Interesting discussion. In epidemiology, adjusting for confounding in a multiple logistic regression model is fairly standard; agree with Bruce (comment #1). But I note some confusion on 1) the naming of dependent and independent variables; 2) standard error; 3) the role of simple versus more complex modeling.

1. The dependent variable (Y) is being a patient or control. There are 2 independent variables (2 Xs): protein A and variable B. A is a continuous variable, and is of primary interest. B is a potential confounder. The simplest analysis to adjust for confounding is to fit a model: Y~A+B. Using logistic regression makes that we have a regression coefficient which can be easily translated to an odds ratio (exp(coef)). The interpretation of the odds ratio is per unit of A. One way to naturally express the effect size is to make the odds ratio relate to the 75 versus the 25 percentile value: an interquartile odds ratio.

The adjusted odds ratio can be compared to the unadjusted odds ratio from the model Y~A. The difference between odds ratios is usually interpreted to reflect confounding, i.e. the correlation between A and B. This correlation can also be studied in an intermediate analysis; simple is a scatter plot with Pearson correlation coefficient r, or if B is dichotomous: logistic regression with B as dependent and A as independent variable.

2. In the logistic regression model Y~A+B, the standard error for the coefficient of A will be larger than in the model Y~A. But in the first model, we estimate an adjusted coefficient, and in the second model an unadjusted coefficient. The first is a conditional estimate, where it is known that the expected value is more extreme than the unconditional estimate, if there is no correlation between A and B. So, if B is not a confounder, there is actually more statistical power in the adjusted model than in the unadjusted analysis, provided that B is related to the dependent variable (being a patient rather control). Another intermediate step is hence to check that B is related to Y: Y~B.

3. The simplest form for the continuous variable A is linear. But for laboratory measurements such as a protein, a log transformation may also be reasonable. Other common transformations to consider are square and cubic. A large set of transformations is captured by fractional polynomials. Other smooth, flexible functions include splines, where restricted cubic splines are particularly attractive in my view. Again, the odds ratio can be calculated to relate to the 75 versus the 25 percentile value: the interquartile odds ratio.

Categorization, specifically dichotomization, is to be avoided, since natural process are smooth rather than discrete. Statistical interactions of the form AxB may be considered, but require rather large data sets for reliable estimation. In epidemiology, most will stick to assuming an effect of A across all levels of B (no interactions, only main effects in the model Y~A+B). CHAID and other more advanced techniques may have value for exploratory analyses, but are uncommon and require huge sample size for stable estimation.

In some fields, it would be fairly standard practice to report the crude odds ratio (OR) for A (i.e., OR for A from a model with no other predictors), followed by an adjusted OR for A (adjusted for B plus whatever other potential confounders you might wish to include). 

Do you also need to consider the possibility of an AxB interaction? 

Hi,

If you don't want to  predict any variable, but just look at the whole distribution and how they interacts, you can use contingency tables, log-linear models or Poisson regression.

Bear in mind that adding a confounding variable to the model will increase the standard error of estimate for the effect of your predictor of interest. So it is important not just to look at p-values but primarily to look at effect sizes. In small studies, the odds ratio may not change much but the p-value becomes non-significant because of the increase in the standard error of estimate. 

In such a case you may find that the estimate for the odds ratio isn't significantly different to the univariate estimate, in which case all you can say is that the addition of the confounder doesn't change the estimate of the odds ratio significantly but it's no longer significant, and plead for a larger study.

What is your response? What are you trying to predict? What are you using to predict the outcome? 

One way to tell if something is confounded with something else is to put both terms into the model and look at the coefficients. Then go back and put one term in the model. If it changes a lot, the you have a confounding variable. If not, you don't. 

Jurac,

You wrote " how important the effect of the variable B on concentration of protein A is" If so, then concentration of protein A is your dependent variable and you don't need logistic regression.

Dear all, thank all of you for your inspiriting replies!

@ Leonardo

Yes, I need to know how important is the effect of the variable B on concentration of protein A, which is significantly different between controls and cases. If it is important I need to adjust the concentration of protein A accordingly in both controls and cases and run the statistic again.

So first of all, how to answer the question: is there an important effect (if any) of the variable B on concentration of protein A that differs between controls and cases?

Ok. I assume that then concentration is your dependent variable, A, then you have your treatment TRT and the variable B. I assume that A is fairly continuous and not a ratio variable, i.e. bounded by 0 and 1.

In this case, you can just obtain a regression model that will tell you how B affects the concentration, and you could add the interaction of B and TRT to obtain different intercepts if significant.

