negative binomial model with interaction in R - r

Question

i have fit negative binomial model to my data as follows:
> ngbinmodel <- glm.nb( seizure.rate ~ age + treatment, data = epilepsy_reduced)
> summary(ngbinmodel)
Call:
glm.nb(formula = seizure.rate ~ age + treatment, data = epilepsy_reduced,
init.theta = 1.498983674, link = log)
Deviance Residuals:
Min 1Q Median 3Q Max
-2.3510 -0.8790 -0.4563 0.4328 1.8916
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 2.0985089 0.5845392 3.590 0.000331 ***
age -0.0007965 0.0193064 -0.041 0.967092
treatment -0.5011593 0.2405658 -2.083 0.037228 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for Negative Binomial(1.499) family taken to be 1)
Null deviance: 71.217 on 57 degrees of freedom
Residual deviance: 66.875 on 55 degrees of freedom
AIC: 341.12
Number of Fisher Scoring iterations: 1
Theta: 1.499
Std. Err.: 0.362
2 x log-likelihood: -333.119
Now I would like to check if i should include the interaction effect between age and treatment. I have found two methods to do it:
> intearaction_nbm<-addterm(ngbinmodel, . ~ . * age,test="Chisq")
> summary(intearaction_nbm)
Df AIC LRT Pr(Chi)
Min. :1 Min. :339.1 Min. :0.9383 Min. :0.3327
1st Qu.:1 1st Qu.:339.4 1st Qu.:0.9383 1st Qu.:0.3327
Median :1 Median :339.6 Median :0.9383 Median :0.3327
Mean :1 Mean :339.6 Mean :0.9383 Mean :0.3327
3rd Qu.:1 3rd Qu.:339.9 3rd Qu.:0.9383 3rd Qu.:0.3327
Max. :1 Max. :340.2 Max. :0.9383 Max. :0.3327
NA's :1 NA's :1 NA's :1
and
> ngbinmodel_int <- glm.nb( seizure.rate ~ age*treatment, data = epilepsy_reduced)
> summary(ngbinmodel_int)
glm.nb(formula = seizure.rate ~ age * treatment, data = epilepsy_reduced,
init.theta = 1.531539174, link = log)
Deviance Residuals:
Min 1Q Median 3Q Max
-2.3503 -0.8742 -0.3848 0.3403 1.8508
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.51361 0.83920 1.804 0.0713 .
age 0.01914 0.02826 0.677 0.4981
treatment 0.60748 1.12199 0.541 0.5882
age:treatment -0.03893 0.03850 -1.011 0.3119
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for Negative Binomial(1.5315) family taken to be 1)
Null deviance: 72.238 on 57 degrees of freedom
Residual deviance: 66.874 on 54 degrees of freedom
AIC: 342.18
Number of Fisher Scoring iterations: 1
Theta: 1.532
Std. Err.: 0.373
2 x log-likelihood: -332.180
I was expecting to obtain the same result from both of the methods.
How can i access the regression estimates of intearaction_nbm?
why are the outcomes different? According to intearaction_nbm i should include the interaction term (the AIC is lower) but according to ngbinmodel_int i should not include the interaction term (AIC increases).
would discretizing my continuous variable age be advised?

Answer 1

Remark: You should move this post to cross validated.
How can i access the regression estimates of intearaction_nbm?
intearaction_nbm gives you the result of the addition of single term to your model, if you print it, you will have a row per possible additional term (age:treatment, age:another_variable, etc.) giving you the AIC and P-value among other things.
why are the outcomes different?
Not possible to answer without the data, but what I would do is define both models and compare their AIC using: AIC(model_1, model_2). This way I am sure that I am comparing the same quantity. As you know, the AIC is defined up to an additional term, and unless you check how it is computed, you cannot be sure that two different functions in two different packages use the same definition.
would discretizing my continuous variable age be advised?
Not possible to answer without the data...

