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I've been using linear models (lm function in R) to determine if the the expression of certain proteins is associated with disease severity. Sometimes the expression of proteins can also correlate with age, so I've been including that in the models as so:

lm(ProteinA ~ DiseaseSeverity + Age, data = mydata)

But where I'm getting confused is in interpreting the output when the entire model is significant but my coefficient of interest is not significant. For instance, consider these results:

(Intercept): p = 0.01438
DiseaseSeverityMild: p = 0.88170
Age: p = 0.00278
F-statistic: 6.418 on 2 and 34 DF,  p-value: 0.004319

In this case, the overall model is significant, and the intercept is significant, and age is significant, but DiseaseSeverity is not significant.

  1. Am I correct in my understanding that I therefore cannot say that ProteinA is a significant correlate of DiseaseSeverity? Or, how would you interpret these results?

  2. Does the fact that the intercept is significant mean anything, either in general or in the context of answering my question of "Is ProteinA expression related to DiseaseSeverity?"?

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  • $\begingroup$ What are the coefficient values - just out of curiosity? As someone who works with health care data, the concept of disease severity/latent state of a patient is often difficult to represent. The issue may be rooted in how severity is reflected/controlled in the model - especially when modeled with demographics that often predict disease states very well. $\endgroup$ – Ryan Morton Jun 28 '17 at 21:51
  • $\begingroup$ Hi Ryan. t-value for DiseaseSeverityMild was -0.15 and the t-value for Age was 3.226. Happy to hear your input! $\endgroup$ – Jay Jun 29 '17 at 14:54
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  1. My interpretation is that Age predicts ProteinA after controlling for the effect of DiseaseSeverity. However, DiseaseSeverity does not predict ProteinA after controlling for Age. (I am not able to say more specifically whether or not it is a positive or negative relationship, since you only posted p-values). I would also not word it like "is a significant correlate." Instead, I would say "predictor." You are doing multiple regression, so you are not looking at correlations. The results may not be causal (that is, measuring association but not effect), but I still would not use the word "correlate" in this scenario, because it is confusing to the reader.

  2. The intercept being significant just means that your intercept (the predicted value of ProteinA when DiseaseSeverity = 0 and Age = 0) is significantly different from 0. That is what it means in general, and I would say that it does not really relate to your research question considering the relationship between ProteinA and DiseaseSeverity

The whole model is significant because Age is a significant predictor of ProteinA. DiseaseSeverity makes no difference after controlling for Age.

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  • $\begingroup$ Hi Mark, First of all thanks so much for reformatting my post. I was having some difficulty getting the text not to just cram all together, which is why I didn't paste the entire output from R. I will say that the t-value for DiseaseSeverityMild was -0.15 and the t-value for Age was 3.226. So that suggests that as age increases, proteinA increases... but what does it mean for a binary factor like DiseaseSeverity? And of course, thank you very much for your thoughtful and extension response. I really appreciate your help! $\endgroup$ – Jay Jun 28 '17 at 20:11
  • $\begingroup$ With a binary predictor like DiseaseSeverity, you could just say that there is no significant mean difference between the two levels after controlling for Age. $\endgroup$ – Mark White Jun 28 '17 at 20:12
  • $\begingroup$ I should go into a little bit more detail: The lm function in R will automatically re-code dichotomous character factors (like what you have) to be 0 and 1. The reference category is coded 0 (usually the first appearing alphabetically, unless you refactor it), while the comparison category is coded 1 (in your case, Mild). Then the coefficient is just interpreted like any other regression coefficient: A one-unit-change in the predictor (DiseaseSeverity) leads to a change equal to whatever the unstandardized regression coefficient is. This pans out to being just a mean difference. $\endgroup$ – Mark White Jun 28 '17 at 20:16
  • $\begingroup$ Thanks again... I just wanted to emphasize one point you made in your explanation: "Age predicts ProteinA after controlling for the effect of DiseaseSeverity. However, DiseaseSeverity does not predict ProteinA after controlling for Age". Does this mean that in these linear models, the significance of each coefficient is assessed while controlling for the other coefficient? Is the same true for all linear models (like LMMs and GLMMs)? Cheers! $\endgroup$ – Jay Jun 28 '17 at 20:29
  • $\begingroup$ Yes. Please see Section 3.2 here: www-bcf.usc.edu/~gareth/ISL/ISLR%20Seventh%20Printing.pdf $\endgroup$ – Mark White Jun 28 '17 at 20:30

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