HTHT

You are interested in the question "whether presence of factor B affects concentration of protein A in patients?" in my opinion you should treat concentration of protein A as a continuous dependent variable and carry out a simple regression analysis keeping type of group (patient-1, control-0) as a dichotomous independent variable and factor B a co-variate. I hope the results of this regression will answer your query.

If concentration of protein (your Factor A) is indeed the dependent variable, then I agree with Ajit.  I would further suggest that you inspect a scatter-plot with X = your Factor B, and with different colours (or markers) and fit-lines for the two groups.  This plot will allow you to visually assess a couple of things:  1) Whether the functional relationship between B and A is linear, and 2) Whether it makes sense to force the slopes for the two groups to be equal.  If the slopes appear to be different, you could include the Group x Factor B product term in your model (thus allowing each group to have a unique slope). 

Have you considered an entirely different approach? CHAID (chi-square-automatic-interaction-detection) is also a classification model as are all logistic regression models. The key difference being "interaction detection". While LR will give you Odds Ratios, it will never (as noted above in previous comments) handle multi-collinear issues that appear when you introduce A X B interactive terms as a step in the model. Interactive terms are useful to know about, and sometimes they do have a combined effect that is stronger than either of them alone. Sometimes the interactive term is significant, and the model improves when you retain only the interactive term, and drop the two independent variables that make it up, from the model.

The ultimate goal of an ideal LR model is good area under the ROC (high sensitivity-specificity), acceptable Hosmer-Lemeshow, significant Omnibus block, step, and model, and iterative convergence in 20 or fewer iterations, and having all of these accomplished by a model that retains only significant variables in the final step of the model equation.

If you run a CHAID model, using your same Dependent Variable as the LR model uses, and use the same set of predictor variables that were used in the LR model, and the same experimenwise alpha (0.05?) for tree branching threshold, if any of your predictor variables are scale level of measure, the CHAID model Tree will "bifurcate" at the "detected" significant cut points. Use these cut points to transform any continuous IV predictors in the LR, into dichotomous predictor IVs and use them instead in the LR. You may be surprised to see multi-collinear issues vanish, and the ROC of the LR model go up dramatically, because what ROC measures is how well the model "discriminates" Yes from No, given the predictive factors being used to calculate the model "c-statistic" (probability). The issue is that the LR model is based on the use of the "nearly lnear" portion of the logit function on either side of the model cut-point threshold, and a continuous variable may be "non-linear" in its higher order dimensions along that same area of the logit that is being used "to discriminate" probabilities of Yes vs. No. The net result is your whole continuous variable is thrown out as insignificant, whereas if you were able to cut it into two or more "significant segments", it can be kept in the model.

For a real "wavy" function as a predictor variable, you may be able to demonstrate magic, in that certain segments have a positive beta weight, and a significant Odds Ratio over 1.0, and certain segments of that same continuous variable exhibit a negative beta weight, and a significant Odds Ratio of less likelihood and lower probability of Dependent Variable = True. That is an interaction that is posed by many "non-linear" chaotic functions in all manner of biological and physical data. I do not know if it will work for your experimental model, but it may be at least worth a try.

The suggestion to look at a scatter plot is a very good one. I would add, look at a three dimensional scatter plot, and plot your Dependent Variable in it as a "grouping variable" (color, shape, texture, etc.) and if your analytical package will allow it, plot the "sub-group centroids" that lie at the intersections of the three axes plotted point "cloud" for the two states of the dependent variable (yes/no) for the LR and CHAID models. If you can, grab the frame of the resulting 3d centroid plot and move around through all three axes of rotation, to visualize the relative proximity of the (hopefully two distinct centroids) and how they are related to each other. Sometimes this will tell you in a "one chunk cognitive glance" that all of the plotted points overlap in the two clouds (the two DV subgroup centroids are almost on top of each other) and there is no hope disentangling those three continuous predictor variables to resolve the Dependent variable. They are "fully entangled" and  need to be either dropped from the model (as described by the LR stepwise model improvement method described previously) or "fixed" (by addition, multiplication, log transform, z-scoring, subtraction, or some other time honored method of addressing the accumulation of alpha error due to multi-collinear effects. Sometimes, in the scatter plot, if you change axial scales from linear to Log, it can also help identify "pockets of variation density" in a cloud like that, and that indicates multiple modality. If you have a multimodal distribution, you will always be better off creating a dummy code for each mode, so that the effect of each mode can have an independent chance of expression in the LR model. Perhaps only one of the modes is a significant predictor of the DV Target.  

