Table 14.15 Asymptotic Partial Likelihood Inference from the Regression Models with Different Link Functions for the Data in Example 14.9 95% Confidence Interval for Odds Ratio Regression Standard Chi-Square Odds Variable Coefficient Error Statistic p Ratio Lower Upper Model with L ogit L ink Function INTERCPT 98.419 1.792 22.061 :0.0001 DBP 0.044 0.018 5.673 0.0172 1.05 1.01 1.08 LACR 0.343 0117 8.627 0.0033 1.41 1.13 1.79 LINSUL 0.870 0.287 9.191 0.0024 2.39 1.38 4.27 Hosmer—Lemeshow test statistic 18.9460 0.0152 Model with Inverse Normal Link Function INTERCPT 94.532 0.953 22.597 :0.0001 DBP 0.023 0.010 5.302 0.0213 LACR 0.186 0.065 8.060 0.0045 LINSUL 0.445 0.156 8.146 0.0043 Hosmer—Lemeshow test statistic 7.386 0.4956 Model with Log-Log Link Function INTERCPT 97.740 1.530 25.589 :0.0001 DBP 0.038 0.016 5.919 0.0150 LACR 0.305 0.096 10.153 0.0014 LINSUL 0.785 0.241 10.592 0.0011 Hosmer—Lemeshow test statistic 17.415 0.0261 Example 14.10 Consider the data in Example 14.9 as nonstratified data, Table 14.15 gives the results from the regression models defined in (14.2.26), (14.2.28), and (14.2.30) by using the stepwise selection method. Based on Hosmer—Lemeshow test statistics, the regression model with the inverse normal link function gives a good fit to the data (p : 0.4956), whereas the other two models do not (p : 0.0152 and p : 0.0261). All three models identify DBP, LACR, and LINSUL as significant covariates for the development of diabetes. The following SAS, SPSS, and BMDP codes may be used to generate the results in Table 14.15. SAS code: data w1; infile ‘c:!ex14d2d6.dat’ missover; input age ageg sex sbp dbp lacr hdl linsul smoke dms dm sn; run; 412         title ‘‘Regression model with the logit link function-generalized logistic regression’’; proc logistic data : w1 descending; model dm : age sex sbp dbp lacr hdl linsul smoke / selection : s lackfit link : logit; run; title ‘‘Regression model with the inverse normal link function‘; proc logistic data : w1 descending; model dm : age sex sbp dbp lacr hdl linsul smoke / selection : s lackfit link : probit; run; title ‘‘Regression model with the log-log link funtion‘; proc logistic data : w1 descending; model dm : age sex sbp dbp lacr hdl linsul smoke / selection : s lackfit link : cloglog; run; SPSS code for the model in (14.2.28) with the forward selection method: data list file : ‘c:!ex14d2d6.dat’ free / age ageg sex sbp dbp lacr hdl linsul smoke dms dm sn. Logistic regression dm with age sex sbp dbp lacr hdl linsul smoke htn /method : fstep /print : all. BMDP code for procedure LR and the model in (14.2.28): /input file : ‘c:!ex14d2d6.dat’ . variables : 12. format : free. /variable names : age, ageg, sex, sbp, dbp, lacr, hdl, linsul, smoke, dms, dm, sn. Use : age, sex to smoke. /regress depend : dm. Interval : age, sex to smoke. Method : mlr. /print cell: used. /end 14.3 MODELS FOR POLYCHOTOMOUS OUTCOMES The regression models in Section 14.2 can be extended to handle outcomes that have more than two categories. These categories may be nominal, for example, different types of heart disease or psychological conditions; or ordinal, for example, different levels of glucose intolerance or different severity of communi- cation disorders. An outcome variable with more than two possibilities is called polychotomous or polytomous. In this section we discuss first the model for     413 nominal polychotomous outcomes (generalized logistic regression model), then the model for ordinal polychotomous outcomes (ordinal regression model). Details regarding these models can be found in Aitchison and Silvey (1957), McCullagh (1980), Green (1984), McCullagh and Nelder (1989), Hosmer and Lemeshow (1989, 2000), Cox and Snell (1989), Afifi and Clark (1990), Agresti (1990), Collett (1991), and Ananth and Kleinbaum (1997). 