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Department of Neurology Neurosciences Centre, and Clinical Epidemiology Unit, All India Institute of Medical Sciences, New Delhi Delhi, India

Abstract
Let us begin with a story. Some years ago, banks came up with a new concept for fixed deposits. The concept was to give interest on interest. This was called ‘compound interest’. Before this concept, the interest was calculated only on the principal – and the approach was called ‘simple interest’. One may call them ‘simple interest model’ and ‘compound interest model’. As you can see, these are two different concepts underlying the computation of the maturity amount. In the simple interest model, there is only one source of interest, i.e. the principal, whereas in the compound interest model, there are two sources of interest: the principal as well as the interest earned periodically.

## Fixed and Random Effects Models

Let us begin with a story. Some years ago, banks came up with a new concept for fixed deposits. The concept was to give interest on interest. This was called ‘compound interest’. Before this concept, the interest was calculated only on the principal – and the approach was called ‘simple interest’. One may call them ‘simple interest model’ and ‘compound interest model’. As you can see, these are two different concepts underlying the computation of the maturity amount. In the simple interest model, there is only one source of interest, i.e. the principal, whereas in the compound interest model, there are two sources of interest: the principal as well as the interest earned periodically.
Computation is based on different formulae such that the formula for simple interest uses only the principal as the source of interest and in the compound interest model, there are two sources of interest – first, the principal and, second, the interest earned over certain fixed periods (say, every 3 months).
The practical difference between the two is that the maturity amount is more in the compound interest than in the simple interest. Thus, you can see that the differences between simple interest model and compound interest model are conceptual, computational and practical. Similarly, the differences between fixed and random effects models may be described as:

• Conceptual
• Computational
• Practical
Conceptually, in the fixed effects model, the studies are believed to estimate the same underlying true effect, and all the differences in the effect estimates are due only to differences in the sample size. In random effects model, the studies estimate an effect that in itself is assumed to be a random variable with a certain distribution. Alternatively, in random effects model, the studies conceptually represent a sample of a large (infinite) number of possible studies on the questions, whereas in fixed effects model, the studies at hand represent the total number of studies on the question. Put another way, we consider the sample as the only reason for differences in results among the studies, whereas in random effects model, we consider a sample as one reason for differences but also consider that there are other reasons for the differences that are not known.
Computationally, in fixed effects model, only one source of differences in the effect measure is considered, that is, the sample size, whereas in the random effects model, two sources of differences are considered, viz. the sample size and the variation of the effects between the studies.
Practically, the random effects model yields a wider confidence interval than the fixed effects model, and the point estimate may be only slightly different. Moreover, in the fixed effects model, the small studies get relatively more weight than in random effects model. Some experts argue for using mixed models, in which known reasons for differences in effects are considered and treated as ‘fixed’, and in addition consideration is given to some reasons as yet unknown, for the differences. I have not seen any meta-analysis in medicine using the mixed model, and commonly used software does not have this feature.