# (Expected) decrease from baseline to follow-up - linear regression?

I have a patient group (n=30) who all have a specific kidney disease. I have their kidney function at baseline and their kidney function at follow-up 1 year later, and some confounders: age, gender, and smoking-status (never, former, or yes).

I have the following hypothesis: The patients’ decrease in kidney function from baseline to follow-up are caused by a natural/expected decrease, and thus, not their disease.

I thought I would make a linear regression to show this, but I’m not so sure anymore - or maybe I’m just not sure how to interpret it. I’ve tried doing it like this:

regress kidney_function_FU age gender smoking kidney_function_base


This gives me the following output:

kidney_function_FU Coef. (95%CI) p-value
age -0.47 (-0.77; -0.18) 0.003
1.gender -2.88 (-11.48; 5.73) 0.496
2.smoking 12.45 (2.06; 22.84) 0.021
3.smoking 0.03 (-9.26; 9.33) 0.994
kidney_function_base 0.64 (0.30; 0.98) 0.001
_cons 56.08 (14.3; 20.21) 0.011

R-squared = 0.72 Adj. R-squared = 0.68

So, I guess R-squared is quiet large, meaning that these variables (confounders + baseline kidney function) explains ≈ 70% of the (decreased) kidney function at follow-up. However, I'm not sure how I could really ever accept my hypothesis - this I could only do, I guess, if R-squared was close to 100?

Is it my interpretation that is wrong, or my choice of analysis?

• The R-squared is high because all patients have kidney disease at the start of the period and they also have kidney disease at the end of the period. I don't see how you can tackle your hypothesis at all with the data that you have. Commented Apr 25, 2023 at 19:45
• What you would need to investigate this is to have another group of people who are matched with the patients in terms of age, gender, smoking and -- importantly -- kidney function. And then follow them prospectively for a length of time. Commented Apr 25, 2023 at 19:46

I think we should have a closer look at your hypothesis:

The patients’ decrease in kidney function from baseline to follow-up are caused by a natural/expected decrease, and thus, not their disease.

Given that you are interested in kidney disease vs. natural decrease causing the (extent of) decrease in kidney function, you are interested in a causal question (i.e., does x cause y, or does kidney disease cause more decrease in kidney function as compared to no kidney disease).

In a hypothetical world, we would want to know the decrease in kidney function of an individual when they would have had kidney disease vs. when they would not have had kidney disease. However, an individual has kidney disease or does not have kidney disease. We cannot observe the outcome under both states of the exposure, or in other words, one of the outcomes is counter to fact. Because we do not know these counterfactual outcomes in individuals, we cannot estimate a so called individual causal effect.

Luckily, we can try to get an estimate of a different causal effect; the average causal effect. An average causal effect would be the effect of x on y on a group level, where we compare individuals who are exposed (who do have kidney disease) versus those who are unexposed (who do not have kidney disease). If we meet some assumptions, we could validly estimate the effect of having vs. not having kidney disease on the decrease in kidney function. These assumptions are the following:

• Exchangeability: the two groups you are comparing (kidney disease vs. no kidney disease) have the same distribution of risk factors for the outcome; their chance of their kidney function decreasing by x mL/min/1.73m2 is equal, outside the possibility of kidney disease influencing this.
• Positivity: each individual has a non-zero probability of having kidney disease or not having kidney disease
• Consistency: when we say kidney disease and no kidney disease, it is sufficiently clear what we mean. Kidney disease could mean diabetic nephropathy, glomerulonephritis, etc. In your case you said a specific kidney disease, so possibly within that disease there are further variations, but those variations should not matter for the outcome. An example of no consistency would be studying 'low BMI' as a cause of death, where low BMI could mean high metabolism, amputation of a leg, malnutrition, or other things, that all have very different prognoses for death.

To meet these assumptions, we can choose a design of a study (e.g., a randomized clinical trial, although it is not ethical and rather difficult to randomize individuals to kidney disease or not) and choose specific analyses. Given that a randomized clinical trial would likely not be accepted by any medical ethical review board, you could choose to use observational, non-experimental data. However, in observational data, exchangeability often does not hold and requires analyses such as regression or more advanced methods to approach (although impossible to be certain about).

