You need to place your code in a $(document).ready(function(){});
<script>
$(document).ready(function(){
$('#txtLob').change(function () {
if ($('#txtLob').val() == "United Traditional") {
$('#txtSubLob').remove();
$('#lineOfBusiness').append("<select name='txtSubLob' id='txtSubLob'><option value='None'>None</option><option value='United CAPSIL'>United CAPSIL</option><option value='United LMF'>United LMF</option><option value='United Vantage'>United Vantage</option><option value='United DI Rider'>United DI Rider</option></select>");
} else if ($('#txtLob').val() == "Universal Life") {
$('#txtSubLob').remove();
$('#lineOfBusiness').append("<select name='txtSubLob' id='txtSubLob'><option value='None'>None</option><option value='CAPSIL'>CAPSIL</option><option value='Vantage'>Vantage</option><option value='TAG'>TAG</option><option value='VUL'>VUL</option></select>");
} else if ($('#txtLob').val() == "Companion Traditional") {
$('#txtSubLob').remove();
$('#lineOfBusiness').append("<select name='txtSubLob' id='txtSubLob'><option value='None'>None</option><option value='Companion CAPSIL'>Companion CAPSIL</option><option value='Companion LMF'>Companion LMF</option></select>");
} else if ($('#txtLob').val() == "Health Reinsurance Ceded") {
$('#txtSubLob').remove();
$('#lineOfBusiness').append("<select name='txtSubLob' id='txtSubLob'><option value='None'>None</option><option value='NHALR_DLR'>NHALR_DLR</option><option value='CI Ceded'>CI Ceded</option><option value='Western and Southern'>Western and Southern</option></select>");
} else if ($('#txtLob').val() == "Med Supp Assumed") {
$('#txtSubLob').remove();
$('#lineOfBusiness').append("<select name='txtSubLob' id='txtSubLob'><option value='None'>None</option><option value='Internal ADM MS Assmd'>Internal ADM MS Assmd</option><option value='External ADM MS Assmd'>External ADM MS Assmd</option></select>");
} else {
$('#txtSubLob').remove();
}
}).trigger('change');
});
</script>
Fiddler does this for you, so that's why it works there but not on your page.