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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, 
                family = "binomial"(link='logit'))

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?

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  • $\begingroup$ In R, try the confint function on the glm objects, which will make likelihood profile intervals. You need to load the package MASS $\endgroup$ Nov 17 at 2:21
  • $\begingroup$ What is your sample size? The output from the quasibinomial model could indicate underdispersion $\endgroup$ Nov 17 at 2:24
  • $\begingroup$ @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. $\endgroup$
    – j.rahilly
    Nov 17 at 6:45
  • $\begingroup$ 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 $\endgroup$ Nov 17 at 11:52
  • $\begingroup$ Have you considered a Poisson family with an offset? $\endgroup$
    – Bryan
    Nov 18 at 19:26

1 Answer 1

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First, only 24 observation (as given in a comment, not in the post itself) is woefully little for logistic regression, so you should not be surprised for the wide confidence intervals. More information about sample size requirements for logistic regression: Sample size for logistic regression?

The reason for the shorter confidence intervals from the quasibinomial model is simple:

(Dispersion parameter for quasibinomial family taken to be 0.11154)

which indicates underdispersion, which is quite unusual. You should compare the lengths of the confidence intervals, and compare the ratio of the lengths to the estimated dispersion parameter.

But first of all, ask yourself what could have caused the underdispersion. This can be caused by competition, which makes for dependence, like in counts of territorial birds where I have seen underdispersion. Since you have a time series, maybe there is negative autocorrelation. But it might also be an artifact of small sample size.

You didn't tell us enough contextual information to help there ...

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