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					STAT312: Applied Regression
         Methods




           http://www.mysmu.edu/faculty/zlyang/
                                                Course Contents

                          1   Linear Regression


                          2   Transformed Linear Regression


                          3   Logistic Regression



                          4   Multinomial Logistic Regression



                          5   Loglinear Model



STAT312, Term II, 10/11                 2                       Zhenlin Yang, SMU
                          Chapter 1: Introduction
           Contents
            History of regression analysis
            Basic concepts
            Applications - Examples
            Sampling models
            Sampling distributions
            Statistical inference
            Large sample inference
            Matrix algebra
STAT312, Term II, 10/11       3               Zhenlin Yang, SMU
                          Chapter 1: Introduction
                            History of Regression Analysis

  The earliest form of regression was the method of least
  squares, which was published by Legendre in 1805, and
  by Gauss in 1809. The term “least squares” is from
  Legendre’s term, moindres carrés. However, Gauss
  claimed that he had known the method since 1795.


  Legendre and Gauss both applied the method to the
  problem of determining, from astronomical observations,
  the orbits of bodies about the Sun. Euler had worked on
  the same problem (1748) without success. Gauss
  published a further development of the theory of least
  squares in 1821, including a version of the Gauss–Markov
  theorem.

STAT312, Term II, 10/11      4                    Zhenlin Yang, SMU
                          Chapter 1: Introduction
                            History of Regression Analysis

  The term "regression" was coined in the nineteenth
  century to describe a biological phenomenon, namely
  that the progeny of exceptional individuals tend on
  average to be less exceptional than their parents and
  more like their more distant ancestors.

  Francis Galton, a cousin of Charles Darwin, studied this
  phenomenon and applied the slightly misleading term
  "regression towards mediocrity" to it. For Galton,
  regression had only this biological meaning, but his work
  was later extended by Udny Yule and Karl Pearson to a
  more general statistical context. Nowadays the term
  "regression" is often synonymous with "least squares
  curve fitting".

STAT312, Term II, 10/11      5                    Zhenlin Yang, SMU
                          Chapter 1: Introduction
                                   Some Basic Concepts

  A variable represents some characteristic of a social
  phenomenon.

  A continuous variable is a variable that assumes values in
  an interval.

  A discrete variable is a variable that assumes distinct
  values.

  A random variable is a variable that is used to represent
  possible outcomes of some random phenomenon, e.g., results
  of tossing a coin, results of rolling a die, etc.

  Response/dependent variable: variable of interest in the
  study.

STAT312, Term II, 10/11        6                          Zhenlin Yang, SMU
                          Chapter 1: Introduction
                                    Some Basic Concepts

   Explanatory/independent variable: variable used to
   “explain” or “study” the variable of interest.

   Regression Analysis: a method for investigating functional
   relationships among variables.

   Categorical variables having ordered scales are called
   ordinal variables.

   Categorical variables having unordered scales are called
   nominal variables.

   Categorical variables are often referred to as qualitative
   variables, and numerical-valued variables are called
   quantitative variables.

STAT312, Term II, 10/11         7                       Zhenlin Yang, SMU
                          Chapter 1: Introduction
                                 Some Basic Concepts




STAT312, Term II, 10/11      8                Zhenlin Yang, SMU
                          Chapter 1: Introduction
                                  Applications - Examples
    Example 1.1 Where to locate a new motor inn?
        La Quinta Motor Inns is planning an expansion.
        Management wishes to predict which sites are likely
         to be profitable.
        Several areas where predictors of profitability can
         be identified are:
           • Competition
           • Market awareness
           • Demand generators
           • Demographics
           • Physical quality


STAT312, Term II, 10/11       9                      Zhenlin Yang, SMU
                                            Chapter 1: Introduction
                                                         Applications - Examples
 Predictors of
                                              Profitability                   Operating Margin
 profitability:
                            Market
   Competition                                 Customers              Community          Physical
                           awareness




