Daniel W.W. Biostatistics: A Foundation for Analysis in the Health Sciences
Подождите немного. Документ загружается.
the procedure goes to the next step. If no variable can be removed, the procedure tries
to add a variable. An F statistic is calculated for each variable not then in the model.
Of these variables, the one with the largest associated F statistic is added, provided its
F statistic is larger than the specified cutoff value (4 if some other value is not speci-
fied). The regression equation is refitted for the new model, the results are printed, and
the procedure goes on to the next step. The procedure stops when no variable can be
added or deleted.
The following example illustrates the use of the stepwise procedure for selecting
variables for a multiple regression model.
EXAMPLE 11.3.1
A nursing director would like to use nurses’ personal characteristics to develop a regres-
sion model for predicting the job performance ( JOBPER). The following variables are
available from which to choose the independent variables to include in the model:
We wish to use the stepwise procedure for selecting independent variables from those
available in the table to construct a multiple regression model for predicting job
performance.
Solution: Table 11.3.1 shows the measurements taken on the dependent variable,
JOBPER, and each of the six independent variables for a sample of
30 nurses.
X
6
= initiative 1INIT2
X
5
= problem-solving skills 1PROB2
X
4
= communication skills 1COMM2
X
3
= ambition 1AMB2
X
2
= enthusiasm 1ENTH2
X
1
= assertiveness 1ASRV2
11.3 VARIABLE SELECTION PROCEDURES 557
TABLE 11.3.1 Measurements on Seven Variables
for Examples 11.3.1
Y
45 74 29 40 66 93 47
65 65 50 64 68 74 49
73 71 67 79 81 87 33
63 64 44 57 59 85 37
83 79 55 76 76 84 33
45 56 48 54 59 50 42
60 68 41 66 71 69 37
73 76 49 65 75 67 43
X
6
X
5
X
4
X
3
X
2
X
1
(
Continued
)
We use MINITAB to obtain a useful model by the stepwise proce-
dure. Observations on the dependent variable job performance (JOBPER)
and the six candidate independent variables are stored in MINITAB
Columns 1 through 7, respectively. Figure 11.3.1 shows the appropriate
MINITAB procedure and the printout of the results.
To obtain the results in Figure 11.3.1, the values of F to enter and F
to remove both were set automatically at 4. In step 1 there are no variables
to be considered for deletion from the model. The variable AMB (Column
4) has the largest associated F statistic, which is
Since 94.8676 is greater than 4, AMB is added to the model. In step 2 the
variable INIT (Column 7) qualifies for addition to the model since its
associated F of is greater than 4 and it is the variable with
the largest associated F statistic. It is added to the model. After step 2
no other variable could be added or deleted, and the procedure stopped.
We see, then, that the model chosen by the stepwise procedure is a two-
independent-variable model with AMB and INIT as the independent vari-
ables. The estimated regression equation is
■
y
N
= 31.96 + .787x
3
- .45x
6
1-2.22
2
= 4.84
F = 19.742
2
= 94.8676.
558
CHAPTER 11 REGRESSION ANALYSIS: SOME ADDITIONAL TECHNIQUES
Y
74 83 71 77 76 84 33
69 62 44 57 67 81 43
66 54 52 67 63 68 36
69 61 46 66 64 75 43
71 63 56 67 60 64 35
70 84 82 68 64 78 37
79 78 53 82 84 78 39
83 65 49 82 65 55 38
75 86 63 79 84 80 41
67 61 64 75 60 81 45
67 71 45 67 80 86 48
52 59 67 64 69 79 54
52 71 32 44 48 65 43
66 62 51 72 71 81 43
55 67 51 60 68 81 39
42 65 41 45 55 58 51
65 55 41 58 71 76 35
68 78 65 73 93 77 42
80 76 57 84 85 79 35
50 58 43 55 56 84 40
87 86 70 81 82 75 30
84 83 38 83 69 79 41
X
6
X
5
X
4
X
3
X
2
X
1
To change the criterion for allowing a variable to enter the model from 4 to some
other value K, click on Options, then type the desired value of K in the Enter box. The
new criterion F statistic, then, is K rather than 4. To change the criterion for deleting a
variable from the model from 4 to some other value K, click on Options, then type the
desired value of K in the Remove box. We must choose K to enter to be greater than or
equal to K to remove.
