Statistical Methods for Survival Data Analysis 3rd phần 9 ppsx

Based on Hosmer—Lemeshow test statistics, the regression model with the inverse DBP, LACR, and LINSUL as significant covariates for the development ofdiabetes.. title ‘‘Regression model w

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

Hosmer—Lemeshow test statistic 18.9460 0.0152

Model with Inverse Normal Link Function

INTERCPT 94.532 0.953 22.597
0.0001

Hosmer—Lemeshow test statistic 7.386 0.4956

Model with Log-Log Link Function

INTERCPT 97.740 1.530 25.589
0.0001

Hosmer—Lemeshow test statistic 17.415 0.0261

Example 14.10 Consider the data in Example 14.9 as nonstratified data,

(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

DBP, LACR, and LINSUL as significant covariates for the development ofdiabetes

The following SAS, SPSS, and BMDP codes may be used to generate theresults 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;

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;

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.

/input file : ‘c:ex14d2d6.dat’

The regression models in Section 14.2 can be extended to handle outcomes thathave more than two categories These categories may be nominal, for example,different types of heart disease or psychological conditions; or ordinal, forexample, 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

nominal polychotomous outcomes(generalized logistic regression model), then

(1990), Collett (1991), and Ananth and Kleinbaum (1997)

Generalized Logistic Regression Models

Let YG 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 YG : k denote that YG belongs to the kth category and k:1,2, , m Suppose that for each

of n subjects, p independent variables xG :(xG, xG, , xGN) are measured These variables can be either qualitative or quantitative Let P(YG :k xG) be the probability that YG:k given the p measured covariates xG; then

KIP(YG:kxG):1 Without loss of generality, using the last catalog as the

reference, the generalized logistic regression model

log P(YG :k xG)

P(YG: m xG) : aI; N

H bIHxGH 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 uIG:aI ;NHbIHxGH Similar to (14.2.1) and (14.2.2),

probability of being in the kth category is

P(YG:k xG): exp(uIG)

l(a, a, , aK\, b, b, , bK\):logL : log L

G P(YG :kG xG) (14.3.3)

where P(YG :kG xG) is given in (14.3.2) and bI:(bI, , bIN), k :1,

estimation and hypothesis testing procedures for the coefficients are similar to

those in the logistic regression model for dichotomous outcomes Strictly

Therefore, the interpretation of the coefficients in these models needs to beclarified Let us consider modeling the relationship between gender and

logP(YG :1 SEXG) P(YG :3 SEXG) : a;b·SEXG

logP(YG :2 SEXG) P(YG :3 SEXG) : a;b·SEXG

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,

( f /n)/(b/n)

f a be

However, if only the data from the normal and CHD participants are used,

f a

Table 14.16 Nominal Cross-Classification of

Cardiovascular (CVD) Status by Gender

SEX CVD

or the ratio of the odds of a male having CHD to the odds of a female having

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

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 STROKEparticipants are used The same interpretation also holds for coefficients of

coefficient for a continuous covariate is the odds ratio of a 1-unit increase inthe covariate assuming that other covariates are the same

Example 14.11 We use the data in Example 14.9 and assume that DM

referent category be NFG For simplicity, only two covariates, systolic blood

log P(i th participant is DM)

P(i th participant is NFG): logP(YG :1 xG)

P(YG :3 xG) : 97.648 ; 0.026SBPG;1.047LINSULG

log P(i th participant is IFG)

P(i th participant is NFG): logP(YG :2 xG)

P(YG :3 xG) : 94.949 ; 0.011SBPG;0.876LINSULG

logP(i th participant is DM)

P(i th participant is IFG): logP(YG :1 xG)

Thus, the odds ratio is 1.03 [exp(0.026)] times (or 3% higher) for a 1-unitincrease in SBP, and 2.85 [exp(1.047)] times (or 185% higher) for a 1-unitincrease 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 themodel for IFG versus NFG SBP is not significant in the model for IFG versus

the examples in Chapter 7, 9, 11, and 12 to perform additional statisticalinferences For instance, we can test whether the coefficients for SBP in the first

the model for DM versus NFG is equal to that in the model for IFG versus

X5 :(b9b)/(v;v 92v), 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

hypoth-esis H: b9b: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 IFGversus NFG

The following SAS, SPSS, and BMDP codes can be used to obtain theresults 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;

model y : sbp linsul

/ ml covb;

contrast ‘Equal coefficients for SBP’ all—parms 0 0 1 91 0 0;

contrast ‘Equal coefficients for LINSUL’ all—parms 0 0 0 0 1 91;

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.

