# Bnomial (Logit) regression for proportion/percentage data

I have run a binomial (logit) regression on some proportion data as the dependent variable in an Interrupted Time Seies Analysis [see below]:

rrfit2a <- glm(Subject Refused Ratio ~ Quarter + int2 +
time_since_intervention2 , df,


Summary outcome:

Call:
glm(formula = Subject Refused Ratio ~ Quarter + int2 +
time_since_intervention2,
family = binomial(link = "logit"), data = df)

Deviance Residuals:
Min        1Q    Median        3Q       Max
-0.82923  -0.22180  -0.01419   0.20225   0.55371

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept)              -0.53235    1.10630  -0.481    0.630
Quarter                  -0.02561    0.11651  -0.220    0.826
int2                      0.90200    1.87742   0.480    0.631
time_since_intervention2  0.05982    0.33073   0.181    0.856

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 3.5315  on 23  degrees of freedom
Residual deviance: 2.5198  on 20  degrees of freedom
AIC: 34.374

Number of Fisher Scoring iterations: 4


I want to report confidence intervals for the model and currently do so using the margins package:

 summary(margins(rrfit2a))
factor     AME     SE       z      p   lower  upper
int2  0.2056 0.4201  0.4893 0.6246 -0.6178 1.0289
Quarter -0.0058 0.0265 -0.2205 0.8254 -0.0577 0.0460
time_since_intervention2  0.0136 0.0752  0.1813 0.8561 -0.1337 0.1610


Confidence intervals suggest in excess of 1 in some instances - which I don't think can be right. Maybe I'm misunderstanding the model or outcome or exponentiation?

However, I found what appear to be much more "realistic" confidence intervals using a quasibinomial.

rrfit1a <- glm(Subject Refused Ratio ~ Quarter + int2 +
time_since_intervention2 , df, family = "quasibinomial")

Call:
glm(formula = Subject Refused Ratio ~ Quarter + int2 + time_since_intervention2,
family = "quasibinomial", data = df)

Deviance Residuals:
Min        1Q    Median        3Q       Max
-0.82923  -0.22180  -0.01419   0.20225   0.55371

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)              -0.53235    0.36948  -1.441    0.165
Quarter                  -0.02561    0.03891  -0.658    0.518
int2                      0.90200    0.62701   1.439    0.166
time_since_intervention2  0.05982    0.11045   0.542    0.594

(Dispersion parameter for quasibinomial family taken to be 0.11154)

Null deviance: 3.5315  on 23  degrees of freedom
Residual deviance: 2.5198  on 20  degrees of freedom
AIC: NA

Number of Fisher Scoring iterations: 4


The quasibinomial fits the model equally, but provides much lower confidence intervals.

 summary(margins(rrfit1a))
factor     AME     SE       z      p   lower  upper
int2  0.2056 0.1403  1.4651 0.1429 -0.0694 0.4805
Quarter -0.0058 0.0088 -0.6604 0.5090 -0.0232 0.0115
time_since_intervention2  0.0136 0.0251  0.5430 0.5871 -0.0356 0.0628


There did not appear to be overdispersion in the original binomial (logit).

Basically I want to know if it would be wrong of me to use the quasibinomial? Are the lower confidence intervals potentially less accurate than the original binomial (logit) or does it just better account for the variance? Is there anything wrong with using the quasibinomial on proporion/percentage data if there is no overdispersion?

• In R, try the confint function on the glm objects, which will make likelihood profile intervals. You need to load the package MASS Nov 17 at 2:21
• What is your sample size? The output from the quasibinomial model could indicate underdispersion Nov 17 at 2:24
• @kjetil b halvorsen We have a small sample of only 24 observations. Recorded dispersion for both the binomial (logit) and quasibinomial were identical at 0.01111 - I guess both showing underdispersion. Nov 17 at 6:45
• Please add new information in comments as edit to the post. We want posts to be self-contained, and comments are easily overlooked. But 22 obs are very little for logistic regression Nov 17 at 11:52
• Have you considered a Poisson family with an offset? Nov 18 at 19:26