Though the stepwise selection procedure is a common technique employed by
researchers, other methods are available. Following is a brief discussion of two such
tools. The final model obtained by each of these procedures is the same model that was
found by using the stepwise procedure in Example 11.3.1.
Forward Selection This strategy is closely related to the stepwise regression
procedure. This method builds a model using correlations. Variables are retained that
meet the criteria for inclusion, as in stepwise selection. The first variable entered into
the model is the one with the highest correlation with the dependent variable. If this vari-
able meets the inclusion criterion, it is retained. The next variable to be considered for
inclusion is the one with the highest partial correlation with the dependent variable. If it
11.3 VARIABLE SELECTION PROCEDURES 559
Dialog box: Session command:
Stat ➤ Regression ➤ Stepwise MTB > Stepwise C1 C2–C7;
SUBC> FEnter 4.0;
Type C1 in Response and C2–C7 in Predictors. SUBC> FRemove 4.0.
Stepwise Regression
F-to-Enter: 4.00 F-to-Remove: 4.00
Response is C1 on 6 predictors, with N = 30
Step 1 2
Constant 7.226 31.955
C4 0.888 0.787
T-Ratio 9.74 8.13
C7 -0.45
T-Ratio -2.20
S 5.90 5.53
R-Sq 77.21 80.68
FIGURE 11.3.1 MINITAB stepwise procedure and output for the data of Table 11.3.1.
meets the inclusion criteria, it is retained. This procedure continues until all of the inde-
pendent variables have been considered. The final model contains all of the independent
variables that meet the inclusion criteria.
Backward Elimination This model-building procedure begins with all of the
variables in the model. This strategy also builds a model using correlations and a prede-
termined inclusion criterion based on the F statistic. The first variable considered for
removal from the model is the one with the smallest partial correlation coefficient. If this
variable does not meet the criterion for inclusion, it is eliminated from the model. The
next variable to be considered for elimination is the one with the next lowest partial cor-
relation. It will be eliminated if it fails to meet the criterion for inclusion. This proce-
dure continues until all variables have been considered for elimination. The final model
contains all of the independent variables that meet the inclusion criteria.
EXERCISES
11.3.1 Refer to the data of Exercise 10.3.2 reported by Son et al. (A-6), who studied family caregiving
in Korea of older adults with dementia. The outcome variable, caregiver burden (BURDEN), was
measured by the Korean Burden Inventory (KBI) where scores ranged from 28 to 140 with higher
scores indicating higher burden. Perform a stepwise regression analysis on the following independ-
ent variables reported by the researchers:
CGAGE: caregiver age (years)
CGINCOME: caregiver income (Won-Korean currency)
CGDUR: caregiver-duration of caregiving (month)
ADL: total activities of daily living where low scores indicate the elderly perform activities
independently.
MEM: memory and behavioral problems with higher scores indicating more problems.
COG: cognitive impairment with lower scores indicating a greater degree of cognitive impair-
ment.
SOCIALSU: total score of perceived social support (25–175, higher values indicating more
support). The reported data are as follows.
