Slide 1

Well, I would like to have some interaction with you about the difficult topic of effect modification. Effect modification is a difficult thing for many epidemiologists but I am sure that you can make it today. So I will start with a small example. It’s very common knowledge that older people on dialysis have a higher mortality compared to young people. We know that. We also know that diabetes patients on dialysis have higher mortality than the non-diabetics. So we could ask ourselves whether these effects are really independent or if they work together? So that if you are both old and have diabetes perhaps you have a kind of extra effect.

Slide 2

So let’s take this as an example. We want to know whether diabetes gives a kind of ’extra‘ effect in the elderly. Now what do you think, does it? Yes? Perhaps when you have them both you are even more inclined to die. Well, the ‘extra’ effect I put here between quotes because what I mean with it is that the combination of diabetes and high age gives a higher mortality than estimated from the combination of the independent effects.

Slide 3

So if it was independent, then they just go together but perhaps there is something more. So if it would give an ’extra’ effect, then we could say that the effect of diabetes is bigger in the older patients than in the younger patients. So the effect is not the same in the younger and the older patients. So we could also rephrase that and say that the effect of diabetes on mortality is modified by age. In the younger people diabetes has a smaller effect perhaps than in the older people so we call this effect modification. You could also say that age and diabetes are not independent risk factors for mortality, or they interact. Now, this is what the topic is all about.

Slide 4

So think about this ’extra’ risk. There are many ways to phrase that. People talk about interaction or synergy or effect modification, like I prefer, or they talk about ‘not independent’ risk factors. They are all difficult concepts. So perhaps the easiest thing is to start thinking about suppose there is no interaction, suppose the effect of diabetes is the same in old and young people. Now, how would the data then look like? So first consider the no ’extra’ risk. It is the same in all ages.

Slide 5

So I have here diabetes patients yes and no, only young people on dialysis, and look, here are the incidence, the mortality rates, (non-cardiovascular mortality rates) per 1000 person years at risk. So in young people without diabetes 10 per 1000 die per year. This is a rough estimate coming from a paper of Van Dijk. If you do have diabetes when you are young, then for non-cardiovascular mortality it is 60. Now, you could say this effect of diabetes in the younger people is + 50. 50 more cases to die per 1000. Or you could say it is 6 times as high. Two ways of expressing this effect.

Slide 6

Now, this is in the young people. So what would this look like in the old people? We know that older people without diabetes have a much higher mortality rate of 150 more or less. So what would we expect here? Suppose there is no ‘extra’ effect, so suppose diabetes has the same effect in younger compared to older people. Then what would we expect here? Would we expect +50 like here or would we expect x 6 just like here? Do you think it is x 6? I guess for both points of view you can make an argument for that. So you could say if I find 200 there, then in both groups I have +50 and if I find 900 then in both groups I have x 6. So what is no effect? We saw that it would be easy to say that the effect of diabetes is the same in young and in old people. But it is not so clear whether 200 is the same effect or 900 is the same effect….

Slide 7

Now, suppose we observe indeed 900 in this group. Then on a relative scale you would say the effect is the same, diabetes works independent of age, so the conclusion then would be there is no interaction, right? If I find 900, they do more than just add the effects. So on a relative scale the conclusion on the same data would be there is interaction.

Slide 8

Likewise, if I suppose I would find only 200, then on an additive scale I would say there is no interaction, the effect of diabetes is the same in younger and in older people but at the same time on a relative scale it is not the same.

Slide 9

So the conclusion whether there is interaction or not depends on the way you look at it. It depends on the model you use, an additive model or a relative model, a multiplicative model. So it is not so clear.
Some people even say, (don’t tell anybody) that the conclusion whether there is interaction depends only on the conclusion you like best.

Slide 10

Now, we have to decide for ourselves what model is most true. That is not so easy but what I’m going to do is this 2x2 table. I’m going to re-write it in a different way and perhaps that helps us. So I have here in my young people without diabetes 10 per 1000 die, and the interpretation is that ten people die anyway no matter they have no diabetes and they are young, they die anyway. So you can give those ten people names. You can say well this is Tim, ten people like Tim, and he dies anyway being young and no diabetes. Now, there are other people who are also young but who have diabetes and there are 60 that die but within those 60 there were 10 who died anyway. So there are 50 more. So those 50 you can give them different names. There are other people those 50. Likewise when you have older people without diabetes, I find 150 to die per year but within those 150 I have my 10 who would die anyway. People like Tim who die anyway. So you have to subtract them so there are 130 additional cases. Now, you can say, what would I expect in the older patients with diabetes? I expect those ten who died anyway plus the 50 who died only because of diabetes and 140 who died only because of old age. So you have to add it up and then you arrive at 200. So this is a way to make it likely that perhaps effects do add up.

