# Statistical Analysis of Heart Rate Data by alextt

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```									Statistical Analysis of Heart Rate Data

Meihui Guo
Dept. of Applied Math., Nation Sun Yat-sen Univ.
Kaohsiung, Taiwan
Jan 10, 2006
Outline
Introduction
Classification    Congenital Heart Data Set
 Statistical Methodology
(a) Classification Procedure
(b) Logistic Analysis
 Empirical Results

Monitoring        Patients after Surgery
 Important Risk Factors for VICU Patients
(a) Heart Rate Variability
(b) LF/HF Ratio Statistic
 Application
Introduction

 CHD (congenital heart disease) : 1% of the live births

 VSD (a hole in the heart) : 30% of the CHD
increased load to the heart；circulatory disturbances

 CHF (congestive heart failure) : prominent circulatory
disturbances, increasing heart rate, increasing respiratory
rate, difficulty in breathing, swelling of legs, enlarged liver…
Introduction
 Detection Method：subjective and/or complex

 Long Term ECG (electrocardiograph) Data

 New Intervals： PR' and RT intervals

 ECG Data collected by Computer

 Median and log (stdev) of PR ' , RR and RT intervals
Data Collection
 Sampling rate：1/500 second
 Sampling duration：10 minutes
Approximate three hundred thousands points in a data set

 ECG data sets of lead II of 89 children
(from one month old to nine years old)
Table 1. List of Data

Status    Normal     VSD without CHF     VSD with CHF
Data No.     30                28              31
Classification Group

Subject

G4 Sick
G1
G2
Normal                         G3
VSD
VSD+CHF
 Classification vector：
X 1  ( logs ( PR ' ), m( PR ' ), log s( RR), m( RR), log s ( RT ), m( RT ))'
m() : median
log s() : log transform of the stdev
X  X1

 Classification function (Mahalanobis distance)：
Di ( X )  ( X  i )'  i1 ( X  i )
i and  i : mean and covariance matrix of the i - th group
ˆ
Di ( X ) : Moment estimator

ˆ          ˆ
Dij ( X )  Di ( X )  D j ( X )               Subject
Dij  0                   Dij  0

Gi                           Gj
 Best classifica tion vector X i : by backward procedure

 Retrospective study and resubstitution method：
Table 2. Classification Table for Two Groups
Actual group     Predicted group        Observation no.
Group 1 Group 2
Group 1          n11        n12              n1
Group 2          n21        n22              n2

n22                 n11
(i) Largest sensitivity     and specificity
n2                  n1
n11n22
(ii) Largest odds ratio         with both sensitivit y and
n12 n21
specificit y greater than 0.7
 The best classification vectors :
X 1  the vector of six interval statistics
X 2  (m( PR ' ), log s ( PR ' ), m( RR), m( RT ), log s ( RT ))'
X 3  (m( PR ' ), m( RR), m( RT ), log s ( RT ))'

s,
Table 5. Specificities, sensitivitie odds ratio of X1,X 2 ,X 3
Comparison pair Class. vector Specificity Sensitivity Odds ratio
G 1 v.s. G 2        X1             0.767        0.786        12.05
G 1 v.s. G 3        X2             0.80         0.839          20.8
G 1 v.s. G 4        X1             0.733        0.813          12
G 1 v.s. G 4       X1, X 2         0.733        0.881          20.4
G 2 v.s. G 3        X3             0.75         0.903          28
 Classify the three groups :

(a ) Three way classifica tion         (b) Two stage classifica tion procedure

Subject                                                 Subject
D12 ( X 1 )  0
or D13 ( X 2 )  0                  otherwise
min Di ( X 1 )

G1                                              Normal                        Sick
G2            G3
D23 ( X 3 )  0           D23 ( X 3 )  0

Table 6. Correct classification rates                                G2                   G3
Classification method Group 1 Group 2 Group 3
ˆ ˆ ˆ
min{ D1 , D2 , D3 }         0.73     0.53           0.81
Two stage method             0.73     0.57           0.87

 Dispersion situation of D12 ( X 1 ) and D13 ( X 2 ) of Normal v.s. Sick             c
 Dispersion situation of D23 ( X 1 ) of G2 v.s. G3.
X 1  the vector of six interval statistics
X 2  (m( PR ' ), log s ( PR ' ), m( RR), m( RT ), log s ( RT ))'
Analysis of Variance                           X 3  (m( PR ' ), m( RR), m( RT ), log s ( RT ))'

