Log-Rate models are used in analyzing rates of individuals who are exposed to a risk of having a certain characteristic. The explanatory variables could be categorical or in a continuous scale. In finding a Log-Rate Model, parameters are estimated and goodness-of-fit are studied to carefully extract the best model to fit our data. Here we revisit three aspects of Log-Rate Models using the data set give at the end of the paper. The three aspects are parameter estimation, goodness-of-fit of the model, and marginal effect of the factors

Baumann, William A.

Heien, Herbert C.

English

Institutional Repository for Minnesota State University, Mankato

Journal of Undergraduate Researchat Minnesota State University,MankatoVolume 3 Article 22003Inferences in Log-Rate ModelsHerbert C. HeienMinnesota State University, MankatoWilliam A. BaumannMinnesota State University, MankatoFollow this and additional works at: http://cornerstone.lib.mnsu.edu/jurPart of the Statistical Models CommonsThis Article is brought to you for free and open access by the Undergraduate Research Center at Cornerstone: A Collection of Scholarly and CreativeWorks for Minnesota State University, Mankato. It has been accepted for inclusion in Journal of Undergraduate Research at Minnesota State University,Mankato by an authorized administrator of Cornerstone: A Collection of Scholarly and Creative Works for Minnesota State University, Mankato.Recommended CitationHeien, Herbert C. and Baumann, William A. (2003) "Inferences in Log-Rate Models," Journal of Undergraduate Research at MinnesotaState University, Mankato: Vol. 3, Article 2.Available at: http://cornerstone.lib.mnsu.edu/jur/vol3/iss1/2Student Agreement:I am submitting my research article to be published in the JUR (The Journal of Undergraduate Researchat Minnesota State University, Mankato), an electronic journal of the Minnesota State UniversityUndergraduate Research Center.I/We certify have followed the accepted standards of scientific, creative, and academic honesty andethics.I understand that my article submission will be blind-reviewed by faculty reviewers who will recommendacceptance for publication; acceptance with revisions; or reject for publication.I understand that as author, I retain the right to present any part of the research in any form in otherpublications.The JUR has the right to reproduce and reprint published submissions for instructional or promotionalpurposes.For complete details, see Journal of Undergraduate Research at Minnesota State University, Mankato policiespage.Mentor Agreement:I have reviewed the submission, and I support its inclusion in the JUR (The Journal of UndergraduateResearch at Minnesota State University, Mankato). I understand that I will be acknowledged as thefaculty mentor for the student author(s). To the best of my knowledge, the student has followed theaccepted standards of scientific, creative, and academic honesty and ethics.Inferences in Log-Rate Models   Herbert C. Heien (herbert.heien@mnsu.edu)   William A. Baumann (william.baumann@mnsu.edu)  and  Mezbahur Rahman (mezbahur.rahman@mnsu.edu)   Department of Mathematics and Statistics  Minnesota State University, Mankato, MN 56002, USA     Abstract Log-Rate models are used in analyzing rates of individuals who are exposed to a risk of having a certain characteristic.  The explanatory variables could be categorical or in a continuous scale.  In finding a Log-Rate Model, parameters are estimated and goodness-of-fit are studied to carefully extract the best model to fit our data.  Here we revisit three aspects of Log-Rate Models using the data set give at the end of the paper.  The three aspects are parameter estimation, goodness-of-fit of the model, and marginal effect of the factors.   1. Introduction  In the categorical data analysis literatures, the survival models and/or the hazard rate models are treated differently than standard logit models. In general, these models are termed as Rate Models or Log-Rate Models.  In its simplest form, a rate is defined as the number of individuals or observations possessing a particular characteristic divided by the total amount of exposure to the risk of having such a characteristic. The Rate Models can easily be connected to the standard Poisson Models. Then the Poisson Models are directly related to the Exponential Models by making conversion of rates per unit interval with the waiting time until the first occurrence.  Here we use a Log-Rate Model to determine the likelihood of premarital births in adolescent populations. 1Heien and Baumann: Inferences in Log-Rate ModelsPublished by Cornerstone: A Collection of Scholarly and Creative Works for Minnesota State University, Mankato, 2003 Let nttt  , ... ,, 21 be the waiting times of n individuals, and assume the distribution function to be )Pr()( tTtF  with probability density function ).