# Additive effect, how to calculate?

#### drgunnar

##### New Member
Dear users,

I hope you have some good ideas about the following Medical dilemma.

I'm conducting a study about Three drugs, A + B + C.

Drug A have been shown to reduce mortality by 20 % alone.
Drug B have been shown to reduce mortality by 10 % alone.
Drug C have been shown to reduce mortality by 90 % alone.

In my study the Three drugs are used together and now i want to calculate the additive effect.

90+10+20 (%) = 110 % so that is not correct.

How would you calculate the additive effect ?

Best regards
Gunnar

#### hlsmith

##### Less is more. Stay pure. Stay poor.
This topic is typically presented for two treatments or variables when examining their tandem benefit. You would insert the individual risks for mortality into a 2x2 table as cell A (0,0), B(1,0), C(0,1), D(1,1), with the first cell A being risk of mortality with no drugs; second cell B being risk of mortality in drug A only; cell C being risk of only drug B; and cell D being the risk of mortality when both drug A and B are present.

Then one option is to:

p11 - p10 - p01 + p00 and see if it is > 0.

or if (B -A) + (C - A) equal or greater than (D - A).

Did your patients only receive all three treatments? You do not have patients with no treatments or just a single group with dual treatment? You are looking to use available data on the individual treatments from other samples, correct?

#### hlsmith

##### Less is more. Stay pure. Stay poor.
Drug A have been shown to reduce mortality by 20 % alone.
Drug B have been shown to reduce mortality by 10 % alone.
Drug C have been shown to reduce mortality by 90 % alone.
Do you know the actual risk for mortality for these three drugs? These results appear to be for Drug A versus no Drug A ([risk for mortality in drug A patients / risk for mortality in no drug A patients] = 0.80),... etc. These are not survival analysis data are they (were they generated with time to event analysis [e.g. proportional hazards regression])?

Also, do you know the risk for mortality in patients on no drug and is this background risk the same in all samples (if data coming from multiple samples)?

#### drgunnar

##### New Member
Thanks for your reply but this is not the answer i´m looking for. No in my case I only have the data I told you.

I have a population (1000) of men 65 years old with cardiovascular diseases. I also have Controls (1000) healthy 65 years old men. I know that the patients with cardiovascular diseases have twice as high mortality compared with the Controls.

Now I have data from old studies that the 3 drugs (a+b+c) have been shown to reduce all cause mortality alone by the percentage mentioned above. Now i want to add the effect of all 3 drugs in my study.

In my World and also what has been proven (research) correct is this.
(1-0.2) x (1-0.1)x (1-0.9) = x then 1-x

#### noetsi

##### No cake for spunky
If you run a regression model you will get the unique effect of any drug controlling for the others (which means in practice the impact on mortality caused exclusively by one of the drugs).

#### hlsmith

##### Less is more. Stay pure. Stay poor.
You are using multiplication, which will likely not place you on an additive scale. What is the reduced mortality for your study patients on Drugs A, B, and C? Is your study prospective and are you using relative risks, time to event, or odds ratios. Are you controlling for any covariates?

What I was trying to get at is whether the reduced mortality from drug A, B, or C individually, from other studies, are coming from comparable patient samples as your sample. In particular, I doubt they are comparing a drug treatment group to a healthy comparison group without any risk for ASCVD. Also, was their study design comparable, did they prospectively follow patients and report relative risks or odds ratios based on proportional hazards regression (time to event analyses), which likely controlled for covariates if treatment assignment was not randomized. Lastly, is it possible to determine the background risk, for I am assuming, a cardiac event related mortality in the comparison studies that examined the effectiveness of drugs A or B or C. If the background risk is not comparable to your study or if they had a different study design or controlled for covariates, or had randomization, or a comparison group at risk for ASCVD then in my world, a calculation for additive effects would be VERY biased and not conscientiously appropriate to report without a lengthy limitations clause.

But it might be fun to make this calculation, just for the sake of entertainment!

#### hlsmith

##### Less is more. Stay pure. Stay poor.
Acknowledging that you likely cannot make this calculation for many reasons (at least not until you clarify the context of your question and what data you have), for fun see the following.

You use the phrase "reduce mortality risk by 20%". Is this a risk difference?

Risk difference: (risk for mortality for patients on drug A - risk for mortality for patients not on drug A). If so, we will call this RD1.

Now I believe (not positive, since I have not done this with 3 treatments) that you can do the following if you have the RD2 (risk difference for drug B), RD3, and RD1,2,3(your risk difference value):

RD1,2,3 - RD1 - RD2 - RD3, which is the risk difference additivity. If the value equals "0" then that is what would be expected, given they all contribute a certain affect. if the value is > "0" then there is additive interaction. If it is < "0", subtractive relationship or possible indirect affects occurring or overlap in the biological mechanism.

P.S., The one place that I have apprehensions (just a little), would be if you need to subtract out RD1,2; RD1,3; and RD2,3; which would not be available. It might be interesting to also calculate those values to see if an additive interaction is present when only two drugs are present and it may help to deconstruct the RD1,2,3.