      Rooms                Nearest         Office         College       Income           Disttwn
                                           space         enrollment
     Number of            Distance to                                    Median       Distance to
     hotels/motels        the nearest                                    household    downtown.
     rooms within         La Quinta inn.                                 income.
     3 miles from
     the site.                                  Data:       EX1-01.XLS               EX1-01.TXT
STAT312, Term II, 10/11                             10                                   Zhenlin Yang, SMU
                               Chapter 1: Introduction
                                           Applications - Examples
  Example 1.2. The table below cross classifies 1091 respondents to the
  1991 General Social Survey by their gender and their belief in an
  afterlife.
     Table 1.1 Cross Classification of Belief in Afterlife by Gender
                                             Belief in Afterlife
     Gender                  Yes                No or undecided
     Females                 435                        147
     Males                   375                        134

   Purpose of the Study: whether an association exists between gender
   and belief in afterlife. Is one sex more likely than the other to belief in
   an afterlife, or is belief in afterlife independent of gender?
   This is a typical 22 contingency table, where an analysis of association
   between two factors (variables) is of interest.

STAT312, Term II, 10/11               11                            Zhenlin Yang, SMU
                                 Chapter 1: Introduction
                                           Applications - Examples
   Example 1.3. The data given below comes from a randomized, double-
   blind clinical trial investigating a new treatment for rheumatoid arthritis.
   Investigators compared the new treatment with a placebo. The response
   measured was whether there was no, some, or marked improvement in
   the symptoms of rheumatoid arthritis.
    Table 1.2 Rheumatoid Arthritis Data                       This is a 2 23
  Gender          Treatment       Improvement                 contingency table,
                               None Some Marked      Total    where interest lies
  Female          Test Drugs    6     5       16     27       in the association
  Female          Placebo       19    7        6     32       between treatment
  Total                         25    12      22     59       and degree of
  Male            Test Drugs    7     2        5     14
                                                              improvement,
  Male            Placebo       10    0        1     11       adjusting for
                                                              gender effect.
  Total                         17    2        6     25

STAT312, Term II, 10/11               12                           Zhenlin Yang, SMU
                              Chapter 1: Introduction
                                          Applications - Examples
  Example 1.4. (Education Expenditure) Per capita expenditure on
  public education can be affected by (1) per capita personal income, (2)
  number of residents per thousand under 18 years of age, (3) number of
  people per thousand residing in urban areas, and (4) the geographical
  region. The data have been collected for each of the 50 states in U.S., in
  1960, 1970, and 1975.
  The problems of interest can be:
  • How is education expenditure related to the factors listed above?
  • Does the geographical region make a difference on education
    expenditure?
  • Is the relationship between education expenditure and other variables
    constant over time?

STAT312, Term II, 10/11              13                          Zhenlin Yang, SMU
                                  Chapter 1: Introduction
                                              Applications - Examples
   Example 1.5. (Probability of Bankruptcies) Detecting ailing financial
   and business establishments is an important function of audit and
   control. Systematic failure to do audit and control can lead to grave
   consequences, such as the saving-and-loan-fiasco of the 1980s in the
   United Stats, and current financial crises. The data P322.txt gives some
   of the operating financial ratios of 33 firms that went bankrupt after 2
   years and 33 that remained solvent during the same period:
              X1 = Retained Earnings/Total Assets
              X2 = Earning Before Interest and Taxes/Total Assets
              X3 = Sales/ Total Assets
              Y = 0 if bankrupt after two years, and 1 if solvent after two
   years.
   Question: given a firm’s characteristics, what is the chance that this
   firm remains solvent after two years?
STAT312, Term II, 10/11                  14                          Zhenlin Yang, SMU
                              Chapter 1: Introduction
                                          Applications - Examples
   Example 1.6. (Chemical Diabetes) To determine the treatment and
   management of diabetes it is necessary to determine whether the patient
   has chemical diabetes of overt diabetes. The data P331.txt is from a
   study to determine the nature of chemical diabetes. The measurements
   were taken on 145 non-obese volunteers who were subject to the same
   regimen. Many variables were measured, but only three considered:
   insulin response (IR), the steady state of plasma glucose (SSPG), which
   measures insulin resistance, and relative weight (RW). The diabetes
   status of each subject was recorded. The clinical classification (CC)
   categories were overt (1), chemical diabetes (2), and normal (3).

   Question: given a subject’s characteristics, what is the chance that
   he/she has overt diabetes, or has chemical diabetes, or is normal?