(tf  The hazard rate is denoted by ),(t  and can be viewed as the instantaneous probability of an event in the interval  ttt , , given the event has not occurred before time t.  Formally, the hazard rate is defined by the following limit:  ].Pr[1lim)(00tT|ttTttttt                                                           (1)  The probability of an event not occurring up to time t is given by the function  .)()(-1]Pr[)(  t duuftFtTtS                                                          (2)   Assuming the waiting times are exponentially distributed, equation (2) may be written as   ).exp()( ittS                                                                                           (3)  The hazard rate is defined by the ratio  ,)(1)()()()(  tFtftStfti                                                                         (4)  and the general hazard rate model may be written as   ),...exp()( 110 pipiiiTi xx   βx                                                    (5)  where p ,...,, 10 are unknown constants as the rate is determined by several regressors.  This exponential hazard rate model can be estimated using a Poisson regression model for counts.  The interested reader may see Powers and Xie (2000) pp. 154-156 for a brief explanation.  In a time interval of length t, the probability of d events is given by  .!)exp()()Pr(dtt,t |dd         (6)  Because the mean number of events in the time interval is  t , for the thi  individual, the expected number of events in the time interval is  ).exp( iTiiiii tt βx        (7)  2Journal of Undergraduate Research at Minnesota State University, Mankato, Vol. 3 [2003], Art. 2http://cornerstone.lib.mnsu.edu/jur/vol3/iss1/2Taking the log (log stands for natural logarithm) of the Poisson means results in the log-linear regression model  ,)log()log( iTiii t βx                                                                     (8)  where ]....,,,,,1[ 321 pT xxxxx   The plan of the paper is as follows:  In sections 2 and 3 we discuss the estimation procedures and goodness-of-fit of the models.  Marginal effects and their inferences are discussed in section 4.  In section 5, we use a real life data to demonstrate the use of a log-rate model, and we write a brief conclusion and give future research potentials in section 6.   2. Estimation of Parameters  We wish to use the method of maximum likelihood to estimate the parameters of this model.  Generally, the estimators are obtained by maximizing the logarithm of the likelihood function.  The likelihood is defined as  ,)!/()}(exp{)}({L1niiiTiidiTii dtt i βxβx                                                 (9)                                      where id is the number of occurrence of the event of interest in the time interval it and n is the sample size. The log-likelihood (log stands for natural logarithm) function is   log L =  iiiTiiiTiii dttd )}.!(log)(exp])[log({ βxβx                              (10)  The system of likelihood equations  )())(log( βUββ  L =   })exp({ 0xβx  iiiTiii td                                       (11)  are nonlinear and hence we use the standard Newton-Raphson numerical solution method (see Scarborough (1979), pp. 201-203) to solve them. The second derivatives, which are used in iteration to find maximum likelihood estimates, are given by   )())(log(2βIβββ  TL =iTiiiTiit }.)exp({ xxβx                                (12)  At the thk  iteration, the estimates are obtained using the equation  3Heien and Baumann: Inferences in Log-Rate ModelsPublished by Cornerstone: A Collection of Scholarly and Creative Works for Minnesota State University, Mankato, 2003    ,ˆˆˆˆ )1(1)1()1()(   kkkk βUβIββ                                           (13)  where )0(βˆ is obtained using the least squares method by regressing the y on the x’s in a linear regression set up as the link function is defined in (8).  The iteration is stopped when the consecutive iteration values are close and/or the log-likelihood values are maximized (see Powers and Xie (2000) pp. 61-63for details).   3. Goodness-of-fit  Log L cannot be used alone as an index of fit because it is dependent on the size of the sample.  Different values of log L result when competing models, models that differ in the number of parameters, are fit to the same data.  The number of parameters, in general, should be more than one, and significantly less than the number of observations.  To assess model fit, we need to know how one model fits relative to another.  An indicator of model fit which measures the extent to which the current model deviates from a more generalized model is given by the likelihood-ratio statistic:                                                                                           ),log(log2log2G 2 fcfc LLLL                                              (14)  where log cL is the log-likelihood of the current model, and log fL is the log-likelihood of the more generalized model.  