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Does c’ Test Help, Anytime? -- On Communication Fallacy of “Effect to Mediate”
Unformatted Document Text:  Does c’ Test Help, Anytime? -- On Communication Fallacy of “Effect to Mediate” Page 16 of 34 P(a*b=0) < τ (0<τ<1) (Ineq. 2) 2 P(c=0) < τ (0<τ<1) (Ineq. 3) a*b*c > 0 (Ineq. 4) We use regression equations to estimate a, b, c and c’ (Baron and Kenny, 1986): M = i 1 + aX + e 1 (Eq. 1) Y = i 2 + c’ X + e 2 (Eq. 2) Y = i 3 + cX+ bM + e 3 (Eq. 3) We may call Eq. 2 “c’ equation” as its only function has been to test the significance of c’. Replacing M in Eq. 3 with the right side of Eq. 1, we have: Y = i 3 + cX+ b(i 1 + aX + e 1 ) + e 3 = (bi 1 +i 3 ) + (a*b+c)X + (be 1 +e 3 ) (Eq.4) Compare Eq. 2 with Eq. 4. We have: c’ = a*b + c (Eq. 5) 3 i 2 = bi 1 + i 3 (Eq. 6) e 2 = be 1 + e 3 (Eq. 7) As a*b and c bear the same sign (Ineq. 4), “a*b+c=0” iff “a*b=0 and c=0,” hence P(a*b+c=0) = P(a*b=0 and c=0) (Eq. 8) 2 Baron and Kenny (1986) prescribed separate a and b tests, then recommended Sobel test (1982, 1986) for a*b as an alternative, which is mathematically more accurate but was computationally more difficult at the time. Since then bootstrap has been shown to be more accurate than Sobel test (Preacher and Hayes, 2004). 3 MacKinnon, Warsi and Dwyer (1995) derived this equation.

Authors: Zhao, XinShu., Chen, Qimei. and Tong, Bing.
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Does c’ Test Help, Anytime? -- On Communication Fallacy of “Effect to Mediate” 
Page 16 of 34 
 
P(a*b=0)
 
< τ    
(0<τ<1)   (Ineq. 
2) 
2
  
P(c=0) < τ  
(0<τ<1)   (Ineq. 
3) 
a*b*c 
    (Ineq. 
4) 
 
We use regression equations to estimate  abc and c’ (Baron and Kenny, 1986): 
 
M
 
i
1
 + aX + e
1
 
   (Eq. 
1) 
Y = i
+ c’ X + e
2
 
   (Eq. 
2) 
Y = i
3
 + cX+ bM + e
3
 
 
   (Eq. 
3) 
 
We may call Eq. 2 “c’ equation” as its only function has been to test the significance of c’.  
 
Replacing M in Eq. 3 with the right side of Eq. 1, we have: 
 
Y = i
3
 + cX+ b(i
aX + e
1
) + e
3
 = (bi
1
+i
3
) + (a*b+c)X + (be
1
+e
3
)   
(Eq.4) 
 
Compare Eq. 2 with Eq. 4. We have: 
 
c’ a*b + c   
 
 
 
(Eq. 5)
3
 
i
2
  
bi
i
 
 
 
 
(Eq. 6) 
e
2
  
be
1
+
 
e
3
   
 
 
 
(Eq. 7) 
 
As a*b and c bear the same sign (Ineq. 4), “a*b+c=0” iff “a*b=0 and c=0,” hence 
P(a*b+c=0) = P(a*b=0 and c=0)  
 
(Eq. 8)
 
 
                                                 
2
  
Baron and Kenny (1986) prescribed separate and b tests, then recommended Sobel test (1982, 1986) for a*b 
as an alternative, which is mathematically more accurate but was computationally more difficult at the time.  Since then 
bootstrap has been shown to be more accurate than Sobel test (Preacher and Hayes, 2004).   
3
  
MacKinnon, Warsi and Dwyer (1995) derived this equation. 


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