The most basic technique that researchers have used to show that vaccines do not cause autism entails comparing the amount of thimerosal that children have received over their lifetime and testing whether those with more exposure are more likely to have developed autism afterward.
For example, a study done by the U.S. Centers for Disease Control and Prevention in 2010 looked at the medical records of 256 children with ASD (autism spectrum disorder) and 752 children without ASD who were born during the same years. Researchers also conducted interviews with the children's mothers, found out the lot number of the vaccines that the children each received, and determined the amount of thimerosal exposure. This study found that children who had the highest percentages of thimerosal exposure below the age of 20 months had no higher chances of developing ASD than children who had the lowest percentages of thimerosal exposure.
Similar research was conducted over hundreds of thousands of children by the World Health Organization and Institute of Medicine in 2004, and both of them concluded that there is no relationship between autism rates and thimerosal exposure.
When thimerosal was blamed for the recent "autism crisis," vaccine manufacturers were pressured into removing the thimerosal from most of their vaccines. Interestingly enough, this change became the perfect material for researchers who were looking into the connection between vaccines and autism. For example, thimerosal was eliminated from vaccines in Denmark in 1992, in Canada in 1996, and in the U.S. in 2001. (It now exists only in some flu shots, but not in typical vaccinations.) Studies — such as one published in Archives of General Psychiatry — found that the number of autism cases continued to rise long after the thimerosal was removed from the vaccines.
For example, one 2006 study at Canada's McGill University examined 28,000 children in Quebec who were exposed to varying dosages of the MMR (measles, mumps, and rubella) vaccine, as well as vaccines that contained themerosal due to differences in immunization schedules. Children born between 1987 and 1991 had mediocre levels of thimerosal exposure, with children born between 1992 and 1995 being exposed to much higher doses due to added vaccines. In 1996, thimerosal was removed from all Canadian vaccines, so children born after that point were not exposed to thimerosal-containing vaccines at all.
Despite these differences in thimerosal exposure, the number of autism cases increased nearly linearly — or at the same rate — throughout the period studied. Researchers concluded that there was no relationship between thimerosal exposure and autism rates, and they publicized similar findings about the MMR vaccine.
In February of 2009, a panel of "special masters" ruled that thimerosal and the MMR vaccine did not cause autism in three different children, all of whom had claimed that they deserved compensation from the Vaccine Injury Compensation Program. Since 2001, there had been thousands of petitions for compensation filed by parents of autistic children, and the panel had been researching the topic since 2007. While the panel did not say that research definitively proves vaccines do not cause autism, it did maintain that there was insufficient evidence to implicate the vaccines in the development of autism.
USA Today. "More Evidence That Vaccines Don't Cause Autism Arises From Study." https://www.usatoday.com/yourlife/health/medical/autism/2010-09-19-thimerosal-vaccines_N.htm
WebMD. "Study: Vaccines Don't Cause Autism." https://www.webmd.com/parenting/news/20060706/study-vaccines-dont-cause-autism
Parents. "Health Update: More Proof That Vaccines Don't Cause Autism." https://www.parents.com/baby/health/autism/health-update-more-proof-that-vaccines-dont-cause-autism/?page=2
CNN Health. "Vaccines Didn't Cause Autism, Court Rules." https://articles.cnn.com/2009-02-11/health/autism.vaccines_1_childhood-vaccines-thimerosal-autistic-disorders?_s=PM:HEALTH