Ordinal Regression Models

If the outcomes involve a rank ordering, that is, the outcome variable isordinal, several multivalued regression models are available Readers interested

following discussion, we introduce the most frequently used model, the tional odds model In this model, the probability of an outcome below or equal

propor-to a given ordinal level, P(Y

Let YG be the outcome of the ith subject Assume that YG can be classified into

m ordinal levels Let YG:k if YG is classified into the kth level and

k : 1, 2, , m Suppose that for each of n subjects, p independent variables

xG :(xG, xG, , xGN) are measured These variables can be either qualitative

or quantitative If the logit link function defined in Section 14.2.3 is used, similar

as having only two outcomes [(Y

logistic regression models Thus, interpretation of the coefficients, bH, such as

is similar to that in a logistic regression model

Let k, , kL 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(YG:kG xG)’s from the n subjects, that is,

l(a, a, , aK\, b, b, , bN) :logL :log L

G P(YG :kG xG) (14.3.8)

where P(YG :kG xG) is as given in (14.3.7) The maximum likelihood estimation

and hypothesis-testing procedures for the coefficients are similar to those

and formula corresponding to(14.3.5)—(14.3.7) are

The log-likelihood function based on these two models can be obtained by

replacing P(YG:kG xG) in (14.3.8) with the respective expressions above.

Example 14.12 Now consider the NFG, IFG, and DM categories inExample 14.9 that represent three levels of severity in glucose intolerance DM

(impaired fasting glucose) as FPG between 110 and 125 mg/dL, and NFG

procedure LOGISTIC with all the covariates The SAS program allows users

case, we use the stepwise selection method, and the results are given in the firstpart of Table 14.18 The stepwise method identifies SBP and LINSUL as

nondiabetes(NFG; IFG)] the estimated model in (14.3.5) is

or remaining NFG For example, the probability of developing IFG is

P(YG :2 xG):P(participant i is IFG)

As noted earlier, the coefficients in these models can be interpreted as those

in the ordinary logistic regression model for binary outcomes In this example,the higher SBP and LINSUL are, the higher the odds of having DM than ofnot having DM, or the higher the odds of having either DM or IFG than ofbeing NFG The odds ratio is 1.02 [exp(0.019)] times (or 2% higher) for a

152% higher) for a 1-unit increase in LINSUL assuming that SBP is the same.From the table, SBP and LINSUL are related significantly to the diabeticstatus in all models with different link functions

SAS and SPSS can also be used for the other two link functions: the inverse

of the cumulative standard normal distribution and the complementary log-log

link functions introduced in Section 14.2.3 Table 14.18 includes the resultsfrom models with these two link functions The results are very similar to thoseobtained using the logit link function.

The following SAS, SPSS, and BMDP codes can be used to obtain theresults in Table 14.18

SAS code:

data w1;

infile ‘c: ex14d2d6.dat’ missover;

input age ageg sex sbp dbp lacr hdl linsul smoke dms dm sn;

run;

title ‘‘Ordinal regression model with logic link function’’;

proc logistic data : w1 descending;

model dms : age sex sbp dbp lacr hdl linsul smoke

/ selection : s lackfit link : logit;

run;

title ‘‘Ordinal regression model with inverse normal link function‘;

proc logistic data : w1 descending;

model dms : age sex sbp dbp lacr hdl linsul smoke

/ selection : s lackfit link : probit;

run;

title ‘‘Ordinal regression model with complementary log-log link function’’;

proc logistic data : w1 descending;

model dms : age sex sbp dbp lacr hdl linsul smoke

/ selection : s lackfit link : cloglog;

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.

plum y with sbp linsul

/link : logit

/print : fit history parameter.

plum y with sbp linsul

/link : probit

/print : fit history parameter.

plum y with sbp linsul

/link : cloglog

/print : fit history parameter.

BMDP PR code for the logit link function only:

/input file : ‘c:ex14d2d6.dat’

variables : 12.

format : free.

/variable names : age, ageg, sex, sbp, dbp, lacr, hdl, linsul, smoke,

Hosmer and Lemeshow discuss broad application of the method, includingmodel-building strategies and interpretation and presentation of analysisresults In addition to the papers and books cited in this chapter, other works

Applications of the logistic regression model can easily be found in variousbiomedical journals

EXERCISES

patients in Table 3.10

(a) Construct a summary table similar to Table 3.11

(b) Construct a table similar to Table 3.12

(c) Use the chi-square test to detect any differences in retinopathy rates

(d) On the basis of these 40 patients, identify the most important riskfactors using a linear logistic regression method.

