The purpose of a case-control study is to identify risk factors. If you wish to calculate the population attributable risk associated with these risk factors, then you need information on the population distribution of these risk factors, but this is often a secondary outcome of the analysis.

Attempting to make the prevalence of cases the same in your case-control study as it is in the general population will result in very considerable methodological inefficiency. The most powerful design is one in which there is a 1:1 case-control ratio. As the prevalence of the cases decreases, the statistical power declines too. So unless cases are very rare, this is an inefficient strategy.

Thanks Ronan for your opinion.

1) I did not get what you meant by "population distribution of these risk factors". In case that you meant the sensitivity and specificity of the risk factor, then how to interpret that in case of contineous variable. Please look at the file attaced.  Here is an imaginary example of what I am planning to do. The lesion size x, in cm, is my contineous variable and the number of cases and controls are plotted for each value of x. Now, the total number of cases in this trial is 32 and the control is only 18. ( the proportion of the cases in this sample is 64% while the true prevalence of the disease in the population is 5%). Can I then do a logistic regression with a curve fitting of those data in order to get the predictive value of the disease as a function of (x)? will that regression be a reliable estimate of the true population?

2) If I go for a 1:1 case-control ratio for my retrospective study, there might be wrong estimation of the probabiliy of the disease ( positive predictive value) at different values of the independeny variables ( indicators) because a 1:1 ratio assumes a 50% prevalence of that disease which is not true in the real life.

There is no such thing as a 'wrong estimation of positive predictive value'. The estimation of positive predictive value is conditional on a given prevalence of the disease. Sensitivity and specificity do not assume prevalence, since they apply to two different populations: the cases and non cases respectively. 

Have a look at Fagan's nomogram (http://www.hsl.unc.edu/services/tutorials/ebm/nomogram.pdf) which illustrates the pre-test and post-test probabilities of disease for a given likelihood ratio.

Dear Adel,

in a case control study you should not try and re-create the disease distribution in the population. You identify cases (those who developed th eoutcome of interest), then you identify controls (those who did not develop the outcome of interest) and then cyou compare then in terms of exposure to whatever risk factors. You can calculate how many cases and controls you need by doing a sample size calculation, which depends on the effect size of the risk factor of interest- in general terms, the more people in your study the better (the smaller the confidence intervals or the smaller the effect you can detect). Do NOT try and select cases or controls to try and model population- this will introduce bias in your study and invalidate your results. I shoudl add that while case-controls studies are quite cheap and realtively quick to perform, they are also the most prone to bias- in other words they are very easy to mess up. If you can get your hands on a book called "Case control studies- design conduct and analysis" by James Schlesselman- they explain all you need to know about these studies

Thanks Edelstein,

What I have read recently is that the case control study gives odd ratio (OR) which is equivalent to the relative risk in cohort study assuming the disease is rare. The OR can be interpreted as the slop of the logit curve in logistic regression ( this is clear because the odd in general is an exponential function of an arbitrarily covariate value (x) multiplied by an coefficient Bi where i =0 for a reference value and 1,2,... for other covariates) and OR becomes then an exponential function of the difference(dx) between the covariate value and the reference value multiplied by B1. So the logit is equal to, In OR= B1dx).

The problem is OR can not give the absolute value of the probability which is required in my case. Because the logistic regression in case control study can only provide B1 not B0, in other words the value of the slop of logit not the intercept. 

So how to get B0 ?

This doesn't overcome the problem that the distribution of the risk factors in your study may not be comparable with their distribution in the population, and so a knowledge of their prevalence becomes critical. 

The distribution of the risk factor in my study is considered as an estimate of the risk factor in the population. This is true especially if the sample size is large.

If there is a survey of a representative sample, one can determine the risk of disease during the execution of a logistic regression. The risk of disease is given by the equation:

P = 1 / (1 + e- (logit P)).

   with:

Logit P = α + β1 variable 1 + β2 variable 2+ β3 variable variable 3 ....

This is not possible with a case-control study. Moreover, you can choose up to four controls per case to increase the power of your study, not more.

To clarify the point about many-to-one matching. It's not so much that you cannot do it, but beyond 2 controls per case, the gains in statistical power become small and after 4:1 matching they are so negligible as to be trivial.

Extra controls tell you more and more about the non-disease population but your study is hampered by the lack of knowledge of the disease population.

Dear Friends, This is an interesting discussion. Some questions:

From which source population (study base) do your cases arise from? You may want to sample controls, independent of the exposure of interest, only among those who would have be identified as a case in your study if they would have developed the outcome.

Depending on the sampling (and, with exclusive sampling, the rarity of the outcome), you can estimate the risk ratio (think you refer to it as relative risk) or the incidence rate ratio from the case-control odds ratio.

Indeed, a control-to-case ratio of 4-to-1 may be appropriate in the majority of instances. However, 10 or more controls per case may be indicated if exposure in infrequent, large effect size, several risk factors assessed simultaneously.

Kind regards, René

Research Study Design Lecture: Modern Case-Control Studies. Rene Schade

Hello Adel,

If I understand your question correctly, you would like to get the value of or know the interpretation of B0 or intercept.

In case-control studies, the intercept from logistic regression is not interpretable.

In a cohort study, the intercept is meaningful and interpretable as "baseline risk".

I reiterate the reference referred to you by another colleague above:

"Case control studies- design conduct and analysis" by James Schlesselman"

In terms of balance - my understanding is that there is no problem here. It is common in case comparison studies where you sample on the cases and the comparisons to have more of the comparisons as they are easier/ cheaper to collect information on. Eg there may be only 95 cases of the rare outcome (the case) but getting a thousand comparison from the general population is easy (But they must have the same opportunity to become a case.).However, there is little gain in efficiency once you have 5 times as many controls/comparisons as cases; so it is not worth going beyond that number when designing your study. So with 95 cases it is worth having 5*95 comparisons if they are easy to recruit/collect and up to that point they will increase the power of the study to detect effects.

See p168 of Clayton and Hills Statistical models in epidemiology for the theory behind this advice.

This paper show how to analyse case - comparison responses in a logit model

http://www.oxfordjournals.org/our_journals/tropej/online/ma_chap11.pdf

The estimate of B0 is the loge odds of the probability of being a case (when your predictor values are all zero)  as such it is determined by your design (reflecting the extent you have gone for balance) and not by the real world. It is related to the ratio of cases to comparisons in the sample and not cases to non-cases in the population.

For non - epi examples see

Lacy, M. 1997. “Efficiently Studying Rare Events: Case-Control Methods for Sociologists”. Sociological Perspectives 40:129

There is also a literature on the analysis of choice-based samples,

Kanaroglou, P S (1994) Sampling and discrete choice analysis Professional Geographer, 46(3) 359-368

O'Kelly, ME (1999) Trade-area models and choice-based samples: methods Environment and Planning A31(4) 613-627.

Early, D W (1999) A micro-analysis of homelessness: an empirical investigation using choice-based sampling Journal of Housing Economics 8(4), 312-327