Hi Lakshmi,
I doesn't appear that your XAML was successfully added to the post. Do you have an EditorStyle set for your check editor? And if so, did you bind the Value property directly to your underlying data property? If you did you'll need to set the Mode to TwoWay so that when you click on the checkbox it updates the underlying data property triggering your ConvertBack method. I'd need to see your XAML to confirm this though.
Pleae find the xaml below, I have applied TwoWay binding at both places:
<Setter Property="Content" >
<Setter.Value>
<MultiBinding Converter="{StaticResource ApprovedByConverter}" Mode="TwoWay" UpdateSourceTrigger="PropertyChanged">
<Binding Path="DataItem.ApprovedByUserId" UpdateSourceTrigger="PropertyChanged" Mode="TwoWay"/>
<Binding Path ="DataContext.UserManagementService" RelativeSource="{RelativeSource AncestorType=UserControl}" Mode="OneWay"/> </MultiBinding>
</Setter.Value>
</Setter>