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location: FAQ / CombiningZandT

Summing z values and summing t values

Following Winer (1971) and Rosenthal (1987) the sum of z values divided by the square root of the number of terms being summed follows a Normal distribution. This approach known as Stouffer's method may be computed using this spreadsheet. For, 1 $$\leq$$ k $$\leq$$ K and k-th z value, $$z_text{k}$$

Combined z = sum over k z(k)/sqrt(K)

Also, note that since chi-square(1) = z2 a sum of chi-squares each having one degree of freedom (df) may be obtained by square rooting each chi-square(1) and entering as a z-value.

Mosteller and Bush(1954) and Rosenthal (1987) also use the formula presented here to show how t statistics (each with degrees of freedom over 2) may be summed to obtain a combined t value. These computations may be performed using this spreadsheet. For 1 $$\leq$$ k $$\leq$$ K we have the k-th t ratio, t(k), with degree of freedom df(k)

Combined t = sum over k t(k) / Sqrt [ Sum over k df(k)/(df(k)-2) ]

Also, note that since F(1,df) = t2(df) a sum of F statistics whose first degree of freedom is one may be evaluated by square rooting and treating as a t statistic shose degrees of freedom equal the second F statistic degree of freedom.

t statistics with more than 30 degrees of freedom closely resemble z values so there will be little difference in results summing t statistics with over 30 degrees of freedom and summing z values.

References

Mosteller F, Bush RR (1954) Selected quantitative techniques. In: Lindzey G, ed. Handbook of Social Psychology, Cambridge, Mass: Addison-Wesley, pp 289-334.

Rosenthal R (1987) Judgement studies design analysis and meta-analysis. CUP:Cambridge.

Winer, B.J. (1971) Statistical principles in experimental design (2nd ed.). New York: McGraw-Hill. (1971 and 1991 editions of this book are available in the CBU library)