#### hlsmith

##### Less is more. Stay pure. Stay poor.
In my World and also what has been proven (research) correct is this.
(1-0.2) x (1-0.1)x (1-0.9) = x then 1-x
Well, for another clarification, 'if reduce mortality by 20%' was in reference to relative risk of 0.80, for patients on drug A relative to patients not on drug A. Then see following:

Relative Excess Risk due to Interaction (RERI): RR(of patients on A, B, C) - RR(patients on A) - RR(patients on B) - RR(patients on C) + 1

Attributable Proportion due to Interaction (AP): RERI / RR(of patients on A, B, C)

and

Synergy Index: (RR(patients on A, B, C) - 1) / ( (RR(patients on A) - 1) + (RR(patients on B) - 1) + (RR(patients on C) -1)) )

P.S., Also not sure if these equations require the subtraction of the AB, AC, BC patients or not - since I have not done these with 3 groups before. Also, you would need all studies to be comparable to do these calculations. Ideally, all data would come out of the same sample and treatments would be randomized. I posted this because the Synergy Index was close to your posted formula. However, it is on the multiplicative scale not additive scale, which may not be apparent in the formula.

P.S.S., Alright, I keep working then spacing out thinking about this question. I think my reservation (e.g., P.S.) could be answered by seeing what gets used in the denominator of three-way multiplicative interactions that are in logistic regression models. I may look that up later tonight or tomorrow.

#### GretaGarbo

##### Human
Is this "drgunnar" the same person as "drgunnar84"?

You are both talking about "In my World".

In the post by "drgunnar84", Miner said:

"There are usually just a few best approaches to solving a problem when you have sufficient information. However, when you are given a vague description, you can only provide a vague answer."

Dr Gunnar! If I come to you as a patient and I say: "I have some pain. Give me medicine!" Do you then have sufficient information to decide about a treatment?

What we have tried to say is that you must be much more specific and less vague.

#### Karabiner

##### TS Contributor
90+10+20 (%) = 110 % so that is not correct.
How would you calculate the additive effect ?
It is not possible. The result can be anything within the
boundaries from 90% to 100%.

For example, if B is a medication which only has
an effect where already A has an effect (only that B is
much less effetive than A), then A+B would not be
able to increase survival, compared to A-only.

But if B is a medication which has an effect only
in those patients where A in't effective, then
A+B would increase survival, compared to A-only.

Or something in between.

Without further data on how much the effects of both
medications overlap, one can only describe the boundaries

Just my 2pence

K.

#### ArtK

##### New Member
It is not possible. The result can be anything within the
boundaries from 90% to 100%.

For example, if B is a medication which only has
an effect where already A has an effect (only that B is
much less effetive than A), then A+B would not be
able to increase survival, compared to A-only.

But if B is a medication which has an effect only
in those patients where A in't effective, then
A+B would increase survival, compared to A-only.

Or something in between.

Without further data on how much the effects of both
medications overlap, one can only describe the boundaries

Just my 2pence

K.
Isn't the situation even more "impossible" than that? What if A and C together
are less effective than C taken alone, for example? How can anyone know about
combination effects of medications without actually conducting studies?

Art

#### drgunnar

##### New Member
Thanks for all complicated answers, i will try again:

Imagine a wizard, he will give me 3 magic pills. He says that the pills are independant of each other, does not work in the same way but will all to the same work. He says that the pills do not interact to each other. He can't explain how the tablets work, all he can say is that it will decrease the lenght of the index finger (by the person who will eat it).

The wizard shows me 3 tablets, they all look the same, and says the following.

tablet A have been shown to reduce the size by 20 % alone.
tablet B have been shown to reduce size 10 % alone.
tablet C have been shown to reduce size by 90 % alone.

How will i calculate the additive effect?

Let us say my index finger is 100 cm (try to imagine it)

1. 100 x 0,9 = 90. 90 x 0,8 = 72. 72x 0,1 =7,2 cm
2. any other ideas?

#### Karabiner

##### TS Contributor
This is a bit annoying. Instead of now telling stories about wizards and
magic pills, you could have told us from the start that you assume
complete independence, i.e. you don't deal with a real medical
problem (in which such an assumption no-one would sensibly make),
but a made-up problem. Of course, if you have three completely
independent multiplicators, the calculation is easy.

#### hlsmith

##### Less is more. Stay pure. Stay poor.
That was a nice analogy, but it flat out does not address your original question. I am also pretty sure that your new question would never come up on your board examination. The biggest reason being you absolutely cannot just add studies together without examing comparability, that is beyond silly.

But you now have an answer to your non-evidence based medicine question, statins and antihypertensives for all.

To get a remote idea of what you should have been looking at, I would reccommend reviewing meta-analysis literature. They just don't jam a bunch of studies together without examining sources of biases and hetergeneity, and those Cochrane Reviews should be your highest level of evidence.