It seems to me that, in the statistical literature, data, models, and inference seem to conflate the terminology of dependent and independent.

Case 1: a cross sectional sample of weight and height in an simple random sample of adolescent women. Take weight as the response for simplicity. The weight "depends" on height and vice versa, but the data, model, and inference is considered "independent" because the residuals are conditionally independent of each other and of the regressor in the linear model?

Case 2: a cross-sectional stratified sample of low income and high income women's weight/height. Weight of women who are in one stratum are more correlated to women in the other stratum, but not after conditioning on height which is a mediating factor of SES. Are the data, the model, and inference considered "independent" but simply that they need to be weighted to be valid?

Case 3: a longitudinal study of height in adolescent women measured monthly over 4 consecutive years. Suppose we are interested in modeling rate of expected growth in a differential equation. So we fit a first order lagged autoregressive model. I inspect the distribution of residuals and it confirms my hypothesis that the residuals are a Gaussian process, e.g. after conditioning on lagged response, height has no autoregressive or exchangeable effect between or within women (highly unlikely though that may be). Are the data, the model, and inference considered "independent" given you are not modeling height but rather the change in height over time?

  • $\begingroup$ I think the problem is insisting on casting all problems into this particular framework. It's not a universal one. Consider instead thinking about what are the random variables in the problem and which ones you're conditioning on. A single variable ($y_{t-1}$ say) may play both "roles", being a random variable of interest at one time and being conditioned on at another time. $\endgroup$ – Glen_b Jun 18 at 7:02
  • $\begingroup$ @Glen_b for some context, I consider this problem because it seems to be a consistent sticking point during scholarly review. More than a few times, a statistical reviewer has come back with the comment, "These data are time series data and so linear regression is not appropriate. Random effects or GEE should be used instead." I don't think that's a correct deduction. Seems symptomatic of teaching "reflexive" statistics. If the residuals are conditionally independent, whether the exposure is a lag or an exogenous variable, it doesn't seem far off to say you are modeling "independent" data. $\endgroup$ – AdamO Jun 21 at 18:16

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