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# Comparing group means to a control group mean

Rutherford (2001, p.90) mentions that comparing two or more conditions which have had the same value differenced from them (e.g. a control group score) induces a correlation of 0.5 between these conditions (see here.)

This is something which is not assumed in between subjects designs where each group represents different subjects. So for example, subtracting the control score from each group score and performing a one-way ANOVA on the differences (A - Control) and (B - Control), where we have a control group and treatment groups, A and B, ignores this induced correlation.

He suggests instead using Dunnett’s test on an ANOVA involving the group means (e.g. for the Control, A and B groups) possibly as a set of planned comparisons although Dunnett's test is usually done as a post-hoc test following an ANOVA with an overall significant main effect of group to guard against obtaining false positive results. Dunnett's test is also described in Howell (1997). It is also computed for one-way between subjects designs in SPSS. Dunnett uses a modified t statistic, td, as its critical value which is also given in an appendix in Howell’s book.

One computes a t statistic of form

$$\frac{\mbox{treatment group mean – control group mean}}{\sqrt{\frac{2 MSE}{n}}}$$

where MSE is the mean square error term for the main effect of group (assuming there is no interaction) obtained from the error term in a one-way ANOVA and n is the number (assumed equal) in each group and compares this to the t statistic, td. A harmonic mean could be used if the group sizes are 'almost' equal (see Howell, 1997). The MSE corresponds to SS(subjects)/df(subjects) for a between subjects factor and SS(subjects x group)/df(subjects x group) if the groups are within subject. The SS(subjects x group) term may be worked out as given here.

Notice that the form of the above t statistic is a very similar to the Tukey test but it uses a different critical value. Dunnett's test can also be used in repeated measures ANOVA.

See also Kirk (1995) who describes th use of Dunnett’s test amongst others.

R code for evaluating the p-value for Dunnett's observed t is here. Alternatively the critical value, td, for a given alpha may also be obtained using R code illustrated here and this latter code is reproduced here.

You can also perform Dunnett's test on repeated measures data one-way ANOVA by formulating a between subjects ANOVA (see here). You would need to replace the 'TUKEY' keyword given there with 'DUNNETT' which gives the syntax as below which assumes the control group is the highest valued group and the data needs to be entered in 'long format' as for a between subjects ANOVA with a response, a within subjects factor, wit, and a subjects factor, sub.

UNIANOVA
y  BY wit sub
/RANDOM = sub
/METHOD = SSTYPE(3)
/INTERCEPT = INCLUDE
/CRITERIA = ALPHA(.05)
/POSTHOC = WIT (DUNNETT)
/DESIGN = wit sub.

References

Kirk, R. E. (1995) Experimental Design: Procedures for the Behavioral Sciences. Pacific Grove: Brooks/Cole.

Howell, D. C. (1997) Statistical Methods for Psychologists. Wadsworth,Belmont,CA. Probably in other editions as well.

Rutherford, A. (2001) ANOVA and ANCOVA a GLM approach. Wiley:New York

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