Statistical Methods in Epidemiology

Handout 10

36-461  Using BMDP LR for Logistic Regresssion  Fall, 1995

Background

The data are from a prospective study whose purpose was to characterize the spontaneous recovery of language functioning in patients during the first 3 to 4 months following a stroke. Details of the design and results of this study may be found in Holland et al. (1985) and Holland, Greenhouse, Fromm, and Swindell (1989). Briefly, patients admitted to a hospital with the diagnosis of stroke and who met the study inclusion criteria were seen daily (6 days/week) for 15 minutes beginning 48 to 72 hours poststroke and continuing until hospital discharge. The daily visits consisted of semistructured conversational interactions and systemic scoring of the patient’s communication behavior. At hospital discharge, patients were formally tested using the Western Aphasia Battery (WAB). If the WAB indicated aphasic dysfunction (AQ < 93.8), the patient was seen and assessed at 1 month postdischarge. If this second WAB demonstrated continued difficulty the patient was assessed again at 2 months postdischarge. The data set to be discussed here consists of 61 patients who had unilateral (right- or left-hemisphere) stroke, ischemic or hemorrhagic etiology, and impaired language functioning at time of hospital discharge. We will use these data to investigate the correlates of language recovery during the two month period post-hospital discharge. Definitions and the codes for the variables of interest are presented in Table 1.

The multiple logistic regression model will allow us to investigate simultaneously the effect of a number of different variables on the likelihood of spontaneous language recovery. A feature of the logistic regression model is that it allows us to use explanatory variables that are either quantitative or discrete. Using the variables defined and coded in Table 1 for the stroke recovery study, the specification of the deterministic component of the multiple linear logistic regression model on the logit scale is

ln(\frac{π}{1−π}) = β_{0} + β_{1} · (SIDE) + β_{2} · (GENDER) + β_{3} · (AGE) + β_{4} · (LOSD) + β_{5} · (W AB1). (1)

Note that if we were to dichotomize length of stay, age and WAB1 then a multiple crossclassification table of these data would consist of 2^{6} = 64 cells or subgroups and that with 61 patients such a table would have numerous sparse or empty cells which would be a problem for a chi-square analysis. An advantage of a statistical model such as the one given in equation (1), however, is that the model provides a structure that smooths the fit of the probability of a response to all the cells in such a table.

Table 0. Example of what the data file ’stroke.dat’ looks like

Side  Gender  Age  LOSD  WAB1  Status
  0     1      74   17   92.5    1
  1     1      75   30   87.8    0
  0     1      72   17   65.9    0
  0     1      60   21   86.4    1
  0     0      69   11   90.0    1
  0     0      62   13   6.2     0

Table 1. Variables and Codes from the Stroke Recovery Study

Variable  Description  Values/Codes
STATUS  Status at last  0 = Not Recovered
observation  1 = Recovered
SIDE  Side of lesion  0 = Left

1 = Right

GENDER  0 = Female

1 = Male

AGE  Age at stroke  In years

onset

LOSD  Length of hospital In days

stay

WAB1  WAB at  Range: 0 - 100

hospital discharge

Interpretation of Regression Coefficients

Figure 2 on the next page presents part of the BMDP output listing the estimates of the regressions coefficients and their standard errors. We see from the column of Figure 2 labelled coeff/S.E. that the coefficients for gender, length-of-stay, and the WAB score at discharge are all significantly related to the probability of spontaneous language recovery at the 0.05 level. The coefficient for length-of-stay is negative indicating that the longer patients are in the hospital the smaller the chance of spontaneous language recovery relative to patients who have shorter lengths-of-stay even after controlling for the effect of the WAB1 score. The coefficient for WAB1 is positive as we would expect. Males, after controlling for the other variables in the model, are exp(2.88) = 17.8 times as likely to recover spontaneous language function as females. A 95% confidence interval for β_{2}, the adjusted log odds ratio of recovery for males relative to females, however, is large and ranges from 0.47 to 5.29. We also note that side of stroke and subject’s age do not have a statistically significant association with spontaneous language recovery after controlling for the effects of the other variables. As part of the BMDPLR output, the results from several goodness-of-fit tests are given. Details about these tests can be found in Hosmer and Lemeshow (1989, Chapter 5).