Slide 11

In fact, Ken Rothman, a famous epidemiologist, used this line of reasoning to say normally in biology most of the time we assume that risk factors do add up.
Now, Rothman says it like this and there are some other ways of explaining why perhaps factors add up. Now, suppose you have a wall here, you see my big wall here and I try to push the wall for you but it’s a very firm wall so I can’t push it, I can’t make it fall down but perhaps if you help me and we push together and then we can make it. Why can we do it together? It is just mathematics, just physics and forces add up. You could say perhaps because we know each other we do more than just adding up, perhaps we do more, we stimulate each other so we can push harder when we do it together than I could do alone. That is then interaction. We call that interaction if people interact and do more than they could do just separately. So I guess it’s plausible to say normally things add up but perhaps you should decide for yourself and in the specific context you have to make a decision yourself.

Slide 12

Now, this is the same table with those patients. If I do have the crude rates, I can add it up and I arrive at 200 while adding up. This is what I just did but I can also do it with relative risk. I have a relative measure so this was x 6 and this was x 15. What would I expect as a relative risk based on additivity? I expect 20 and why is that? Because this is the one who dies anyway. 6 times as many die because of diabetes only, so that’s 5 additional. Here it’s 15 -1,  it’s 14 additional and when you add that up you arrive again at 20. So, also if you don’t have the crude data but only the relative risk, with this small equation you can arrive at the same effect on an additive scale.
So if I observed a relative risk higher than 20, then I would say there’s interaction.

Slide 13

Now, a real world example from a study in Norway, American Journal of Kidney Disease. They wanted to know whether obesity, smoking and physical inactivity are risk factors for chronic kidney disease. We all know that. But they wondered: are the men more vulnerable? Do they have an additional problem because of being male?

Slide 14

So again the same kind of table. You start with females who are physically active and those have the lowest risk, so that’s a kind of reference. So a relative risk of 1. Then we have females who are not active and they have a higher chance to develop CKD, a 2.6 relative risk. The males who are active have a slightly higher risk than the females, 1.22. So, the males who are not active they observed in this study 2.38. Now, we want to know is this more than expected based on this additivity? Now, you can make the calculation for yourself. If you do, you arrive at 2.86. How do I do that? That is the one here, the 1.64 additional effect here, the 0.22 additional effect over here and when you add it you arrive at this. So you see that in fact, the males have a somewhat lower relative risk than expected based on adding up the separate effects. You could also say well this is more or less the same so anyway the men are not more vulnerable to lack of physical activity than the females. You can also subtract those two and make a measure out of it. So that’s a small decrease from additivity.

Slide 15

Now you can also look at the other risk factors like the activity we already saw. You can look at smoking and do it in the same way and then it’s really almost the same as adding up. You can also do it with obesity, then it’s only slightly higher for the males than the females but if you look at those three together, they arrive at the conclusion that men are not more vulnerable. Perhaps for many of us that’s quite a nice conclusion…

Slide 16

So, this concept is not very difficult that there’s additive effects and relative effects and that you can look at it differently. But you have to realise that comparing risk differences is not the same as comparing relative risks. So if it’s not so difficult why then is it so difficult in the literature and confusing? Why are there so many errors in the literature? Now, that is because of the way usually data are analysed. If you do it with tables like this then you see it’s quite easy and you can do it just on the back of a small piece of paper. But when you do it with a computer, then you are prone to running into trouble and I’ll show you why.

Slide 17

This is the most basic form of analysis, linear regression we see that people with higher age have higher diastolic blood pressure. This is kind of regression formula that some of you know. Now, if I do that separately for people with a high BMI and a low BMI, I see here that at any certain age people with high BMI have a high diastolic pressure. We know that. BMI increases your blood pressure. When it’s this line then the effect of BMI is the same independent of age. So you just add in your formula something for the BMI category. Now, it could also be that in reality this is the effect and then you have a steeper slope and then you say well the effect of body mass index is here only small and here it’s larger in total. So this is what we call interaction because BMI does more than add up to age, it is dependent  on your age what the effect of BMI is and you can make a kind of interaction term in your model to denote this extactly.

Slide 18

So here when you do a linear regression on diastolic pressure and age and BMI, you can easily see with this that there is an interaction but what you do then is look at whether it does more than just adding up. The model assumes that normally it adds up and if you include an interaction, then it does more perhaps than adding up.