Interesting discussion. In epidemiology, adjusting for confounding in a multiple logistic regression model is fairly standard; agree with Bruce (comment #1). But I note some confusion on 1) the naming of dependent and independent variables; 2) standard error; 3) the role of simple versus more complex modeling.

1. The dependent variable (Y) is being a patient or control. There are 2 independent variables (2 Xs): protein A and variable B. A is a continuous variable, and is of primary interest. B is a potential confounder. The simplest analysis to adjust for confounding is to fit a model: Y~A+B. Using logistic regression makes that we have a regression coefficient which can be easily translated to an odds ratio (exp(coef)). The interpretation of the odds ratio is per unit of A. One way to naturally express the effect size is to make the odds ratio relate to the 75 versus the 25 percentile value: an interquartile odds ratio.

The adjusted odds ratio can be compared to the unadjusted odds ratio from the model Y~A. The difference between odds ratios is usually interpreted to reflect confounding, i.e. the correlation between A and B. This correlation can also be studied in an intermediate analysis; simple is a scatter plot with Pearson correlation coefficient r, or if B is dichotomous: logistic regression with B as dependent and A as independent variable.

2. In the logistic regression model Y~A+B, the standard error for the coefficient of A will be larger than in the model Y~A. But in the first model, we estimate an adjusted coefficient, and in the second model an unadjusted coefficient. The first is a conditional estimate, where it is known that the expected value is more extreme than the unconditional estimate, if there is no correlation between A and B. So, if B is not a confounder, there is actually more statistical power in the adjusted model than in the unadjusted analysis, provided that B is related to the dependent variable (being a patient rather control). Another intermediate step is hence to check that B is related to Y: Y~B.

3. The simplest form for the continuous variable A is linear. But for laboratory measurements such as a protein, a log transformation may also be reasonable. Other common transformations to consider are square and cubic. A large set of transformations is captured by fractional polynomials. Other smooth, flexible functions include splines, where restricted cubic splines are particularly attractive in my view. Again, the odds ratio can be calculated to relate to the 75 versus the 25 percentile value: the interquartile odds ratio.

Categorization, specifically dichotomization, is to be avoided, since natural process are smooth rather than discrete. Statistical interactions of the form AxB may be considered, but require rather large data sets for reliable estimation. In epidemiology, most will stick to assuming an effect of A across all levels of B (no interactions, only main effects in the model Y~A+B). CHAID and other more advanced techniques may have value for exploratory analyses, but are uncommon and require huge sample size for stable estimation.

Thanks to all of you for contributing to this topic. I have read all comments with high interest. Unfortunately, answers appear faster than I can study the background for the previous ones. However, from what I have learned so far here, from the underlying background of the comments, and from a discussion I have had with a bio-statistician I would conclude at this moment:

1) Multiple logistic regression is not appropriate because at this moment I’m not trying to predict the outcome (the diagnosis as such is not the primary aim) based on the Protein A concentration and potentially a confounding variable B. At this moment I just know, that the Protein A concentration is significantly higher in cases than in controls (Mann Whitney test, P value = 0.0061 [median and IQR: 65.90 (20.70-503.4) vs. 34.25 (20.93-86.03)] and I’m trying to find out if potentially confounding variable B (gestational age in this case) may contribute to this observed difference significantly. If so, what would be the Protein A concentration adjusted for gestational age.

2) Multiple linear regression, best in form of ANCOVA, seems to be the most appropriate for answering the question. In this case the dependent variable (Y) will be the protein A concentration (continuous variable with non-normal distribution), independent variable X1 (will be the diagnosis (case=1, control=0) and X2 will be the potential confounder B (continuous variable with non-normal distribution).

3) Continuous variables with non-normal distribution should be transformed in order to get normal or at least close-to-normal distribution. But how to de-transform predicted Protein A concentration adjusted for gestational age calculated from the model?

I would be grateful for any additional comments to mine conclusions. I have also deposited the crude data at http://ShareGadget.com/141214570, feel free to analyze it in your way.

Jurah, regarding your 3rd point, note that for ordinary least squares (OLS) models, it is the ~errors~ (not the outcome variable) that are assumed to be normally distributed.  For further discussion and commentary, click the link below.  HTH.

https://www.researchgate.net/post/Is_linear_regression_valid_when_the_outcome_dependant_variable_not_normally_distributed?_tpcectx=profile_answers