s=0.25
m=0, s=0.5
m=0, s=1
m=0, s=2
0.4
0.2
0
0
1
2
x
3
0
1
x
2
3
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David LeBlond, PhD 2/2017
Exponential
Story: An electrical system experiences occasional random blackouts.
The long term mean blackout rate, b, is constant. Measure x, the time
between blackouts.
Leonard Euler
1773, Swiss mathematician
first to use letter “e”
pdf: D = EXPONDIST(x, b, FALSE)
cdf: P = EXPONDIST(x, b, TRUE)
qf: x = -LN(1-P)/b
1
0.4
0.3
Cumulative distribution
Probability density
b=0.05
b=0.1
b=0.2
0.2
b=0.4
0.1
0.8
0.6
b=0.05
b=0.1
b=0.2
b=0.4
0.4
0.2
0
0
0
5
10
x
15
0
5
x
10
15
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David LeBlond, PhD 2/2017
Gamma
My Gamma (& Gampa)
1965, Battle Creek, MI
Story: Same as Exponential, except measure x, the time required for a
≥ 1 blackouts to occur.
pdf: D = GAMMADIST(x, a, 1/b, FALSE)
cdf: P = GAMMADIST(x, a, 1/b, TRUE)
qf: x = GAMMAINV(P,a,1/b)
1
a=1, b=0.4
a=2, b=0.4
a=4, b=0.4
a=8, b=0.4
0.3
0.2
Cumulative distribution
Probability density
0.4
0.1
0.8
a=1, b=0.4
a=2, b=0.4
a=4, b=0.4
a=8, b=0.4
0.6
0.4
0.2
0
0
0
5
10
x
15
0
5
x
10
15
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David LeBlond, PhD 2/2017
Woloddi Weibull
1939, Sweedish engineer
Explosions to study ocean floor
Weibull
Story: Same as Exponential except the long term mean blackout rate
b is not constant but increases (c > 1, “wear out”) or decreases (c< 1,
“infant mortality”) over time. Measure x, the time between blackouts.
pdf: D = WEIBULL(x, c,1/b , FALSE)
cdf: P = WEIBULL(x, c, 1/b , TRUE)
qf: x = (-LN(1-P))^(1/c)/b
1
c=1, b=0.4
c=2, b=0.4
c=4, b=0.4
c=8, b=0.4
1
0.8
0.6
0.4
0.2
0
Cumulative distribution
Probability density
1.2
0.8
0.6
c=1, b=0.4
c=2, b=0.4
c=4, b=0.4
c=8, b=0.4
0.4
0.2
0
0
1
2
3
x
4
5
0
1
2
x
3
4
5
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David LeBlond, PhD 2/2017
Beta
Piere Simon de Laplace
1814, French mathematician
“rule of succession” based on beta
Story: A very large lot of widgets contains K defects. Widgets are sampled
randomly from the lot until “a” defectives have been found. Let b = K-a+1.
Measure x, the fraction of the lot tested.
pdf: D=EXP(GAMMALN(a+b)-GAMMALN(a)-GAMMALN(b))*x^(a-1)*(1-x)^(b-1)
cdf: P = BETADIST(x ,a ,b ,0 ,1 )
qf: x = BETAINV(P, a, b, 0, 1)
Probability density
2
1.5
1
0.5
1
Cumulative distribution
a=2, b=4
a=4, b=2
a=1, b=1
a=0.5, b=0.5
2.5
0.8
0.6
a=2, b=4
a=4, b=2
a=1, b=1
a=0.5, b=0.5
0.4
0.2
0
0
0
0.2
0.4
0.6
x
0.8
1
0
0.2
0.4
x
0.6
0.8
1
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David LeBlond, PhD 2/2017
Rev. Thomas Bayes
1763, London
“Bayes Rule”
Uniform
Story: All values of X between a and b are equally likely to occur.
pdf: D =1/(d-c)
cdf: P = c+x*(d-c)
qf: x = (P-c)/(d-c)
1
d c
SDPOP
0
c
d c

