Statistical Methods for Psychology

(Michael S) #1
problems (as measured by Achenbach’s Youth Self-Report Form [YSR] and by the Child
Behavior Checklist [CBCL]). The study represented an effort to understand risk factors for
emotional/behavioral problems in adolescents. Among the analyses of the study was the
set of intercorrelations between these variables at Time 1. These correlations are presented
in Table 12.4.
Most standard correlation programs print out a t statistic for each of these correlations.
However, we know that with 21 hypothesis tests, the probability of a Type I error based on
that standard t test, if all null hypotheses were true, would be high. It would still be high if
only a reduced set of them were true. For this reason we will apply the modified Bonferroni
test proposed by Larzelere and Mulaik. There are two ways to apply this test to this set of
correlations. For the first method we could calculate a t value for each coefficient, based on

(or take the t from a standard computer printout) and then proceed exactly as we did for the
Holm procedure). Alternatively, we could operate directly on the two-tailed pvalues asso-
ciated with the t test on each correlation. These pvalues can be taken from standard com-
puter printouts, or they can be calculated using commonly available programs. For
purposes of an example, I will use the p-value approach.
Table 12.5 shows the correlations to be tested from Table 12.4 as well as the associated
pvalues. The pvalues have been arranged in increasing numerical order. (Note that the
sign of the correlation is irrelevant—only the absolute value matters.)
The right-hand column gives the value of required for significance. For example, if
we consider 21 contrasts to be of interest,. By
the time we have rejected the first four correlations and wish to test the fifth largest, we are
going to behave as if we want a Bonferroni tadjusted for just the
remaining correlations. This correlation will be tested at
.
Each correlation coefficient is tested for significance by comparing the pvalue associ-
ated with that coefficient with the entry in the final column. For example, for the largest
correlation coefficient out of a set of 21 coefficients to be significant, it must have a proba-
bility (under ) less than .00238. Because the probability for r 5 .69 is given as
.0000 (there are no nonzero digits until the sixth decimal place), we can reject and
declare that correlation to be significant.
Having rejected for the largest coefficient, we then move down to the second row,
comparing the obtained pvalue against p 5 .00250. Again we reject H 0 and move on to the

H 0


H 0


H 0 : r= 0

a¿=a>(k 2 i 1 1)=.05> 17 =.00294

212511 = 2124 = 17


k 2 i 1 1 5

a¿=a>(k 2 i 1 1)=.05> 21 =.00238

a¿

t=

r 1 (N 2 2)

3 (1 2 r^2 )

382 Chapter 12 Multiple Comparisons Among Treatment Means


Table 12.4 Correlations among behavioral and stress measures
(1) (2) (3) (4) (5) (6) (7)
Mother
(1) Stress 1.00 .69 .48 .37 .02 .30 .03
(2) Symptoms 1.00 .38 .42 .12 .39 .19
Father
(3) Stress 1.00 .62 .07 .22 .07
(4) Symptoms 1.00 .00 .24 .20
Adolescent
(5) Stress 1.00 .11 .44
(6) CBCL 1.00 .23
(7) YSR 1.00

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