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1
PPLOT
PPLOT produces probability plots of one or more sequence or time series variables. The variables can be standardized, differenced, and/or transformed before plotting. Expected normal values or deviations from expected normal values can be plotted. PPLOT can be used to investigate whether the data are from a specified distribution: normal, lognormal, logistic, exponential, Weibull, gamma, beta, uniform, Pareto, Laplace, half normal, chi-square and Student’s t.
Notation
The following notation is used throughout this chapter unless otherwise stated:
X Sample mean S Sample standard deviation
LX Sample mean for^ ln^ 1 6 x^ i
LS Sample standard deviation for ln 1 6 x i
x (^) i Value of the i th observation x (^) ( ) i The i th smallest observation Ri Corresponding rank for x (^) i n Sample size frdist ( x (^) i ) (^) Fractional rank of x (^) i for the specified distribution function a (^) dist ( xi ) (^) Score for the specified distribution function
α Location parameter
β Scale parameter γ (^) Shape parameter ν (^) Degrees of freedom
1 This procedure was introduced in SPSS 7.0 and replaces the NPPLOT procedure of earlier releases.
Fractional Ranks
Based on the rank Ri for the observation x (^) i , the fractional rank fr (^) dist ( xi ) is computed and used to estimate the expected cumulative distribution function of X. One of four methods can be selected to calculate the fractional rank frdist ( x (^) i ) :
fr x
R n
R n
R n R n
dist i
i
i
i i
%
&
K K K
'
K K K
Blom
Rankit
Tukey Van der Waerden
Scores
The score of the specified distribution for case i is defined as
a (^) dist ( x (^) i ) = Fdist −^1 ( frdist ( x (^) i )) i = 1 , K, n
where Fdist −^1 is the inverse cumulative specified distribution function.
P-P Plot
For a P-P plot, the fractional rank and the cumulative specified distribution function Fdist are plotted:
2^ frdist^ 1 6 x^ i ,^ Fdist^ 1 6 x^ i 7 i^ =^1 ,^ K, n
Student’s t ( ν ) v 0 > 05 is the degrees of freedom specified by the user.
Uniform( a,b ) a is a lower bound and b is an upper bound.
Weibull( β , γ ) β 0 > 05 is a scale parameter and γ 0 > 05 is a shape parameter.
Estimates of the Parameters
The estimates for parameters for each distribution are defined below.
Beta( β 1 , β 2 ) $^
β (^1 )
% & '
( )
X
X X
S
scale parameter
β^ $^ ( ) (^2 1 )
% &
K
'K
( )
K
*K
X
X X
S
3 8 scale parameter
Chi-square( ν ) ν is the degrees of freedom specified by the user.
Exponential( β ) β =$
X
scale parameter
Gamma( γ , β ) γ =$
X
S
2 2 shape parameter
β =^ $ X S^2
scale parameter
Half Normal( β ) β =$^ x (^) 12 +... + x (^) n^2 scale parameter
Laplace( α , β ) α$ = X location parameter
β =^ $ S
2
2
scale parameter
Logistic( α , β ) α$ = X location parameter
β^ $ π
=
S
, π = 31415927. scale parameter
Lognormal β
∧ − = exp( L X ) scale parameter
γ
∧ = LS shape parameter
Normal( α , β ) α$ = X location parameter
β =^ $ S scale parameter
Pareto( β ,b ); β =$^ min (^) ; x 1 (^) , K, x (^) n @ scale parameter
$ ln $
b LX
4 9^ β
index of inequality
Student’s t ( ν ) v is the degrees of freedom specified by the user.
Uniform( a,b ) a $^ = min ; x 1 (^) , K, x (^) n @ lower bound
b^ $^ x , , x = max (^) ; 1 K n @ upper bound
Weibull( β , γ ) $ ( )
β =
− −
= −
=
∑
∑
U Y nU Y
U U
i i i
n
i i
n
1 2 1
scale parameter
γ^ $^ = ( −(( −β ) / β))
− ∧ − ∧ exp Y U shape parameter
where Yi = ln 4 − ln 21 − frdist 1 6 x (^) i 79 and U (^) i =ln1 6 xi
References
Kotz, S., and Johnson, N. L., eds. 1988. Encyclopedia of statistical sciences. John Wiley & Sons, Inc.: New York.