CGAGE CGINCOME CGDUR ADL MEM COG SOCIALSU BURDEN
41 200 12 39 4 18 119 28
30 120 36 52 33 9 131 68
41 300 60 89 17 3 141 59
35 350 2 57 31 7 150 91
37 600 48 28 35 19 142 70
42 90 4 34 3 25 148 38
49 300 26 42 16 17 172 46
39 500 16 52 6 26 147 57
49 309 30 88 41 13 98 89
560 CHAPTER 11 REGRESSION ANALYSIS: SOME ADDITIONAL TECHNIQUES
(Continued)
CGAGE CGINCOME CGDUR ADL MEM COG SOCIALSU BURDEN
40 250 60 90 24 3 147 48
40 300 36 38 22 13 146 74
70 60 10 83 41 11 97 78
49 450 24 30 9 24 139 43
55 300 18 45 33 14 127 76
27 309 30 47 36 18 132 72
39 250 10 90 17 0 142 61
39 260 12 63 14 16 131 63
44 250 32 34 35 22 141 77
33 200 48 76 33 23 106 85
42 200 12 26 13 18 144 31
52 200 24 68 34 26 119 79
48 300 36 85 28 10 122 92
53 300 12 22 12 16 110 76
40 300 11 82 57 3 121 91
35 200 8 80 51 3 142 78
47 150 60 80 20 18 101 103
33 180 19 81 20 1 117 99
41 200 48 30 7 17 129 73
43 300 36 27 27 27 142 88
25 309 24 72 9 0 137 64
35 250 12 46 15 22 148 52
35 200 6 63 52 13 135 71
45 200 7 45 26 18 144 41
36 300 24 77 57 0 128 85
52 600 60 42 10 19 148 52
41 230 6 60 34 11 141 68
40 200 36 33 14 14 151 57
45 400 96 49 30 15 124 84
48 75 6 89 64 0 105 91
50 200 30 72 31 3 117 83
31 250 30 45 24 19 111 73
33 300 2 73 13 3 146 57
30 200 30 58 16 15 99 69
36 250 6 33 17 21 115 81
45 500 12 34 13 18 119 71
32 300 60 90 42 6 134 91
55 200 24 48 7 23 165 48
50 309 20 47 17 18 101 94
37 250 30 32 13 15 148 57
40 1000 21 63 32 15 132 49
40 300 12 76 50 5 120 88
49 300 18 79 44 11 129 54
37 309 18 48 57 9 133 73
47 250 38 90 33 6 121 87
41 200 60 55 11 20 117 47
EXERCISES 561
(Continued)
CGAGE CGINCOME CGDUR ADL MEM COG SOCIALSU BURDEN
33 1000 18 83 24 11 140 60
28 309 12 50 21 25 117 65
33 400 120 44 31 18 138 57
34 330 18 79 30 20 163 85
40 200 18 24 5 22 157 28
54 200 12 40 20 17 143 40
32 300 32 35 15 27 125 87
44 280 66 55 9 21 161 80
44 350 40 45 28 17 142 49
42 280 24 46 19 17 135 57
44 500 14 37 4 21 137 32
25 600 24 47 29 3 133 52
41 250 84 28 23 21 131 42
28 1000 30 61 8 7 144 49
24 200 12 35 31 26 136 63
65 450 120 68 65 6 169 89
50 200 12 80 29 10 127 67
40 309 12 43 8 13 110 43
47 1000 12 53 14 18 120 47
44 300 24 60 30 16 115 70
37 309 54 63 22 18 101 99
36 300 12 28 9 27 139 53
55 200 12 35 18 14 153 78
45 2000 12 37 33 17 111 112
45 400 14 82 25 13 131 52
23 200 36 88 16 0 139 68
42 1000 12 52 15 0 132 63
38 200 36 30 16 18 147 49
41 230 36 69 49 12 171 42
25 200 30 52 17 20 145 56
47 200 12 59 38 17 140 46
35 100 12 53 22 21 139 72
59 150 60 65 56 2 133 95
49 300 60 90 12 0 145 57
51 200 48 88 42 6 122 88
54 250 6 66 12 23 133 81
53 30 24 60 21 7 107 104
49 100 36 48 14 13 118 88
44 300 48 82 41 13 95 115
36 200 18 88 24 14 100 66
64 200 48 63 49 5 125 92
51 120 2 79 34 3 116 97
43 200 66 71 38 17 124 69
54 150 96 66 48 13 132 112
29 309 19 81 66 1 152 88
Source: Gwi-Ryung Son, R.N., Ph.D. Used with permission.
562 CHAPTER 11 REGRESSION ANALYSIS: SOME ADDITIONAL TECHNIQUES
11.3.2 Machiel Naeije (A-7) identifies variables useful in predicting maximum mouth opening (MMO,
millimeters) for 35 healthy volunteers. The variables examined were:
AGE: years
DOWN_CON: downward condylar translation, mm
FORW_CON: forward condylar translation, mm
Gender: 0 Female, 1 Male
MAN_LENG: mandibular length, mm
MAN_WIDT: mandibular width, mm
Use the following reported measurements to perform a stepwise regression.
