Statistics Toolbox

Example: N-Way ANOVA with Large Data Set

In the previous example we used anova2 to study a small data set measuring car mileage. Now we study a larger set of car data with mileage and other information on 406 cars made between 1970 and 1982. First we load the data set and look at the variable names.

• load carbig
  whos
    Name               Size         Bytes  Class
    Acceleration     406x1           3248  double array
    Cylinders        406x1           3248  double array
    Displacement     406x1           3248  double array
    Horsepower       406x1           3248  double array
    MPG              406x1           3248  double array
    Model            406x36         29232  char array
    Model_Year       406x1           3248  double array
    Origin           406x7           5684  char array
    Weight           406x1           3248  double array
    cyl4             406x5           4060  char array
    org              406x7           5684  char array
    when             406x5           4060  char array
  

We will focus our attention on four variables. MPG is the number of miles per gallon for each of 406 cars (though some have missing values coded as NaN). The other three variables are factors: cyl4 (four-cylinder car or not), org (car originated in Europe, Japan, or the USA), and when (car was built early in the period, in the middle of the period, or late in the period).

First we fit the full model, requesting up to three-way interactions and Type 3 sums-of-squares.

• varnames = {'Origin';'4Cyl';'MfgDate'};
  anovan(MPG,{org cyl4 when},3,3,varnames)
  ans =
      0.0000
         NaN
           0
      0.7032
      0.0001
      0.2072
      0.6990
  
  
  

Note that many terms are marked by a "#" symbol as not having full rank, and one of them has zero degrees of freedom and is missing a p-value. This can happen when there are missing factor combinations and the model has higher-order terms. In this case, the cross-tabulation below shows that there are no cars made in Europe during the early part of the period with other than four cylinders, as indicated by the 0 in table(2,1,1).

• [table,factorvals] = crosstab(org,when,cyl4)
  table(:,:,1) =
      82    75    25
       0     4     3
       3     3     4
  table(:,:,2) =
      12    22    38
      23    26    17
      12    25    32
  factorvals = 
      'USA'       'Early'    'Other'
      'Europe'    'Mid'      'Four' 
      'Japan'     'Late'          []
  

Consequently it is impossible to estimate the three-way interaction effects, and including the three-way interaction term in the model makes the fit singular.

Using even the limited information available in the ANOVA table, we can see that the three-way interaction has a p-value of 0.699, so it is not significant. We decide to request only two-way interactions this time.

• [p,tbl,stats,termvec] = anovan(MPG,{org cyl4 
  when},2,3,varnames);
  termvec'
  ans =
       1     2     4     3     5     6
  
  
  

Now all terms are estimable. The p-values for interaction term 4 (Origin*4Cyl) and interaction term 6 (4Cyl*MfgDate) are much larger than a typical cutoff value of 0.05, indicating these terms are not significant. We could choose to omit these terms and pool their effects into the error term. The output termvec variable returns a vector of codes, each of which is a bit pattern representing a term. We can omit terms from the model by deleting their entries from termvec and running anovan again, this time supplying the resulting vector as the model argument.

• termvec([4 6]) = []
  termvec =
       1
       2
       4
       5
  anovan(MPG,{org cyl4 when},termvec,3,varnames)
  
  
  

Now we have a more parsimonious model indicating that the mileage of these cars seems to be related to all three factors, and that the effect of the manufacturing date depends on where the car was made.

Example: N-Way ANOVA with Small Data Set

Multiple Linear Regression