STAT312, Term II, 10/11              15                          Zhenlin Yang, SMU
                                         Chapter 1: Introduction
                                                   Applications - Examples
   Example 1.7. (Political Ideology and Party Affiliation) The data
   given below, from a General Social Survey, relates political ideology to
   political party affiliation.

             Table 1.3 Political Ideology and Part Affiliation Data
                                                     Political Ideology
                   Political    Very      Slightly     Moderate     Slightly        Very
    Gender          Party      Liberal    Liberal                 Conservative   Conservative

    Female      Democratic       44         47           118          23              32
                Republican       18         28            86          39              48
    Male        Democratic       36         34            53          18              23
                Republican       12         18            62          45              51


   Question: Who is more conservative, democrats or republicans, males
   of females?

STAT312, Term II, 10/11                       16                                 Zhenlin Yang, SMU
                                              Chapter 1: Introduction
                                                                       Sampling Models
  Normal Distribution: A random variable X is said to have a normal
  distribution, denoted by X ~ N(, 2), if its probability density function
  (pdf) is of the form
                                  1        (x  )2 
                f ( x;  ,  )       exp          
                                 2         2 2
                                                     
   This is a continuous distribution with mean  and variance 2.
    0.4                                                      0.2

                                    =1, 1.5, 2                                        =1.0
    0.3                                                     0.15



    0.2                                                      0.1



    0.1                                                     0.05



      0                                                        0
           4      6       8   10    12   14       16               0    5   10   15         20


STAT312, Term II, 10/11                                17                             Zhenlin Yang, SMU
                                            Chapter 1: Introduction
                                                                           Sampling Models
   Binomial Distribution: A random variable X is said to have a binomial
   distribution, denoted by X ~ Bin(n,p), if its probability function is:

                                 n!
                    p ( x)              x (1   ) n x , x  0,1, 2, , n
                             x!(n  x)!
   a discrete distribution with mean n and variance n(1- ).
          0.2                                                    0.2




                                                          p(x)
   p(x)




                    n=20, p=0.5
                                                                                    n=20, p=0.3



          0.1                                                    0.1




          0.0                                                    0.0

                0                 10            20                     0       10                     20
                                           x                                                      x

STAT312, Term II, 10/11                              18                                   Zhenlin Yang, SMU
                                         Chapter 1: Introduction
                                                                   Sampling Models
   Poisson Distribution: A random variable X is said to have a Poisson
   distribution, denoted by X ~ Poi(), if its probability function is:
                            e   x
                    p( x)           , x  0,1, 2, ,
                               x!
   a discrete distribution with mean , called the Poisson rate, and
   variance 2.



                                                        p(x)
      p(x)




                                                       0.15
     0.10                      mean=10
                                                                            mean=6

                                                       0.10


     0.05
                                                       0.05



     0.00                                              0.00
             0            10                 20
                                         x                     0       10                 x   20


STAT312, Term II, 10/11                           19                                 Zhenlin Yang, SMU
                                                  Chapter 1: Introduction
                                                                      Sampling Models
   Chi-squared Distribution: A random variable X is said to have a Chi-
   squared distribution with  degrees of freedom if its pdf is of the form

                       x ( 2 ) 2 exp(  x 2)                      a continuous distribution
              f ( x)           2
                                               , x  0,             with mean 2 and variance
                             2 ( 2)                               4, denoted by  
                                                                                     2



                =3
   0.2                                                                 2  Z12  Z 2    Z2
                                                                                     2


 0.15                                                                  where Z1 , Z2, …, Z
                      =6                                              are iid standard normal
   0.1                           =12                                  random variables.
 0.05                                             =20
                                                                       It is a distribution used
      0                                                                in statistical inference
          0       5         10          15   20    25     30   35

STAT312, Term II, 10/11                                  20                          Zhenlin Yang, SMU
                                   Chapter 1: Introduction
                                                                  Sampling Models
  Student’s t distribution: A continuous r.v. T is said to
  have a Student’s t distribution with v df, denoted by T ~ tv,
  if its pdf has the form:
                                             ( v 1) 2
                     [(v  1) 2]  t 2 
              f(t) =              1                    ,  < t < , v > 0
                      (v 2) v     v

   • The t distribution is
   symmetric around zero
   • If Z ~ N(0, 1) and Y ~  2 (v )
   and if Z and Y are indep.,
      T =Z          Y v ~ tv
   • It approaches to N(0, 1) as v
   
STAT312, Term II, 10/11                       21                                Zhenlin Yang, SMU
                                   Chapter 1: Introduction
                                                      Sampling Models
   F Distribution: If X1 ~  (v1 ) and X2 ~ 
                               2                2
                                                    (v 2 )   are independent,
   then the r.v.         X 1 v1
                   Y=
                          X 2 v2
   follows an F–distribution, with v1 df in the numerator and v2 df in the
   denominator.


