difference between lrm's "Model L.R." and anova's "Chi-Square"

I am running lrm() with a single factor. I then run anova() on the fitted
model to obtain a p-value associated with having that factor in the model.

I am noticing that the "Model L.R." in the lrm results is almost the same
as the "Chi-Square" in the anova results, but not quite; the latter value
is always slightly smaller.

anova() calculates the p-value based on "Chi-Square", but I have
independent evidence that "Model L.R." is the actual -2*log(LR), so should
I be using that?

Why are the values different?

prob_a <- inv.logit(rnorm(1,0,1))
prob_b <- inv.logit(rnorm(1,0,1))
data <- data.frame(
factor=c(rep("a",500),rep("b",500)),
outcome=c(sample(c(1,0),100,replace=T,prob=c(prob_a,1-prob_a)),
          sample(c(1,0),100,replace=T,prob=c(prob_b,1-prob_b))))
fit <- lrm(outcome~factor,data)

fit           # gives "Model L.R." e.g. 8.23, 11.76, 6.89...
anova(fit)    # gives "Chi-Square" e.g. 8.19, 11.69, 6.85...

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anova (anova.Design) computes Wald statistics.  When the log-likelihood
is very quadratic, these statistics will be very close to log-likelihood
ratio chi-square statistics.  In general LR chi-square tests are better;
we use Wald tests for speed.  It's best to take the time and do
lrtest(fit1,fit2) in Design, where one of the two fits is a subset of
the other.

Frank Harrell

Quoting Frank E Harrell Jr <f.harrell at vanderbilt.edu>:

anova (anova.Design) computes Wald statistics.  When the log-likelihood
is very quadratic, these statistics will be very close to log-likelihood
ratio chi-square statistics.  In general LR chi-square tests are better;
we use Wald tests for speed.  It's best to take the time and do
lrtest(fit1,fit2) in Design, where one of the two fits is a subset of
the other.

Frank Harrell

Thanks, this is great, but in my case, there's just one factor,

fit1 <- lrm(outcome~factor,data)

and I'm having trouble constructing the subset 'null model', as e.g.

fit2 <- lrm(outcome~1,data)

returns an error message.

How do I construct a null model with lrm() so that I can use lrtest() to test a
model with only one predictor?

I apologize for asking what must be a very simple question but I have been
unable to find the answer by searching R-help.

Thanks,
Dan

P.S. Second point: I have another case where I use lmer(), and there the null
model includes a random effect so I don't get the problem above. It looks like
with lmer objects anova() uses LLR, not Wald. Is that right?