This may be a stupid question but why are you comparing against a normal if your response is binary? I would try logistic regression. I assume your DV is smoking(yes,no). If I assumed correctly try that. If not, ask a different question. Good luck, David

Initial response is what David said, though usually I suggest providing more information about the existing question.

Logit transformation?

Discriminant analysis?

Probit analysis?

Question: Do you have binary data (yes/no) or do you have percentages? I surveyed 50 universities and smoking at the first university was 38%, at another it was 27.32%, and so forth. I had data like the latter case where I measured disease presence in an insect at seven field locations. Step 1 in the analysis was to calculate prevalence. I then used those data for all other analyses. Just because you have binary data does not mean that it must be used as binary data if the experimental design and analysis methodologies support converting the binary data to prevalence as the first step.

Firstly, thanks to all for answers.

I also have same question that David done: why normal residuals if distribution is binomial? I do not know why, but it is used by several people to diagnose models. So, my question also is: is this graphic important? Perhaps this is my main question in this topic.

A have done other adjustment tests, mathematical tests, and they are saying me that model is well adjusted. Hosmer and Lemeshow, ROC curve and pseudo R² de Nagelkerke shows the model is good. So... Must I ignore this graphic and go ahead?

I do not want to use logistic regression because my prevalence of smokers is high for using an odds ratio (OR) as link function. It is related in some articles that OR super-estimates associations:

BARROS, A. J.; HIRAKATA, V. N. Alternatives for logistic regression in cross-sectional studies: an empirical comparison of models that directly estimate the prevalence ratio. BMC Medical Research Methodology, v. 3, n. 1, p. 21, 20 dez. 2003.

COUTINHO, L. M. S.; SCAZUFCA, M.; MENEZES, P. R. Métodos para estimar razão de prevalência em estudos de corte transversal. Revista de Saúde Pública, v. 42, n. 6, p. 992–998, dez. 2008.

Poisson, as these references shows, can be used to estimate prevalence ratios (PR).

My variable is binary (Y/N). So, it can be explained as a percentage: 38.11% of students, YES, are smokers. Modeling would explain if this percentage is significantly greater in males, for example.

How could I use probit link function? Is this applicable in glm models? Which family should I use, so? 

Marcos, when we experience over dispersion cases, such adjustment can be made using negative binomial distribution when running regression analysis! try it and let us know the outcome

agreed with David and  Marcos. better to use logistic or probit regression model but you have to think about your sample size. it should be sufficient. 

Though I agree that a modified Poisson approach would result in a better estimate of risk than the odds ratio, I'm a little curious to know why you believe you have overdispersion in your outcome variable (e.g. smoking). It could simply be that there is a relatively high prevalence of smoking among high school students.

Thank all for answers, again,

Yes, Vincent, I should use negative binomial distribution IF my data was over-dispersed. But this is not my case: my data are SUB-dispersed. So I noticed that quasi-Poisson would be the better, according to my readings. By the way, I had tried also negative-binomial, and graphic was the same.

I still do not know how could I use probit link function and use coefficients as a validated epidemiological indicator. I would be glad if someone could suggest some references about it. As I have informed, and I gave some references about it, logistic regression is not a good idea.

About Chad comments, I firstly describe my outcome variable. I observe mean and variance and it can already say something about equidispersion. After modeling, I apply a chi-square test to compare residual deviance versus degree of freedom. In software R, this is made as follows:

> with(MODEL, cbind(res.deviance = deviance, df = df.residual,

p = pchisq(deviance, df.residual, lower.tail=FALSE)))

where:

MODEL is a glm model fitted.

If p-value is over 0.05, than we have equi-dispersion, and Poisson could be used. If it is < 0.05, so we can have sub or over-dispersion, and descriptive results about outcome variable is significantly different.

Some other colleagues suggested me that qq-norm graphic can be not too important. So, I am going ahead with quasi-Poisson model.

The place I encounter probit analysis is for modeling insect mortality. Insects are treated with different doses of insecticide and the number alive and number dead are recorded. One then models the proportion that die as a function of dose. One goal in designing this sort of experiment is to have treatments that cover the range in mortality from 100% to 0%.

In reading more of the discussion, I am not certain that probit analysis will work in this case.

I have never heard of a 35% prevalence being somehow "too high" for logistic regression.  I have used it for modeling very low or very high prevalences.  It sounds like you may be getting your advice from people who don't know what they're talking about.  Unfortunately, there is a lot of that in the social sciences.

After reading this thread again, I don't think Marcos's problem is with the "high" prevalence of smoking but with his model not converging. Using a modified Poisson approach (I think you may be referring to it as "quasi Poisson") would help you if the non-convergence is due to low numbers after adjustment. See Zou, 2004 (https://www.ncbi.nlm.nih.gov/pubmed/15033648) for more details on the model methodology. However, non-convergence could also indicate that you are over-adjusting your model so be careful.

Thanks again for answers.

About Robertson's commentary, I have sent above some references about what is a "high prevalence" for logistic regression. It is showed that prevalence of outcome >10% is sufficient to take odds ratios much higher than prevalence ratios.

About Logan's answer, I can say: YES, I am following Zou's considerations. But I am modifying some aspects. I am using robust variance to calculate better prevalence ratios confidence intervals. But I am following also an important Poisson's distribution presuppose: mean must be equal to variance. As my outcome variable has a variance too smaller than mean, so I am following some recommendations on using quasi-Poisson for sub-dispersed data.

But I have modeled with Poison, logistic, negative binomial... And ALL of these show me the same normal residuals graphic. This is the point.