On the other hand, if the null hypothesis that 
  
    
      
        
          a
          
            j
          
        
        =
        0
      
    
    {\displaystyle a_{j}=0}
   cannot be rejected, then equivalently the hypothesis of no causal effect of 
  
    
      
        
          x
          
            j
          
        
      
    
    {\displaystyle x_{j}}
   on y cannot be rejected.
As a further kind of explanation, Aristotle identified the final cause, specifying a purpose or criterion of completion in light of which something should be understood. Aristotle himself explained,

Cause means
(a) in one sense, that as the result of whose presence something comes into being—e.g., the bronze of a statue and the silver of a cup, and the classes which contain these [i.e., the material cause];
(b) in another sense, the form or pattern; that is, the essential formula and the classes which contain it—e.g. the ratio 2:1 and number in general is the cause of the octave—and the parts of the formula [i.e., the formal cause]. (c) The source of the first beginning of change or rest; e.g. the man who plans is a cause, and the father is the cause of the child, and in general that which produces is the cause of that which is produced, and that which changes of that which is changed [i.e., the efficient cause]. (d) The same as "end"; i.e. the final cause; e.g., as the "end" of walking is health. For why does a man walk? "To be healthy", we say, and by saying this we consider that we have supplied the cause [the final cause]. (e) All those means towards the end which arise at the instigation of something else, as, e.g., fat-reducing, purging, drugs, and instruments are causes of health; for they all have the end as their object, although they differ from each other as being some instruments, others actions [i.e., necessary conditions].