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OTE/SPH
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August 31, 2006
JWBK119-12
Logistic Regression Approach 187
Table 12.9 Logistic regression table for model with single categorical explanatory variable.
95% CI of odds ratio
Odds
Predictors Coefficient SE Coeff. Z p-value ratio Lower Upper
Constant ˆ α S = 1.87 0.76 2.46 0.014

Dummy
Spine variable
Condition combinations

c S1 c S2
2 1 0 ˆ β S1 =−2.28 1.19 −1.92 0.055 0.10 0.01 1.05
3 0 1 ˆ β S2 =−1.74 0.84 −2.06 0.039 0.18 0.03 0.92

1, 2 and 3 can thus be evaluated by considering the three combinations of (c S1 , c S2 )at
(0,0), (1,0) and (0,1) respectively. The linear functions for the log odds evaluated for
each spine condition are listed in Table 12.10.
The odds ratios evaluated at each setting of the categorical variable for spine con-
ditions 2 and 3 shown in Table 12.9 are odds ratios in comparison with the case of the
horseshoe crab having spine condition 1. The odds ratio is evaluated by taking the
antilog of the logit functions shown in Table 12.10. The odds ratios corresponding to
spine condition 2 and 3 are respectively

ˆ
ˆ
exp α S + ˆ β S1 − ˆα and exp α S + ˆ β S2 − ˆα S .
These odds ratios compare the relative differences in odds of finding satellite crabs
for female horseshoe crabs with different spine conditions against a reference spine
condition, which is spine condition 1 in this case. The confidence intervals for these
odds ratio can be approximated using the asymptotic standard error of the parameter
estimates. Using MINITAB, these confidence intervals are evaluated as (0.01, 1.05)
and (0.03, 0.92) for the odds ratio spine conditions 2 and 3 respectively. From these
confidenceintervals,thereappearstobenosignificantdifferenceintheoddsoffinding
satellite crabs for female horseshoe crabs having spine conditions 1 and 2. However,
the estimated odds of finding satellite crabs for female horseshoe crabs with spine
condition 3 appears to be at most 0.92 times different from that of horseshoe crabs
with spine condition 1.

Table 12.10 Logit funtions relevant to each spine condition.

Dummy variable
combinations

π Pres (c S1 ,c S2 )
Spine Condition c S1 c S2 log
1−π (c S1 ,c S2 )
Pres
1 0 0 ˆ α S
2 1 0 ˆ α S + ˆ β S1
3 0 1 ˆ α S + ˆ β S2