14.3.1 Models for Nominal Polychotomous Outcomes: Generalized Logistic Regression Models Let Y G denote the outcome for individual i. The outcome can be one of the m nominal categories, such as different cell types of lung cancer. Let Y G : k denote that Y G belongs to the kth category and k : 1, 2, , m. Suppose that for each of n subjects, p independent variables x G : (x G , x G , , x GN ) are measured. These variables can be either qualitative or quantitative. Let P(Y G : k " x G ) be the probability that Y G : k given the p measured covariates x G ; then  K I P(Y G : k " x G ) : 1. Without loss of generality, using the last catalog as the reference, the generalized logistic regression model log P(Y G : k "x G ) P(Y G : m " x G ) : a I ; N  H b IH x GH k : 1, 2, , m 9 1 (14.3.1) can be used to study the association of the covariates x to the outcome. To simplify the notation, let u IG : a I ;  N H b IH x GH . Similar to (14.2.1) and (14.2.2), the model in (14.3.1) assumes that the dependence on the covariates of the probability of being in the kth category is P(Y G : k "x G ) :  exp(u IG ) 1 ;  K\ H exp(u HG ) k : 1, 2, , m 9 1 1 1 ;  K\ H exp(u HG ) k : m (14.3.2) This model reduces to the logistic regression model in (14.2.1) and (14.2.2) when m : 2. Let k  , , k L be the outcomes observed for the n subjects. Then the log-likelihood function based on the n outcomes observed is the logarithm of the product of all P(Y G : k G " x G )’s from the n subjects, that is, l(a  , a  , , a K\ , b  , b  , ,b K\ ) : log L : log  L  G P(Y G : k G " x G )  (14.3.3) where P(Y G : k G " x G ) is given in (14.3.2) and b I : (b I , , b IN ), k : 1, 2, , m 9 1. There are a total of (m 9 1)(p ; 1) unknown coefficients. The estimation and hypothesis testing procedures for the coefficients are similar to 414         those in the logistic regression model for dichotomous outcomes. Strictly speaking, the models in (14.3.1) are not logistic regression models if m 9 2. Therefore, the interpretation of the coefficients in these models needs to be clarified. Let us consider modeling the relationship between gender and cardiovascular disease status, NORMAL, STROKE, and CHD (coronary heart disease). Let the outcome variable Y be defined as Y : 1 if CHD, :2if STROKE, and :3 if NORMAL, and the covariate SEX defined as SEX : 1 if male and :0 if female. Then the two models according to (14.3.1) are log P(Y G : 1 " SEX G ) P(Y G : 3 " SEX G ) : a  ; b  · SEX G log P(Y G : 2 " SEX G ) P(Y G : 3 " SEX G ) : a  ; b  · SEX G It is clear that neither of them is a logistic regression model. In the following, we show how to interpret the coefficients b  and b  in these models. From the first model, log P(Y : 1 " SEX : 1)/P(Y : 3 " SEX : 1) P(Y : 1 " SEX : 0)/P(Y : 3 " SEX : 0) : log P(Y : 1 " SEX : 1) P(Y : 3 " SEX : 1) 9 log P(Y : 1 " SEX : 0) P(Y : 3 " SEX : 0) : (a  ; b  ) 9 a  : b  and thus P(Y : 1 " SEX : 1)/P(Y : 3 " SEX : 1) P(Y : 1 " SEX : 0)/P(Y : 3 " SEX : 0) : exp(b  ) (14.3.4) Now let us cast the data into a 3;2 contingency table as in Table 14.16. The left side of (14.3.4) can be estimated by ( f/n  )/(b/n  ) (e/n  )/(a/n  ) : fa be However, if only the data from the normal and CHD participants are used, fa be : [ f/(b ; f )]/[b/(b ; f )] [e/(e ; a)]/[a/(e ; a)]     415 Table 14.16 Nominal Cross-Classification of Cardiovascular (CVD) Status