Answer 2

Let us consider the dataset quine and the following model with only main effects for Eth and Lrn factors:
library(MASS)
negbin_no_int <- glm.nb(Days ~ Eth + Lrn, data = quine)
summary(negbin_no_int)
# Coefficients:
# Estimate Std. Error z value Pr(>|z|)
# (Intercept) 3.0367 0.1334 22.764 < 2e-16 ***
# EthN -0.5520 0.1597 -3.457 0.000546 ***
# LrnSL 0.0388 0.1611 0.241 0.809661
extractAIC(negbin_no_int)
# [1] 3.000 1112.576
The model with the interaction term between the two factor is:
negbin_with_int <- glm.nb(Days ~ Eth * Lrn, data = quine)
summary(negbin_with_int)
# Coefficients:
# Estimate Std. Error z value Pr(>|z|)
# (Intercept) 2.9218 0.1503 19.446 <2e-16 ***
# EthN -0.3374 0.2100 -1.607 0.108
# LrnSL 0.2929 0.2307 1.269 0.204
# EthN:LrnSL -0.4956 0.3201 -1.549 0.122
extractAIC(negbin_with_int)
# [1] 4.000 1112.196
The statistical significance of the interaction term is p=0.122.
Now we compare the two models using addterm:
interaction_nbm <- addterm(negbin_no_int, . ~ . + Eth:Lrn, test="Chisq")
print(interaction_nbm)
# Model:
# Days ~ Eth + Lrn
# Df AIC LRT Pr(Chi)
# <none> 1112.6
# Eth:Lrn 1 1112.2 2.3804 0.1229
The AICs given by addterm are the same calculated using extractAIC.
If you want to see the regression estimates of addterm, you can add a summary(print(nfit)) inside the function, as follows:
myaddterm <- function (object, scope, scale = 0, test = c("none", "Chisq"),
k = 2, sorted = FALSE, trace = FALSE, ...)
{
if (missing(scope) || is.null(scope))
stop("no terms in scope")
if (!is.character(scope))
scope <- add.scope(object, update.formula(object, scope))
if (!length(scope))
stop("no terms in scope for adding to object")
ns <- length(scope)
ans <- matrix(nrow = ns + 1L, ncol = 2L, dimnames = list(c("<none>",
scope), c("df", "AIC")))
ans[1L, ] <- extractAIC(object, scale, k = k, ...)
n0 <- nobs(object, use.fallback = TRUE)
env <- environment(formula(object))
for (i in seq_len(ns)) {
tt <- scope[i]
if (trace) {
message(gettextf("trying + %s", tt), domain = NA)
utils::flush.console()
}
nfit <- update(object, as.formula(paste("~ . +", tt)),
evaluate = FALSE)
nfit <- try(eval(nfit, envir = env), silent = TRUE)
print(summary(nfit))
ans[i + 1L, ] <- if (!inherits(nfit, "try-error")) {
nnew <- nobs(nfit, use.fallback = TRUE)
if (all(is.finite(c(n0, nnew))) && nnew != n0)
stop("number of rows in use has changed: remove missing values?")
extractAIC(nfit, scale, k = k, ...)
}
else NA_real_
}
dfs <- ans[, 1L] - ans[1L, 1L]
dfs[1L] <- NA
aod <- data.frame(Df = dfs, AIC = ans[, 2L])
o <- if (sorted)
order(aod$AIC)
else seq_along(aod$AIC)
test <- match.arg(test)
if (test == "Chisq") {
dev <- ans[, 2L] - k * ans[, 1L]
dev <- dev[1L] - dev
dev[1L] <- NA
nas <- !is.na(dev)
P <- dev
P[nas] <- MASS:::safe_pchisq(dev[nas], dfs[nas], lower.tail = FALSE)
aod[, c("LRT", "Pr(Chi)")] <- list(dev, P)
}
aod <- aod[o, ]
head <- c("Single term additions", "\nModel:", deparse(formula(object)))
if (scale > 0)
head <- c(head, paste("\nscale: ", format(scale), "\n"))
class(aod) <- c("anova", "data.frame")
attr(aod, "heading") <- head
aod
}
interaction_nbm1 <- myaddterm(negbin_no_int, . ~ . + Eth:Lrn, test="Chisq")
The output is:
Call:
glm.nb(formula = Days ~ Eth + Lrn + Eth:Lrn, data = quine, init.theta = 1.177546225,
link = log)
Deviance Residuals:
Min 1Q Median 3Q Max
-2.5770 -1.0470 -0.3645 0.3521 2.7227
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 2.9218 0.1503 19.446 <2e-16 ***
EthN -0.3374 0.2100 -1.607 0.108
LrnSL 0.2929 0.2307 1.269 0.204
EthN:LrnSL -0.4956 0.3201 -1.549 0.122
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for Negative Binomial(1.1775) family taken to be 1)
Null deviance: 182.93 on 145 degrees of freedom
Residual deviance: 168.18 on 142 degrees of freedom
AIC: 1114.2
Number of Fisher Scoring iterations: 1
Theta: 1.178
Std. Err.: 0.146
2 x log-likelihood: -1104.196