Table 3.1 Let ‘‘response’’ be defined as stable, partial response, orcomplete response

(a) Compare each of the five skin test results of the responders withthose of the nonresponders

(b) Use a linear logistic regression method to identify the most ant risk factors related to response

import-(i) Consider the five skin tests only

(ii) Consider age, gender, and the five skin tests

melanoma, and six skin tests) in Exercise 3.3 and Exercise Table 3.3.Identify the most important prognostic factors that are related toremission Use both univariate and multivariate methods

complete response, partial response, or stable disease) Include gender,age, nephrectomy treatment, lung metastasis, and bone metastasis asindependent variables

(a) Identify the most significant independent variables

(b) Obtain estimates of odds ratios and confidence intervals whenapplicable

X Show that the log odds ratio for X:x;m versus X:x is mb, where b is the logistic regression coefficient.

re-gression model for CVD by using the stepwise selection method toselect risk factors among the same factors as those noted at the bottom

of Table 12.7 Compare the results obtained with those in Table 12.7

that is, the sampling probability is independent of the risk factors x,

14.10 Consider the data in Table 12.4 Fit the generalized logistic regression

by using the SAS CATMOD, SPSS NOMREG, or BMDP PRprocedure Select risk factors among those noted at the bottom ofTable 12.7 using the stepwise selection method in the BMDP PRprocedure Compare the results with those given in Table 13.5

DM versus NFG, with SEX as the covariate, by using the data from

binary outcome IFG versus NFG, with SEX as the covariate, by using

covariate, and discuss your findings

A P P E N D I X A

Newton Raphson Method

is a numerical iterative procedure that can be used to solve nonlinearequations An iterative procedure is a technique of successive approximations,

and each approximation is called an iteration If the successive approximations approach the solution very closely, we say that the iterations converge The

maximum likelihood estimates of various parameters and coefficients discussed

in Chapters 7, 9, and 11 to 14 can be obtained by using the Newton—Raphson

method In this appendix we discuss and illustrate the use of this method, firstconsidering a single nonlinear equation and then a set of nonlinear equations

preferably, and then the first approximate iteration is given by

Example A.1 Consider the function

f (x) : x 9 x ; 2

428

Figure A.1 Graphical presentation of the Newton—Raphson method for Example A.1.

method The first derivative of f (x) is

 ) :2 and f (x) :2 Thus, the first iteration, following (A.1), gives

Figure A.1 gives the graphical presentation of f (x) and the iteration.

It should be noted that the Newton—Raphson method can only find the real

shown in Figure A.1;the other two are complex roots

The Newton—Raphson method can be extended to solve a system of

equations with more than one unknown Suppose that we wish to find values

of x, x, , xN such that

f(x, , xN) :0 f(x, , xN) :0

$

fN(x, , xN) :0 Let aGH be the partial derivative of fG with respect to xH;that is, aGH:*fG/*xH.

The matrix

J:

a % aN a % aN

bN % bNN

be the corresponding values of the functions f, , fN, that is,

are close enough to zero or when differences in the x values at two consecutive

iterations are negligible

Example A.2 Suppose that we wish to find the value of x and x such that

x

terminates and the solution of the two simultaneous equations is x :91, x:2.

The number of iterations required depends strongly on the initial values

chosen In Example A.2, if we use x

to find the solution Interested readers may try it as an exercise

A P P E N D I X B

Statistical Tables

433

Table B-1 Normal Curve Areas

Source: Abridged from Table 1 of Statistical Tables and Formulas, by A Hald, JohnWiley & Sons,

1952 Reproduced by permissionof JohnWiley & Sons.

434

Table B-2 Percentage Points of the
-Distribution

Source: ‘‘Tables of the Percentage Points of the
-Distribution,’’ by Catherine M Thompson,

Biometrika, Vol 32, pp 188—189 (1941) Reproduced by permissionof the editor of Biometrika.

435

437

the examples in Chapter 7, 9, 11, and 12 to perform additional statisticalinferences... statisticalinferences For instance, we can test whether the coefficients for SBP in the first

the model for DM versus NFG is equal to that in the model for IFG versus

X5 :(b9b)/(v;v 92 v),... coefficients for SBP’ all—parms 0 91 0;

contrast ‘Equal coefficients for LINSUL’ all—parms 0 0 91 ;

run;

SPSS code:

data