by Gender SEX CVD Status (Y) Female (0) Male (1) NORMAL (3) ab STROKE (2) cd CHD (1) ef —— Total n  n  which is an estimate of P(CHD " Male)/[1 9 P(CHD " male)] P(CHD " female)/[1 9 P(CHD " female)] or the ratio of the odds of a male having CHD to the odds of a female having CHD. Therefore, the exp(b  ) obtained from the first model can be interpreted as an estimate of the ratio of the odds of a male having CHD to the odds of a female having CHD if only the data from the normal and CHD participants are used. Similarly, exp(b  ) obtained from the second model can be interpreted as an estimate of the ratio of the odds of a male having STROKE to the odds of a female having STROKE if only the data from the normal and STROKE participants are used. The same interpretation also holds for coefficients of continuous covariates in the models of (14.3.1); that is, an exponentiated coefficient for a continuous covariate is the odds ratio of a 1-unit increase in the covariate assuming that other covariates are the same. Example 14.11 We use the data in Example 14.9 and assume that DM (Y : 1), IFG (Y : 2), and NFG (Y : 3) are three nominal categories. Let the referent category be NFG. For simplicity, only two covariates, systolic blood pressure (SBP) and log insulin (LINSUL), are included. Table 14.17 gives the results from fitting these covariates to the model (14.3.1). log P(i th participant is DM) P(i th participant is NFG) : log P(Y G : 1 " x G ) P(Y G : 3 " x G ) :97.648 ; 0.026SBP G ; 1.047LINSUL G log P(i th participant is IFG) P(i th participant is NFG) : log P(Y G : 2 " x G ) P(Y G : 3 " x G ) :94.949 ; 0.011SBP G ; 0.876LINSUL G 416         Table 14.17 Asymptotic Partial Likelihood Inference from the Generalized Logistic Regression Model for Example 14.11 95% Confidence Interval Order of for Odds Ratio Coefficients Regression Standard Chi-Square Odds bySAS k Variable Coefficient Error Statistic p Ratio Lower Upper DM vs. NFG b  1 INTERCP 97.648 1.648 21.530 :0.0001 b  1 SBP 0.026 0.010 6.300 0.0121 1.03 1.01 1.05 b  1 LINSUL 1.047 0.304 11.870 00006 2.85 1.57 5.17 IFG vs. NFG b  2 INTERCP 94.949 1.427 12.020 00005 b  2 SBP 0.011 0.010 1.410 0.2346 1.01 0.99 1.03 b  2 LINSUL 0.876 0.262 11.150 0.0008 2.40 1.44 4.01 DM vs. IFG INTERCP 92.699 1.850 2.130 0.1445 SBP 0.015 0.012 0.480 0.2239 1.02 0.99 1.04 LINSUL 0.171 0.333 1.270 0.6066 1.19 0.62 2.28 H  : b  : b  1.48 0.2239 H  : b  : b  0.27 0.6066 417 Consequently, log P(i th participant is DM) P(i th participant is IFG) : log P(Y G : 1 " x G ) P(Y G : 2 " x G ) : log P(Y G : 1 " x G ) P(Y G : 3 " x G ) 9 log P(Y G : 2 " x G ) P(Y G : 3 " x G ) : (97.648 ; 4.949) ; (0.026 9 0.011)SBP G ; (1.047 9 0.876)LINSUL G :92.699 ; 0.015SBP G ; 0.171LINSUL G Thus, the odds ratio is 1.03 [exp(0.026)] times (or 3% higher) for a 1-unit increase in SBP, and 2.85 [exp(1.047)] times (or 185% higher) for a 1-unit increase in LINSUL from the model for DM vs. NFG. The odds ratio is 2.40 [exp(0.876)] times (or 140% higher) for a 1-unit increase in LINSUL from the model for IFG versus NFG. SBP is not significant in the model for IFG versus NFG (p : 0.2346). Neither SBP nor LINSUL is significant in the model for DM versus IFG (p : 0.2239 and p : 0.6066, respectively). One can also follow the examples in Chapter 7, 9, 11, and 12 to perform additional statistical inferences. For instance, we can test whether the coefficients for SBP in the first two models are equal (whether the odds ratio for a 1-unit increase of SBP in the model for DM versus NFG is equal to that in the model for IFG versus NFG), that is, H  : b  9 b  : 0 (where the subscripts 3 and 4 are the orders of the coefficients given by SAS). From (11.2.13), under H  , Wald’s statistic, X 5 : (b  9 b  )/(v  ; v  9 2v  ), has an asymptotic chi-square distribution with 1 degree of freedom, where v  and v  are the estimated variance of b  and b  , respectively, and v  is the estimated covariance of b  and b  . From Table 14.17, the hypothesis is not rejected (p : 0.2239). Similarly, the hypoth- esis H  : b  9 b  : 0 is not rejected (p : 0.6066); that is, there is insufficient evidence to say that the change in odds ratio for a 1-unit increase in LINSUL in the model for DM versus NFG is not equal to that in the model for IFG versus NFG. The following SAS, SPSS, and BMDP codes can be used to obtain the results in Table 14.17. SAS code: data w1; infile ‘c:!ex14d2d6.dat’ missover; input age ageg sex sbp dbp lacr hdl linsul smoke dms dm sn; y : 4-dms; run; title ‘‘Generalized logistic regression model’’; proc catmod data : w1; direct sbp linsul; 418         model y : sbp linsul / ml covb; contrast ‘Equal coefficients for SBP’ all — parms 0 0 1 9100; contrast ‘Equal coefficients for LINSUL’ all — parms 0000191; run; SPSS code: data list file : ‘c:!ex14d2d6.dat’ free / age ageg sex sbp dbp lacr hdl linsul smoke dms dm sn. Compute y : 4-dms. nomreg y with sbp linsul /print : fit history parameter lrt. BMDP PR code: /input file : ‘c:!ex14d2d6.dat’ . variables : 12. format : free. /variable names : age, ageg, sex, sbp, dbp, lacr, hdl, linsul, smoke, dms, dm, sn. Use : age, sex to smoke. /transform y : 4-dms. /group codes(y) : 1, 2, 3. Names(y) : DM, IFG, NFG. /regress depend : y. Level : 3. Type : nom. Interval : age, sex to smoke. enter : .05, .05. remove ::.05, .05. /print cell: model. /end 14.3.2 Model for Ordinal Polychotomous Outcomes: Ordinal Regression Models If the outcomes involve a rank ordering, that is, the outcome variable is ordinal, several multivalued regression models are available. Readers interested in these models are referred to McCullagh and Nelder (1989), Agresti (1990), Ananth and Kleinbaum (1997), and Hosmer and Lemeshow (2000). In the following discussion, we introduce the most frequently used model, the propor- tional odds model. In this model, the probability of an outcome below or equal to a given ordinal level, P(Y -k), is compared to the probability that it is higher than the level given, P(Y 9k). Let Y G be the outcome of the ith subject. Assume that Y G can be classified into m ordinal levels. Let Y G : k if Y G is classified into the kth level and     419 k : 1, 2, , m. Suppose that for each of n subjects, p independent variables x G : (x G , x G , , x GN ) are measured. These variables can be either qualitative or quantitative. If the logit link function defined in Section 14.2.3 is used, similar to the logistic regression model (14.2.3), we consider the following models: logit(P(Y G - k " x G )) : log P(Y G -k " x G ) 1 9 P(Y G - k " x G ) : a I ; N  H b H x GH k : 1, 2, , m 9 1 (14.3.5) or, equivalently, let u IG : a I ;  N H b H x GH , P(Y G - k " x G ) : exp(a I ;  N H b H x GH ) 1 ; exp(a I ;  N H b H x GH ) : exp(u IG ) 1 ; exp(u IG ) k : 1, 2, , m 9 1 (14.3.6) Therefore, P(Y G : k " x G ) : P(Y G - k " x G ) 9 P(Y G - k 9 1 " x G ) :  exp(u G ) 1 ; exp(u G ) k : 1 exp(u IG ) 1 ; exp(u IG ) 9 exp(u I\G ) 1 ; exp(u I\G ) k : 2, , m 9 1 1 9 exp(u K\G ) 1 ; exp(u K\G ) k : m (14.3.7) If m : 2, that is, there are only two outcome levels, (14.3.7) reduces to the logistic regression model in (14.2.3). The models in (14.3.5) can be thought of as having only two