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| = Summing z values and summing t values = | |
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| = Combining z values and t values with large number of degrees of freedom = | 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 $$\leq$$ k $$\leq$$ K and k-th z value, $$z_text{k}$$ |
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| Following (for example Howell (2002) p.144) the square of a Z value follows a chi-square distribution on 1 degree of freedom. Moreover the sum of N Z values follows a chi-square distribution on N degrees of freedom. | $$ \mbox{Combined z} = \frac{ \sum_text{k} z_text{k}}{\sqrt{K}}$$ |
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| Since a T value with a large number of degrees of freedom (say df > 30) closely follows a standard Normal distribution the above also applies to combining T statistics with large numbers of degrees of freedom. | Also, note that since chi-square(1) = $$z^text{2}$$ 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. |
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| Both these approaches are useful in combining results from multiple testing e.g. t statistics assessing brain activities at different time points. | 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 $$\leq$$ k $$\leq$$ 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) |
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 [attachment:combineZ.xls spreadsheet]. For, 1 $$\leq$$ k $$\leq$$ 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 (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 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 $$\leq$$ k $$\leq$$ 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)