12
d
Cumulative distribution
Probability density
Standard Uniform has c=0 & d=1 (same as Beta with K=a=b=1)
1
0
c
d
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David LeBlond, PhD 2/2017
Student’s t
William Sealy Gossett (“Student”)
1908, Dublin
Guinness Brewery
Story: A sample of size v+1 is taken from a normal population with true mean
m and true sigma s. The sample average and SD are calculated. Let x =
(average-m)/SD/SQRT(v+1).
pdf: D=EXP(GAMMALN((v+1)/2)-GAMMALN(v/2))/SQRT(PI()*v)/(1+x^2/v)^((v+1)/2)
cdf: P = IF(x<0,TDIST(-x,v,1),1-TDIST(x,v,1))
qf: x = IF(P<0.5,-TINV(2*P,v),TINV(2*(1-P),v))
x’ = m+s*x ~ LSSt. substitute (x’-m)/s for x and D/s for D to obtain the pdf cdf
and qf for the LSSt distribution.
1
Probability Density
0.4
Cumulative distribution
v=1
v=2
v=4
0.3
v=8
v=16
0.2
v=32
Std Normal
0.1
0
0.8
v=1
v=2
v=4
v=8
v=16
v=32
Std Normal
0.6
0.4
0.2
0
-6 -5 -4 -3 -2 -1
0
x
1
2
3
4
5
6
-6
-5
-4
-3
-2
-1
0
x
1
2
3
4
5
6
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David LeBlond, PhD 2/2017
Friedrich Helmert, discoverer
1875, German Geologist
Lunar crater
Karl Pearson
1900, England
“Chisquare test”
Chisquare
Story: A sample of size v+1 is taken from a normal random
variable with known standard deviation, s. The sample variance
(SD^2) is calculated. Measure x = v*(sample variance)/s^2.
pdf: D = GAMMADIST(x, v/2, 2, FALSE)
cdf: P = 1-CHIDIST(X,v)
qf: X = CHIINV(1-P,v)
1
0.15
Cumulative distribution
Probability density
0.2
v=4
v=10
v=20
0.1
0.05
0
0
5
10
15
20
x
25
30
35
0.8
v=4
0.6
v=10
0.4
v=20
0.2
0
0
5
10
15
20
25
30
35
x
25
David LeBlond, PhD 2/2017
F
Sir Ronald Fisher
~1925, England
Invented ANOVA (and everything else)
A random sample of size v1+1 is taken from a normal population and the sample
variance (V1) is calculated. A second independent random sample of size v2+1 is
also taken from the same population and the sample variance (V2) is calculated.
Measure x = V1/V2 .
pdf: D = EXP( GAMMALN((v1+v2)/2)-GAMMALN(v1/2 )-GAMMALN(v2/2 ) )
*v1^(v1/2)*v2^(v2/2)*x^(v1/2-1)/(v2+v1*x)^(v1/2+v2/2)
cdf: P = 1-FDIST(X, v1, v2)
qf: x = FINV(1 - P,v1, v2)
1.4
1
Cumulative distribution
Probability density
1.2
v1=4, v2=40
v1=10, v2=40
v1=40, v2=40
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
x
2
2.5
3
0.8
0.6
v1=4, v2=40
v1=10, v2=40
0.4
v1=40, v2=40
0.2
0
0
0.5
1
1.5
x
2
2.5
3
26
David LeBlond, PhD 2/2017
Why Monte-Carlo
simulation?
Consider …
N = some messy f(A…M)
We know (or guess) the
PDs of A…M.
We know (or guess) f()
Goal: distribution of N?
typical manufacturing process
Solution:
• Generate random “draws” for variates A…M
• Calculate N=f(A…M) for each draw
• The calculated N’s will be draws from the target PD
27
David LeBlond, PhD 2/2017
Simulating continuous variates
uniform pdf
1
RAND()
0
1
cdf of target
distribution
0
qf of target distribution
Inverse cdf method:
just substitute RAND()
for P in the qf
pseudo-random draws from
target distribution
e.g. x is a draw from a normal distribution with mean m and sigma s
x = NORMINV(RAND(), m, s)
28
David LeBlond, PhD 2/2017
Simulating count variates
29
David LeBlond, PhD 2/2017
Simulating count variates
0.4
0.4
0.3
Observed
Proportion
Theoretical pmf
0.2
0.1
0.3
0.2
0.1
0
10
8
6
4
0
2
0
Proportion or
Probability mass
Histogram
Count
30
David LeBlond, PhD 2/2017
Creating a histogram in EXCEL
2
1
3
31
David LeBlond, PhD 2/2017
Creating a histogram in EXCEL
4
32
David LeBlond, PhD 2/2017
Creating a histogram in EXCEL
9
5
6
7
8
33
David LeBlond, PhD 2/2017
Creating a histogram in EXCEL
wala…!
Histogram
12
Frequency
10
8
6
4
2
400
380
360
340
320
300
280
260
240
220
200
180
160
0
Bin
34
David LeBlond, PhD 2/2017
Distribution Shape
Skewness
Kurtosis
Platykurtic
Leptokurtic
Student, 1927
35
David LeBlond, PhD 2/2017
Summary statistics
Statistic
Estimates true…
n
EXCEL formula
