# How to calculate the confidence interval of the ratio of two means (paired data)

#### FitzChivalry

##### New Member
Hi there,

I am writing a systematic review and need to calculate the 95% confidence intervals of the ratio of means from whatever the clinical trials I'm reviewing give me.

The problem is: none of the studies actually report this information, and all I am given usually is the mean, standard deviation or standard error of the mean. As all studies are either sequential or crossover clinical trials, all data are paired.

I have found this calculator which uses a theorem I'm not familiar with called Fieller's theorem to work this out given the mean, SD or SEM, but apparently it only works for independent samples or non-paired data:

Does anyone in this community know of a way I can achieve this for paired data? I have read that you can do a paired t test of ratios but you need the raw data to take the log of all scores then the mean of these logs to get the geometric mean - but I only have what's published in the papers, i.e. mean, SD or SEM.

Thank you in advance for helping me with this - I'm desperate to get this finished!

Kind regards,

Shane

#### Disvengeance

##### New Member
I'm not sure what you mean by "ratio of means from whatever the clinical trials I'm reviewing." Could you be more specific about what you are trying to calculate from what?

#### FitzChivalry

##### New Member
Sorry about that. The trials all look at the effect of a high vegetable diet on drug metabolising enzyme activity. The 'means' I mentioned are the mean values of the drug-metabolising metric chosen for all participants before the intervention and the mean value of this metric after the intervention. Examples include the area under the concentration-time curve (AUC) for a drug or it's metabolite from 0 to 4 h post dose, etc.. Conventionally, the studies will present these AND the mean difference with a p-value. The studies have varying endpoints and I don't wish to analyse them directly via meta-analysis.

My aim is to make a Forest plot to visually summarise some sort of point estimate with common ground across the studies and its variability all in one place. Even though the endpoints may be different, I figured that if I present the ratio of the drug-metabolising metric before and after the study intervention, then regardless of methodology and metric chosen, I can visually present the size of the effect with a 95% confidence interval, as the ratio of before/after seems more meaningful than the exact value of the difference (harder to compare across different methodologies and endpoints).

Essentially I want to do what this person did:

http://www.ncbi.nlm.nih.gov/pubmed/?term=dolton+fruit+juice

If you can access the full-text for this paper, the Forest plot in Figure 2, pg 626 is what I'm trying to do. He didn't use the plot for it's usual purpose in meta-analysis - instead he compared many different studies using different interventions with different fruit juices and used this as a tool to view all studies in one place and quickly assess the size of the effect and it's confidence interval. If you can't access it, I could take a screenshot of the figure and post it here assuming I don't get sued by the journal for copyright infringement and so on.

It might be obvious that I have no idea what I'm doing! By way of background I'm a pharmacist that has gone back to do a PhD in clinical pharmacology and my knowledge of statistics is very cursory at an SPSS beginner level after a few short courses etc..

Any and all help is appreciated!

#### FitzChivalry

##### New Member
Hi, I'm not sure if you got the extra information I posted; if not it may be because I didn't click reply to your post but rather I think I just posted again on the thread.

I have since found a version of Fieller's theorem in a paper by Franz (2007) suitable for estimating the confidence interval of a ratio of two paired measurements (like in my crossover trials). However, the covariance of the two measurements is needed for this formula - I have a new question:

Can I calculate the covariance of [measurement before intervention] and [measurement after intervention] without the raw data using some combination of the mean, SD or SE of both?

Thank you

#### BGM

##### TS Contributor
Can I calculate the covariance of [measurement before intervention] and [measurement after intervention] without the raw data using some combination of the mean, SD or SE of both?
The general answer is no unless you have some specific model assumption behind.

Next time if you are doing similar multivariate analysis, you should keep the raw data if possible; if only moments are available, at least one should keep the cross moment to infer the dependency structure.

#### FitzChivalry

##### New Member
I would be keen to know more if there is a way - I have no access to the raw data unfortunately as it's a systematic review of the literature; I only have what the authors of each study publish, which is usually the mean, SD or SE of the baseline measurement, post-intervention measurement or the SD/SE of the difference in the means of these measurements.

I found this: http://handbook.cochrane.org/chapte...dard_deviations_for_changes_from_baseline.htm

They detail a way to impute the correlation coefficient for paired differences - can this statistic be used in any way to impute/estimate the covariance of the two measurements?

Sorry and thank you I am in way over my head statistically here.

#### hlsmith

##### Less is more. Stay pure. Stay poor.
curieroy, is this a new questions?

#### BGM

##### TS Contributor
Sorry just read this lately. To OP:

They detail a way to impute the correlation coefficient for paired differences - can this statistic be used in any way to impute/estimate the covariance of the two measurements?
So the basic model you have is like

$$X - Y = Z$$

And thus

$$Var[X] + Var[Y] - 2Cov[X, Y] = Var[Z]$$

$$\Rightarrow Cov[X, Y] = \frac {Var[X] + Var[Y] - Var[Z]} {2}$$

Since $$Cov[X, Y] = Corr[X, Y]SD[X]SD[Y]$$, we also have

$$Corr[X, Y] = \frac {Var[X] + Var[Y] - Var[Z]} {2SD[X]SD[Y]}$$

which is shown in the link you provided.

So you can use this, as long as this is your underlying model, and all 3 variances are available to you. The reason why I state it is not possible to obtain the covariance in the previous post is that I do not know you also have the variance of the difference.