The data in this example are measurements of 159 fish caught in Finland’s Lake Laengelmaevesi; this data set is available
from the Puranen. For each of the seven species (bream, roach, whitefish, parkki, perch, pike, and smelt), the weight, length, height, and
width of each fish are tallied. Three different length measurements are recorded: from the nose of the fish to the beginning
of its tail, from the nose to the notch of its tail, and from the nose to the end of its tail. The height and width are recorded
as percentages of the third length variable. The fish data set is available from the `Sashelp`

library.

The following step uses PROC CANDISC to find the three canonical variables that best separate the species of fish in the `Sashelp.Fish`

data and create the output data set `outcan`

. When the NCAN=3 option is specified, only the first three canonical variables are displayed. The ODS EXCLUDE statement excludes
the canonical structure tables and most of the canonical coefficient tables in order to obtain a more compact set of results.
The TEMPLATE and SGRENDER procedures create a plot of the first two canonical variables. The following statements produce
Figure 31.1 through Figure 31.6:

title 'Fish Measurement Data'; proc candisc data=sashelp.fish ncan=3 out=outcan; ods exclude tstruc bstruc pstruc tcoef pcoef; class Species; var Weight Length1 Length2 Length3 Height Width; run; proc template; define statgraph scatter; begingraph / attrpriority=none; entrytitle 'Fish Measurement Data'; layout overlayequated / equatetype=fit xaxisopts=(label='Canonical Variable 1') yaxisopts=(label='Canonical Variable 2'); scatterplot x=Can1 y=Can2 / group=species name='fish' markerattrs=(size=3px); layout gridded / autoalign=(topright); discretelegend 'fish' / border=false opaque=false; endlayout; endlayout; endgraph; end; run; proc sgrender data=outcan template=scatter; run;

PROC CANDISC begins by displaying summary information about the variables in the analysis. This information includes the number of observations, the number of quantitative variables in the analysis (specified with the VAR statement), and the number of classes in the classification variable (specified with the CLASS statement). The frequency of each class is also displayed.

Figure 31.1: Summary Information

Class Level Information | ||||
---|---|---|---|---|

Species | Variable Name |
Frequency | Weight | Proportion |

Bream | Bream | 34 | 34.0000 | 0.215190 |

Parkki | Parkki | 11 | 11.0000 | 0.069620 |

Perch | Perch | 56 | 56.0000 | 0.354430 |

Pike | Pike | 17 | 17.0000 | 0.107595 |

Roach | Roach | 20 | 20.0000 | 0.126582 |

Smelt | Smelt | 14 | 14.0000 | 0.088608 |

Whitefish | Whitefish | 6 | 6.0000 | 0.037975 |

PROC CANDISC performs a multivariate one-way analysis of variance (one-way MANOVA) and provides four multivariate tests of the hypothesis that the class mean vectors are equal. These tests, shown in Figure 31.2, indicate that not all the mean vectors are equal (p < 0.0001).

Figure 31.2: MANOVA and Multivariate Tests

Fish Measurement Data |

The CANDISC Procedure

Multivariate Statistics and F Approximations | |||||
---|---|---|---|---|---|

S=6 M=-0.5 N=72 | |||||

Statistic | Value | F Value | Num DF | Den DF | Pr > F |

Wilks' Lambda | 0.00036325 | 90.71 | 36 | 643.89 | <.0001 |

Pillai's Trace | 3.10465132 | 26.99 | 36 | 906 | <.0001 |

Hotelling-Lawley Trace | 52.05799676 | 209.24 | 36 | 413.64 | <.0001 |

Roy's Greatest Root | 39.13499776 | 984.90 | 6 | 151 | <.0001 |

NOTE: F Statistic for Roy's Greatest Root is an upper bound. |

The first canonical correlation is the greatest possible multiple correlation with the classes that can be achieved by using a linear combination of the quantitative variables. The first canonical correlation, displayed in Figure 31.3, is 0.987463. Figure 31.3 also displays a likelihood ratio test of the hypothesis that the current canonical correlation and all smaller ones are zero. The first line is equivalent to Wilks’ lambda multivariate test.

