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Consider this scenario: scientists hypothesize that a particular disease occurs when levels of a particular hormone are high. They gather data: 1000 people with the disease, 1000 without, and measure their hormone levels.

Assuming the data has normal distribution, the scientists can do a t-test or one-way anova to test if the difference in hormone levels between the disease and non-disease group is significantly different from zero. (I guess this would be a one-sided t-test). In R, the Anova would be expressed like

model <- lm( HormoneLevel ~ Disease, ...)

where Disease is 0/1 according to disease|non-disease, and HormoneLevel is a continuous value (amount per litre or something).

HOWEVER, can the test also be done in the other direction, with HormoneLevel as the independent variable?

model <- lm( Disease ~ HormoneLevel, ...)

Perhaps this makes more sense conceptually as a regression, since the scientists believe that the hormone level may be a cause of the disease.

So the question: is it valid and desirable to switch the dependent and independent variables in this way? If so, are there restrictions on when it can be done?

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    $\begingroup$ If you want to build a valid model for Disease ~ HormoneLevel , you might use logistic regression, such as glm(Disease ~ HormoneLevel, family = binomial(link="logit")). $\endgroup$ – Sal Mangiafico Jul 9 '18 at 16:12
  • $\begingroup$ ok, yes. However my question is more about the validity and advantage/disadvantage (if valid) of flipping the independent and dependent variables. $\endgroup$ – sportscan Jul 12 '18 at 12:40
  • $\begingroup$ As you say, if you think that the hormone causes the disease or that the hormone can predict the disease, then the model Disease ~ Hormone makes sense. There's no reason to think that Hormone ~ Disease is the default or natural model to use. If there's no prediction or causation intended, correlation could be used instead. $\endgroup$ – Sal Mangiafico Jul 12 '18 at 12:57
  • $\begingroup$ Thank you. So if we use a logistic regression, we can get a prediction of Disease from Hormone, divide our data into training (,validation), test, and report some sort of accuracy. But is there a way to report a p-value or "significance", as is possible by formulating it in the other direction? $\endgroup$ – sportscan Jul 13 '18 at 9:23
  • $\begingroup$ logistic regression in R gives the usual GLM output - including asymptotic p-values for individual variables and null and residual deviance, from which an asymptotic overall test could be performed. This is covered in most statistical treatments of generalized linear models, and in several questions on site e.g. this one - stats.stackexchange.com/questions/108995/… $\endgroup$ – Glen_b -Reinstate Monica Jul 13 '18 at 10:45

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