# Reviewer questioning my stats, need a second opinion (multiple linear regression)

I just got reviews for my first article and one of the reviewer is questioning my stats and he made me doubt about it. I cross-posted on reddit and one redditor suggested me to come here for a second opinion (http://tinyurl.com/pqzt).

Here’s a quick rundown of my study: 150 patients. I was interested in how well exposure to a certain toxin (main IV, predictor) could predict scores on a clinical questionnaire (DV). I used a standard linear regression model ("enter" method) and included other predictors that are known to affect scores on my DV (age, education, disease duration and motor disability). All variables are on a continuous scale. I’ve found that my model is significant, so is my main predictor (toxin exposure), along with age and education. There is no colinearity problem as per the VIF index. If that helps, here's the histogram and the plots of the residuals: http://imgur.com/vsb

One reviewer is questioning my regression model because he says that since about 70 people out of 150 had an exposure value of 0 (non-exposed), my model is only fit for to those who were exposed to the toxin (about 80). My understanding of the regression model and the number of degree of freedom in the ANOVA (140ish) table makes me thinks he is wrong. He also said that the preliminary correlations I ran between were not fit for the whole sample because 70 people had an exposure value of “0”, even though I used the whole sample for the analysis.

I played around with my data to get a better grasp of the problem. When I ran the same regression model (enter method) on those with exposure only (n=80), my predictor "toxin" fell short of significance. I believe the lack of statistical power could be to blame (6 predictors with a “n” of 80 and combined with the "weak" effect size). Then I went back to the whole sample and I added a dichotomous variable (exposed or not exposed), but neither the exposure status (yes or no) nor the total exposure (scale from 0 to 100) were significant with the “enter method” (all variables entered simultaneously). However, using a stepwise method, the toxin exposure level variable (main predictor) was now significant again.

So please, can you confirm that I am right/wrong, and do you have any advice on how to write this to the editor/reviewer so my paper doesn’t get rejected in the second round? Thanks!

• As an aside - the problems with stepwise regression are quite well documented (see stats.stackexchange.com/questions/20836/…). The p-values that come out of stepwise regression need to be corrected - don't rely on them at face value. – NickB2014 Jul 17 '15 at 7:09
• I think your reviewer don't understand linear regession. – Deep North Jul 17 '15 at 8:42
• Since every molecule known to man is almost surely present in almost anyone's body, you could replace zero exposure by randomly chosen values within a small interval, say $(10^{-25}, 10^{-24})$ mol/Kg, and repeat the analysis :-). If that's not convincing enough, repeat a few hundred times for a sensitivity study (no more emoticons needed here, I hope). It would be more biologically realistic and it would destroy the reviewer's statistical objections (to the extent they hinge on "no exposure" and having $70$ people with the same exposure). – whuber Jul 17 '15 at 13:19
• Would it be possible to add a regression plot of clinical scores ~ toxins to your question? – Ruthger Righart Jul 22 '15 at 9:56
• The links provided do not work, at least for me! – Gavin M. Jones Jul 22 '15 at 13:36

If you think there is a discontinuity in the effect of the exposure at an exposure (toxin level) of zero, you can test a more general hypothesis using at least 2 predictors: an indicator of toxin > 0 and something like log(toxin + 1). The 2 d.f. "chunk" test for the combined effects of these two predictors tests the null hypothesis that toxin level is associated with the outcome, allowing for a discontinuity at zero. You can get the chunk test using a general contrast with 2 d.f. or by omitting both variables and doing the "difference in $R^2$" test.

The reviewer is incorrect.

It is very important to make sure that you have chosen the right model for the clinical outcome score. You are assuming the score is a continuous variable without a great number of ties, and that the residuals from the model have a Gaussian distribution.

Avoid any removal of variables on the basis of $P$-values.

It would be important to (1) see the regression plots of the relation between toxin and clinical score and (2) to know in more detail what your experimental treatment consisted of. I created an oversimplified data example in R to illustrate the problem.

Data example:

data1<-data.frame(tox=c(0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3), clin=c(10,10,10,10,10,10,10,10,10,20,30,40,20,30,40,20,30,40))
model1<-lm(data1$clin ~ data1$tox)

data2<-data.frame(tox=c(1,1,1,2,2,2,3,3,3), clin=c(20,30,40,20,30,40,20,30,40))
model2<-lm(data2$clin ~ data2$tox)

par(mfrow=c(1,2))
plot(data1$clin ~ data1$tox, xlim=c(0,4), ylim=c(0,40), xlab="toxin", ylab="clinical score")
abline(model1, col="blue")
plot(data2$clin ~ data2$tox, xlim=c(0,4), ylim=c(0,40), xlab="toxin", ylab="clinical score")
abline(model2, col="yellow")


Model1 would show a significant regression model. However, the effect may entirely disappear if we removed clinical scores at tox=0 values, as shown in model2. It would be essential to know how the scatterplot of data looks like when you removed tox=0. In this case it would be hard to believe there is some linear (or higher level) relation between the dose of toxin and clinical scores.

It may however still be worthwhile to perform a group comparison, no-toxin (tox=0) versus toxin (tox>0). From a research methodological point of view it would be important to know what tox=0 really means. What kind of treatment did patients receive at tox=0 and tox>0? If a placebo treatment was used (i.e., the only difference is that tox>0 received really a toxin and tox=0 received something fake) then a simple group comparison may be still valid to test the effect of no-toxin vs toxin.

Try to turn your toxin exposure into a categorical predictor and run the same model. And if the IV still significant run a reduced model with only the significant predictors (toxin exposure, along with age and education) as a continuous predictor for the 80 participant who has exposed to the toxin.

I think your reviewer has a valid concern of the 70 non exposure participant. Alternatively you can randomly sample some participant from your 70 non-exposures and try to run the same model.