Figure 31.3: Canonical Correlations

Fish Measurement Data |

The CANDISC Procedure

Canonical Correlation |
Adjusted Canonical Correlation |
Approximate Standard Error |
Squared Canonical Correlation |
Eigenvalues of Inv(E)*H = CanRsq/(1-CanRsq) |
Test of H0: The canonical correlations in the current row and all that follow are zero | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Eigenvalue | Difference | Proportion | Cumulative | Likelihood Ratio |
Approximate F Value |
Num DF | Den DF | Pr > F | |||||

1 | 0.987463 | 0.986671 | 0.001989 | 0.975084 | 39.1350 | 29.3859 | 0.7518 | 0.7518 | 0.00036325 | 90.71 | 36 | 643.89 | <.0001 |

2 | 0.952349 | 0.950095 | 0.007425 | 0.906969 | 9.7491 | 7.3786 | 0.1873 | 0.9390 | 0.01457896 | 46.46 | 25 | 547.58 | <.0001 |

3 | 0.838637 | 0.832518 | 0.023678 | 0.703313 | 2.3706 | 1.7016 | 0.0455 | 0.9846 | 0.15671134 | 23.61 | 16 | 452.79 | <.0001 |

4 | 0.633094 | 0.623649 | 0.047821 | 0.400809 | 0.6689 | 0.5346 | 0.0128 | 0.9974 | 0.52820347 | 12.09 | 9 | 362.78 | <.0001 |

5 | 0.344157 | 0.334170 | 0.070356 | 0.118444 | 0.1344 | 0.1343 | 0.0026 | 1.0000 | 0.88152702 | 4.88 | 4 | 300 | 0.0008 |

6 | 0.005701 | . | 0.079806 | 0.000033 | 0.0000 | 0.0000 | 1.0000 | 0.99996749 | 0.00 | 1 | 151 | 0.9442 |

The first canonical variable, `Can1`

, shows that the linear combination of the centered variables `Can1`

=–0.0006 `Weight`

– 0.33 `Length1`

2.49 `Length2`

+ 2.60 `Length3`

+ 1.12 `Height`

– 1.45 `Width`

separates the species most effectively (see Figure 31.4).

Figure 31.4: Raw Canonical Coefficients

Fish Measurement Data |

The CANDISC Procedure

Raw Canonical Coefficients | |||
---|---|---|---|

Variable | Can1 | Can2 | Can3 |

Weight | -0.000648508 | -0.005231659 | -0.005596192 |

Length1 | -0.329435762 | -0.626598051 | -2.934324102 |

Length2 | -2.486133674 | -0.690253987 | 4.045038893 |

Length3 | 2.595648437 | 1.803175454 | -1.139264914 |

Height | 1.121983854 | -0.714749340 | 0.283202557 |

Width | -1.446386704 | -0.907025481 | 0.741486686 |

PROC CANDISC computes the means of the canonical variables for each class. The first canonical variable is the linear combination
of the variables `Weight`

, `Length1`

, `Length2`

, `Length3`

, `Height`

, and `Width`

that provides the greatest difference (in terms of a univariate F test) between the class means. The second canonical variable provides the greatest difference between class means while being
uncorrelated with the first canonical variable.

Figure 31.5: Class Means for Canonical Variables

Class Means on Canonical Variables | |||
---|---|---|---|

Species | Can1 | Can2 | Can3 |

Bream | 10.94142464 | 0.52078394 | 0.23496708 |

Parkki | 2.58903743 | -2.54722416 | -0.49326158 |

Perch | -4.47181389 | -1.70822715 | 1.29281314 |

Pike | -4.89689441 | 8.22140791 | -0.16469132 |

Roach | -0.35837149 | 0.08733611 | -1.10056438 |

Smelt | -4.09136653 | -2.35805841 | -4.03836098 |

Whitefish | -0.39541755 | -0.42071778 | 1.06459242 |

Figure 31.6 displays a plot of the first two canonical variables, which shows that `Can1`

discriminates among three groups: (1) bream; (2) whitefish, roach, and parkki; and (3) smelt, pike, and perch. `Can2`

best discriminates between pike and the other species.

Figure 31.6: Plot of First Two Canonical Variables