outcomes [(Y - k) versus (Y 9 k)] and therefore are logistic regression models. Thus, interpretation of the coefficients, b H , such as the exponentiated coefficient [exp(b H )] for a discrete or a continuous covariate is similar to that in a logistic regression model. Let k  , , k L be observed outcomes from n subjects. Then the log-likelihood function based on the n outcomes observed is the logarithm of the product of all P(Y G : k G " x G )’s from the n subjects, that is, l(a  , a  , , a K\ , b  , b  , , b N ) : log L : log  L  G P(Y G : k G " x G )  (14.3.8) where P(Y G : k G " x G ) is as given in (14.3.7). The maximum likelihood estimation and hypothesis-testing procedures for the coefficients are similar to those discussed previously. If the probit link function in (14.2.27) is used, the models 420         and formula corresponding to (14.3.5)—(14.3.7) are \(P(Y G - k " x G )) : a I ; N  H b H x GH k : 1, 2, , m 9 1 P(Y G - k " x G ) : (u IG ) k : 1, 2, , m 9 1 P(Y G : k " x G ) : P(Y G - k " x G ) 9 P(Y G - k 9 1 " x G ) :  (u G ) k : 1 (u IG ) 9 (u I\G ) k : 2, , m 9 1 1 9 (u K\G ) k : m If the complementary log-log link function in (14.2.29) is used, the models and formula corresponding to (14.3.5)—(14.3.7) are log[9log(1 9 P(Y G -k " x G ))] : a I ; N  H b H x GH k : 1, 2, , m 9 1 P(Y G - k " x G ) : 1 9 exp[9exp(u IG )] k : 1, 2, , m 9 1 P(Y G : k " x G ) : P(Y G - k " x G ) 9 P(Y G - k 9 1 " x G ) :  1 9 exp[9exp(u G )] k : 1 exp[9exp(u I\G )] 9 exp[9exp(u IG )] k : 2, , m 9 1 exp[9exp(u K\G )] k : m The log-likelihood function based on these two models can be obtained by replacing P(Y G : k G " x G ) in (14.3.8) with the respective expressions above. Example 14.12 Now consider the NFG, IFG, and DM categories in Example 14.9 that represent three levels of severity in glucose intolerance. DM (diabetes) is defined as fasting plasma glucose (FPG) .126 mg/dL, IFG (impaired fasting glucose) as FPG between 110 and 125 mg/dL, and NFG (normal fasting glucose) as FPG : 110 mg/dL. Thus, it is reasonable to consider the outcome variable as ordinal. Let the outcome variable Y : 1if DM, 2 if IFG, and 3 if NFG. We fit the models in (14.3.5) using the SAS procedure LOGISTIC with all the covariates. The SAS program allows users to use a variable selection method (forward, backward, and stepwise). In this case, we use the stepwise selection method, and the results are given in the first part of Table 14.18. The stepwise method identifies SBP and LINSUL as significant independent variables. For k : 1 [i.e., we compare diabetes with     421 [...]... 91 :     J: 91 0 91 1 J\ : 91 0 91 1 Iteration 1 Following (A.3), we obtain x : 0 9 [ (91 ) (91 ) ; 0 (91 )] : 91  x : 1 9 [ (91 ) (91 ) ; 1 (91 )] : 1  With these values, f  : 1, f  : 91 , and   J: 93 91 2 1 J\ : 91 91 2 3 Iteration 2 From (A.3) we obtain x : 91 9 [ (91 )(1) ; (91 ) (91 )] : 91  x : 1 9 [(2)(1) ; (3) (91 )] : 2  With these values, f  : 0 and f  : 0 Therefore, the iteration procedure... x 9 2x 9 1 : 0     x 9 x ; x 9 2 : 0    -  432 In this case, p : 2: f : x ; x x 9 2x 9 1      f : x 9 x ; x 9 2     Since * f /*x :2x ;x 9 2, * f /*x :x , * f /*x :3x 91 , and * f /*x : 1,             the Jacobian matrix is J: 2x ; x 9 2 x    3x 9 1 1  (A.4) Let the initial estimates be x : 0, x : 1, f  : 91 , and f  : 91 :     J: 91 0 91 ... model for ordinal polychotomous outcomes in BMDP PR is defined as log P(Y 9 k " x ) N G G :  ;   x :u  I H GH IG 1 9 P(Y 9 k " x ) H G G k : 1, 2, , m 9 1 Compared with (14.3.5),  : 9a , k : 1, 2, , m 9 1; I I 2, , p  : 9b , j : 1, H H Bibliographical Remarks