However, in your question, you mention the data is all on patients with that specific kidney disease, therefore we cannot draw a causal conclusion: we are missing a group without kidney disease to compare the kidney function with and can therefore not draw a causal conclusion.

You could describe the decrease in kidney function and look for other data on a healthy population, but this quickly impedes any causal inference. If you just want to describe the decrease in kidney function, you could indeed use a linear regression model, although you then make the assumption the decrease in kidney function over time follows a linear pattern.

To summarize: you are asking a causal question (does having the kidney disease change the decrease in kidney function when compared to no kidney disease [i.e., natural course]), but do not have the data to answer this causal question.

On a side note: your interpretation of the R-squared is correct: your model explains ~70% of variation in the outcome.

If you do have a control group without kidney function available, you should still be wary of the following:

• Make sure you have an adequate sample size, n = 30 is rather small and will impede many conclusions due to statistical uncertainty
• Be careful in choosing when to start follow-up: there are many biases as a result of picking the wrong moment (see references for more on this)
• Clearly define the question you are interested in: what groups do you want to compare

Here are some references which talk about the above and go more into depth on the topic of causal inference:

• On biases when picking the wrong start of follow-up and other easy to avoid biases when working with observational data: Fu EL, van Diepen M, Xu Y, Trevisan M, Dekker FW, Zoccali C, Jager K, Carrero JJ. Pharmacoepidemiology for nephrologists (part 2): potential biases and how to overcome them. Clin Kidney J. 2020 Dec 14;14(5):1317-1326. doi: 10.1093/ckj/sfaa242. PMID: 33959262; PMCID: PMC8087121.
• Using observational data to emulate a randomized controlled trial: Hernán MA, Robins JM. Using Big Data to Emulate a Target Trial When a Randomized Trial Is Not Available. Am J Epidemiol. 2016 Apr 15;183(8):758-64. doi: 10.1093/aje/kwv254. Epub 2016 Mar 18. PMID: 26994063; PMCID: PMC4832051.
• A book by one of the leaders in causal inference with intuitive and mathematical explanations for drawing causal conclusions: Hernán MA, Robins JM (2020). Causal Inference: What If. Boca Raton: Chapman & Hall/CRC.
• If the book by Hernán is a bit much, Westreich offers a lighter introduction to causal inference: Westreich D (2020). Epidemiology by Design: A Causal Approach to the Health Sciences.
• Although your answer introduces causal analysis well and succinctly, most of it is not relevant to the question: getting a kidney disease is not at all like choosing to give a patient treatment A or treatment B or no treatment. A counterfactual to the fact "Person X developed kidney disease" makes no sense. Commented Apr 25, 2023 at 19:41
• @dipetkov I agree that my answer might have been a bit overenthusiastic. I must however respectfully disagree with your statement that kidney disease as a cause makes no sense. It is true that we often think of causes as interventions (e.g., give drug A vs. B), but causal inference is not limited to interventions (albeit that states are less likely to meet the consistency assumption). In support of my statement I attach the following paper: Pearce N, Vandenbroucke JP. Educational note: types of causes. Int J Epidemiol. 2020 Apr 1;49(2):676-685. doi: 10.1093/ije/dyz229. PMID: 31711141. Commented Apr 25, 2023 at 19:47
• I admit I'm not a doctor. There might be different causes for kidney disease. I'm still not sure causal analysis is applicable to the situation this question describes in principle. It's definitely not applicable in this case because the OP only has patients with the kidney disease in their study. Commented Apr 25, 2023 at 19:47
• @dipetkov The paper is on how we can define causes (and is freely available). The first lines from the abstract: We explore the different types of causes that are commonly investigated by epidemiologists. We first distinguish between causes which are events (including actions) and causes which are states. Second, we distinguish between modifiable and non-modifiable states. This yields three types of causes: fixed states (non-modifiable), dynamic states (modifiable) and events (including actions). The comments are restrictive in discussing this, but I'm open to talking more with you about it. Commented Apr 25, 2023 at 19:50
• Thank you for the paper (& for providing a proper citation which many fail to do). I'll try to take a look & I assume the OP would benefit as well. Commented Apr 25, 2023 at 19:53