Table 3. p values of the interval statistics

None are                      log s ( PR ' ) m(PR ' ) log s ( RR) m(RR)                log s ( RT ) m(RT )
significant.   G1 v.s. G4      0.6182       0.2306           0.0483       0.1250        0.0614        0.1793
G1 v.s. G2      0.7265       0.5926           0.7790       0.8239        0.2329        0.7800
G1 v.s. G3      0.2934       0.0131           0.0014       0.0033        0.0638        0.0094
G2 v.s. G3      0.1560       0.0012           0.0090       0.0017        0.3622        0.0085

Table 4. p values of D12 ( X 1 ), D13 ( X 2 ) and D23 ( X 3 )
D12 ( X 1 )         D13 ( X 2 )       D23 ( X 3 )
significant.    G1 v.s. G4          0.0001              0.0003
G1 v.s. G2          0.0003
G1 v.s. G3                              0.0001
G2 v.s. G3                                                 0.0001
Logistic Analysis
 P(Y  1 X 1 ) 
   Log-odds ln 
 P(Y  0 X 1 ) 

Y  1: a subject is allocated to
(a) G2 : G1 v.s. G2
(b) G3 : G1 v.s. G3 or G2 v.s. G3
(c) G4 : Normal v.s. Sick group

 Assume P(Y  0)   0 and P(Y  1)   1
f1 ( X 1 ) 1
p (Y  1 X 1 ) 
f 0 ( X 1 ) 0  f1 ( X 1 ) 1
f j ( X 1 ) : conditional prob. of X 1 given Y  j ( j  0,1)

P (Y  1 X 1 )        1       f1 ( X 1 )
 Log odds          ln(                )  ln( )  ln(            )
P (Y  0 X 1 )        0       f0 ( X1 )
 Three Logistic Regression Models
P (Y  1 X 1 )
(I) Linear model ln(                )   0  1 ' X 1   .                                  (1)
P (Y  0 X 1 )
P (Y  1 X 1 )
(II) Quadratic model ln(                          )   0  1 ' X 1   2 ' QX   . (2)
P (Y  0 X 1 )
QX : quadratic and cross product terms of X 1
1                                          1
e.g. f1 ( X )   2        2   q ( D2 ( X )) and f 0 ( X )  1        2   q ( D1 ( X ))
f1 ( X 1 )      1 2         q ( D2 ( X 1 )
ln(            )   ln( )  ln(                ).
f0 ( X1 )       2 1         q ( D1 ( X 1 )
If f 0 () and f1 () are multivaria te Normal,
f1 ( X 1 )      1        1
ln(              )   ln( 2 )  D12 ( X 1 )
f0 ( X1 )       2 1      2
P (Y  1 X 1 )              ˆ
(III ) Proposed model ln(                )   0  1 D12 ( X 1 )                           (3)
P (Y  0 X 1 )
 Receiver Operating Characteristic (ROC) curves
Introduction
 Important Risk Factors for VICU Patients
(a) Heart Rate Variability
(b) LF/HF Ratio Statistic
 Laboratory and Clinical Studies :

---Sympathetic
Mancia et al. (1999) : Cats SAD Operation (切除手術),
LF decreases
Hojgaard et al. (1998) : Human  -blockade (藥物阻斷),
LF/HF decreases
---Parasympathetic
Rimoldi et al. (1990) : Dogs Atropine (藥物抑制),
HF decrease
Laitinen et al. (1999) : Human short term blood pressure
variability with Sympathovagal balance, use LF/HF as
index of Sympathovagal balance

Winchell et al. (1996) : Surgical ICU population
(i) low overall HRV
(ii) low LF/HF ratio : Sympathetic/Parasympathetic Balance
associated with increased mortality for patients in ICU.

 Low Frequency (LF) Power
(i) Sympathetic activity only
(ii) A predominance of sympathetic activity with a para-
sympathetic component.

 High Frequency (HF) Power –Parasympathetic activity
 LF/HF : an index of sympathovagal balance
Heart Rate Variability
(1) EWRMS
* Heart rates at time t : {Yt , 1  t  n}
* Assume Y1   ,  , Yn   , ~ MN (0,  Y P )
2

* Let
S n  (1  r ) S n 1  r (Yn   ) 2
2              2

n
  r (1  r ) n  k (Yn   ) 2  (1  r ) n S 0
2

k 1

 Y'RY  (1  r ) n S 0
2

Y  (Yn , Yn 1 ,  , Y1 )' ,
R  diag[r , r (1  r ), , r (1  r ) n 1 ].
* For large n,
2               n
Sn
~ Q    j  2 (1)
Y
2                  j
j 1

 j ' s are the n characteristic roots of U  PR

2
Sn
* The c.d . f . of           is given by
   2
Y

1 1  sin  (u )
F ( x)                du
2  0 u (u )