The linear logistic regression method is discussed extensively in Cox ( 197 0), Cox and Snell ( 198 9), Collett ( 199 1), Kleinbaum ( 199 4),... remaining NFG For example, the probability of developing IFG is P(Y : 2 " x ) : P(participant i is IFG) G G exp (95 .485 ; 0.019SBP ; 0 .92 5LINSUL ) G G : 1 ; exp (95 .485 ; 0.019SBP ; 0 .92 5LINSUL ) G G exp (96 .753 ; 0.019SBP ; 0 .92 5LINSUL ) G G 9 1 ; exp (96 .753 ; 0.019SBP ; 0 .92 5LINSUL ) G G Thus, for a person whose systolic blood pressure is 140 mmHg and whose log insulin is 3, the probability of developing... SBP LINSUL 1 2 96 .753 95 .485 0.0 19 0 .92 5 Regression Coefficient 1.183 1.151 0.007 0.213 INTERCP1 INTERCP2 SBP LINSUL 93 .97 1 93 .240 0.011 0.530 0.677 0.664 0.004 0.123 INTERCP1 INTERCP2 SBP LINSUL 95 .626 94 .562 0.014 0.715 26.831 32.571 22.708 6.114 18.803 0.0001 0.0001 0.0001 0.0134 0.0001 26.261 34.415 23. 790 6.311 18.674 0.0001 0.0001 0.0001 0.0120 0.0001 1.02 2.52 Odds Ratio 0 .91 5 0. 894 0.006 0.162...  -  430 Third iteration: 0.7456 x : 91 .6364 ;  : 91 .5304  7.0334 f (x ) : 90 .054   f (x ) : 6.0264   Fourth iteration: 0.054 x : 91 .5304 ;  : 91 .52144  6.0264 f (x ) : 90 .00036   f (x ) : 5 .94 43   Fifth iteration: 0.00036 : 91 .52138 x : 91 .52144 ;   5 .94 43 f (x ) : 0.0000017   At the fifth iteration, for x : 91 .52138, f (x) is very close to zero If the stopping rule... G : log 1 9 P(Y - 1 " x ) P(participant i is nondiabetic) G G : 96 .753 ; 0.019SBP ; 0 .92 5LINSUL G G For k : 2, the estimated model in (14.3.5) is log P(Y - 2 " x ) P(participant i is either DM or IFG) G G : log 1 9 P(Y - 2 " x ) P(participant i is NFG) G G : 95 .485 ; 0.019SBP ; 0 .92 5LINSUL G G According to (14.3.7), we can estimate the probability of developing DM, IFG, or remaining NFG For example,... of analysis results In addition to the papers and books cited in this chapter, other works on the subject include Anderson ( 197 2), Mantel ( 197 3), Prentice ( 197 6), Prentice and Pyke ( 197 9), Holford et al ( 197 8), and Breslow and Day ( 198 0) Applications of the logistic regression model can easily be found in various biomedical journals EXERCISES 14.1 Consider the study presented in Example 3.5 and the data. .. x axis [ f (x) : 0] between 91 and 92 This gives us a good hint of an initial value of x Suppose that we begin with x : 91 ;   f (x ) : 2 and f (x ) : 2 Thus, the first iteration, following (A.1), gives     2 x : 91 9 : 92   2 and f (x ) : 94 and f (x ) : 11 Following (A.2), we obtain the following:     Second iteration: 4 x : 92 ; : 91 .6364   11 f (x ) : 90 .7456   f (x ) : 7.0334... (x ) or d : x 9 x   is in the neighborhood of I I I\ 10\ or 10\ Example A.1 Consider the function f (x) : x 9 x ; 2 428 -  4 29 Figure A.1 Graphical presentation of the Newton—Raphson method for Example A.1 We wish to find the value of x such that f (x) : 0 by the Newton—Raphson method The first derivative of f (x) is f (x) : 3x 9 1 Since f (91 ) : 2 and f (92 ) : 94 , graphically . Silvey ( 195 7), McCullagh ( 198 0), Green ( 198 4), McCullagh and Nelder ( 198 9), Hosmer and Lemeshow ( 198 9, 2000), Cox and Snell ( 198 9), Afifi and Clark ( 199 0), Agresti ( 199 0), Collett ( 199 1), and Ananth. IFG) : exp (95 .485 ; 0.019SBP G ; 0 .92 5LINSUL G ) 1 ; exp (95 .485 ; 0.019SBP G ; 0 .92 5LINSUL G ) 9 exp (96 .753 ; 0.019SBP G ; 0 .92 5LINSUL G ) 1 ; exp (96 .753 ; 0.019SBP G ; 0 .92 5LINSUL G ) Thus, for a. " x G ) P(Y G : 3 " x G ) 9 log P(Y G : 2 " x G ) P(Y G : 3 " x G ) : (97 .648 ; 4 .94 9) ; (0.026 9 0.011)SBP G ; (1.047 9 0.876)LINSUL G :92 . 699 ; 0.015SBP G ; 0.171LINSUL G Thus,