As he seeks re-election, Mr. Trump has showcased his role in fulfilling the Republican judicial agenda. One afternoon last November, he gathered an array of Republican leaders and conservative judicial activists to celebrate his success.

“I’ve always heard, actually, that when you become president, the most — single most — important thing you can do is federal judges,” he said.

How the Numbers Were Calculated

In examining the president’s judicial appointments, The Times compiled two databases of information about judges who were named to the U.S. Court of Appeals by Mr. Trump and his predecessors.

One focused on the professional and political backgrounds of judges appointed by Mr. Trump, Mr. Obama and Mr. Bush. The other analyzed published opinions in the court’s 12 regional circuits to gain insights into ruling patterns and rates of dissent.

The Biographies

The Times compared the appellate judges’ experiences outside the court. All told, there were 168 appointees — 51 by Mr. Trump, 55 by Mr. Obama and 62 by Mr. Bush.

The database drew primarily on biographical questionnaires the appointees had submitted to the Senate Judiciary Committee, obtained from staff members, the Congressional Record and other sources. They listed jobs and internships held since college, judicial clerkships, club memberships, affiliations with political campaigns and other information. Some judges volunteered more detail than others.

Separately, campaign finance data was compiled from two sources: the National Institute on Money in Politics, which has access to state donations since 2000 and federal ones since 2010, and the Center for Responsive Politics, which tracks federal donations beginning in 1989. In searching the donations, The Times sometimes found matches by using variations of judges’ names, including maiden names, as well as other relevant information like employment.

Calculations of partisan donations were based on federal contributions to political candidates or causes of the same party as the judge’s appointing president. Past political activity was measured more broadly and included work for politicians of any party; volunteer or paid work for political campaigns of any party; memberships affiliated with any party; donations to campaigns of any party; participation as a candidate for any party; references to “Republican” or “Democrat” in any answer in the questionnaire; and work in a political post in the federal government, including political duties assigned to a federal employee.

The age of judges on their appointment date was based on years of birth provided by the Federal Judicial Center, the official clearinghouse for court research.

The Rulings and Dissents

The database includes more than 10,000 opinions published from 2017 through last year in the 12 regional circuit courts. The 13th appeals court, the Federal Circuit, hears mostly intellectual property cases and has no Trump appointees.

The case list was published by the Federal Judicial Center. Only cases designated “published, written and signed” were included in the analysis, because they carry the weight of precedence and represent the most legally impactful work. For consistency, all of the cases involved a standard three-judge panel with a named opinion author.

For every case, The Times parsed the text of the opinion to identify the judges, whose names are redacted from the judicial center’s data. Additional information about the judges was obtained by joining the case data to a separate biography data set kept by the center.

The data was analyzed in two ways: first, to determine how often cases involved a dissent, and second, to determine how often individual judges agreed or disagreed with their two colleagues on a panel.

On the case level, the data showed that when a judge named by Mr. Trump served in a pivotal role — as the author of an opinion on a panel with only one Democratic appointee, or as the only Republican appointee on a panel — the rate of dissent increased significantly.

For individual judges, the analysis split each panel into three pairings. If the case was unanimously decided, all judges were deemed to have agreed. If one judge dissented, that judge was deemed to have disagreed with the other two. While judges appointed by presidents of different parties were more likely to disagree than judges appointed by presidents of the same party, the difference was far more pronounced for many, though not all, of the new Trump appointees, the analysis found.

There were caveats to the findings. Some of the circuits have a higher dissent rate overall, for example, and some circuits appear more often in the database because they conduct a higher share of their work in the form of published opinions.

Even accounting for those factors, the findings were supported by a separate regression analysis, which accounted for other variables, including the circuit hearing the case, the topic before the court, the type of appeal and whether the ruling affirmed or overturned a decision by a lower court.

Hillary Flynn and Jaclyn Peiser contributed reporting. Jack Begg contributed research.