AGE DOWN_CON FORW_CON Gender MAN_LENG MAN_WIDT MMO
21.00 4.39 14.18 1 100.86 121.00 52.34
26.00 1.39 20.23 0 93.08 118.29 51.90
30.00 2.42 13.45 1 98.43 130.56 52.80
28.00 .18 19.66 1 102.95 125.34 50.29
21.00 4.10 22.71 1 108.24 125.19 57.79
20.00 4.49 13.94 0 98.34 113.84 49.41
21.00 2.07 19.35 0 95.57 115.41 53.28
19.00 .77 25.65 1 98.86 118.30 59.71
24.00 7.88 18.51 1 98.32 119.20 53.32
18.00 6.06 21.72 0 92.70 111.21 48.53
22.00 9.37 23.21 0 88.89 119.07 51.59
21.00 3.77 23.02 1 104.06 127.34 58.52
20.00 1.10 19.59 0 98.18 111.24 62.93
22.00 2.52 16.64 0 91.01 113.81 57.62
24.00 5.99 17.38 1 96.98 114.94 65.64
22.00 5.28 22.57 0 97.86 111.58 52.85
22.00 1.25 20.89 0 96.89 115.16 64.43
22.00 6.02 20.38 1 98.35 122.52 57.25
19.00 1.59 21.63 0 90.65 118.71 50.82
26.00 6.05 10.59 0 92.99 119.10 40.48
22.00 1.51 20.03 1 108.97 129.00 59.68
24.00 .41 24.55 0 91.85 100.77 54.35
21.00 6.75 14.67 1 104.30 127.15 47.00
22.00 4.87 17.91 1 93.16 123.10 47.23
22.00 .64 17.60 1 94.18 113.86 41.19
29.00 7.18 15.19 0 89.56 110.56 42.76
25.00 6.57 17.25 1 105.85 140.03 51.88
20.00 1.51 18.01 0 89.29 121.70 42.77
27.00 4.64 19.36 0 92.58 128.01 52.34
26.00 3.58 16.57 1 98.64 129.00 50.45
23.00 6.64 12.47 0 83.70 130.98 43.18
25.00 7.61 18.52 0 88.46 124.97 41.99
22.00 5.39 11.66 1 94.93 129.99 39.45
31.00 5.47 12.85 1 96.81 132.97 38.91
23.00 2.60 19.29 0 93.13 121.03 49.10
Source: Machiel Naeije, D.D.S. Used with permission.
EXERCISES 563
11.3.3 One purpose of a study by Connor et al. (A-8) was to examine reactive aggression among chil-
dren and adolescents referred to a residential treatment center. The researchers used the Proactive/
Reactive Rating Scale, obtained by presenting three statements to clinicians who examined the
subjects. The respondents answered, using a scale from 1 to 5, with 5 indicating that the state-
ment almost always applied to the child. An example of a reactive aggression statement is,
“When this child has been teased or threatened, he or she gets angry easily and strikes back.”
The reactive score was the average response to three statements of this type. With this variable
as the outcome variable, researchers also examined the following: AGE (years), VERBALIQ
(verbal IQ), STIM (stimulant use), AGEABUSE (age when first abused), CTQ (a measure of
hyperactivity in which higher scores indicate higher hyperactivity), TOTALHOS (total hostility
as measured by an evaluator, with higher numbers indicating higher hostility). Perform stepwise
regression to find the variables most useful in predicting reactive aggression in the following
sample of 68 subjects.
REACTIVE AGE VERBALIQ STIM AGEABUSE CTQ TOTALHOS
4.0 17 91 0 0 0 8
3.7 12 94 0 1 29 10
2.3 14 105 0 1 12 10
5.0 16 97 0 1 9 11
2.0 15 97 0 2 17 10
2.7 8 91 0 0 6 4
2.0 10 111 0 0 6 6
3.3 12 105 0 0 28 7
2.0 17 101 1 0 12 9
4.3 13 102 1 1 8 11
4.7 15 83 0 0 9 9
4.3 15 66 0 1 5 8
2.0 15 90 0 2 3 8
4.0 13 88 0 1 28 8
2.7 13 98 0 1 17 10
2.7 9 135 0 0 30 11
2.7 18 72 0 0 10 9
2.0 13 93 0 2 20 8
3.0 14 94 0 2 10 11
2.7 13 93 0 1 4 8
3.7 16 73 0 0 11 11
2.7 12 74 0 1 10 7
2.3 14 97 0 2 3 11
4.0 13 91 1 1 21 11
4.0 12 88 0 1 14 9
4.3 13 90 0 0 15 2
3.7 14 104 1 1 10 10
3.0 18 82 0 0 1 7
4.3 14 79 1 3 6 7
1.0 16 93 0 0 5 8
4.3 16 99 0 1 21 11
564 CHAPTER 11 REGRESSION ANALYSIS: SOME ADDITIONAL TECHNIQUES
(Continued)
REACTIVE AGE VERBALIQ STIM AGEABUSE CTQ TOTALHOS
2.3 14 73 0 2 8 9
3.0 12 112 0 0 15 9
1.3 15 102 0 1 1 5
3.0 16 78 1 1 26 8
2.3 9 95 1 0 23 10
1.0 15 124 0 3 0 11
3.0 17 73 0 1 1 10