The likelihood ratio statistic has a Chi-Square distribution with 12 KK  degrees of freedom, where 2K and 1K denote the number of parameters in the more generalized model and the current model, respectively.  A comparative study of different choices of general models can be seen in Simonoff (1998).   4. Marginal Effects          For the log-rate model, the marginal effects can be thought of as the relative risk associated with a certain variable.  The overall mean effect in (7) is  ).exp()( βxβx TT                                                                   (15)  Then the marginal effect due to the thk  factor can be considered as  ).exp(x))(( βxxβx TkkTk           (16)  The estimate of k can be computed as  4Journal of Undergraduate Research at Minnesota State University, Mankato, Vol. 3 [2003], Art. 2http://cornerstone.lib.mnsu.edu/jur/vol3/iss1/2)ˆexp(xˆ βxTkk             (17)  where kx is the mean of the thk  factor values in the sample and Tx is the vector of the means of the factor values in the sample.  The estimate of the variance for kˆ can be obtained using the delta method (see Ramsey and Schafer (2002) pp. 328-329 for details) as follows: the first derivative of kˆ with respect to s'ˆi are    .,...,2,1,0for  ),ˆexp(xxˆˆˆ piTikikki  βx       (18)  Then the approximate variance of kˆ can be written as  ).ˆ,ˆ(Cov)ˆ(V0 0jikjpipjkik            (19)  The estimate of the variance can be obtained as     ,ˆˆˆ)ˆ(Vˆ 10 0  ljkjpipjkik βI        (20) where     1ˆ ljβI  is the thlj  element of the matrix    1ˆ βI  in (13).   5. Application  We consider data analyzed by Powers and Xie (2000). The data studied was occurrence-exposure data on premarital births to young women participating in the National Longitudinal Survey of Youth from 1979 to 1988.  It provided retrospective information on the dates and occurrence of first birth and first marriage, and was categorized using three age intervals, race, and family structure   Information on those experiencing premarital births was recorded and is displayed in Table 1.  In Table 1, “E” represents the number of person-months from age 14 to age at premarital birth, and “D” represents the cell-specific number of events.  The approach taken here is that which is presented in Simonoff (1998).  Different log-rate models are compared to find the best model.  A saturated model and its nested models are defined.  We then use the deviations between the maximized log-likelihood from each model to perform a series of Chi-square tests so as to ascertain which model fits best as described in the Section 3.  The models used were: 5Heien and Baumann: Inferences in Log-Rate ModelsPublished by Cornerstone: A Collection of Scholarly and Creative Works for Minnesota State University, Mankato, 2003 )25(           )ln(E )ln()24()ln(E )ln()23()ln(E )ln()22()ln(E )ln()21()ln(E )ln(44330i4422110i3322110i443322110i9988776655443322110ixxxxxxxxxxxxxxxxxxxxxiiiii                                                                                                 . and  ,,,,person white-non a ngrepresenti variableindicator an is  structure,family intact -non a ngrepresenti variableindicator an is  ,20-18 interval age  thengrepresenti variableindicator an  is  ,18-16 interval age  thengrepresenti variableindicator an is where4394284173263154321xxxxxxxxxxxxxxxxxxx     Using SPSS, MATLAB, and Mathematica to perform the iterations necessary for the maximum likelihood method, the following results were obtained:  Model Fitted Model Maximum Log-Likelihood IterationsNeeded (21) 9876543211232.01558.01517.00305.000278.00466.17286.06380.23927.19593.6)Eln()ln(xxxxxxxxxii -4164.86 5 (22) 43210.9049950.5612.43271.3501980798.6)Eln()ln(xxxxii -4165.94 4 (23) 3210.66472.41481.33102460.6)Eln()ln(xxxii -4234.20 4 (24) 4210.97662.40761.34155705.6)Eln()ln(xxxii -4179.19 4 (25) 43 0.8713930.5034357251.5)Eln()ln( xxii   -4486.82 3  Using these results, we tested the competing models using the likelihood-ratio statistic as described in section 3 in order to determine goodness-of-fit.  To perform the tests, we started by testing the saturated model (21) against the main factors model (22), and then tested the main factors model against its nested counterparts.  The results of the Chi-square tests, performed with  = 0.05, are as follows:  6Journal of Undergraduate Research at Minnesota State University, Mankato, Vol. 3 [2003], Art. 2http://cornerstone.lib.mnsu.edu/jur/vol3/iss1/2Test 2G  d.f. Critical Value Conclusion (22) vs. (21) -2[(-4165.94)-(-4164.86)] = 2.16 6 12.59 Adequate fit for (20) (23) vs. (22) -2[(-4234.20)-(-4165.94)] = 136.52 1 3.84 Adequate fit for (20) (24) vs. (22) -2[(-4179.19)-(-4165.94)] = 26.5 1 3.84 Adequate fit for (20) (25) vs. (22) -2[(-4486.82)-(-4165.94)] = 641.76 2 5.99 Adequate fit for (20)  The main factors model (22) compared to the saturated model (with all the interactions) (21) had adequate fit.  