[R] Profile confidence intervals and LR chi-square test

[Inman, Brant A. M.D.]

System: R 2.3.1 on Windows XP machine.

I am building a logistic regression model for a sample of 100 cases in
dataframe "d", in which there are 3 binary covariates: x1, x2 and x3.

----------------

> summary(d)
 y      x1     x2     x3    
 0:54   0:50   0:64   0:78  
 1:46   1:50   1:36   1:22  

> fit <- glm(y ~ x1 + x2 + x3, data=d, family=binomial(link=logit))

> summary(fit)

Call:
glm(formula = y ~ x1 + x2 + x3, family = binomial(link = logit), 
    data = d)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-1.6503  -1.0220  -0.7284   0.9965   1.7069  

Coefficients:
            Estimate Std. Error z value Pr(>|z|)  
(Intercept)  -0.3772     0.3721  -1.014   0.3107  
x11          -0.8144     0.4422  -1.842   0.0655 .
x21           0.9226     0.4609   2.002   0.0453 *
x31           1.3347     0.5576   2.394   0.0167 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 137.99  on 99  degrees of freedom
Residual deviance: 120.65  on 96  degrees of freedom
AIC: 128.65

Number of Fisher Scoring iterations: 4

> exp(fit$coef)
(Intercept)         x11         x21         x31 
  0.6858006   0.4429233   2.5157321   3.7989873 
---------------

After reading the appropriate sections in MASS4 (7.2 and 8.4 in
particular), I decided to estimate the 95% confidence intervals for the
odds ratios using the profile method implemented in the "confint"
function. I then used the "anova" function to perform the deviance
chi-square tests for each covariate.

---------------
> ci <- confint(fit); exp(ci)
Waiting for profiling to be done...
                2.5 %    97.5 %
(Intercept) 0.3246680  1.413684
x11         0.1834819  1.048154
x21         1.0256096  6.314473
x31         1.3221533 12.129210

> anova(fit, test='Chisq')
Analysis of Deviance Table

Model: binomial, link: logit

Response: y

Terms added sequentially (first to last)


     Df Deviance Resid. Df Resid. Dev P(>|Chi|)
NULL                    99    137.989          
x1    1    5.856        98    132.133     0.016
x2    1    5.271        97    126.862     0.022
x3    1    6.212        96    120.650     0.013
----------------

My question relates to the interpretation of the significance of
variable x1.  The OR for x1 is 0.443 and its profile confidence interval
is 0.183-1.048.  If a type I error rate of 5% is assumed, this result
would tend to suggest that x1 is NOT a significant predictor of y.
However, the deviance chi-square test has a P-value of 0.016, which
suggests that x1 is indeed a significant predictor of y. How do I
reconcile these two differing messages? I do recognize that the upper
bound of the confidence interval is pretty close to 1, but I am certain
that some journal reviewer will point out the problem as inconsistent.

Brant Inman