3.3 11 105 0 0 23 5
4.0 11 89 0 0 27 8
1.7 9 88 0 1 2 8
2.3 16 96 0 1 5 7
4.7 15 76 1 1 17 9
1.7 16 87 0 2 0 4
1.7 15 90 0 1 10 12
4.0 12 76 0 0 22 10
5.0 12 83 1 1 19 7
4.3 10 88 1 0 10 5
5.0 9 98 1 0 8 9
3.7 12 100 0 0 6 4
3.3 14 80 0 1 3 10
2.3 16 84 0 1 3 9
1.0 17 117 0 2 1 9
1.7 12 145 1 0 0 5
3.7 12 123 0 0 1 3
2.0 16 94 0 2 6 6
3.7 17 70 0 1 11 13
4.3 14 113 0 0 8 8
2.0 12 123 1 0 2 8
3.0 7 107 0 0 11 9
3.7 12 78 1 0 15 11
4.3 14 73 0 1 2 8
2.3 18 91 0 3 8 10
4.7 12 91 0 0 6 9
3.7 15 111 0 0 2 9
1.3 15 71 0 1 20 10
3.7 7 102 0 0 14 9
1.7 9 89 0 0 24 6
Source: Daniel F. Connor, M.D. and Lang Lin. Used with permission.
11.4 LOGISTIC REGRESSION
Up to now our discussion of regression analysis has been limited to those situations
in which the dependent variable is a continuous variable such as weight, blood pres-
sure, or plasma levels of some hormone. Much research in the health sciences field
11.4 LOGISTIC REGRESSION 565
is motivated by a desire to describe, understand, and make use of the relationship
between independent variables and a dependent (or outcome) variable that is discrete.
Particularly plentiful are circumstances in which the outcome variable is dichotomous.
A dichotomous variable, we recall, is a variable that can assume only one of two
mutually exclusive values. These values are usually coded for a success and
for a nonsuccess, or failure. Dichotomous variables include those whose two
possible values are such categories as died–did not die; cured–not cured; disease
occurred–disease did not occur; and smoker–nonsmoker. The health sciences profes-
sional who either engages in research or needs to understand the results of research
conducted by others will find it advantageous to have, at least, a basic understanding
of logistic regression, the type of regression analysis that is usually employed when
the dependent variable is dichotomous. The purpose of the present discussion is to
provide the reader with this level of understanding. We shall limit our presentation to
the case in which there is only one independent variable that may be either continu-
ous or dichotomous.
The Logistic Regression Model Recall that in Chapter 9 we referred to
regression analysis involving only two variables as simple linear regression analysis. The
simple linear regression model was expressed by the equation
(11.4.1)
in which y is an arbitrary observed value of the continuous dependent variable. When
the observed value of Y is the mean of a subpopulation of Y values for a given value
of X, the quantity , the difference between the observed Y and the regression line (see
Figure 9.2.1) is zero, and we may write Equation 11.4.1 as
(11.4.2)
which may also be written as
(11.4.3)
Generally the right-hand side of Equations 11.4.1 through 11.4.3 may assume any value
between minus infinity and plus infinity.
Even though only two variables are involved, the simple linear regression model
is not appropriate when Y is a dichotomous variable because the expected value (or mean)
of Y is the probability that and, therefore, is limited to the range 0 through 1,
inclusive. Equations 11.4.1 through 11.4.3, then, are incompatible with the reality of the
situation.
If we let then the ratio can take on values between 0 and
plus infinity. Furthermore, the natural logarithm (ln) of can take on valuesp>11 - p2
p>11 - p2p = P1Y = 12,
Y = 1
E
1y
ƒ
x2
= b
0
+ b
1
x
m
y
ƒ
x
= b
0
+ b
1
x
P
m
y
ƒ
x
,
y = b
0
+ b
1
x +P
Y = 0
Y = 1
566 CHAPTER 11 REGRESSION ANALYSIS: SOME ADDITIONAL TECHNIQUES