Several meta-analyses have assessed this question by computing the Standardized Mean Difference or SMD statistic. The SMD is a measure that allows us to compare different studies. For context, the effect of stimulant medication for treating ADHD is about 0.9. SMDs less than 0.3 are considered low, between 0.3 to 0.6 medium, and anything greater than high.
A 2004 meta-analysis combined the results of fifteen studies with a total of 219 participants and found a small association(SMD = .28, 95% CI .08-.49) between consumption of artificial food colors by children and increased hyperactivity. Excluding the smallest and lowest quality studies further reduced the SMD to .21, and a lower confidence limit of .007 also made it barely statistically significant. Publication bias was indicated by an asymmetric funnel plot. No effort was made to correct the bias.
A 2012 meta-analysis by Nigg et al. combined twenty studies with a total of 794 participants and again found a small effect size (SMD =.18, 95% CI .08-.29). It likewise found evidence of publication bias. Correcting for the bias led to a tiny effect size at the outer margin of statistical significance (SMD = .12, 95% CI .01-.23). Restricting the pool to eleven high-quality studies with 619 participants led to a similarly tiny effect size that fell just outside the 95% confidence interval (SMD = .13, CI =0-.25, p = .053). The authors concluded, "Overall, a mixed conclusion must be drawn. Although the evidence is too weak to justify action recommendations absent a strong precautionary stance, it is too substantial to dismiss."
In 2013 a European ADHD Guidelines Group consisting of 21 researchers (Sonuga-Barke et al.) performed a systematic review and meta-analysis that examined the efficacy of excluding artificial colors from the diets of children and adolescents as a treatment for ADHD. While many interventions showed benefits in unblinded assessments, only artificial food color exclusion and, to a lesser extent, free fatty acid supplementation remained effective under blinded conditions. The findings suggest that eliminating artificial food dyes may meaningfully reduce ADHD symptoms in some children, though it should be noted that the positive results were mostly seen in children with other food sensitivities.
The research to date does suggest a small effect of artificial food colors in aggravating symptoms of hyperactivity in children, and a potential beneficial effect of excluding these substances from the diets of children and adolescents, but the evidence is not very robust. More studies with greater numbers of participants, and better control for the effects of ADHD medications, will be required for a more definitive finding.
In the meantime, given that artificial food colors are not an essential part of the diet, parents could consider excluding them from their children's meals, since doing so is risk-free, and the cost (reading labels) is negligible.