= COUNT(range)
xbar
mean
= AVERAGE(range)
SD
sigma
= STDDEV(range)
g1
skewness
= SKEW(range)
g2
excess kurtosis
= KURT(range)
Pth quantile
qf
= PERCENTILE(range,P)
36
David LeBlond, PhD 2/2017
Normality tests
Shapiro-Wilk, Ryan-Joiner
•based on a variance ratio (W)
•powerful but don’t work well if data are rounded
•not informative
Kolmogorov-Smirnov, Lilliefors
•based on largest difference between theoretical and empirical cdf
•not very powerful, not informative
Anderson-Darling, Cramer von Mises
•based on squared difference between theoret and empirical cdf
•very powerful, maybe too powerful, not informative
Normal Probability (Q-Q) Plot
•based on comparison of theoretical and empirical qf
•not a statistical test, but very informative
D’Agostinos K2, Jarque-Bera, Skewness-Kurtosis-Omnibus
•based on testing for Skewness=0, Kurtosis=3, or both
•powerful and informative
•in Stata, PRISM, Distribution Analyzer
37
David LeBlond, PhD 2/2017
Beware Kolmogorov-Smirnov
Look normal?
Histogram
12
Frequency
10
Probability Plot of X
8
Normal
6
99.9
4
2
400
380
360
340
320
95
90 Bin
300
280
240
Percent
220
200
180
160
260
99
0
80
70
60
50
40
30
20
Mean
StDev
N
KS
P-Value
10
5
1
0.1
100
150
200
250
X
300
250.0
41.44
62
0.104
0.091
350
?
400
38
David LeBlond, PhD 2/2017
Beware Shapiro-Wilk
Raw
4.64869244
12.4021887
9.67430794
8.158978
11.4408614
9.48448461
13.5148202
.
.
.
Goodness-of-Fit Test
Rounded
5
12
10
8
11
9
14
.
.
.
Goodness-of-Fit Test
Shapiro-Wilk W Test
Shapiro-Wilk W Test
W
Prob<W
W
Prob<W
0.982650
0.2126
0.972848
0.0367*
?
39
David LeBlond, PhD 2/2017
A test for skewness
n
= COUNT(range)
Skewness
= SKEW(range)
A
= (n-2)*Skewness/SQRT(n*(n-1))
B
= A*SQRT( (n+1)*(n+3)/6/(n-2)
C
= 3*(n^2+27*n-70)*(n+1)*(n+3)/(n-2)/(n+5)/(n+7)/(n+9)
D
= -1+SQRT(2*(C-1))
E
= 1/SQRT(LN(SQRT(D)))
F
= SQRT(2/(D-1))
Z1
= E*LN(B/F+SQRT((B/F)^2+1))
p-value
= 2*(1-NORMDIST(ABS(Z1),0,1,TRUE))
)
40
David LeBlond, PhD 2/2017
A test for kurtosis
n
= COUNT(range)
ExcessKurtosis
= KURT(range)
G
= (n-2)*(n-3)*(ExcessKurtosis)/(n+1)/(n-1)+3*(n-1)/(n+1)
H
= 3*(n-1)/(n+1)
I
= 24*n*(n-2)*(n-3)/(n+1)^2/(n+3)/(n+5)
J
= (G - H)/SQRT(I)
K
= SQRT( 6*(n+3)*(n+5)/n/(n-2)/(n-3) )*6*(n^2-5*n+2)/(n+7)/(n+9)
L
= 6+(8/K)*( 2/K+SQRT(1+4/K^2)
Z2
= (1-2/9/L-((1-2/L)/(1+J*SQRT(2/(L-4))))^(1/3))/SQRT(2/9/L)
p-value
= 2*(1-NORMDIST(ABS(Z2),0,1,TRUE))
)
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David LeBlond, PhD 2/2017
An omnibus test for both
skewness and kurtosis
p-value = CHIDIST(Z1^2+Z2^2,2)
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David LeBlond, PhD 2/2017
Example: some non-normal data
Raw Data
N
18
xbar 7.17
SD
6.24
Skewness
Excess Kurtosis
6
Frequency
5
4
3
2
1
Bin
24
20
16
12
8
4
0
0
1.62
1.79
p-value (Skewness) 0.006
p-value (Kurtosis) 0.117
p-value (Omnibus) 0.006
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David LeBlond, PhD 2/2017
Example: Transformation to normality
Transformed = -5.2 + 1.21*ASINH((Raw – 0.18)/0.14)
(Johnson “SU” transformation)
Transformed Scale
N
18
xbar 0
SD
1.03
Skewness
Excess Kurtosis
7
Frequency
6
5
4
3
-0.03
0.24
2
1
Bin
2.
5
M
or
e
2
1.
5
1
0.
5
0
-1
-0
.5
-2
-1
.5
-2
.5
0
p-value (Skewness) 0.96
p-value (Kurtosis) 0.64
p-value (Omnibus) 0.90
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David LeBlond, PhD 2/2017
Identifying a transformation
(Distribution Analyzer Software*)
10
Distributions
Kurtosis
Normal
Beta
Exponential
Weibull
Gamma
Johnson Su
Lognormal
Normal
Uniform
Impossible Area
Untransformed (S=1.62, K=4.79)
Transformed (S=0.08, K=3.12)
7
4
Original Raw Data
1
-3
-2
-1
0
1
Skewness
* Taylor Enterprises, www.variation.com
2
3
Transformation
identified
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David LeBlond, PhD 2/2017
Bonus: Excel matrix functions
 a b c   g h i  a  g b  h c  i 
d e f    j k l   d  j e  k f  l 

