Goodness-of-fit for fixed effect logit model using 'bife' package

Question

I am using the 'bife' package to run the fixed effect logit model in R. However, I cannot compute any goodness-of-fit to measure the model's overall fit given the result I have below. I would appreciate if I can know how to measure the goodness-of-fit given this limited information. I prefer chi-square test but still cannot find a way to implement this either.

    ---------------------------------------------------------------                 
    Fixed effects logit model                   
    with analytical bias-correction                 

    Estimated model:                    
    Y ~ X1 +X2 + X3 + X4 + X5 | Z                   

    Log-Likelihood= -9153.165                   
    n= 20383, number of events= 5104                    
    Demeaning converged after 6 iteration(s)                    
    Offset converged after 3 iteration(s)                   

    Corrected structural parameter(s):                  

        Estimate    Std. error  t-value Pr(> t) 
    X1  -8.67E-02   2.80E-03    -31.001 < 2e-16 ***
    X2  1.79E+00    8.49E-02    21.084  < 2e-16 ***
    X3  -1.14E-01   1.91E-02    -5.982  2.24E-09    ***
    X4  -2.41E-04   2.37E-05    -10.171 < 2e-16 ***
    X5  1.24E-01    3.33E-03    37.37   < 2e-16 ***
    ---                 
    Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1                  

    AIC=  18730.33 , BIC=  20409.89                     


    Average individual fixed effects= 1.6716                    
    ---------------------------------------------------------------                 

Answer 1

Let the DGP be

n <- 1000
x <- rnorm(n)
id <- rep(1:2, each = n / 2)
y <- 1 * (rnorm(n) > 0)

so that we will be under the null hypothesis. As it says in ?bife, when there is no bias-correction, everything is the same as with glm, except for the speed. So let's start with glm.

modGLM <- glm(y ~ 1 + x + factor(id), family = binomial())
modGLM0 <- glm(y ~ 1, family = binomial())

One way to perform the LR test is with

library(lmtest)
lrtest(modGLM0, modGLM)
# Likelihood ratio test
#
# Model 1: y ~ 1
# Model 2: y ~ 1 + x + factor(id)
#   #Df  LogLik Df  Chisq Pr(>Chisq)
# 1   1 -692.70                     
# 2   3 -692.29  2 0.8063     0.6682

But we may also do it manually,

1 - pchisq(c((-2 * logLik(modGLM0)) - (-2 * logLik(modGLM))),
           modGLM0$df.residual - modGLM$df.residual)
# [1] 0.6682207

Now let's proceed with bife.

library(bife)
modBife <- bife(y ~ x | id)
modBife0 <- bife(y ~ 1 | id)

Here modBife is the full specification and modBife0 is only with fixed effects. For convenience, let

logLik.bife <- function(object, ...) object$logl_info$loglik

for loglikelihood extraction. Then we may compare modBife0 with modBife as in

1 - pchisq((-2 * logLik(modBife0)) - (-2 * logLik(modBife)), length(modBife$par$beta))
# [1] 1

while modGLM0 and modBife can be compared by running

1 - pchisq(c((-2 * logLik(modGLM0)) - (-2 * logLik(modBife))), 
           length(modBife$par$beta) + length(unique(id)) - 1)
# [1] 0.6682207

which gives the same result as before, even though with bife we, by default, have bias correction.

Lastly, as a bonus, we may simulate data and see it the test works as it's supposed to. 1000 iterations below show that both test (since two tests are the same) indeed reject as often as they are supposed to under the null.