     • This distributional
     result is often used to
     construct F test in
     linear regression
     models.


STAT312, Term II, 10/11               22                               Zhenlin Yang, SMU
                          Chapter 1: Introduction
                                  Sampling Distributions

  The Central Limit Theorem
    If a random sample X1, …, Xn is drawn from any
     population, the sampling distribution of the
     sample mean x is approximately normal for a
     sufficiently large sample size.
    The larger the sample size, the more closely
     the sampling distribution of x will resemble a
     normal distribution.

STAT312, Term II, 10/11      23                 Zhenlin Yang, SMU
                              Chapter 1: Introduction
                                       Sampling Distributions
    In more detail:
    Let X represent a population, and X1, …, Xn be a
    random sample drawn from this population. Then,
     1.  X   X
                         2
     2.     2
              X
                          x
               n
     3. If X is normal, X is normal. If X is nonnormal
          X is approximat ely normally distribute d for
         sufficient ly large sample size (n  30).

STAT312, Term II, 10/11           24                      Zhenlin Yang, SMU
                                Chapter 1: Introduction
                                         Sampling Distributions
  Sampling Distribution of a Sample Proportion
      ˆbe
  Let p the proportion of “successes” in a sequence of
   n Bernoulli trials.

      From the laws of expected value and variance, it can be
                       ˆ              ˆ
       shown that E( p ) = p and Var(p) = p(1-p)/n
      If both np ≥ 5 and np(1–p) ≥ 5, then

                                p p
                                 ˆ
                          Z 
                                p (1  p )
                                    n

      Z is approximately standard normally distributed.
STAT312, Term II, 10/11             25                 Zhenlin Yang, SMU
                             Chapter 1: Introduction
                                       Sampling Distributions
 Sampling Distribution of the Difference Between
 Two Sample Means
      The distribution of x1  x 2 is normal if
             The two samples are independent, and
             The parent populations are normally distributed.

      If the two populations are not both normally
       distributed, but the sample sizes are 30 or
       more, the distribution of x1  x 2 is
       approximately normal.


STAT312, Term II, 10/11           26                     Zhenlin Yang, SMU
                                    Chapter 1: Introduction
                                                   Sampling Distributions
     Applying the laws of expected value and
      variance we have:

             E ( X 1  X 2 )  E ( X 1 )  E ( X 2 )  1   2
                                                                  12       2
                                                                             2
             Var ( X 1  X 2 )  Var ( X 1 )  Var ( X 2 )             
                                                                  n1        n2

     We can define:
                                    ( x1  x2 )  ( 1   2 )
                              Z
                                             12       2
                                                        2
                                                   
                                            n1         n2

STAT312, Term II, 10/11                    27                                    Zhenlin Yang, SMU
                                         Chapter 1: Introduction
                                                       Sampling Distributions
  Sampling Distribution of the difference between
  two Sample Proportions
   From the laws of expected value and variance, it can be shown
    that
           E( p1  p 2 )  E( p1 )  E( p 2 )  p1  p 2
              ˆ    ˆ          ˆ         ˆ
                                                                   p1 (1  p1 ) p 2 (1  p 2 )
                   Var ( p1  p 2 )  Var ( p1 )  Var ( p 2 ) 
                         ˆ    ˆ             ˆ            ˆ                     
                                                                        n1           n2
   If both n1p1 ≥ 5, n1(1-p1) ≥ 5, n2p2 ≥ 5, n2(1-p2) ≥ 5, then
                                    p1  p2  ( p1  p2 )
                                    ˆ    ˆ
                     Z
                              p1 (1  p1 ) n1  p 2 (1  p2 ) n2
   Z is approximately standard normally distributed.