 
 

= Arange + Brange
a d 
a b c
b e 

d e f 




 c f 
T
= TRANSPOSE(Arange)
g h i j
a
b
c


  ag  bk  co ah  bl  cp ai  bm  cp aj  bn  cr 
k
l
m
n
d e f  

 

 o p q r  dg  ek  fo dh  el  fp di  em  fp dj  en  fr 


= MMULT(Arange,Brange)
1.
2.
3.
4.
5.
6.
a b
d e

 g h
c
f 
i 
1
= MINVERSE(Arange)
Name the ranges for the input matrices
Determine the “shape” (rows&columns) of the result
Select the cell range for the result matrix
Type the matrix formula
Press CNTL+SHFT+ENTER
If desired, name the new range for use in further matrix formulas
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David LeBlond, PhD 2/2017
Matrix Example: simple linear regression
Goal:
Estimate Slope and intercept

 
̂  X X  X Y
T
1
T
=MMULT(MMULT(MINVERSE(MMULT(TRANSPOSE(Xrange),Xrange)),
TRANSPOSE(Xrange)),Yrange)
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David LeBlond, PhD 2/2017
Key messages
• Understanding PDs is critical for quality decisions
• pdf, pmf, cdf, qf EXCEL formulas
• Histograms in EXCEL
• Monte-Carlo simulation in EXCEL
• Skewness-Kurtosis normality test in EXCEL
• Matrix calculations in EXCEL
•Taylor Enterprises “Distribution Analyzer”
• PPT on ASQ FD&C web site
• questions? [email protected]
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David LeBlond, PhD 2/2017
References
1. Evans M et al (1993) Statistical distributions 2nd edition, John Wiley,
NY
2. D’Agostino RB et al (1990) A suggestion for using powerful and
informative tests of normality American Statistician 44(4) 316-321
3. Distribution analyzer software and other good information at Wayne
Taylor’s web site: www.variation.com/da
4. LeBlond, D (2008) Data, Variation, Uncertainty, and Probability
Distributions, Journal of GxP Compliance, Vol. 12, No. 3, pp 30-41.
5. LeBlond, D (2008) Using Probability Distributions to Make Decisions,
Journal of Validation Technology, Spring 2008, pp 2 – 14.
6. LeBlond, D (2008) Estimation: knowledge building with probability
distributions, Journal of GxP Compliance, Vol. 12 (4), 42-59.
7. LeBlond, D (2008) Estimation: knowledge building with probability
distributions – Reader Q&A, Journal of Validation Technology, Vol.
14(5), 50-64.
8. LeBlond D (2009) Hypothesis Testing: Examples in Pharmaceutical
Process and Analytical Development, Journal of GxP Compliance, to
be published
9. Neter J, et al (1996) Applied Linear Statistical Models, 4th edn, Irwin
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David LeBlond, PhD 2/2017