ISSUE: How to make a generalization of "beta coefficients"?
META ANALYSIS: The function of meta analysis is to make a generalization of various findings by "many" researchers on the same (almost the same) subject matter. Therefore, whatever field the subject mater may e in, the objective of meta analysis is the same: what general statement can we make about "these findings"?
BETA COEFFICIENT of what? from where? how they were derived? ... in what field? These questions are moot. Assume that you have these "beta coefficients" given as a set of data; let's call the set of "beta coefficients" = Zi: (z1, z2, ..., zn). Our objective is to verify if a general statement could be made about Xi; (beta coefficient series from various research publications). For this, we will use WEIBULL DISTRIBUTION. Our objective is to find H(t) and S(t).
Step 1. With Zi, find F(t) where;
(1) F(t) = (i - 0.30) / (n + 0.4) .... assume that n < 100, otherwise
(2) F(t) = i / (n + 1) .... if the sample size is n > 100
Step 2. Construct QQ plot, first determine X array and Y array in order to construct Weibull linear equation by:
(3) Xi = ln(ln(1 / (1 - F(t))))
(4) Yi = ln(Zi)
Now we have Xi and Yi. Apply conventional definition that Xi = independent variable and Yi = dependent variable. With any statistical program, derive the linear equation: Y = a + bX. If you do not have software, then do it by hand by following these simple basic statements:
I = nΣXY - (ΣX)(ΣY)
II = nΣX2 - (ΣX)2
III = nΣY2 - (ΣY)2
The slope is determined by: b = I / II and the Y-intercept is given by:
(5) a = Y^ - bX^
... where Y^ = mean of Y array and X^ = mean of X array. At this point, you have constructed a linear regression equation: Y = a + bX.
Step 3. Find Weibull Statistics.
(6) B = 1 / b (note: this is Weibull beta, NOT your beta coefficients)
(7) Eta = ea
(8) CDF = 1 - e-A
... where A = (X / B)1/B ... where X = value you want to evalue
(9) PDF = JKL
(9.1) J = (B / etaB)
(9.2) K = XB-1
(9.3) L = e-M ... where M = (X/eta)B
Now find the reliability of the system:
(10) R = 1 - CDF
This R is the system reliability. How reliable is the system to generate or produce your "Zi = beta coefficients" that you have.
(11) H(t) = (B / eta)(X / eta)B-1
This H(t) is the instantaneous failure of the system. You can say that of all the research articles that you review, i.e. these are your "system", you expect to see its normal failure rate as H(t) value. This is the value of the counter-argument that your meta-analysis of "beta coefficient" would fail immediately. The opposite of it is the survival rate: S(t) which is simply:
(12) S(t) = 1 - H(t)
You may use this S(t) value as an equivalence to what is commonly called "effect size" in general meta analysis language and H(t) is the opposite.
WHAT ABOUT THE TIME FACTOR: H(t) and S(t) is time dependent---so too were your articles published. The data were not themselves time dependent. True, but they are produced at different time; for purpose of viewing the "production of research publication" as a generator of data for the meta analysis, it is suffice for this approach.
REFERENCES: See attached.
Thanks Paul I am working on your suggestions and will let you know once I am done
Hi Nishant,
I do not know the details of your study, but I am assuming you are interested in combining beta coefficients from different studies into a pooled estimate. I am attaching 2011 JAMA paper which combined the adjusted regression coefficients from the Cox PH model. The authors could not combine hazard ratios directly due to within-study variable waiting time for adjuvant chemo therapy, hence their proposed solution. Hope you find this helpful. Note the Greenland paper they cite for the calculation of standard error for the beta coefficient.
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