The Authoritarian personality is characterised by strict adherence to conventional values and ablief in absolute obedience or submission to authority. It is a disposititional explanation for why some individuals need little pressure in order to obey.
Milgram believed that obedience was due to situtaional factors, he found that proximity of the victim and location of the study were the main causes of differences in rates in obedience, not variations in personality. Relying purely of authoritarianism as an explanation for obedience is inflexible and doesn't take these other factors into account.
Elms and MIlgram found differences in characteristics of the authoritarian personality and those of fully obedient participants. They found those fully obedient had healthy postive relationships with there parents rather than the overly strict family environment assciated with authoritarianism, suggesting this isn't the only explanation for obedience.