Slide 19

Now, you can also do the same with another outcome and that is not diastolic blood pressure per se, but the risk to develop or to have a high blood pressure and this risk normally is denoted not by the risk but by the logarithm of the odds of the risk. Well just a kind of transformation. Then it’s the same kind of line but now the y axis is on a logarithmic scale. So again, you can look at high BMI and the effect is the same, suppose, and then you just add something but you add it on a logarithmic scale. We know on a logarithmic scale it’s different. Things tend to multiply instead of adding up. So if you look at the interaction now with an interaction in the model, then you see interaction but it’s more than an additive effect on a log scale. So on a normal scale it’s more than multiplying!

Slide 20

Here, for the people who like formulas (and for the people who don’t just skip this slide). Here in linear regression if there is no interaction, you just add the effects of age and BMI, and in logistic regression you add it on a log scale and if you get rid of the log, then it turns out to be multiplication. So if there’s no interaction in the model, linear regression assumes additive effects and logistic regression assumes multiplicative effects. The problem is that it’s not only with logistic regression but the same with Cox regression we all do for survival analysis. That’s exactly the same with the log scale.

Slide 21

So with an interaction term in the model, with linear regression you test more than additivity, the supra-additive effect. But with the logistic regression and Cox regression you test the supra-multiplicative effect, so it has something else. So the choice of the regression model you use, linear or logistic, determines the kind of interaction effect we study. That is a bit strange because you never realise that the kind of model you use makes that you have a different interpretation of the interaction term. We thus agree that you should decide yourself whether you study an additive or a multiplicative model. Now the computer decides it for you. This is not what we want.

Slide 22

So back to the real world examples. This is a study published in Circulation a couple of years ago on the effects of apolipoprotein size as a predictor of mortality in end stage renal disease, the CHOICE study, and in the abstract there is this sentence; ”The association between size and mortality was lower in the white patients, 1.3 than in black patients, 1.7”. Now, why do you think that would be the case?

Slide 23

This is what they found, a very big study so the hazard ratio of death of the small size versus the large size in the whites is 1.3, in the blacks is 1.7.

Slide 24

If we re-write this table and do it like this again, so the blacks with the high size proteins have a relative risk of 1. The blacks with a small have times 1.7. For the whites the relative risk is 1.3 but if you do it in a correct way, what would you expect based on additivity? You add it up again like we did and you arrive at 3.20. So what we see in whites is not a lower effect but the same effect or even slightly more. So their conclusion that the effect was bigger in the blacks than in the whites is not correct, if you look at it on an additive scale. At least this should have been discussed in the discussion, the interpretation. This is not what is done in the paper, so in this paper in Circulation in the discussion they have to write complicated things about why could these differences be. It’s very easy, it’s only because they have a different baseline rate.

Slide 25

Now, another example published in Kidney International a few years ago, the association between mental health, physical function especially, and haemodialysis mortality. In the abstract or in the discussion they write: against our hypothesis about the relation between physical function and age, there was a clear dose-response for physical function in the younger people but not in the older people. So the effect of physical function as a predictor of mortality is much smaller in the older people than in the younger people. Now, again, why would that be? Why would the effect of a risk factor be smaller in old people compared to young people? That depends on the way you look at it.

Slide 26

This is the graph they did and it’s a bit difficult to see but you see here almost 1 in the young people and a high and a low physical function score and it’s almost 3, so three times and here it’s less than three times and especially here. So that’s why they are saying the effect is smaller on a relative scale. But if you look at the difference, then you see there are two additionals from 1-3 and here it is from 4-6. So indeed it’s the same effect. So, there’s no significant interaction anyway if you look at it in a proper way. So be critical on how they do it, if they arrive at a conclusion about interaction.

Slide 27

Now in conclusion interaction is easier to see as a kind of effect modification because if you say interaction then your statistician will just plug it into the model and then you are lost because you don’t know the way it’s done. So if you say it’s an effect modification then you can say goodbye to all the effects that are modified. You have to decide whether you want to look at these effects on an additive scale or on a relative scale. You have to decide what model you want to have and many people go along with Rothman who says most of the time it’s additive, but feel free to make a different decision but make this decision yourself and not the computer model, so don’t let the computer fool you. And this problem arises not only with the logistic but also with the Cox regression. So I hope I have a modified a little bit the way this kind of stuff affects you. I hope I have modified it a little bit, so now you understand better what kind of effect this has on you.