STAT312, Term II, 10/11                          28                                    Zhenlin Yang, SMU
                          Chapter 1: Introduction
                                      Sampling Distributions
 Sampling Distribution of Sample Variance
   The statistic (n  1) s 2  2 has a Chi-squared
   distribution with df = n-1, if the population
   is normally distributed.

         d.f. = 5                (n  1) s 2
                          2                  ,   d. f .  n  1
                                         2



                                 d.f. = 10




STAT312, Term II, 10/11          29                           Zhenlin Yang, SMU
                          Chapter 1: Introduction
                                   Sampling Distributions
  Sampling Distribution of the Ratio of Two
  sample Variances              2   2
                                       s1  1
     Define the statistic:           F 2 2
     where the two samples
                                       s2  2
     are drawn from two Normal populations

   The sampling distribution of this statistic
   is an F distribution with df n1 = n1–1 for the numerator
   and df n2 = n2–1 for the denominator.
STAT312, Term II, 10/11       30                    Zhenlin Yang, SMU
                           Chapter 1: Introduction
                                        Statistical Inference
    Any numerical feature of a population, such as mean and
     variance, is called a parameter.
    Statistical Inference deals with drawing
     generalizations about population parameters from an
     analysis of information contained in the sample data.
     Studying the whole population is usually impractical;
     that is why we study a part of it. Inferences include:
    Point Estimation: obtain a guess or an estimate for
     the unknown true parameter value.
    Interval Estimation: obtain an interval of plausible
     values for the parameter, and determine the accuracy of
     the procedure.
    Testing hypothesis: decide whether the value of the
     parameter is equal to some pre-assumed value.

STAT312, Term II, 10/11         31                      Zhenlin Yang, SMU
                                Chapter 1: Introduction
                                               Statistical Inference
• Point Estimation
    Let f(x; ) be the pdf with parameter (vector) . Let X1, X2, …, Xn be a
    random sample drawn from f(x; ).
    Point estimation of  is to find a statistic such that its value computed
    from the sample data would reflect value of  as closely as possible.
    Such a statistic is called an estimator of  and a specific value of the
    estimator computed form sample data is called an estimate of .
    Maximum Likelihood Estimation Method:
    The joint pdf of X1, X2, …, Xn when regarded as a function of , is
    called the likelihood function of :
                  L( )  f ( x1; ) f ( x2 ; ) f ( xn ; )
                                   ˆ
    The value of  , denoted by  , that maximizes L() is called the
    maximum likelihood estimator, or the MLE.
STAT312, Term II, 10/11               32                           Zhenlin Yang, SMU
                                     Chapter 1: Introduction
                                                    Statistical Inference
  Example 1.8. Bernoulli sampling: f ( x, p)  p x (1  p)1 x , x  0,1

           L( p)  p  xi (1  p) n xi , 0  p  1         p   xi n
                                                              ˆ


 Example 1.9. Normal sampling: Xi ~ N(, 2)

                                                 n ( xi   ) 2 
                                            n
                                     1 
                      L(  ,  )         exp               
                              2

                                    2              2 2     
                                                 i 1           
                      1 n                   1 n
                     X i  X , and    ( X i  X ) 2
                   ˆ                    ˆ 2

                      n i 1                n i 1
   Other methods: least square, method of moment, Bayesian estimator, etc.
   Properties:    unbiasness, relative efficiency, etc.
STAT312, Term II, 10/11                    33                        Zhenlin Yang, SMU
                                 Chapter 1: Introduction
                                             Statistical Inference
 • Confidence Interval
  Let L(X) and U(X) be functions of X = (X1, X2, …, Xn) such that
              P[L(X) <  < U(X)] = 1 – a
  Then the interval {L(X), U(X)} is called a 100(1–a)% confidence
  interval (CI) for , L(X) and U(X) the lower and the upper confidence
  limits, and (1–a) the confidence coefficient associated with the interval.

  It is an approximate CI if the above equality holds only approximately.