1 n              1
where           (u )   tan ( i u )  xu
-1

2 i 1           2
n
 (u )   (1   u )
1
2   2       4
i
i 1
(1) EWMV
* Let n  (1   ) n 1  n
*     S n  (1  r ) S n 1  r (Yn  n ) 2
2              2

 Y' ( I  M )' R ( I  M )Y  (1  r ) n S 0
2

Y  (Yn , Yn 1 ,  , Y1 )' ,
0   (1   )  (1   ) 2   (1   ) n 1 
                                         n2 
 0             (1   )   (1   ) 
                                           
M                                               
                     0        (1   ) 
                             0               
                                              

                                     0        

2       n
Sn
* For large n,             ~   j  2 (1)
Y
2
j 1
j

where  j ' s are the characteristic roots of U  P( I  M )' R( I  M )
Spectral Analysis of Heart Rate Variability

* Heart rates at time t :{Yt , 1  t  n}

* Spectrum of Yt :

1
f ( )     ( 0  2  k cos(k )), 0    
2         k 1

Autocovariancce Function :  k  Cov(Yt , Yt  k )

* Measure of Sympathetic/Parasympathetic Balance
 LF/HF
* A Conventional Ratio Statistic RA  ln( 1
1
n1   jLF
ˆ
f ( j )
)
n2   iHF
fˆ ( i )

ˆ ( )  1 (ˆ0  2 ˆk cos(k )) sample spectrum
f
2          k 1

2j
n1 : number of Fourier frequencies ( n , j  0,1, 2,  , [ n ]) in the LF range
2
n2 : number of Fourier frequencies in the HF range
RA  Log ratio of two weighted sum of Chi  square
* Distribution of RA ?

* Newly Proposed Statistic :
ˆ
( jLF f ( j )) n1
1

RG  ln
ˆ ( i )) n2
1
( iLF f
Lemma
The ratio statistic
ˆ
( jLF f ( j )) n1
1

RG  ln
ˆ ( i )) n2
1
( iLF f
is distributed asympotically as
1            1

 ln W j  n iHFln Wi  
n1 jLF        2

where Wi ' s are  2 (2) r.v.' s and   E ( RG) which depends on n1 , n2
and f ( i ) for i  LF and i  HF .
Theorem   Let
1                   1
X
n1
 ln W j 
jLF
 ln Wi
n2 iHF
where Wi ' s are i.i.d.  2 (2) r.v.' s, then
(i) the characteristic function of X is
n1                 n2
       i              i 
 (t )  (1  ) (1  ) ,
      n1             n2 
(ii) the c.d . f . of X is
1 1   (t ) sin( (t )  tx)
F ( x)                                dt ,
2     0            t
t          n1
t       n2

 (t )  (                  ) (
2
)2
n1 sinh(t / n1 ) n2 sinh(t / n2 )

t              t
 (t )   (n2 tan ( ))  n1 tan ( )).
1          1

k 1       n2 k           n1k
Application : VICU Patients

 Data Collection
--- Kaohsiung Vetern General Hospital :
Vascular Intensive Care Unit
--- HP dhppi, PcAnywhere, Excel files
--- Long term one-minute heart rate data after operation
--- Classification of Patients :
Recover, Short Term and Long Term CHF, Dead

 HRV Monitoring
---EWRMS & EWMV Control Charts
Application : VICU Patients

 LF/HF Monitoring

 30 minute heart rates
 LF Range : 0.04  0.15 Hz, n1  3
 HF Range : 0.15  0.4 Hz, n2  8
1         ˆ ( j )  1 
 RG   jLF ln f                    ˆ
ln f ( i )
3                    8 iHF
 X  RG  RG, Var( X )  0.75393
Application : VICU Patients
Table 1. Quantiles of Standardized X

Probabilities 0.135%    0.5%    2.5%    97.5%   99.5% 99.865%
Quantiles     -3.177   -2.614 -1.872   1.627   2.121   2.417

 Shewhart Control Chart for Standardized X
UCL  (99.865%, 99.5%, 97.5%)
CL  0
LCL  (0.135%, 0.5%, 2.5%)
Risk factors and recover conditions

(1) Number of out of control points in LF/HF charts
every four hours
(2) Number of out of control points in HRV charts
every four hours
(3) Number of (1) + (2) in three groups

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