Model (22) had adequate fit compared to all the other models. Thus, we decided to qualify the main factors model (22) as the adequate model for this data.  The marginal effects are computed as described in the Section 4.  The marginal effect for the first factor, age interval 16-18, is calculated as .858.3)ˆexp(ˆ 11    This means, the target population of age group 16-18 has a 3.858 times higher rate of premarital births than the other age groups.  A similar interpretation can be made for the marginal effect for the second factor, age interval 18-20, which is )ˆexp(ˆ 22   .390.11  The marginal effect for the third factor, non-intact family structure, is ,752.1)ˆexp(ˆ 33    which means that non-intact families have a 1.752 times higher rate of premarital births than intact families and the marginal effect for the fourth factor, non-white, is ,472.2)ˆexp(ˆ 44   which means that non-white families have a 2.472 times higher rate of premarital births than white families.  The variances of the marginal effects are computed using the delta method mentioned in section 4.  The computed variance-covariance matrix for the estimates of the parameters is,       .0.0070      0.0007-   0.0001    0.0001     0.0050-0.0007-    0.0052     0.0002    0.0001     0.0025-0.0001     0.0002     0.0109    0.0078     0.0080-0.0001     0.0001     0.0078    0.0107     0.0080-0.0050-   0.0025-   0.0080-   0.0080-   0.0132  ˆˆV1  I   The partial derivatives of the marginal effects are computed as  Marginal Effect k ˆ  16-18 18-20 Non-Intact Non-White3.85811.3901.7522.472 7Heien and Baumann: Inferences in Log-Rate ModelsPublished by Cornerstone: A Collection of Scholarly and Creative Works for Minnesota State University, Mankato, 2003Then, using (20), the variances are computed as ,0413.0)ˆV( 1  ,1241.0)ˆV( 2    ,0091.0)ˆV( 3  and .0173.0)ˆV( 4   These estimates of variances would be used in testing and in finding confidence intervals for the corresponding marginal effects to see the significance of the marginal effect estimates.   6. Conclusion  In conclusion, we see that the older age group has a higher rate of premarital births. Non-white have 2.472 times higher incidence of premarital births.  Non-intact families have a 1.752 higher birth rate than intact families.    Such procedures can be applied to any risk exposure data in which are categorical and/or quantitative in nature, such as studies of rare diseases in different cross-sections of the society.  For too many iterations in the parameter estimation procedure, estimates might have higher variations and the log-likelihood function is not maximized.   Table 1  Intact Non-intact  White   Non-white White Non-white Age D E D E D E D E 14-16 17 13220 33 13838 10 7332 68 12827 16-18 39 10266 104 9823 42 5417 160 8516 18-20 43 3552 112 3331 42 1599 128 2594   7. References              Cox, D. R. and Snell, E. J. (1989), Analysis of Binary Data, 2nd edition. London:          Chapman and Hall.  Efron, Bradley (1983), Estimating the Error Rate of a Prediction Rule: Improvement on Cross-Validation, Journal of the American Statistical Association,Vol.78 (382), pp.  316-331.  Efron, Bradley (1986), How Biased is the Apparent Error Rate of a Prediction Rule?, Journal of the American Statistical Association, pp. 461-470.  McCullagh, P. and J. A. Nelder (1989), Generalized Linear Models, 2nd edition. New York: Chapman and Hall.   Powers, Daniel A. and Xie, Yu (2000), Statistical Methods for Categorical Data  Analysis. San Diego, CA: Academic Press. 8Journal of Undergraduate Research at Minnesota State University, Mankato, Vol. 3 [2003], Art. 2http://cornerstone.lib.mnsu.edu/jur/vol3/iss1/2 Rahman, Mezbuhar, Cortes, Judy and Pardis, Cyrus (2001), A Note on Logistic Regression, Journal of Statistics & Management Systems, Vol. 4, No. 2, pp. 175-187.  Ramsey, Fred L. and Schafer, Daniel W. (2002), The Statistical Sleuth: A Course in Methods of Data Analysis, 2nd edition. Pacific Grove, CA: Duxbury.  Ryan, Thomas P. (1997), Modern Regression Methods, New York: Wiley.  Scarborough, James B. (1979), Numerical Mathematical Analysis. Baltimore, MD: The Johns Hopkins Press.  Siminoff, Jeffrey S. (1998), Logistic Regression, Categorical Predictors, and Goodness-of-Fit: It Depends on Who You Ask, The American Statistician, Vol. 52, No.1, pp. 10-14.  Van Houwelingen, J. C. and S. Le Cessie (1990), Predictive Value of Statistical Models, Statistics in Medicine, 1303-1325.     9Heien and Baumann: Inferences in Log-Rate ModelsPublished by Cornerstone: A Collection of Scholarly and Creative Works for Minnesota State University, Mankato, 2003

Inferences in Log-Rate Models

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Inferences in Log-Rate Models

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