Figure 1. BMDP LR Command file

/ CONTROL   COLUMN=72.    /END
/ PROBLEM  TITLE IS 'LANGUAGE RECOVERY FOLLOWING STROKE'.
/ INPUT    VARIABLES = 6.
          FORMAT IS FREE.
          FILE = 'stroke.dat'.
/ VARIABLE NAMES ARE SIDE, GENDER, AGE, LOSD, WAB1, STATUS.
           USE = 1,2,3,4,5,6.
/ REGRESS  DEPEND IS STATUS.
           INTERVAL IS  SIDE, GENDER, AGE, LOSD, WAB1.
           MODEL = SIDE, GENDER, AGE, LOSD, WAB1.
           START = IN, IN, IN, IN, IN. CONSTANT = IN.
           MOVE  = 0, 0, 0, 0, 0.
/ PRINT    CELLS = NONE.  NO HIST.
           NO PLOT. CASE=0. NEWS.
/ PLOT     XVAR =PREDPROB.
           YVAR =STDRESID.
/ END

Figure 2. Partial Output from the BMDP LR Analysis of the Stroke Data

STEP NUMBER   0
---------------
                         LOG LIKELIHOOD =   -14.494
GOODNESS OF FIT CHI-SQ   (2*O*LN(O/E))  =    28.988  D.F.=  55  P-VALUE= 0.999
GOODNESS OF FIT CHI-SQ (HOSMER-LEMESHOW)=    34.610  D.F.=   8  P-VALUE= 0.000
GOODNESS OF FIT CHI-SQ   ( C.C.BROWN )  =     8.452  D.F.=   2  P-VALUE= 0.015
                                        STANDARD
     TERM                COEFFICIENT       ERROR   COEFF/S.E.  EXP(COEFFICIENT)
SIDE                       -1.3694       1.111      -1.233      0.2543
GENDER                      2.8773       1.267       2.271       17.77
AGE                       -0.69369E-02  0.3070E-01 -0.2259      0.9931
LOSD                      -0.19617      0.9664E-01  -2.030      0.8219
WAB1                       0.16453      0.7300E-01   2.254       1.179
CONSTANT                   -11.078       6.380      -1.736      0.1545E-04

CORRELATION MATRIX OF COEFFICIENTS
----------------------------------
              SIDE      GENDER    AGE       LOSD      WAB1      CONSTANT
SIDE             1.000
GENDER          -0.284     1.000
AGE             -0.127     0.201     1.000
LOSD             0.210    -0.535    -0.192     1.000
WAB1            -0.311     0.536     0.062    -0.305     1.000
CONSTANT         0.260    -0.509    -0.334     0.130    -0.931     1.000

Analysis of the Fit of the Model

Figure 3 presents a residual analysis for this logistic regression.

One relatively easy method for the analysis of the fit of the logistic regression model is based on calculating the predicted probability of spontaneous language recovery based on the model in (1) compared to the observed probability of language recovery Specifically, substitute the estimated regression coefficients given in Figure 2 into the multiple logistic regression equation in (1). For each patient in the study calculate his or her individual predicted probabilty of language recovery by calculating the linear combination of each patient’s individual set of explanatory variables with the estimated regression coefficients and use the inverse of the logit transformation to solve for the probability of language recovery, π̂. For example, consider a patient whose stroke was on the left side (SIDE=0), was a 61 year old male (GENDER=1, AGE=61), stayed in the hospital 19 days (LOSD=19), and whose WAB at discharge was 81.6 (WAB1=81.6). The estimated probability of spontaneous language recovery based on the model for such a patient is

π̂ = \frac{1}{1 + e^{−(−11.1−1.37·(0)+2.88·(1)−0.0069·(61)−0.196·(19)+0.165·(81.6))}} = 0.77.

This value can be interpreted to mean that 77 patients out of 100 with the above characteristics can be expected to recover normal language functioning spontaneously within the first 2 months post-hospital discharge. To study the goodness of fit of the model, we obtain the frequency distribution of the values of π̂, and determine, say, the quintiles of the distribution. The observed number of patients who had normal language functioning are tallied within each quintile, and the expected number of patients who had normal language functioning (predicted by the model) is determined by summing the π̂’s for all the patients in the quintile. Table 2 presents the expected and observed number of patients who had normal language functioning for each quintile of risk of language recovery. Comparisons of the expected number predicted by the model to the observed number suggests that this model fits the data reasonably well.

A test for whether there is a significant difference between the observed and expected numbers in Table 2 can be constructed based on a Pearson chi-square test. This chi-square test would have degrees-of-freedom equal to the number of groups (e.g., 5 for the number of quintiles) minus 2 (Hosmer and Lemeshow, 1989, chapter 5). In our example, this chi-square statistics is equal to 3.60 with 3 degrees-of-freedom and has a non-significant p-value=0.69, indicating that the model adequately fits the data. A small p-value would indicate a significant lack of fit of the model.

Table 2. Analysis of the fit of the logit model in Equation (1)

Expected Observed
Quintile of risk  Number  Number
Highest  5  10.7  11
4  3.1  2
3  0.2  1
2  0.0  0
Lowest  1  0.0  0
Total  14.0  14

Figure 3. Plot of Standardized Residual vs Predicted Probability

.+.....+.....+.....+.....+.....+.....+.....+.....+.....+.....+.
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      .10         .30         .50         .70         .90
0.0         .20         .40         .60         .80         1.0
                         PREDPROB  10
EXTREME CASES --
CASE NO.  STDRESID    CASE NO.  PREDPROB
59      7.9404        57      0.9785