Recording user response times for a quiz
Just some simple stuff, 
a common.php file for common code used by both files. DRY don't repeat yourself.

Writing a script like you have is fine, the points i have noted will help you if you come back to maintain it in a year or so, and have forgotten how it works. 

The bulk of programming jobs are about maintaining a code base, not creating new stuff, so if you can learn how to make maintenance easier, it will be beneficial for you in the long run. (I wish i had learnt this stuff much earlier on)

I have added inline comments explaining the changes i made

&lt;?php
/* common.php */
require 'classes/Database.php';

session_start();
date_default_timezone_set("UTC");

?&gt;



&lt;?php
/* get_game_data.php
 *
 * user clicks button to start game
 * and triggers ajax request sent to this file
 */

require 'common.php';

// moved into common.php
// session_start();

if($_SERVER['REQUEST_METHOD'] === 'GET')
{
    if(isset($_GET['gamesettings'])){///check certian user defined game options are set

        // moved into common.php
//        require 'classes/Database.php';

        // moved into common.php
//        date_default_timezone_set("UTC");

        $_SESSION['app_start'] = new DateTime('NOW');////store time at which game/first question starts
        $_SESSION['app_index'] = 0;

        // warning if $_GET['app'] not set this will get a warning notice, we can never trust GET/POST data
        // if($_GET['app'] === 'aural') {
        if(isset($_GET['app']) &amp;&amp; $_GET['app'] === 'aural') {

            $db = new Database();

            $gameData = getGameData($db);///just an example
            //etc...
            //query database for game data, i.e. the questions,

            // i prefer to do my internal logic before i output data. this way if something fails i can handle it prior to responding to the user
            $_SESSION['right_answers'] = $gameData['answers'];///store the right answers to varify against user answers later

            echo json_encode($gameData['questions']);//then send back to browser



        }

    }
}
?&gt;



&lt;?php
/* post_game_data.php
 *
 * user answers question in a textbox etc.
 * then clicks a submit button which posts the answer to
 * this file via ajax again
 */

require 'common.php';

// moved into common.php
//session_start();




// moved into common.php
// date_default_timezone_set("UTC");

if($_SERVER['REQUEST_METHOD'] === 'POST')
{
    // what you have done is not bad, i just prefer to have all user supplied data handled in one place, POST/GET
    // so i know that everything beyond here has been sanitized
//    if(
//       isset($_POST['user_answer']) &amp;&amp;
//       isset($_SESSION['app_index'])
//    )

    $userAns = isset($_POST['user_answer']) ? $_POST['user_answer'] : null;

    // good variable names can also help reduce the need for comments
    // which of these two following variable names are easier to understand

    // $index = $_SESSION['app_index'];///current question in the game
    $current_question_index = isset($_SESSION['app_index']) ? $_SESSION['app_index'] : null;

    if($userAns != null &amp;&amp; $current_question_index != null)
    {

        // same here with variable naming
        // if you are going to maintain these scripts over time, you need to make it easier to remember what is going on

        // $len = count($_SESSION['words']);///number of questions in the game
        $num_questions = count($_SESSION['words']);

        // handled above, no need to repeat
        //      $userAns = $_POST['user_answer'];

        $rightAns = $_SESSION['right_answers'];


        // why use a comment here that could change over time when you can make an easily understandable function name instead
        // if(isset($_SESSION['app_start']))////if this is the first question
        if (is_first_question())
        {
            $responseTime = getTimeTaken($_SESSION['app_start']);
            unset($_SESSION['app_start']);

            // $_SESSION['question_start'] = new DateTime('NOW');
            start_question_timer();
        }
        // else if(isset($_SESSION['question_start']))///not the first question        
        else if(is_question_started())///not the first question
        {
            $responseTime = getTimeTaken($_SESSION['question_start']);

           // $_SESSION['question_start'] = new DateTime('NOW');
            start_question_timer();
        }

        // which of these two is going to be easier to understand next year when you come back to maintain your script
        if ($current_question_index &lt; $num_questions) {
    //    if($index &lt; $len)
        {
            $crtAns = $rightAns[$current_question_index];

            // this is probably overkill in this case, but again why write a comment when you can write a nice function name
            // $mark = ($userAns == $crtAns)?1:0;///check user answer against correct answer
            $mark = mark_user_answer($userAns, $crtAns);

            // easy function names to make the code more understandable
            // $_SESSION['app_index']++;
            move_to_next_question();

            // given your if {} elseif {} statement above, $responseTime may not have been set, as there is no default case
            // this should be handled properly
            echo json_encode(array($crtAns, $mark, $responseTime));////send right answer to broswer with mark and response time
        }


    }

}


// move functions below main code as what is most important should come first
// perhaps this function could be called elapsedTimeInSeconds, 
// get infers you are getting a value, and there is no indication what the return type might be
function getTimeTaken($startDTO) {

    // good variable naming, easy to understand what is happening
    $endDTO     = new DateTime('NOW');
    $timeDiff   = $startDTO-&gt;diff($endDTO);
    $timeTaken  = $timeDiff-&gt;format('%s');

    return $timeTaken;
}


function is_first_question() {
    return isset($_SESSION['app_start']);
}

function is_question_started() {
    return isset($_SESSION['question_start'];
}

//
function mark_user_answer($user_answer, $correct_answer) {
    return ($user_answer == $correct_answer) ? 1 : 0;
}

function move_to_next_question() {
    $_SESSION['app_index']++;
}

function start_question_timer() {
    $_SESSION['question_start'] = new DateTime('NOW');}
}

?&gt;