FAQ/CombiningZandT - CBU statistics Wiki

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Combining z values and 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 [attachment:combineZ.xls spreadsheet]. For, 1 <= k <= K and k-th Z value, $$z_text{k}$$

$$ \mbox{Combined Z} = \frac{ \sum_text{k} Z_text{k}}{\sqrt{K}}$$

Also, note that since chi-square(1) = $$z^text{2}$$ a sum of chi-squares each having one degree of freedom may be obtained by square rooting each chi-square(1) and entering as a z-value.

Mosteller and Bush(1954) and Rosenthal (1987) also 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 [attachment:combineT.xls spreadsheet]. For 1<= k <= K and k-th t ratio, $$t_text{k}$$ with degree of freedom $$df_text{k}$$

$$ \mbox{Combined t} = \frac{\sum t_text{k}}{\sqrt{ \frac{ \sum_text{k} df_text{k} }{df_text{k}-2} }}$$

Also, note that since $$F(1,df) = t^text{2}(df)$$ 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. The requisite chapter may be viewed [http://books.google.co.uk/books?id=zLCv3Ca2BJoC&pg=PA212&lpg=PA212&dq=lancaster+%26+1961+%26+%22p+values%22&source=web&ots=dj1zMMHTM3&sig=A8QbbKGJCmc0F7C2JH7F_NfCqkk&hl=en#PPA211,M1 here]

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)