   Bernoulli sampling: an approx. CI for p: p  Z a 2 p(1  p) n
                                            ˆ         ˆ     ˆ

   Normal sampling: an exact CI for :   ta 2 
                                       ˆ         ˆ    n 1

STAT312, Term II, 10/11               34                         Zhenlin Yang, SMU
                               Chapter 1: Introduction
                                               Statistical Inference
   • Test Statistical Hypothesis

   Null Hypothesis (H0): A theory about the values of population
     parameter(s), representing the status quo, accepted until proven
     false.
   Alternative Hypothesis (Ha): A theory that contradicts H0, which is
      favored upon existence of sufficient evidence.
   Test Statistic: A sample statistic used to decide whether to reject H0,
     which a measure of difference between the data and what is
     expected under the null hypothesis.
   Rejection Region: The numerical values of test statistic for which H0
     is rejected. This region is chosen so that the probability is a that it
     will contain the test statistic when H0 is true, thereby leading to a
     wrong rejection (Type I error). It is also referred to as level of
     significance.

STAT312, Term II, 10/11               35                           Zhenlin Yang, SMU
                            Chapter 1: Introduction
                                          Statistical Inference

    Conclusion: If the numerical value of the test statistic falls
      in the rejection region, we reject the H0 and conclude
      that the Ha is true. If the test statistic does not fall in the
      rejection region, we reserve the judgment about which
      H0 is true. An incorrect acceptance of H0 leads to a
      Type II error.
    p–value: the probability (assuming H0 is true) of observing
      a value of the test statistic that is at least as
      contradictory to the null hypothesis as the one
      computed from the data.
    Power of the test: Probability of rejecting a wrong null
      hypothesis.

STAT312, Term II, 10/11           36                       Zhenlin Yang, SMU
                                       Chapter 1: Introduction
                                                      Statistical Inference
   Example 1.10. From past sales records, it is known that 30% of
   the population buys Brand X toothpaste. A new advertising
   campaign is completed, and to test its effectiveness, 1000
   people are asked whether they buy Brand X toothpaste now. If
   334 answer yes, does this indicate that the advertising campaign
   has been successful?

   H0: p = 0.30, Ha: p > 0.30.
             ˆ
   n = 1000, p = 0.334, Z0.05= 1.65. Rejection region: Z > 1.65.
   Test Stat.
                            p  p0
                            ˆ              0.334  0.3
                Z=                       =             = 2.35
                          p0 (1  p0 ) n    0.01449
STAT312, Term II, 10/11                      37                    Zhenlin Yang, SMU
                                    Chapter 1: Introduction
                                 Large Sample Inference Methods
 When the exact sampling distribution of an estimator is unknown,
 statistical inference can only be made approximately, based large
 sample properties of the estimator. Common methods include:
                        ˆ   0 H 0
  Wald Statistics:                ~ N (0,1)
                               ˆ
                       ASE ( ) approx.
                               S ( 0 )       H0
   Score Statistic:                           ~ N (0,1)
                             ASE[ S ( 0 )]      .
                                            approx

                             where S ( 0 )  d log[ L( )] d     0



   LR Statistic:   maximum likelihood when parameters satisfy H0
                              maximum likelihood when parameters unrestrict ed
                                     H0
                           2 log  ~  df
                                        2
                                        .
                                   approx
STAT312, Term II, 10/11                     38                            Zhenlin Yang, SMU
                                 Chapter 1: Introduction
                                                       Matrix Algebra
      Vector:                             Matrix:

          a1                                 a11    a12     a1n 
                                                                  
      a = a 2  , b = b1   b2  bn     A =  a 21   a 22    a2n 
                                                          
                                                                  
         a                                  a               am n 
          n                                  m1     am2           

    Transpose: a   a1     a2  an 
                               10 1          2           23 
    Matrix Multiplication: A = 
                                3 6 , b =
                                              then A  b   
                                               ,
                                               3           24 
                                                         
                            0.10526316 - 0.01754386 
    Matrix Inverse: A 1  
                            - 0.05263158 0.17543860 
                                                     
                                                    

STAT312, Term II, 10/11              39                           Zhenlin Yang, SMU
                          Chapter 1: Introduction
                                  Computer Software: R

     ‘R’ is a computer package which does statistical
      analysis in a rather simple way,
     R is an open source software project and can be
      freely downloaded from:
             http://info.smu.edu.sg/rsite/
             http://cran.r-project.org/
             http://cran.bic.nus.edu.sg/
             http://www.mysmu.edu/faculty/zlyang/
     Other popular software include: Excel, Minitab,
      Matlab, SPSS, SAS, Gauss, S-Plus.

STAT312, Term II, 10/11      40                 Zhenlin Yang, SMU

				
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